Citation
Structural damage assessment and finite element model refinement using measured modal data

Material Information

Title:
Structural damage assessment and finite element model refinement using measured modal data
Creator:
Kaouk, Mohamed
Publication Date:
Language:
English
Physical Description:
xiv, 167 leaves : ill. ; 29 cm.

Subjects

Subjects / Keywords:
Algorithms ( jstor )
Cumulative damage ( jstor )
Damage assessment ( jstor )
Damping ( jstor )
Eigenvectors ( jstor )
Matrices ( jstor )
Property damage ( jstor )
Stiffness ( jstor )
Stiffness matrix ( jstor )
Trusses ( jstor )
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1993.
Bibliography:
Includes bibliographical references (leaves 162-166).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Mohamed Kaouk.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
021778056 ( ALEPH )
AKA9310 ( NOTIS )
30784235 ( OCLC )

Downloads

This item has the following downloads:


Full Text










STRUCTURAL DAMAGE ASSESSMENT AND FINITE ELEMENT
MODEL REFINEMENT USING MEASURED MODAL DATA














By

MOHAMED KAOUK


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


1993

































A Mes Parents

Tout le m6rite de ce travail, s'il en est, vous revient.













ACKNOWLEDGEMENTS


I would like to express my sincere gratitude toward my advisor, Dr. David Zimmerman,

for his expert guidance, friendship, endless encouragement and support. I will forever be

indebted to him for inspiring me in my research and for the priceless education I acquired

from him. I am also grateful for the financial support he provided me during the course of my

graduate studies.

Words are not enough to express my deepest gratitude to my parents for their love, moral

and financial support; it is these things that have made this work possible. I also wish to

acknowledge my sisters and brothers for their continuous encouragement.

I would like to thank the members of my supervisory committee, Drs. Norman Fitz-Coy,

Lawrence Malvern, Bhavani Sankar, and Kermit Sigmon, for their helpful advice.

I wish to thank my colleagues of the Dynamic Systems and Control Laboratory for their

consideration and entertaining discussions. In particular, William Leath and Cinnamon

Larson for their friendship and encouragement.

I would like to thank the staff of the Aerospace Engineering, Mechanics, and

Engineering Science department for their assistance throughout the years, especially Shirley

Robinson for making my life easier during registration, John Young for his prompt responses

in fixing my computer problems, and Ronald Brown for his assistance in the machine shop

and for many stimulating discussions.

I would like to acknowledge the financial support received from Harris Corporation,

NASA/Florida Space Grant Consortium and Florida High Technology Council. I would like

to thank Dr. T. A. Kashangaki of the NASA Langley Research Center and Dr. S. W. Smith of

the University of Kentucky for providing the data of the Eight-Bay Truss used in this study.

Last, but not least, I am grateful to my good friends Joel Payabyab and Fadel Abdallah

for their continuous support.
















TABLE OF CONTENTS


paga

ACKNOWLEDGEMENTS ................................................................................ iii
LIST OF TABLES ................................................................................................. ix
LIST OF FIGURES ............................................................................................... x

A B ST R A C T ............................................................................................................ xiii
CHAPTERS

1 INTRODUCTION ................................................................................. 1

1.1 Finite Element Model Refinement ................................................. 1
1.1.1 O verview ........................................................................... 1
1.1.2 Literature Survey ............................................................. 3
1.2 Structural Damage Assessment ....................................................... 6
1.3 Objective of the Present Study ....................................................... 8

2 MATHEMATICAL PRELIMINARIES AND PRACTICAL
ISSUES TO THE PROBLEMS OF MODEL
REFINEMENT AND DAMAGE DETECTION ............................... 11
2.1 Introduction ..................................................................................... 11
2.2 The Eigenvalue Problem of Discrete Systems ................................ 11
2.2.1 Undamped Models .............................................................. 15
2.2.2 Proportional Damped Models ............................................ 16
2.3 Experimental Modal Analysis ....................................................... 16
2.4 Analytical/Experimental Model Dimensions Correlation ............. 19
2.4.1 Model Reduction Methods .................................................. 19
2.4.1.1 Static Reduction ................................................. 21
2.4.1.2 IRS Reduction ....................................................... 22
2.4.1.3 Dynamic Reduction ............................................... 22








2.4.2 Eigenvector Expansion Methods ........................................ 23
2.4.2.1 Dynamic Expansion ............................................ 23
2.4.2.2 Orthogonal Procrustes Expansion ........................ 24
2.5 Eigenvector Orthogonalization ..................................................... 26
2.5.1 Optimal Weighted Orthogonalization .................................. 27
2.5.2 Selective Optimal Orthogonalization .................................. 27
2.6 Load Path Preservation .................................................................. 28

3 INVERSE / HYBRID PROBLEM APPROACH FOR FINITE
ELEMENT MODEL REFINEMENT ................................................ 31

3.1 Introduction ................................................................................... 31
3.2 Theoretical Formulation ................................................................. 31
3.3 Numerical Illustrations .................................................................. 35
3.3 Sum m ary ......................................................................................... 37

4 SYMMETRIC EIGENSTRUCTURES ASSIGNMENT
MODEL REFINEMENT ALGORITHM ....................... 39
4.1 Introduction .................................................................................... 39
4.2 Problem Formulation ..................................................................... 39
4.2.1 Standard Eigenstructure Assignment Formulation ............. 40
4.2.2 Symmetric Eigenstructure Assignment Formulation ......... 42
4.2.3 Best Achievable Eigenvectors ............................................. 44
4.2.4 Selection of Bo : The Subspace Rotation Method .............. 46
4.3 Numerical Illustrations ................................................................. 47
4.3.1 Damage Detection: Kabe's Problem ................................... 48
4.3.1.1 Local to Global Mode Change ............................ 49
4.3.1.2 Consistent Modes ................................................. 52
4.3.2 Model Refinement of a Cantilever Beam:
Experimental Study .......................................................... 55
4.3.2.1 Modal Test Description ......................................... 55
4.3.2.2 Finite Element Model Description ........................ 56
4.3.2.3 Application of SEAMRA ...................................... 57
4.4 Discussion of the SEAMRA's Formulation ................................... 57
4.5 Sum m ary ......................................................................................... 59








5 DAMAGE LOCATION: THE SUBSPACE ROTATION
ALG O RITH M ................................................................................. 60

5.1 Introduction .................................................................................. 60
5.2 The Subspace Rotation Algorithm: The Direct Method ................ 60
5.3 The Subspace Rotation Algorithm: The Angle
Perturbation Method ................................................................. 63
5.4 Practical Issues .............................................................................. 64
5.4.1 Cumulative Damage Vectors ............................................... 64
5.4.2 Eigenvector Filtering Algorithm ......................................... 65
5.5 Sum m ary ......................................................................................... 66

6 THE MINIMUM RANK PERTURBATION THEORY............... 67
6.1 B background .................................................................................. 67
6.2 The Minimum Rank Perturbation Theory:
Theoretical Background ............................................................ 68
6.3 Damage Extent: Undamped Structures ......................................... 72
6.3.1 Damage Extent: Mass Properties ........................................ 73
6.3.2 Damage Extent: Stiffness Properties .................................. 74
6.3.3 Damage Extent: Mass and Stiffness Properties ................... 76
6.3.3.1 Application of the MRPT ...................................... 76
6.3.3.2 Decomposition of Matrix B ................................. 77
6.4 Damage Extent: Proportionally Damped Structures ..................... 79
6.4.1 Damage Extent: Stiffness and Damping Properties ............ 79
6.4.2 Damage Extent: Mass and Damping Properties .................. 82
6.4.3 Damage Extent: Mass and Stiffness Properties ................... 84
6.4.4 Damage Extent: Mass, Damping and Stiffness Properties .... 85
6.5 Damage Extent: Nonproportionally Damped Structures .............. 87
6.5.1 Damage Extent: Damping and Stiffness Properties ........... 89
6.5.2 Damage Extent: Mass and Damping Properties .................. 90
6.5.3 Damage Extent: Mass and Stiffness Properties .................. 92
6.6 Practical Issues .............................................................................. 94
6.6.1 The Concept of "Best" Modes ............................................. 95
6.6.2 Application of the Eigenvector Filtering Algorithm ............ 97
6.7 Sum m ary ....................................................................................... 97










7 VALIDATION AND ASSESSMENT OF THE SUBSPACE
ROTATION ALGORITHM AND THE MINIMUM
RANK PERTURBATION THEORY ............................................. 98

7.1 Introduction .................................................................................. 98
7.2 Kabe's Problem .......................................................................... 98
7.2.1 Problem Description ........................................................... 98
7.2.2 Damage Location ............................................................... 100
7.2.3 Damage Extent ............................................................... 103
7.3 Damage Detection: Fifty-Bay Two-Dimensional
Truss: Undamped FEM ........................................................ 105
7.3.1 Problem Description .......................................................... 105
7.3.2 Damage Location ............................................................... 106
7.3.3 Damage Extent .................................................................. 107
7.4 Experimental Study: The NASA 8-Bay Truss ............................ 112
7.4.1 Problem Description .......................................................... 112
7.4.2 Refinement of the Original FEM ....................................... 115
7.4.3 Damage Location ............................................................... 118
7.4.4 Damage Extent .................................................................. 120
7.4.4.1 The Brute Force Method ..................................... 121
7.4.4.2 The Damage Consistent Method ........................ 121
7.4.4.3 Application of the Eigenvector Filtering
Algorithm ........................................................ 122
7.5 Experimental Study: Mass Loaded Cantilevered Beam .............. 140
7.5.1 Problem Description .......................................................... 140
7.5.2 Analytical and Experimental Models Dimension
C orrelation ................................................................... 141
7.5.3 Refinement of the Original FEM ...................................... 141
7.5.4 Damage Location .............................................................. 142
7.5.5 Damage Extent .................................................................. 143
7.6 Fifty-Bay Two-Dimensional Truss: Proportionally
D am ped FEM ............................................................................. 144
7.6.1 Problem Description .......................................................... 144
7.6.2 Damage Location ............................................................... 145
7.6.3 Damage Extent ................................................................... 146








7.7 Eight-Bay Two-Dimensional Mass-Loaded
Cantilevered Truss ................................................................... 148
7.7.1 Problem Description .......................................................... 148
7.7.2 Proportionally Damped Configuration: Damage of
Small Order of Magnitude ............................................... 149
7.7.2.1 Damage Location ................................................. 150
7.7.2.2 Decomposition of Matrix B ................................. 150
7.7.2.3 Damage Extent .................................................... 152
7.7.3 Undamped Configuration: Damage of Large
Order of Magnitude ......................................................... 153
7.7.3.1 Noise Free Eigendata ........................................... 153
7.7.3.2 Noisy Eigendata .................................................. 154
7.8 Sum m ary ........................................................................................ 156

8 CONCLUSION AND SUGGESTIONS FOR FUTURE WORK ......... 159

REFEREN CES .................................................................................................... 162

BIOGRAPHICAL SKETCH ............................................................................... 167















LIST OF TABLES


Table Pa


3.1 Kabe's Problem: Elemental Stiffness Components ................................. 38

4.1 Properties of the Cantilever Beam ............................................................ 56

4.2 Measured Natural Frequencies and Damping Ratios
of the Cantilever Beam ............................................................................ 56

4.3 Measured Mode Shapes of the Cantilevered Beam ................................... 56

7.1 Fifty-Bay Truss: Summary of Damage Detection
Results using the M RPT ........................................................................ Ill

7.2 Mass Properties of the Eight Bay Truss .................................................... 113

7.3 Strut Properties of the Eight Bay Truss ..................................................... 114

7.4 NASA 8-Bay: Truss Damage Case Definitions ..................................... 115

7.5 Comparison of Analytical and Experimental Frequencies ....................... 116

7.6 Summary of the Filtering Process for Single Member Damage Cases ......... 123

7.7 NASA 8-Bay Truss: Summary of the Damage Assessment Results ........... 139

7.8 Mass Loaded Cantilevered Beam Properties ......................................... 140

7.9 Analytical and Experimental Frequencies of the "Healthy" Structure ...... 142

7.10 50-Bay 2-Dimensional Truss: Summary of the Percentage
Error with Respect to the Exact Damage .......................................... 147

7.11 Problem 7.7: Percentage Error of Damage Estimate with
Respect to Exact Dam age ...................................................................... 156














LIST OF FIGURES


Figure Page


1.1 Overview of Finite Element Model Refinement ...................................... 2

1.2 Overview of FEM Model Refinement Process Used for
D am age A ssessm ent ............................................................................ 7
2.1 Components of a Vibration Measurement System for
M odal A analysis .................................................................................. 17
2.2 A simple Experimental Modal Analysis Setup ...................................... 18

2.3 Flow Chart of the Iterative Load Preservation Path Algorithm ............. 29

3.1 Kabe's Problem: Analytical Test Structure .............................................. 36

4.1 Best Achievable Eigenvector Projection ................................................ 45

4.2 Rotation of the Achievable Subspace ..................................................... 47

4.3 K abe's Problem ..................................................................................... 48

4.4 Results for Kabe's Problem using the 1st Mode,
Full Eigenvector ............................................................................... 50

4.5 Results for Kabe's Problem Modes 1, 2, 3 and
Eigenvectors Components 1, 2, 3 ......................................................... 51

4.6 Results for Kabe's Problem using Load Path Preservation,
Modes 1, 2, 3 and Eigenvectors Components 1, 2, 3 ............................. 52

4.7 Results for Kabe's Problem using the 1 st Mode,
Full Eigenvector ................................................................................. 53

4.8 Results for Kabe's Problem Modes 1,2,3 and
Eigenvectors Components 1, 2, 6 ........................................................ 54

4.9 Experimental Cantilever Beam ............................................................. 55

4.10 Experimental and Analytical Frequency Response Function
of the Cantilever Beam ......................................................................... 58








7.1 K abe's Problem ...................................................................................... Y'

7.2 Kabe's Problem: Damage Location Results using the Subspace
Rotation Direct Method with the Eigendata of the 1st Mode ............... 101

7.3 Kabe's Problem: Damage Location Results using the Angle
Perturbation Method with the Eigendata of the 1st Mode ................... 101

7.4 Kabe's Problem: Damage Location Results using Lin's
Algorithm with the Eigendata of the 1st Mode .................................... 102

7.5 Kabe's Problem: Damage Location Results using the Angle
Perturbation Method with the Eigendata of the 1st and 2nd Modes ....... 102

7.6 Kabe's Problem: Damage Location Results using Lin's
Algorithm with the Eigendata of the 1st and 2nd Modes .................... 103

7.7 Kabe's Problem: Damage Extent Results using the MRPT
with the Eigendata of M ode 2 ............................................................... 104

7.8 Kabe's Problem: Damage Extent Results using Baruch's Method ........... 105

7.9 Fifty-Bay Two-Dimensional Truss ...................................................... 106

7.10 Fifty-Bay Truss: Damage Location Results using the Subspace
Rotation Algorithm with the Eigendata of the First Ten Modes .......... 107

7.11 Fifty-Bay Truss: Damage Extent Results using the
MRPT with the Eigendata of Modes 8 and 9 ..................................... 109

7.12 Fifty-Bay Truss: Damage Extent Results using the
MRPT with the Eigendata of the First Ten Modes ............................. 110

7.13 Fifty-Bay Truss: Damage Extent Results using Baruch's Algorithm ....... 110

7.14 The NASA Eight-Bay Hybrid-Scaled Truss: Damage Cases ................ 112

7.15 The NASA 8-Bay Truss: Lacing Pattern ................................................ 113

7.16 NASA 8-Bay Truss: Typical Frequency Response Comparison .............. 117

7.17 NASA 8-Bay Truss: Perturbation to the Original Stiffness
Matrix that Resulted From the Refinement Process .............................. 118

7.18 NASA 8-Bay Truss: Cumulative Damage Vector Associated
w ith C ase F ......................................................................................... 124

7.19 NASA 8-Bay Truss: Damage Assessment of Case A .............................. 125

7.20 NASA 8-Bay Truss: Damage Assessment of Case C .............................. 126










7.21 NASA 8-Bay Truss: Damage Assessment of Case D .................................. z/

7.22 NASA 8-Bay Truss: Damage Assessment of Case E ............................... 128

7.23 NASA 8-Bay Truss: Damage Assessment of Case G ............................... 129

7.24 NASA 8-Bay Truss: Damage Assessment of Case H ............................... 130

7.25 NASA 8-Bay Truss: Damage Assessment of Case I ............................ 131

7.26 NASA 8-Bay Truss: Damage Assessment of Case J ............................ 132

7.27 NASA 8-Bay Truss: Damage Assessment of Case K ............................ 133

7.28 NASA 8-Bay Truss: Damage Assessment of Case L ............................ 134

7.29 NASA 8-Bay Truss: Damage Assessment of Case M ............................ 135

7.30 NASA 8-Bay Truss: Damage Assessment of Case N ............................ 136

7.31 NASA 8-Bay Truss: Damage Assessment of Case 0 ............................ 137

7.32 NASA 8-Bay Truss: Damage Assessment of Case P ............................ 138

7.33 The Mass Loaded Cantilevered Beam .................................................... 140

7.34 Mass Loaded Cantilevered Beam: Damage Assessment ....................... 143

7.35 50-Bay 2-Dimensional Truss .................................................................. 145

7.36 50-Bay 2-Dimensional Truss: Damage Location .................................... 146

7.37 50-Bay 2-Dimensional Truss: Damage Extent ....................................... 148

7.38 The Eight-Bay Two-Dimensional Mass-Loaded Cantilevered Truss ....... 149

7.39 Problem 7.7: Cumulative Damage Location Vector: First Four Modes ...... 150

7.40 Problem 7.7: Cumulative Vectors Associated with the Exact
and Computed Bin, Bd, Bk: First Three Modes .................................... 151

7.41 Problem 7.7: Exact and Computed AMd, ADd, AKd ............................... 152

7.42 Problem 7.7: Cumulative Vector Associated with B, Bm, Bd, Bk
B Computed using Modes 1-4
Bm, Bd, Bk Computed using Modes 3,4 & 5 ..................................... 155

7.43 Problem 7.7: Exact and Computed AMd, ADd, AKd ............................... 157













Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirement for the Degree of Doctor of Philosophy

STRUCTURAL DAMAGE ASSESSMENT AND FINITE ELEMENT
MODEL REFINEMENT USING MEASURED MODAL DATA

By

Mohamed Kaouk

August 1993


Chairperson: Dr. David C. Zimmerman
Major Department: Aerospace Engineering, Mechanics and Engineering Science

This study investigates the problems of model refinement and structural damage

assessment. The essence of the model refinement problem is to adjust finite element models

(FEMs) of structures with the intent of producing a correlation between experimental and

analytical modal properties. Recently, the framework of model refinement has been adopted

to determine the location and extent of structural damage. Damage will result in changes to

the modal properties of the healthy structure. A further refinement of an accurate FEM of the

structure using damaged modal parameters is expected to generate adjustments to the FEM at

locations associated with damage. Analysis of these adjustments can then be used to assess

damage. In this investigation, four algorithms relevant to the subjects of model refinement

and damage detection are presented. The development of a model refinement algorithm with

its basis in inverse theory is first presented. The algorithm, termed the inverse/hybrid

approach, is illustrated in a comparative computer simulated study. Next, an existing

eigenstructure assignment model refinement algorithm is improved to better approach the

damage assessment problem. The enhanced algorithm is evaluated and compared to other

techniques using simulated and experimental data. The algorithm is shown to perform well








in assessing damage and refining FEMs. A damage location algorithm that bypasses the

general framework of model refinement is discussed. The damage location algorithm,

termed the subspace rotation, is similar to the modal force error criteria proposed by several

researchers. Greater insight to the modal force error criteria, along with a new viewpoint that

reduces the effects of measurement noise, is discussed. Furthermore, an efficient damage

extent algorithm based on a minimum rank perturbation theory (MRPT) is developed. The

formulation of the MRPT is consistent with the effect of most structural damage on FEMs.

The characteristics of the subspace rotation algorithm and the minimum rank perturbation

theory are illustrated using simulated and experimental testbeds. The decomposition of the

damage assessment problem into location and extent subproblems is shown to be

advantageous from both for computational efficiency and for engineering insight.














CHAPTER 1
INTRODUCTION




1.1 Finite Element Model Refinement


1.1.1 Overview

An important aspect in the design process of a structure is the evaluation of its

performance under expected dynamic loading conditions. Dynamic performance can be

analyzed by using either analytical or experimental techniques. Experimental analyses are

generally very costly, time consuming and can encounter technical difficulties. One example

of a commonly encountered technical barrier is the ground base reproduction of a weightless

environment for the testing of space structures. This shortcoming and the advent of

computers have sparked a growing interest in the uses of analytical techniques. This type of

analysis utilizes a mathematical model of the actual structure for computer simulated

evaluation of structural performance. Presently, most analytical models used in industry are

finite element models generated by using Finite Element Methods. The accuracy of a finite

element model (FEM) can be improved to some extent by increasing the number of degrees

of freedom (DOFs) included in the model. However, accuracy of the FEM may still be

lacking due to uncertainties in material properties, geometric layout and fabrication induced

errors. Thus, it is essential to "validate" these FEMs prior to their acceptance as a basis for

analysis. One way to validate them is to compare their modal properties (eigenvalues/

eigenvectors) with the measured modal parameters of the actual structure as obtained from

experimental modal analysis (EMA). A FEM is acceptable when these two sets of modal

properties are in agreement. Unfortunately, this agreement rarely occurs. As a result, the








FEM must be adjusted to produce a correlation of analytical and experimental modal

properties. An overview of this procedure is schematically illustrated in Figure 1.1

(Zimmerman and Smith, 1992).


Experimental
Analytical Frequency
Response
Mass Functions
Damping or
Stiffness Hankel
Matrices / Matrices







Modal Parameters j--- Modal Parameters



Yes

Done
(Confidence in
FEM established)



Figure 1.1 Overview of Finite Element Model Refinement
(Zimmerman and Smith, 1992)



In the past, the process of adjusting FEMs was performed on an ad-hoc basis aided by

engineering experience. This practice was naturally time consuming and in most cases

inadequate for large-order, complex FEMs that are typically needed for accurate dynamic

modelling. In the past years, growing interest has been focused on developing systematic

procedures to produce correlated FEMs. These efforts have resulted in the development of a

large number of algorithms. Finite element model adjustment procedures have been








commonly referred to in the literature as, FEM refinement, FEM adjustment, FEM

correction, FEM correlation, and FEM identification.

1.1.2 Literature Survey

Comprehensive literature surveys covering a large portion of the work that addresses the

model refinement problem can be found in the book chapter by Zimmerman and Smith

(1992), and in the papers by Ibrahim and Saafan (1987) and Heylen and Saas (1987). For

completeness and to properly underline the objective of the current study, it is appropriate to

include a brief survey of the development made in this area.

The concept of using experimental modal data in analytical studies was initiated by an

early work presented by Rodden (1967). In his work, Rodden explored the possibilities of

generating mass and stiffness property matrices by using experimentally measured modal

data. The experimentally generated mass and stiffness matrices were nonsymmetric. Brock

(1968) improved the work of Rodden by proposing a strategy to insure symmetry of property

matrices.

The essence of the model refinement concept, as adopted by most researchers, is to

modify the finite element model (FEM) of systems with the intention of producing a

correlation of test and analytical modal parameters. In general, this task has been approached

through two different philosophies. The first amounts to modifying globally the finite

element property matrices of the system. This approach is commonly referred to as the

property matrix update. The alternative approach is to individually correct parameters of

each element of the finite element model. This will be addressed here as the physical

parameter update.

Property matrix update algorithms can be classified into two subclasses of

methodologies: optimal-matrix update and control-based eigenstructure assignment

techniques.








The basic philosophy of the optimal-matrix update is to minimize the correction to the

FEM property matrices to accomplish the analytical/test modal correlation. The pioneering

work in this area can be credited to Baruch and Bar Itzhack (1978). In their formulation, by

assuming that the mass matrix is correct, the refinement of the stiffness matrix of an

undamped FEM is cast as a constrained minimization problem. The objective of their

formulation is to determine the minimal Frobenius-norm symmetric stiffness adjustment

that satisfies the eigenvalue problem in terms of the experimentally measured eigenvalues

and eigenvectors. A computationally efficient closed form solution was developed for the

updated stiffness matrix. Berman and Nagy (1983) extended the Baruch and Bar Itzhack

approach to the refinement of both the mass and stiffness matrices. This same methodology

was further extended by Fuh et al. (1984) to update the mass, damping and stiffness matrices

of damped FEMs. Fuh and his colleagues used cross-orthogonality relationships to correct

the mass and damping matrices and a constrained minimization problem similar to the one

proposed by Baruch and Bar Itzhack to determine the refined stiffness matrix. The problem

of adjusting the mass, damping and stiffness matrices was also attempted by Hanagud et al.

(1984). In their approach, all three property matrices of a nonproportionally damped FEM

are incorporated in the constrained minimization problem.

The previously stated algorithms do not preserve the load path (sparsity pattern) of the

original analytical property matrices. Kabe (1985) proposed a reformulation of the Baruch

and Bar Itzhack algorithm that constrains the updated stiffness matrix to preserve the load

path of the original stiffness matrix. In addition, in his reformulation, he adopted a percent

weighting on the stiffness changes instead of mass matrix weighting as used in the Baruch

and Bar Itzhack algorithm. Kammer (1987) proposed an alternative solution approach to the

problem defined by Kabe that uses projection matrix theory and the Moore-Penrose

generalized inverse. The algorithms formulated by Kabe and Kammer involve an excessive

amount of computational effort. Another alternative and more efficient formulation of the








Kabe problem that utilized a generalization of Marwill-Toint updates was developed by

Smith and Beattie (1991).

The other subclass of property matrix updates is based on the general framework of

control-based eigenstructure assignment algorithms. The essence of this approach is to

determine pseudo-controllers that would assign the experimentally measured modal

parameters to the original analytical FEM. The pseudo-controllers are then translated into

matrix adjustment to the original property matrices. This approach was initially formulated

by Inman and Minas (1990) to adjust the damping and stiffness matrices of the original FEM.

In their formulation, the symmetry of the adjusted property matrices was enforced through

an iterative process that involves a numerical nonlinear optimization process. Zimmerman

and Widengren (1989, 1990) proposed a formulation that replaces the iterative process of

Inman and Minas with a generalized algebraic Riccati equation. Further details about this

approach are treated in Chapter 4.

The alternative philosophy to the model refinement problem is the physical parameter

update. The basic idea of this alternative approach consists of adjusting some or all structural

physical parameters to produce test/analysis modal correlation. Structural parameters are

design variables such as material densities, modulus of elasticity, cross-sectional areas,

element lengths and others. This type of procedure results in corrected FEMs that are

consistent with the framework of the original FEM.

Most methodologies that have adopted this philosophy have used the sensitivity

derivatives of the system eigenvalues and/or eigenvectors with respect to the structural

parameters. Generally, the refinement process amounts to solving for the corrected

structural parameters through an iterative optimization problem directed by the sensitivity

derivatives. Some notable work in this area was accomplished by Collins et al. (1974), Chen

and Garba (1980), Adelman and Haftka (1986), Creamer and Hendricks (1987), Flanigan

(1991), Martinez et al. (1991), to name only a few. In these studies, different sensitivity

formulations and iterative optimization strategies are proposed. A direct approach that








bypasses the use of closed-form sensitivity derivatives was presented in the papers of Hajela

and Soeiro (1990) and Soeiro (1990).

White and Maytum (1976) set forth another alternative methodology that can be

classified in the physical parameter update group. In their approach, adjustments to the

stiffness matrix are viewed as a collection of known submatrices, elements or group of

elements. Each submatrix is multiplied by an unknown scaling factor. Correction of the

original stiffness matrix amounts to determining the scaling factors that would improve the

correlation of analytical and experimental modal properties. The scaling factors can be

physically viewed as functions of the stiffness structural parameters. A large

underdetermined system of equations in terms of the scaling factor is generated. The scaling

factors are then computed by using a pseudo-inverse. An improvement to the White and

Maytum approach is presented in the paper by Lim (1990).



1.2 Structural Damage Assessment


Structures, in general, are prone to structural damage during their service lives, that

could affect their overall performance and could result in catastrophic failures. This is of

critical concern when expensive systems and/or human lives are at stake. On this basis, it is

highly desirable to monitor the structural health of systems such as airplanes, space systems,

bridges, buildings, oil platforms, among others, to prevent such catastrophic events.

Currently, the most common structural health monitoring methods involve visual

inspections supplemented with X-rays, acoustic emission, magnetic resonance and

ultrasonic testing. These approaches can be time consuming, costly and difficult to perform

on inaccessible structural components. Additionally, none of these approaches provide a

quantitative assessment of the magnitude of the damage.

In response to these shortcomings, structural health monitoring approaches based on

the framework of existing model refinement techniques have been recently proposed by

several researchers (Smith and Hendricks, 1987; Chen and Garba, 1988; Ricles and








Kosmatka, 1992; Zimmerman and Kaouk, 1992a,b; Kaouk and Zimmerman, 1993a,b).

These techniques utilize the vibration signature of the pre- and post- damaged structure, in

conjunction with an analytical model of the original structure, to assess both the location and

extent of the structural damage. The pre-damage modal parameters are used to correct

(refine) the original finite element model (FEM) to determine an "accurate" reference

baseline. Once damaged, the post-damage modal properties of the structure are used to

further "refine" the refined analytical model. This results in perturbations to the refined

analytical model. Analysis of the perturbations could indicate the damage location and

extent. An overview of the application of model refinement algorithms in assessing

structural damage is summarized in Figure 1.2.



Undamaged Model Experimental
(Correlated) Frequency
Response
Mass Functions
Damping or
Stiffness Hankel
Matrices Matrices



No



Modal Parameters Modal Parameters



Yes

Done
Structure Healthy



Figure 1.2 Overview of Finite Element Model Refinement
Process Used for Damage Assessment








Notable exceptions to the direct use of FEM refinement algorithms to the damage

detection problem are the work of Lin (1990), Ojalvo and Pilon (1988), and Gysin (1990). In

the work of Lin, a flexibility matrix is determined using experimental data. This matrix is

then multiplied by the original stiffness matrix, with those rows and/or columns that differ

significantly from a row and/or column of the identity matrix indicating which degrees of

freedom have been most affected by the damage. It is then assumed that damage has occurred

in structural elements connecting those degrees of freedom. An overview of the work by

Ojalvo and Pilon (1988), and Gysin (1990) is given in Chapter 5.

Although the problems of damage assessment and model refinement are related to one

another, they have quite different characteristics. In essence, the model refinement concept

is based on the philosophy of the minimum change to the original FEM. Naturally, this

minimum change constraint has a tendency of smearing the changes throughout the entire

FEM. This aspect is inconsistent with the effects of structural damage on FEMs which are

usually localized perturbations of possibly large magnitude. In fact, structural damage often

occurs at discrete locations and only affects a few elements of the FEM.



1.3 Objective of the Present Study


The present study investigates the development of new and promising model refinement

and damage detection methodologies. Although considerable research has been done in

these areas, no methodology has been fully successful in dealing with the refinement or the

damage detection problem of "real life" systems. The main objective of this investigation is

to formulate efficient model refinement algorithms that are consistent with the effect of

structural damage.

In chapter 2, practical concepts and issues related to the general areas of model

refinement are presented. First, the concept of the eigenvalue problem of discrete structural

models is reviewed with emphasis on the associated cross-orthogonality conditions. A brief

discussion of modal analysis follows. The problem associated with incomplete eigenvector








measurements is then investigated and already existing techniques to deal with this problem

are presented. Furthermore, eigenvector orthogonalization techniques useful to a large

number of model refinement algorithms are reported. Finally, an iterative approach to the

problem of load path preservation encountered in a large number of property matrix update

algorithms is discussed.

In chapter 3, the author proposes and formulates a new model refinement algorithm.

The newly developed algorithm, named the inverse/hybrid method, is naturally based on the

inverse problem. The model type under consideration is proportionally damped and the

inaccuracies of the analytical model is assumed to be due to errors in the modelling of the

damping and stiffness properties. A complete hybrid set of modal data is generated by

approximating the unmeasured modal information by the corresponding analytical modes.

Symmetry of the corrected stiffness and damping matrices is enforced by mass

orthogonalization the complete hybrid eigenvector matrix. An orthogonalization procedure

that assigns more confidence on the measured eigenvectors is proposed. A comparative

study of the inverse/hybrid method and the algorithm proposed by Baruch and Bar Itzhack

(1978) shows that both algorithms give similar results. However, it is shown that the

inverse/hybrid approach is less computationally efficient.

Chapter 4 proposes improvements to the symmetric eigenstructure assignment model

refinement algorithm (SEAMRA) formulated by Zimmerman and Widengren (1989). The

author develops a technique to enhance eigenvector assignability. The technique, termed the

subspace rotation method, is based on rotating the achievable eigenvector subspaces into the

experimental eigenvectors. The subspace rotation method results in both a decrease in the

computational burden and an increase in the accuracy of the assigned eigenvector. The

enhanced SEAMRA is then evaluated and compared to other algorithms using both

computer simulated and experimental testbeds. It is shown that the enhanced algorithm is

suitable for damage detection applications.








In Chapter 5, an efficient damage location algorithm that utilizes modal data

information but bypasses the general framework of the model refinement problem is

presented. This location algorithm is an outgrowth of the subspace rotation method used to

enhance eigenvector assignability of the SEAMRA. The proposed location algorithm is

similar to the modal force error criteria presented by several researchers (Ojalvo and Pilon,

1988; Gysin, 1990). Further interpretation of the algorithm operation is given. Additionally,

the author proposes and formulates a new viewpoint that reduces the effect of measurement

noise for certain types of structures. Based on this formulation, an eigenvector filtering

algorithm is also developed.

Chapter 6 presents the formulation of computationally attractive damage extent

algorithms that are based on a minimum rank perturbation theory (MRPT) developed by the

author. The formulation of the MRPT is consistent with the effect of many classes of

structural damage on FEMs. Several MRPT based algorithms are formulated to

accommodate structures with undamped, proportionally damped, and nonproportionally

damped FEMs. For each type of FEM, several damage scenarios are considered.

Discussions of the characteristics and properties of the developed algorithms are presented

along with practical issues that can be used to improve their performance.

In Chapter 7, the algorithms developed in Chapters 5 and 6 are demonstrated and

evaluated using both computer simulated and actual experimental data. The main objective

of these example problems is to illustrate the potential of these algorithms in assessing

structural damage. In all examples, the location of damage is first determined by using the

algorithm presented in Chapter 5. An MRPT based algorithm is then utilized to assess the

extent of the damage. Several key points made throughout the formulation in Chapters 5 and

6 are emphasized. In particular, it is shown that the damage extent calculations can be greatly

enhanced by making use of the damage location algorithm.

In Chapter 8, summaries and conclusions of the issues discussed in this study are

presented along with some suggestions for future work.














CHAPTER 2
MATHEMATICAL PRELIMINARIES AND PRACTICAL ISSUES RELATED TO
THE PROBLEMS OF MODEL REFINEMENT AND DAMAGE DETECTION




2.1 Introduction

In this chapter, general mathematical preliminaries and practical issues relevant to the

areas of model refinement and damage detection are discussed. In Section 2.2, an overview

of the general eigenvalue problem for discrete models is given. Further, the concept of modal

analysis is introduced in Section 2.3. Section 2.4 deals with the concept of incomplete

degrees of freedom measurement. Two alternative approaches are discussed as possible

solutions to the incomplete measurement problem. Two eigenvector orthogonalization

techniques are discussed in Section 2.5. Finally, in Section 2.6, an algorithm to preserve

sparsity in updated property matrices is presented. The concepts discussed in this chapter

will be frequently referred to in the course of the forthcoming chapters.


2.2 The Eigenvalue Problem of Discrete Systems

In practice, most engineering structures are continuous systems with spatially

distributed material properties. The vibration motion of these structures in terms of their

distributed properties is usually governed by one or several partial differential equations.

With complex systems, both the development and the analysis of partial differential

equations of motion are tedious and in many cases impossible. These shortcomings and the

advent of digital computers have motivated the development of approximate modelling of

continuous systems in order to simplify the equations of motion. The general idea behind

these approximations is to represent the exact distributed model of a system by a discrete one.








This concept is known as spatial discretization which eliminates the continuous spatial

dependence of the distributed properties. The discrete model is of finite order and is

described by a finite number of variables known as degrees of freedom (DOFs). The number

of DOFs used in the discrete model depends on the desired accuracy in representing the

continuous model. Commonly, the vibration motion of systems in terms of their discrete

models is described by a set of simultaneous ordinary differential equations that are usually

simpler to develop and analyze than the partial differential equations of the continuous

system. One of the most commonly used approximate discrete modelling techniques is the

finite element method. The model generated by a finite element method is called a finite

element model (FEM). More elaborate discussions of the concepts of continuous and

discrete models as well as finite element methods are covered in detail in the books of

Meirovitch (1986, 1980), Inman (1989), and Hughes (1987).

Commonly, the free vibration motion of a structure in terms of an nth order discrete

model is represented by the following set of simultaneous ordinary differential equations

Mw(t) + Dw(t) + Kw(t) = 0 (2.2.1)

where M, D and K are termed, respectively, the mass, damping and stiffness matrices. They

are models of the mass, damping and stiffness properties of the structure. Since the system

model is order n, these matrices are of dimension nxn and are generally real. The variable

w(t) represents the n displacements of the n-DOF model of the structure. The overdots

represent differentiation with respect to time. The mass matrix, M, is always symmetric

positive definite. The stiffness (K) and damping (D) matrices of nongyroscopic and

noncirculatory systems are symmetric. In general, the modelling of the mass and stiffness

properties of the structure is simpler and more accurate than the modelling of the damping

properties. In the forthcoming discussion, it is assumed that the system under consideration

is nongyroscopic and noncirculatory. The standard solution to Eq. (2.2.1) is of the form


w(t) = vext


(2.2.2)








where v is a constant nxl vector and k is a constant scalar. Substituting Eq. (2.2.2) into Eq.

(2.2.1) and dividing the resultant equation by ext yields the condition

My X2 + Dv X + Kv = 0 (2.2.3)

There are n sets of nontrivial complex conjugate solutions (Xi, y,) to Eq. (2.2.3). Note that

since the property matrices (M, D, K) are real, if (iQ, yi) is a solution set to Eq. (2.2.3), the

complex conjugate of that set is also a solution. The problem of solving for these solutions

is commonly known as the eigenvalue problem and is sometimes referred to as the

characteristic value problem. The scalar Xi and the vector vi are known, respectively, as the

eigenvalue and eigenvector of the ith mode of vibration of the structure. For a general

damped system, eigenvalues and eigenvectors are both complex. Note that Eq. (2.2.3) can

be easily rearranged in the more general mathematical eigenvalue problem format as

[ 0 Inxn i [ ]
M-'K M-'D [Y = ki yi (2.2.4)

where Inxn is the nxn identity matrix. Equation (2.2.4) is called a state space representation

of Eq. (2.2.3).

The eigenvalue and eigenvector can be related to some physical characteristic properties

of structures. For that matter, the ith eigenvalue is written as

Xi = -- ii + j (Ii -- (2.2.5)


where j = 1 Note that in writing this equation it is assumed that the system is

underdamped. The real scalar oi and ti are the natural frequency and damping ratio (or

damping factor), respectively, of the ith mode of the structure. The eigenvector yi indicates

the "shape" of vibration of the ith mode of the structure. The sets of frequencies, damping

ratios and mode shapes are sometimes referred to as modal parameters.

The symmetric nature of the property matrices (M, D, K) constrains the eigenvectors to

satisfy some cross-orthogonality relationships. For the purpose of discussing these







cross-orthogonality relationships, consider the following alternative state space
representation of the eigenvalue problem in Eq. (2.2.3)
[M O][VA VA ][A 0]+ [D K VA VA] = [ (2.2.6)
0 -K V V 0 ?A K 0 V V 0

where V = [ v n ]
A = diag(X1, X2, ..., Xn)
where the overbar denotes the matrix complex conjugate operator. Based on this particular
representation and the fact that the state matrices of Eq. (2.2.6) are symmetric, by proper
normalization of the eigenvectors, the cross-orthogonality relationship associated with the
system are given by

[VA VA M 0][VA = 2nx2n (2.2.7)
V V 0 K V V 2nx2n

VA VAN D K "VA VA] [A 0]
V V K 0 V V 0 X

Equations (2.2.7) and (2.2.8), respectively, clearly imply the following relations

VA] [M _][VA] =[0] (2.2.9)
V 0 K V 0

[VA]'[D K VA (2.2.10)
V K O V 0
T
where [ ] denotes the complex conjugate transpose operator ( [ ] = ). Note that,
contrary to Eqs. (2.2.7-8), no particular normalization of the eigenvector is needed in order
for Eqs. (2.2.9-10) to be satisfied.
Another state space representation of Eq. (2.2.3) is given by







0 M VA VA A 0 M 0] VA VA [0 (2
[M DN V V 0 ] 0 K[ V V 0jj
Based on the same argument discussed earlier, the cross-orthogonality conditions that arise
from this representation are

M ]T[ ][ ] = 2nx2n (2.2.12)
V v M Dr V Vx]

VA VA T[ M ][ VA VA] [A 0]
V V 0 K V V 0O

Again, the following two relationships follow,

VA][ [0 M][VA] = [0] (2.2.14)
V M D V 0

VAI -M ][VA] = [] (2.2.15)
V [ 0 K V 0

2.2.1 Undamped Models
In the modelling of structures, it is often assumed that the damping is negligible and
hence is set to zero. For these type of system models, the eigenvalue problem in matrix form
is given by
MVA2 + KV = [0] (2.2.16)
The matrices V and A are as defined earlier. For undamped systems, the eigenvalues X, are
pure imaginary and the eigenvectors yi are real. Note that the eigenvalues are related to the
system natural frequency by
X2 = co? (2.2.17)
Furthermore, by proper normalization of the eigenvectors, the cross-orthogonality relations
associated with this type of systems are








VTMV = Inxn (2.2.18)

VTKV = diag(wO, o, .2) (2.2.19)

2.2.2 Proportionally Damped Models

When the damping of the structure is accounted for, it is sometimes modelled to be

proportional,

D = aM + p3K (2.2.20)

where a and 03 are real scalars. The eigenvectors of a proportionally damped system are the

same as the eigenvectors associated with the corresponding undamped system. The

cross-orthogonality relationships associated with proportionally damped models are

VTMV = Inxn (2.2.21)

VTDV = diag(2(-o1, 22W2, 2ae0n) (2.2.22)

VTKV = diag(ao, o Wn) (2.2.23)

where all variables have the same definitions as in the previous discussion.

A more detailed development of the eigenvalue problem can be found in the books of

Meirovitch (1986, 1980) and Inman (1989).


2.3 Experimental Modal Analysis

The vibration characteristics of structures can also be measured experimentally. Recall

that the structural vibration characteristics are given by natural frequencies, damping ratios

and mode shapes (eigenvectors). The process of measuring the modal parameters is known

as modal testing or experimental modal analysis. An introductory treatment of the concept

of experimental modal analysis can be found in the book of Inman (1989); a more rigorous

coverage is treated in the book of Ewins (1986).

The hardware components needed in a modal analysis experiment are identified in

Figure 2.1 (Inman, 1989). A schematic of a simple modal vibration measurement test setup








is shown in Figure 2.2. Brief descriptions of some of the components and their functions are

given below.


T := Transducer
SC := Signal Conditioner


Figure 2.1 Components of a Vibration Measurement System
For Modal Analysis (Inman, 1989).



The exciter is used to deliver the driving force that puts the structure in motion. The

two most commonly used exciters are the shaker and the impulse hammer.


Transducers are devices that measure the driving force as well as the response of the

structure. They sense mechanical forces or motions and, then, convert them into

electrical signals. Mechanical forces are usually recorded by a force transducers.

Commonly, the vibration response of structures is measured by accelerometers.

Accelerometers are mounted directly on the structure and, naturally, record the

local accelerations.


Signal conditioners are used to match the signal, received from the transducers, to

the specification of the analyzer. These devices can also be used to amplify the

signals.















Accelerometer


Force
Transducer


Signal
Conditioner


Accelerometer


Signal
Conditioner


Signal Analyzer


Figure 2.2 A Simple Experimental Modal Analysis Setup.

The signal analyzer processes the electrical signal received from the signal

conditioners. The standard type of analyzer allows time domain signals to be

viewed in the frequency domain via a Fast Fourier Transform (FFT) algorithm. In

an FFT, the signals are first filtered, digitized and then transformed into discrete

frequency spectra. The frequency spectra can then be manipulated to compute the

modal properties of the structure.


It is important to note that the experimentally identified modal parameters are usually

affected by unpredictable measurement noise. Typically, natural frequencies are identified

to within 1 to 2% repeatability; damping ratios to within 5 to 15% repeatability, and mode

shapes to within 5 to 10% repeatability. In practice, due to testing limitations, the set of


Impulse
Hammer








structural modal parameters identified experimentally is incomplete with respect to the

analytical model. Experimental incompleteness is manifested in two forms, (i) a limited

number of measured modes of vibration, (ii) a limited number of eigenvector DOF

measurements.



2.4 Analytical and Experimental Model Dimensions Correlation


One major prerequisite common to most model refinement algorithms is to match the

number of degrees of freedom (DOFs) in the experimentally measured eigenvector and in

the discrete analytical model. Two different approaches have been commonly used to

resolve this matching problem when an incomplete set of eigenvector DOFs have been

measured. The first approach consists of reducing the dimension of the discrete analytical

model to the number of the measured DOFs. The other approach is to expand the measured

eigenvector to the size of the analytical model. A good evaluative survey of a number of

analytical model reduction techniques has been compiled by McGowan (1991). The subject

of eigenvector expansion is discussed in fair detail in the papers by Gysin (1990), and

Zimmerman and Kaouk (1992a). In the next two sections, some commonly used model

reduction and eigenvector expansion algorithms are reported and described.

2.4.1 Model Reduction Methods

In this section the general framework of the model reduction concept is first presented.

Then formulations of three commonly used model reduction techniques are summarized.

Mostly, the concept of model reduction has only been studied for undamped models. In this

presentation no attempt has been made to generalize these concepts to damped models.

For the purpose of reporting the general concept of model reduction, consider the

eigenvalue problem associated with an undamped model,


MVA2 + KV = [0]


(2.4.1)








where, as defined earlier, M and K are, respectively, the mass and stiffness matrix; V is the

eigenvector matrix and A is the diagonal eigenvalue matrix. Assume that only a subset of

the eigenvector DOFs has been experimentally measured.

Equation (2.4.1) can be reordered such that the DOFs associated with the measured

DOFs are in the upper rows of the equation,


MoVoA2 + KoVo = [0] (2.4.2)


Vm Kmm Kmu Mmm Mmu
where Vo = Vu Ko = Kum Kuu Mo Mum Mu


The matrices Mo, Ko, and Vo are, respectively, the reordered mass, stiffness and

eigenvector matrices. In the above equation, the subscripts "m" and "u" denote, respectively,

the components associated with the measured and unmeasured DOFs. A transformation

matrix P, that relates matrices Vu and Vm can be defined as

Vu = PVm (2.4.3)

A substitution of this relationship in the reordered eigenvector matrix Vo results in


Vo = p Vm = TVm (2.4.4)


Substituting Eq. (2.4.4) into Eq. (2.4.2) and premultiplying by TT yields the eigenvalue

problem of the reduced model,

MrVmA2 + KrVm = [0] (2.4.5)

where Mr = TTMoT

Kr = TTKoT

where Mr and Kr are the reduced mass and stiffness matrices, respectively. In terms of the

partitioned matrices, the reduced matrices are defined as








Mr = Mmm + pTMum + MmuP + pTMuuP (2.4.6)

Kr = Kmm + PTKum + KmuP + PTKuuP (2.4.7)

At this point, the only condition placed on P is the relationship of Eq. (2.4.3). Naturally,

matrix P can be computed directly from Eq. (2.4.3) if the eigenvectors of the system are

available. This approach is known as the exact reduction method and has been discussed in

the papers by Kammer (1987) and O'Callahan et al. (1989). The exact reduction method

requires solving for a large number of eigenvectors, which can be computationally

expensive. The three reduction methods that are presented in the forthcoming discussions

propose alternative techniques to compute matrix P. The first two do not require the

computation of the system eigenvalue problem. The last one requires the knowledge of one

eigenvalue which is computationally admissible.

2.4.1.1 Static Reduction

This reduction method is often referred to as Guyan (1965) reduction. In the static

reduction, the mass properties associated to the unmeasured DOFs are assumed negligible.

With that assumption, Eq. (2.4.2) can be written as

Mmm 0 Vm 2 Kmm Kmu Vm 0
[Mm O][VMA2 + [mm :] : = (2.4.8)
0 0 Vu Kum Kuu Vu 0

The second row of this matrix equation can then be manipulated as

Vu = K-uKumVm (2.4.9)

From comparing Eq. (2.4.9) to Eq. (2.4.3), it can deduced that the transformation

matrix, P, computed using the Guyan approach is given by

Pg = KualKum (2.4.10)

The reduced mass and stiffness matrices can then be computed by substituting matrix Pg

for matrix P in Eqs. (2.4.6) and (2.4.7). Naturally, the Guyan assumption (Eq. (2.4.8))








suggests that if the mass properties of the omitted DOFs are not small, the accuracy of the

Guyan reduced model could be lacking.

2.4.1.2 IRS Reduction

The improved reduction method (IRS) was formulated by O'Callahan (1989). It is an

improvement over the Guyan reduction in that it accounts for the mass properties of the

unmeasured DOFs. In the formulation of the IRS method, the Guyan reduced model is

corrected to include the mass influence of the unmeasured DOFs. This formulation is

somewhat lengthy and the interested reader is referred back to the paper of O'Callahan

(1989) or the thesis of McGowan (1991). The transformation matrix P computed using the

IRS reduction is

PIRS = Pg + KuU Mum + MuuPg MrKr, (2.4.11)

The reduced IRS model is then computed by substituting matrix PIRS in Eqs. (2.4.6-7).

2.4.1.3 Dynamic Reduction

The dynamic reduction was proposed as another improvement to the Guyan reduction

(Kidder, 1973; Miller, 1980; and Paz, 1984). This reduction utilizes the dynamic equation

associated with a single mode of vibration to compute the transformation matrix P. In this

technique, the transformation matrix P is arrived at by considering the reordered dynamic

equation associated with the ith mode,

[2Mmm + Kmm Mmu + Kmu m] (2.4.12)
I -(2.4.12)
?2Mum + Kum XMuu + Kuu JYu 0

where Xj is the ith eigenvalue; vmi and vYu are, respectively, the measured and unmeasured

eigenvector DOFs associated with the ith mode of vibration. Based on this partition, two

equations can be generated. By using the rows that correspond to the unmeasured DOFs

(lower rows), the following relationship is obtained,








Yu = [Muuux + Kuu] Mum + Kum] I, (2.4.13)

Hence, the transformation matrix associated with the ith mode is defined by

Pd, = [MuuX? + Kuu] [Mumk32 + Kum] (2.4.14)

The reduced mass and stiffness matrices are then computed by using Eqs. (2.4.6) and (2.4.7).
Note that different modes will result in different transformation matrices Pd and, hence,

different reduced mass and stiffness matrices.

2.4.2 Eigenvector Expansion Methods

Alternatively, the dimension of the measured eigenvectors can be correlated to the
dimension of the analytical model by using eigenvector expansion algorithms. The common
basis of these algorithms is the interpolation of the unmeasured eigenvector components. In
the forthcoming sections, two eigenvector expansion algorithms are reviewed.

2.4.2.1 Dynamic Expansion

The dynamic expansion technique (Berman and Nagy, 1983) is one of the most
commonly used eigenvector expansion algorithms. A slight modification of the Berman and
Nagy formulation is presented here to accommodate damped systems (Fuh et al., 1984). In
the formulation of the dynamic expansion, it is assumed that the measured modes satisfy the
eigenvalue problem involving the property matrices of the original model (M, D, K). For the
ith measured mode, this assumption takes the form

(k2iM + XeD + K) ve, = 0 (2.4.15)

where Xe. and ve. are, respectively, the ith experimental eigenvalue and eigenvector. The

matrices M, D and K have the same definitions as in the earlier sections. Assume that only
a subset of the DOFs of eigenvector ve. has been measured. Equation (2.4.15) can be

reordered, as in Section 2.4.1, such that the measured eigenvector DOFs reside in the upper
half of the equation,








Mmm Mmu] Dmm Dmu Kmm Kmu 1 e[ 0ei n211
Mum MuuJ e Dum Duu Kum Kuu LVXeuJ,


where Yemand Vu, are, respectively, the measured and unmeasured DOFs of ve.. The

subscripts "m" and "u" denote measured and unmeasured components. A rearrangement of
Eq. (2.4.16) yields


[Mmmn + keDmm + Kmm Mmu + ke,Dmu + Kmu em,i
I = (2.4.17)
L Mum + XeDum + Kum 2Muu + ke.Duu + Kuu jYeu,

From Eq. (2.4.17), two matrix equations in function of Vem, and veu, can be generated.

By using the equation associated with the second row of the partition, the unmeasured
components of the ith experimental eigenvector are computed to be


eu = [Muu + DXe + Ku] [MumX2J + Dumke, + Kum vemi (2.4.18)

Note that this expansion works on a single mode at a time. Also, notice that it involves
the original analytical model (M, D, K). This implies that the accuracy of the expansion
depends on the validity of the original analytical model.


2.4.2.2 Orthogonal Procrustes Expansion


Another expansion process that has shown great promise is the orthogonal Procrustes
(OP) expansion method presented in the papers of Smith and Beattie (1990) and Zimmerman
and Kaouk (1992a). The technique uses the general mathematical framework of the
orthogonal Procrustes problem (Golub and Van Loan, 1983).
Let Vem be the experimentally measured eigenvector component matrix and Vain be the
corresponding analytical eigenvector component matrix. The essence of the OP expansion is
to find a unitary transformation matrix Pop that closely rotates Vam into Vem. This is
attempted by solving the following problem,








Minimize 11 Vem VamPop IF
(2.4.19)
subject to pTpPop =

The solution to this general problem is discussed in the book of Golub and Van Loan and

is given by

Pop = YZT (2.4.20)

where Y and Z are, respectively the left and right singular matrices of matrix D defined by

(D = VTVem (2.4.21)

Let Vau be the eigenvector matrix associated with the unmeasured DOFs. In the

orthogonal Procrustes expansion, it is assumed that the transformation matrix computed

above also rotates Vau into the unmeasured "experimental" eigenvector component matrix,

Veu, as

Veu = VauPop (2.4.22)

Two different approaches have been defined to generate the expanded experimental

eigenvector matrix. The first is suggested in the paper by Smith and Beattie (1990),

Vam
Ve = Vau Pop (2.4.23)


In this definition, the expanded "experimental" eigenvector matrix is the rotated analytical

eigenvector matrix. The actual experimental measured eigenvector components Vem are

replaced by matrix VamPop. The advantage of the approach is that the resulting
"experimental" eigenvector satisfies the cross-orthogonality conditions (see Section 2.4).

The second viewpoint was proposed by Zimmerman and Kaouk (1992a),

[Vemn
Ve = [VauPopj (2.4.24)


Here, the unaltered eigenvector components measured experimentally are inserted in Ve. In

this viewpoint, if cross-orthogonalization of the expanded experimental eigenvectors is








required, a separated orthogonality algorithm can be used (Section 2.5). In the paper by

Smith at al. (1993), it is shown that for actual model refinement problems, both viewpoints

give equivalent eigenvector expansion results. However, for damage detection problems,

a preliminary study indicates that eigenvectors expanded by using the viewpoint defined in

Eq. (2.4.24) give better assessment of the damage.


2.5 Eigenvector Orthogonalization


Most matrix update algorithms require the measured eigenvectors to satisfy a

cross-orthogonality condition. This is especially true in matrix update algorithms in which

(i) the model of the structure is assumed undamped and the modelling errors are assumed to

be in only one of the two property matrices (M or K is assumed correct) or (ii) the system is

modeled by a proportionally damped model with errors in only two of the three property

matrices (M, D or K). In these situations, in order to insure symmetry of the updated property

matrices, it is required that the expanded experimental eigenvectors be orthogonal with

respect to the property matrix assumed accurate. This situation is encountered in the optimal

update algorithms developed by Baruch and Bar Itzhack (1978), Kabe (1985), Kammer

(1985), and Smith and Beattie (1991), Zimmerman and Kaouk (1992b), Kaouk and

Zimmerman (1993b) among others. In most of these algorithms, it is assumed that the mass

matrix is correct. This assumption is used in a number of the model refinement algorithms

since the inertial properties of structures are known to a good extent. In these cases, one

would expect the expanded experimental eigenvectors to be mass orthogonal. However,

because of measurements errors, this condition rarely occurs. For this reason, a great deal of

effort was focused on the development of mass orthogonalization techniques. Some of the

most notable work in that area was performed by Targoff (1976), Baruch and Bar Itzhack

(1978), and Baruch (1979). In the next two sections, two orthogonalization techniques

(Baruch and Bar Itzhack, 1978; Baruch, 1979) are discussed. Both techniques are mass

orthogonalization techniques; however, with obvious modifications, these techniques can be








adopted to solve the orthogonalization problem of the eigenvectors with respect to the

stiffness or the damping matrices.

2.5.1 Optimal Weighted Orthogonalization

The essence of the standard mass orthogonalization technique is to modify the measured

eigenvectors such that the mass cross-orthogonality condition is satisfied. Baruch and Bar

Itzhack (1978) proposed an elegant solution to that problem. An overview of their problem

statement and solution is given below.

Assuming that Ve is a matrix of expanded experimental eigenvectors that need to be

mass orthogonalized. The present formulation searches for the optimal mass weighted

change of matrix Ve such that the mass cross-orthogonality condition is satisfied. This

problem is cast as

Minimize N (Veo Ve) 1F (2.5.1)

subject to VT M Veo = I (2.5.2)

where N=M112 and M is the mass matrix. By means of a Lagrange multiplier, Eq. (2.5.2)

can be incorporated into Eq. (2.5.1); then the application of the optimality conditions yields

the following expression for Veo,

Veo= Ve(V M Ve)-1/2 (2.5.3)

Before being incorporated into the orthogonalization process, the measured

eigenvectors have to be unit mass normalized, i.e.,

Vej = ve,(vT M ve 12 (2.5.4)

where ve. is the it expanded experimental eigenvector (ith column of Ve).

2.5.2 Selective Optimal Orthogonalization

Some structures exhibit rigid body modes (modes with zero eigenvalues). It is desirable

to preserve these rigid body modes in the refinement process. However, some matrix update








algorithms require the rigid body modes and the experimental eigenvectors to be mass

orthogonal to insure symmetry of the updated property matrices (see Chapter 6). Naturally,

the rigid body modes will be corrupted if they are incorporated along with the expanded

experimental eigenvectors in the above orthogonalization process. Thus, Baruch (1979)

presented a modification of the procedure in Section 2.5.1 to deal with such a problem. The

resulting problem is a selective orthogonalization and is formulated as follows,

minimize 11 N (Veo Ve) 1F (2.5.5)


Subject to Veo M Veo = I (2.5.6)

and Veo M Vr = [0] (2.5.7)

In the above equations, Ve and Vr are, respectively, the expanded experimental eigenvector

matrix and the rigid body mode matrix. Again, the Lagrange multiplier is used, and the

orthogonalized experimental eigenvector matrix that satisfies the conditions in Eqs. (2.5.6)

and (2.5.7) is found to be


Veo= Q(QT M Q)-/2
(2.5.8)
where Q = Ve VrVTMVe

Note that, as in previous process, the expanded experimental eigenvectors have to be

unit mass normalized.


2.6 Load Path Preservation

Many matrix update algorithms introduce additional load paths in their updated models,

i.e., elements of the mass, damping or stiffness matrices that were originally zero may

become nonzero. Whether or not preserving the original load path is a practical problem is

still a matter of current debate. It seems that for damage assessment of truss structures it is

desired to maintain load paths. In the paper of Zimmerman and Kaouk (1992a), an iterative

approach to preserve the load path of the original property matrices was developed. The








approach was presented in the context of the symmetric eigenstructure assignment model

refinement algorithm (discussed in Chapter 4); however, its application can also be extended

to other model refinement algorithms. In Figure 2.3, a flow chart of the iterative load

preservation algorithm is presented. The procedure is illustrated for a general model

refinement scenario in which all three property matrices (M, D, K) are being updated.

However, it can be easily modified to accommodate other refinement problems.


Figure 2.3 Flow Chart of the Iterative Load Path Preservation Algorithm



In the flow chart, the matrices Ma,m, Da,m, and Ka,m are respectively the adjusted

masked mass, damping and stiffness matrices defined by








Ma,m = Ma 0 Mm

Da,m = Da 0 Dm (2.6.2)

Ka,m = Ka 0 Km

where Ma, Da, and Ka are the adjusted mass damping and stiffness matrices. The matrices

Mm, Dm, and Km are the masking matrices associated with the original mass, damping and

stiffness matrix. By definition, the masking matrix, Am, associated with matrix A is given

by

Am(i,j) = 1 if A(i,j) 0
(2.6.1)
Am(i,j) = 0 if A(i,j) = 0

In Eqs. (2.6.2), the operator 0 is the element-by-element (scalar) matrix multiplication. Let

B and C be two nxn matrices, then the element-by-element multiplication of B and C is given

by

S = B 0 C => S(i,j) = B(i,j) C(i,j) i,j = l,..., n (2.6.3)

At every iteration, the norms of the matrix differences between corresponding adjusted

and adjusted masked property matrices are computed. At a given iteration, if the three

computed norms are equal to zero or within user set limits, then the load paths of the original

three property matrices have been exactly achieved or achieved within user state guidelines.

Thus, the procedure is halted, and the refined model consists of the adjusted property

matrices computed at that particular iteration. It should be noted that there is no formal

guarantee of convergence in using this iterative procedure. Experience gained in using the

present algorithm indicates that if the experimental modal data are consistent with the

sparsity pattern, the procedure will converge. Consistent data means that there exist mass,

damping and stiffness matrices that have the same sparsity pattern as the original matrices

and also exhibit the measured test data. Otherwise, if the data are inconsistent, the original

sparsity patterns will not be exactly preserved. In this case, the added load path terms of Ma,

Da and Ka which should be zero will be closer to zero after application of the algorithm.















CHAPTER 3
INVERSE / HYBRID APPROACH FOR FINITE
ELEMENT MODEL REFINEMENT




3.1 Introduction

The inverse eigenvalue problem is concerned with the construction of the property

matrices (mass, damping or stiffness) of a dynamic model using experimentally measured

modal data. These techniques require complete modal properties. Thus, for an n-DOF

model, n natural frequencies, damping ratios and mode shapes (eigenvectors) must be

measured, and the identified mode shapes must be of dimension n. Due to practical testing

limitations, this is rarely accomplished for typical large structures. In this chapter, the

application of the inverse problem is extended to model updating by combining

experimental measurements and original analytical FEM modal information. Again,

refinement implies correlating the measured and analytical modal properties.


3.2 Theoretical Formulation

The dynamic structure under consideration is assumed to be successfully modelled by

an n-DOF proportionally damped nongyroscopic and noncirculatory (symmetric property

matrices) FEM. The free vibration motion of such a dynamic structure can be analytically

represented by a differential equation of the form

Mw(t) + Dw(t) + Kw(t) = 0 (3.2.1)

where the variables M, D, and K are nxn real symmetric matrix models of the mass, damping

and stiffness properties of the structure. The nxl time varying vector w(t) represents the n








displacements of the n-DOF model of the system. The overdots represent differentiation

with respect to time. The eigenvalue problem associated with the differential equation

shown in Eq. (3.2.1) is of the form

Mvy. X + Dv.i X + Kvy = 0 (3.2.2)

where Xi and yi are, respectively, the eigenvalue and eigenvector of the ith mode of vibration.

In this problem, it is assumed that the accuracy of the original FEM is lacking and, hence,

needs improvement. Furthermore, it is assumed that the inaccuracy of the original FEM is

solely due to modeling errors in the stiffness and damping properties.

The model refinement, proposed herein, exploits the cross-orthogonality relations that

arise from the symmetric nature of the property matrices and the proportional damping

assumption. As discussed in Chapter 2, by proper normalization of the eigenvectors these

cross-orthogonality relations have the form

VT M V = Inxn (3.2.3a)

VT D V = diag(2tlow, ,2tn n) = 5: (3.2.3b)

VTKV = diag(01 2, ,on2) = Q (3.2.3c)


V = [ ... V]

where wi and i are the natural frequency and damping ratio, respectively, of the ith mode

of the structure. The matrix Inxn is the nxn identity matrix. It is important to recognize that

Eq. (3.2.3a) represents necessary and sufficient conditions for conserving symmetry and

damping proportionality when updating the stiffness and damping properties of

proportionally damped systems.

Suppose that p (p << n) modes of an existing structure have been experimentally

identified (mode shapes or eigenvectors, frequencies and damping ratios). Assume that the

dimension of measured eigenvectors is equal to the dimension of the FEM, i.e., all n

components of the measured eigenvectors are available. It is widely accepted that in the








absence of specific experimental measurements a good approximation to the unmeasured

modes is their corresponding analytical modal information. With that in mind, a complete

hybrid set of modal data is generated by combining experimental and analytical information

Vea = [Ve Va]

[Qe 0
0 Qa (3.2.4)

le 0
lea = 0 a



where V is the eigenvector matrix; 2 and I are diagonal matrices of frequencies squared and

damping ratios, respectively. The subscripts e and a denote, respectively, experimental and

analytical sets. At this point, the complete "hybrid" set of eigenvectors, Vea, does not satisfy

the cross-orthogonality conditions defined in Eq. (3.2.3a); thus the conditions in Eqs.

(3.2.3b) and (3.2.3c) are not met. One possible solution to this problem is to modify all of

Vea in an optimal way to comply with the orthogonality requirement. This approach treats

all parts of Vea equally, and thus overlooks the fact that the experimental modes are known

with a higher confidence. Naturally, a technique that assigns a higher credibility to the

experimental eigenvectors, Ve, is preferable. This can be achieved by incorporating Vea into

the orthogonalization process group by group in the order of their descending credibility

(experimental then analytical). If the problem is set such that the experimental modes, Ve,

are corrected first, it is clear that the analytical modes, Va, will be subject to larger correction

when incorporated into the orthogonalization process since they will be subject to more

constraints.

The experimental modes, Ve, are orthogonalized by using the orthogonalization

technique formulated by Baruch and Bar Itzhack (1978). The general formulation of this

orthogonalization technique is discussed in Chapter 2. For this particular application, the

problem consists of finding the matrix ,Veo, that satisfies the cross-orthogonality condition,








Vo M Veo = Ipxp (3.2.5)

and that minimizes the weighted Euclidean norm,

(D = |N (Veo Ve) I
(3.2.6)
where N = M1/2

The solution to this problem, as reported in Chapter 2, is


Veo= Ve(V M Ve)-1/2 (3.2.7)

The next step is to invoke the orthogonality requirement on the analytical eigenvector

matrix, Va, by searching for a matrix Vao that satisfies the following two conditions,

Vao M Vao = I(n-p)x(n-p) (3.2.8)


and Vao M Veo = [0] (3.2.9)

while minimizing the objective function,

F =1 N (Vao Va) IIF
(3.2.10)
where N = M1/2


A similar problem was also treated by Baruch (1980) in a different context. A brief

discussion of the solution approach can be found in Chapter 2. The set of eigenvectors, Vao

that satisfies this problem is given by


Vao = Q(QT M Q)-/2
(3.2.11)
Q = Va VeoV oMVa


Clearly, the resultant matrix, Veao = [ Veo Vao ], satisfies Eq. (3.2.3a). The corrected
stiffness and damping matrices are then computed using Eqs. (3.2.2b) and (3.2.2c).








Ka = MVeao Q ea VTaoM (3.2.12a)

Da = MVeao E eaVeLoM (3.2.12a)

where Oea = 2Ileagea


Note that the matrices Ka and Da computed from Eqs. (3.2.12a,b) will be symmetric.

The above formulation suggests that the system modelled by the original mass matrix (M)

and the stiffness (Ka) and damping (Da) matrices computed from Eqs. (3.2.12a,b) will have

eigenvectors Veao, frequencies 4ea, and damping ratios Hea. Some structures exhibit rigid

body modes of vibration. Commonly, it is desirable to preserve these rigid body modes in the

updated model. The above formulation also suggest that the updated model will preserve the

original rigid body modes.

The procedure developed above can be easily contracted to address the case when the

system model does not account for the effects of damping (undamped model). The

contraction can be obtained by setting to zeros matrices D, Oea, and FIea in Eqs. (3.2.2-4).

The computational burden limits the size of the FEM which can be updated (order of

200). Essentially, the limiting factor is that all mode shapes of the structure that are not

available from experimental measurements must be calculated analytically by solving the

eigenvalue / eigenvector problem.


3.3 Numerical Illustration

The system addressed in this investigation is the commonly used eight degrees of

freedom model shown in Figure 3.1. This model was developed by Kabe (1985) to give a

common testbed for the evaluation of the performance of model refinement algorithms. An

original undamped analytical model of the system was generated by using the mass and

stiffness properties shown in Figure 3.1. The elements of the original analytical stiffness

matrix are displayed in the second column of Table 3.1. In this problem, it is assumed that the

original stiffness matrix of the model is incorrect. These inaccuracies were simulated by

























mi =0.001 m8= 0.002 mj=1.0 j=2,..., 7

k1 = 1000 k2 = 10 k3 = 900 k4 = 100 k5 = 1.5 k6 = 2.0


Figure 3.1 Kabe's Problem: Analytical Test Structure.


using incorrect stiffness constants for most of the springs. The elements of the actual correct

stiffness matrix are given in the last column of Table 3.1. Note that the present model

refinement problem is very challenging and because of the large difference between the

stiffness matrices of the original and exact model. In this study, two cases will be considered.

In the first case, it is assumed that only the modal parameters (eigenvalue and eigenvector) of

the first mode were measured. In the other, the modal parameters of first three modes are

assumed to be available. In both cases, the measured eigenvectors are supposed to be full (all

degrees of freedom of the eigenvector(s) are measured).

The main objective of this investigation is to compare the performance of the

inverse/hybrid method to the algorithm proposed by Baruch and Bar Itzhack (1978). The

Baruch and Bar Itzhack model update technique is one of the most commonly used model

refinement algorithm. One of the main reasons for its common use is because the algorithm

is computationally efficient. The updated stiffness matrices generated by using the Baruch

and Bar Itzhack approach for the one mode and three mode cases are shown in the third and








fourth column of Table 3.1, respectively. The fifth and sixth columns of Table 3.1 display the

results of using the inverse/hybrid algorithm for the one mode and three modes cases. For

both cases, it is clear that the performances of both algorithms is lacking in predicting the

exact stiffness matrix. As expected, It can also be seen that both algorithms generate better

results when using three measured modes. A comparison of the results generated using the

Baruch and Bar Itzhack approach and the inverse/hybrid method shows that both algorithms

give the same type of results. This similarity of results was also encountered in other

example problems.


3.4 Summary

A model refinement approach that uses a hybrid set of experimental and analytical

modal properties was formulated. The developed approach, termed the inverse/hybrid

algorithm, was illustrated by using a computer simulated example. Part of the evaluation of

the proposed algorithm was the comparison of its performance with the performance of the

Baruch and Bar Itzhack algorithm. It was found that both algorithms give the same type of

results. However, the computations involved in the inverse/hybrid approach exceed those

involved in the Baruch and Bar Itzhack algorithm. Essentially, the inverse/hybrid approach

requires the computation of all eigenvalues and eigenvectors of the structure that are not

available from experimental measurements. As will be illustrated in the forthcoming

chapter, the Baruch type approaches are not suited for damage assessment applications. For

these reasons, further development of the Inverse/Hybrid algorithm was not investigated and

new formulations (Chapters 4 & 6) were considered.








Table 3.1 Kabe's Problem: Elemental Stiffness Components.


Baruch Inverse/Hybrid
Element # Original Mode 1 Modes 1-3 Mode 1 Modes 1-3 Exact
(1,1) 2.0 2.0 2.0 2.0 2.0 1.5
(1,2) -2.0 -2.0 -3.0 -2.0 -3.0 -1.5
(1,3) 0.0 0.0 -0.1 0.0 -0.1 0.0
(1,4) 0.0 0.0 -0.1 0.0 -0.1 0.0
(1,5) 0.0 0.0 -0.1 0.0 0.1 0.0
(1,6) 0.0 0.0 0.1 0.0 0.1 0.0
(1,7) 0.0 0.0 0.1 0.0 0.0 0.0
(1,8) 0.0 0.0 0.0 0.0 0.0 0.0
(2,2) 1512.0 1508.6 1024.2 1510.6 1024.3 1011.5
(2,3) -10.0 -31.5 -68.5 -21.5 -71.4 -10.0
(2,4) 0.0 -8.9 -9.0 -15.0 -8.4 0.0
(2,5) 0.0 -8.9 -20.9 -15.0 23.8 0.0
(2,6) 0.0 -21.6 38.5 -11.5 35.2 0.0
(2,7) 0.0 -3.9 -9.1 -1.6 -8.7 0.0
(2,8) 0.0 -0.1 0.1 -0.1 0.1 0.0
(3,3) 1710.0 1574.6 1560.8 1624.1 1612.2 1110.0
(3,4) 0.0 -44.9 -49.8 -75.6 -76.2 0.0
(3,5) -200.0 -244.8 -244.1 -275.6 -276.9 -100.0
(3,6) 0.0 -136.5 -123.4 -86.2 -74.5 0.0
(3,7) 0.0 -24.6 50.8 -13.1 46.8 0.0
(3,8) 0.0 -0.4 -0.1 -0.4 0.0 0.0
(4,4) 850.0 1083.1 1087.9 1102.1 1099.9 1100.0
(4,5) -200.0 3.27 25.1 51.8 48.0 -100.0
(4,6) -200.0 -254.3 -242.2 -276.5 -274.9 -100.0
(4,7) 0.0 -10.1 27.0 -17.2 30.8 0.0
(4,8) 0.0 -0.2 -0.1 -0.2 -0.2 0.0
(5,5) 850.0 1082.2 1089.5 1101.4 1101.1 1100.0
(5,6) 0.0 -45.2 -49.5 -76.5 -75.1 0.0
(5,7) 0.0 -10.1 -12.4 -17.2 -11.7 0.0
(5,8) 0.0 -0.2 -0.1 -0.2 -0.2 0.0
(6,6) 1714.0 1576.4 1565.0 1617.5 1610.8 1112.0
(6,7) -10.0 -34.8 -82.6 -23.1 -76.2 -10.0
(6,8) -4.0 -4.4 -4.7 -4.4 -4.5 -2.0
(7,7) 1512.0 1507.5 1027.9 1510.1 1028.0 1011.5
(7,8) -2.0 2.1 -4.1 -2.1 -4.2 -1.5
(8,8) 6.0 6.0 6.0 6.0 6.0 3.5














CHAPTER 4
SYMMETRIC EIGENSTRUCTURE ASSIGNMENT MODEL
REFINEMENT ALGORITHM




4.1 Introduction

Eigenstructure assignment is a control concept used to alter the transient response of

linear systems. This is done by forcing the system to have some predetermined eigenvalues

and eigenvectors. A detailed overview of eigenstructure assignment theories can be found in

the paper by Andry et al. (1983). Inman and Minas (1990), Zimmerman and Widengren

(1989, 1990), and Widengren (1989) have developed model refinement algorithms based on

the mathematical framework of eigenstructure assignment. The basic idea of these model

refinement techniques is to design the pseudo-controller which is required to produce the

measured modal properties (natural frequencies, damping ratios and mode shapes) with the

original finite element model (FEM) of the structure. The pseudo-controller is then

translated into matrix adjustments applied to the initial FEM.

In this work, the eigenstructure assignment based model refinement algorithm proposed

by Zimmerman and Widengren (1989, 1990) is extended to better approach the damage

assessment problem. A subspace rotation algorithm is developed to enhance eigenvector

assignability. Finally, the enhanced algorithm is tested and compared to other techniques on

both "simulated" and actual experimental data.


4.2 Problem Formulation

In this section, a review of the Zimmerman and Widengren (1989, 1990) refinement

technique, which is termed the symmetric eigenstructure assignment model refinement








algorithm (SEAMRA), is presented. This review is essential in order to properly introduce

and discuss the extension and improvement proposed in this work.

4.2.1 Standard Eigenstructure Assignment Formulation

Consider the standard differential equation of motion of an n degrees of freedom

damped, nongyroscopic and noncirculatory structure with control feedback,

Mw(t) + Dw(t) + Kw(t) = Bou(t) (4.2.1)

Again, M, D, and K are n-by-n real symmetric matrix models of the mass, damping and

stiffness properties of the structure. Assume that these matrices were generated using the

finite element method. The nxl time varying vector, w(t), represents the n displacements

of the n-DOF FEM of the system. The overdots represent differentiation with respect to

time. In control terminology, BO is the nxm (m << n) control influence matrix describing

the actuator force distributions and u(t) is the mx 1 vector of output feedback control forces

defined by

u(t) = Fy(t) (4.2.2)

In Eq. (4.2.2), F is the mxr feedback gain matrix and y(t) is the rxl output of sensor

measurements defined by

y(t) = Cow + C1w (4.2.3)

in which Co and C1 are the rxn output influence matrices corresponding to position and

velocity, respectively. A substitution of Eqs. (4.2.2) and (4.2.3) into Eq. (4.2.1) yields

Mw(t) + (D BoFC1)w(t) + (K BoFCo)w(t) = 0 (4.2.4)

It is clear, from Eq. (4.2.4), that the feedback controller results in residual changes, BoFCo

and BoFC1, to the stiffness and damping matrices, respectively. These changes can be

viewed as perturbations to the initial finite element model (FEM) such that the adjusted FEM

matches closely the experimentally measured modal properties. The adjusted FEM consists

of the original mass matrix and the adjusted stiffness and damping matrices given by







Ka K
Da = D


- BoFCo
- BoFC1


(4.2.5)


Assume that modal analysis of the structure under consideration has been performed and that
p modes (p eigenvalues Xei, and p eigenvectors Ve) have been identified. As discussed earlier
in Chapter 2, in practice p is typically much less than n. The feedback gain matrix F, such
that the adjusted FEM eigendata matches the experimental modal parameters, is computed
using standard eigenstructure assignment theories (Andry et al. 1983):


F = [Z A f[C C -1


where


(4.2.6)


A0 Inxn
A M-1K -M-iD


B = M-1K


T = [B P]


A = T-'AT = A]



V = T-1 W W
WA WA


STB = Imxm
0


A = diag(Xe, ,e2, ., ep)


w = [eaivea,, .-,Veap]


Z = S-1[ W A [ A
WA WA


The overbar in the above equations indicates the complex conjugate operator. The vectors

Yea, in matrix W are the expanded "best achievable" eigenvectors associated with the
experimentally measured eigenvectors ve,. An explanation of the concept of "best


T- S
T- =1
S2








achievable" eigenvectors is discussed in Section 4.2.3. The submatrix P of matrix T is

arbitrary as long as T is invertible.

At this point, the variables BO, Co and C1 are still arbitrary. A random selection of these

variables will usually result in nonsymmetric perturbation matrices and, consequently

nonsymmetric adjusted stiffness and damping matrices. This clearly conflicts with the

fundamental symmetry requirement of most structures' FEM. In the formulation of Inman

and Minas (1990), the resulting perturbation matrices from the pseudo-controller are forced

to be symmetric through a nonlinear unconstrained optimization problem. Zimmerman and

Widengren (1989, 1990) proposed a non-iterative and computationally more efficient

approach to satisfy the symmetry requirement. This approach in discussed in the following

section.

4.2.2 Symmetric Eigenstructure Assignment Formulation

The perturbation matrices are symmetric if the following conditions are met,

BoFCo = CF TBo (a)
(4.2.7)
BoFC1 = CF TB (b)

At this point, two additional assumptions are made. As a prerequisite to the existence of the

inverse of some matrices used in the computations, it is assumed that the number of pseudo

sensors and actuators is equal to twice the number of measured modes (m=r=2p). The other

assumption consists of restricting the matrices Co and C1 by the conditions

CO = GoBo (a)
0 (4.2.8)
C1 = GIBT (b)

where Go and G1 are mxm invertible matrices. A substitution of Eq. (4.2.8) into Eq. (4.2.7)

simplifies the symmetry conditions to the following relationships,

FGo = GTFT (a)
(4.2.9)
FGi = G TFT (b)








By using the conditions in Eq. (4.2.9), along with the expression for the feedback gain matrix

(Eq. (4.2.6)), a necessary but not sufficient condition on Go and GI, for symmetric

perturbation matrices, is expressed in the form of a generalized algebraic Riccati equation,

A1X + XA2 + XA3X + A4 = [0] (4.2.10)

where

X-GG'G0

A, = ) T]

A2 =*- 1- 1 *


A3 = *-1-1(a ** a )~-*

A4 = o*a-10-la* Imxm


[w*BO' "AW*BO]
T= a = a= Z- AIV
W Bo AW Bo



The matrices Al, W, Z, and V are defined in Eqs. (4.2.6). The superscript ()-* indicates the

inverse of the complex conjugate transpose matrix. Equation (4.2.10) can be solved for X

by using the techniques described in the papers of Potter (1966) or Martensson (1971). In

general, there exist multiple solutions (X's) to this generalized algebraic Riccati equation.

With all solutions computed, the next step is to decompose these solutions into Go's and GI's.

It is shown in the paper by Zimmerman and Widengren (1989) that for a given solution X,

any selection of Go and GI satisfying X = G- 1Go results in the same adjusted damping

(Da) and stiffness (Ka) matrices. Hence, either GI (or Go) can be chosen arbitrarily, as long

as its inverse exists. Then, Go (or GI) is calculated from the relationship X = GI 'G0.

For each set (Go, G1), a feedback gain matrix F is calculated from Eq. (4.2.6), and the

corresponding adjusted damping (Da) and stiffness (Ka) matrices are computed using Eqs.








(4.2.5). At this point, a rationale is proposed to choose the most meaningful adjusted

damping and stiffness matrices. Among all computed sets (Da, Ka), it is apparent that only

the ones that are real and symmetric are acceptable. When dealing with a model refinement

problem, among all acceptable solutions, the final selection could be made by choosing the

set (Da, Ka) that minimizes the cost function,

J = q |I K Ka IIF + I| D Da IIF (4.2.11)

where q K IF



Clearly, this process selects the set (Da, Ka) that results in a minimum change from the

original set (D, K). The scale factor q in Eq. (4.2.11) is used to give equal weight to the

changes in D and K. For the damage detection problem, there is no unique rationale to

choose the "best" set (Da, Ka). A physically intuitive approach is to use engineering

judgement in selecting the "best" updated model. Thus, all acceptable "adjusted" sets of

solutions should be inspected to determine which best provides information concerning the

state of damage.


4.2.3 Best Achievable Eigenvectors


From standard eigenstructure assignment theory (Andry, et al. 1983), it is shown that the

measured eigenvectors are not always exactly assignable to the adjusted finite element

model. In fact, it can be shown that the measured eigenvectors are assigned exactly if and

only if they lie in their respective achievable subspace. The achievable subspace associated

to the ith mode is defined by

L, = (MX? + DXi + K)-B0 (4.2.12)


where 1i is the measured eigenvalue of the ith mode. When all n components of the

experimental eigenvectors are available, the ith best achievable eigenvectors is defined as the








least square projection of the ith experimental eigenvector Ve, on the ith achievable subspace

Li. This projection is schematically illustrated in Figure 4.1.




Achievable Subspace








Figure 4.1 Best Achievable Eigenvector Projection.



This best achievable eigenvector is given by


yea = LL L] L ve (4.2.13)


When only a subset s of the eigenvector components are measured, s < n, the least square

projection discussed above can be used to simultaneously expand and project the measured

eigenvectors. In this case, the ith expanded best achievable experimental eigenvector is

given by


Yea, = Li i Li LiVe (4.2.14)


where Li are the rows of Li which correspond to the measured eigenvector components.

Notice that the calculation of the p achievable subspace using Eq. (4.2.13) requires p

inversions of an nxn matrix. Although the matrix to be inverted is typically banded, this

may present a practical computational burden when dealing with large FEMs. The next

section discusses an approach that does not require the actual computation of the achievable

subspaces and hence avoids this computational burden.








4.2.4 Selection of Bu : The Subspace Rotation Method

So far, the control influence matrix has not yet been completely defined. The preceding

formulation suggest that different Bo may possibly result in different adjusted FEM. Hence,

it is essential to develop a physically meaningful rationale to select B0.

Zimmerman and Widengren (1989, 1990) proposed an approach, termed the mode

selection method, that consists in selecting Bo such that the unmeasured modes of the

structure are nearly unchanged. In other word, BO is selected such that only the measured

modes of the structure are corrected. This selection technique fixes the achievable subspaces

in which the eigenvectors must lie, and hence places a limitation on the assignment process.

In most studied cases, the experimental eigenvectors were not assigned exactly since their

assignment "success" depends on the locations vis-a-vis the achievable subspaces set by the

selection of B0.

In this work, a new method of selecting BO, termed the subspace rotation method, is

proposed. The subspace rotation method is based on selecting BO such that the measured

eigenvectors lie exactly in the achievable eigenvectors subspaces. This procedure is

illustrated in Figure 4.2 and is accomplished by setting BO as

B0 = [br, br2, ... ,brp I bi, bi, ... bi] (4.2.15)

where br = real [(MX2 + DXk + K)ve]


bi = imaginary [(MX2 + DX, + K)v

where ve. is the eigenvector associated with the jth experimentally measured mode, and it

is assumed that all n components of the experimental eigenvectors are available. This could

be accomplished by any of the procedures discussed in Chapter 2. Clearly, when Bo is

selected as shown in Eq. (4.2.15), the measured expanded eigenvectors lie exactly in the

achievable subspaces defined in Eq. (4.2.12). Hence, there is no need for the projection

operations defined in Eq. (4.2.13). This eliminates the required p inverses of nxn matrices








involved in computing the achievable subspaces. As will be seen in Chapters 5 and 6, the

elements of BO, as defined by Eq. (4.2.15), give an indication to the pseudo-controller about

the extent of modification of each DOF in order for the structure to exhibit the jth measured

eigenvalue and eigenvector.






Rotated Subspace --


Nominal Subspace







Figure 4.2 Rotation of the Achievable Subspace.





4.3 Numerical Illustrations



In this section, the characteristics of the proposed enhancement to the symmetric

eigenstructure assignment model refinement algorithm (SEAMRA) are evaluated and

compared to other refinement techniques for two example problems. The first problem is a

widely-used spring-mass computer simulated example (Kabe, 1985). It is used here for the

purposes of illustrating model refinement for a large local discrepancy, analogous to a

damage detection situation. The phenomena of global/local mode switching and load path

preservation are examined in this problem. The second problem is used to illustrate the

characteristics of the enhanced SEAMRA in updating the finite element model (FEM) of a

laboratory cantilever beam using actual measured modal parameters.








4.3.1 Damage Detection: Kabe's Problem


Kabe's eight degree of freedom spring-mass system is shown in Figure 4.3. The mass

and stiffness properties of the system are included in the figure. This problem presents a

challenging situation for damage detection in that stiffness values of various magnitudes are

included. The model exhibits closely-spaced frequencies and both local and global modes of

vibration.


m =0.001 m = 0.002 m = 1.0

ki = 1000 k2 = 10 k3 = 900 k4 = 100


j =12,.. 7


Figure 4.3 Kabe's Problem.




A variation of Kabe's original problem is used here. Rather than the standard initial

model commonly used, which has incorrect values for all of the connecting springs, only a

single spring constant is changed. This is reflective of the fact that damage may occur as a

large local change in the stiffness of a structural member.








4.3.1.1 Local to Global Mode Change


In the first problem, Kabe's initial model is only incorrect for the spring between

masses 3 and 5. A value of 500, five times that of the exact spring, is assumed in this problem.

Changing the spring value from 500 to 100 also causes a local mode of vibration to be

replaced by a global mode, thus presenting a difficult challenge for damage detection.

Figure 4.4 presents element-by-element stiffness matrix results for applying the Baruch

and Bar Itzhack update (1978) and the symmetric eigenstructure assignment model

refinement algorithm. Baruch Damage indicates that the update was made using Baruch and

Bar Itzhack's algorithm. SEA-M Damage indicates that the update was made using the

SEAMRA with BO selected by using the modal (M) selection method. SEA-SR Damage

indicates that the update was made by using the SEAMRA with BO selected using Subspace

Rotation (SR) method. The x-coordinate on all plots are the indices of a column vector

constructed by storing the upper triangular portion of the stiffness matrix in a column vector.

The y-coordinate on each plots consists of the difference between the updated stiffness

matrix elements and the original stiffness matrix.

In the first case, as shown in Figure 4.4, it is assumed that only the fundamental mode of

vibration is measured, but all eigenvector components have been measured. Thus, no

expansion of eigenvectors is required. It is evident from Figure 4.4 that the Baruch update is

unable to discern the damage, but that both the SEA-M and SEA-SR are able to clearly locate

the damage. In fact, the SEA-SR was able to exactly reproduce the correct stiffness matrix.

This was true independent of which mode was used in the update. Also, it should be noted

that the Baruch update tends to focus elemental changes in the third and fifth row of the

stiffness matrix, indicating the possibility of damage between these degrees of freedom, but

certainly giving no clear indication to the extent of damage. As is evident from the plot, the

Baruch update has spread errors over several elements. Using the algorithm of Lin (1990),

the damage vector is given as a = [1.0 0.93 0.72 0.83 0.70 0.90 0.97 1.0]T, where the

element number corresponds to the structural DOFs and a number less than 1 indicates the



















-500


Indices


Baruch Damage


JUULI I I i


0 10 20
Indices


0 10 20 30
Indices


0 10 20 30
Indices


Figure 4.4 Results for Kabe's Problem using the 1st Mode, Full Eigenvector.



possibility of damage affecting that DOF It is obvious that DOFs 3 and 5 are affected by

damage, but the results also indicate strong damage of DOF 4.

In the second case, as shown in Figure 4.5, it is assumed that the first three modes of

vibration have been measured, but only the first three components of the eigenvectors have

been measured. The eigenvectors components were expanded for the Baruch update using

dynamic expansion (Berman and Nagy, 1983) with subsequent orthogonalization (Baruch

and Bar Itzhack, 1978). The least squares expansion was used for the SEA-M update. The

SEA-SR update utilized the orthogonal Procrustes expansion (Chapter 2). In comparing

Figure 4.5 to Figure 4.4, it is clear that the damage detection capabilities of all three

algorithms have been degraded when using expanded mode shapes, even though more

modes have been measured. However, both the SEA-M and SEA-SR updates give a clear


enr\




















-500W '
0 10 20
Indices

500 __ SEA-M Dan
5001- 1


-500
0


Baruch Damage


3UUr j I "


0-


I I I


10 20 30
Indices


0


10 20
Indices

SEA-SR Dar


0 10 20 30
Indices


Figure 4.5 Results for Kabe's Problem using Modes 1, 2, 3,
and Eigenvectors Components 1, 2, 3.




indication to both the location and extent of damage. Using Lin's algorithm, the damage

vector is given as a = [1.0 0.81 0.75 0.83 0.82 0.79 0.85 1.0]1. It is difficult from

inspection of a to determine the location of damage.

The effect of applying the iterative load path algorithm described in Chapter 2 in the

update procedure is shown in Figure 4.6. For the Baruch update, 100 iterations were

performed. For the SEA-M and SEA-SR updates 2 and 3 iterations respectively, were

performed. The iterations were halted early for both SEA updates because the discrepancy

between the eigenstructure before and after masking was within the numerical precision of

the symmetric eigenstructure assignment software. It is seen that the load path enforcement

further enhances the damage detection capability of both SEA updates.


r f


0n


-









_500 Actual Damage




0----- ---,----


-500
0


10 20 30
Indices


0d


0-


-500 L
0


0 10 20 30
Indices


Baruch Damage


10 20
Indices


20
Indices


Figure 4.6 Results for Kabe's Problem using Load Path Preservation,
Modes 1, 2, 3, and Eigenvectors Components 1, 2, 3.



4.3.1.2 Consistent Modes


In the second problem, the initial model is only incorrect for the spring between masses 4

and 6. A value of 200, two times that of the exact spring, is assumed in this problem. In this

problem, all global and local modes remain global and local modes respectively after

damage. It should be noted that finding a problem with this feature was difficult.

In the first case, as shown in Figure 4.7, it is assumed that only the fundamental mode of

vibration is measured, but all eigenvector components have been measured. It is evident

from Figure 4.7 that the Baruch and SEA-M update are unable to discern the damage, but

that the SEA-SR is able to clearly locate damage. In fact, the SEA-SR was able to exactly

reproduce the correct stiffness matrix. Again, this was true independent of which mode was









Actual Damage Baruch Damage

t100- n 100-


0 0------^-- 0 0 -----jU


-100- -100-

0 10 20 30 0 10 20 30
Indices Indices

SEA-M Damage SEA-SR Damage

100 100





-100 -100-
I II
0 10 20 30 0 10 20 30
Indices Indices

Figure 4.7 Results for Kabe's Problem using the 1st Mode, Full Eigenvectors.



used in the update. It should be noted that the Baruch and SEA-M update tends to focus

elemental changes in the fourth and sixth rows of the stiffness matrix, indicating the

possibility of damage between these degrees of freedom, but certainly giving no clear

indication to the extent of damage. Using Lin's algorithm, the damage vector is given as a =

[1.0 0.98 0.92 0.85 0.87 0.84 0.95 1.0]T. This algorithm does not clearly identify the

damage location.

In the second case, as shown in Figure 4.8, it is assumed that the first three modes of

vibration have been measured, but only DOFs 1,3, and 6 of the eigenvectors have been

measured. In comparing Figure 4.8 to Figure 4.7, it is clear that the damage detection

capability of all three algorithms has again been degraded when using expanded mode

shapes. Only the SEA-SR update gives a clear indication to the location of damage, but is








unable to predict the exact extent. Using Lin's algorithm, the damage vector is given as a=

[0.99 0.18 0.55 0.34 0.52 0.33 0.41 1.0]T. Again, it is difficult from inspection of a to

determine the location of damage. In fact, inspection of a indicates that DOF 2 is the most

likely damaged DOF.


100


0


-100


0 10 20 30
Indices

SEA-M Damage


10 20 30
Indices


01


Baruch Damage


-100l


D 10 20 30
Indices

SEA-SR Damage


100K


0


-100


0 10 20 30
Indices


Figure 4.8 Results for Kabe's Problem using Modes 1, 2, 3,
and Eigenvectors Components 1, 2, 6.





It should be noted that in this problem, it was critical to have the proper DOFs measured.

When the second test case was run with the first three DOFs measured, no algorithm was able

to locate damage. In this case, the eigenvectors components were relatively unaffected by

damage, thus causing substantial error in the eigenvector expansion process.








4.3.2 Model Refinement of a Cantilever Beam: Experimental Study


4.3.2.1 Modal Test Description

The structure used in this investigation is the aluminum cantilevered beam shown in

Figure 4.9. The dimensions and material properties of the beam are given in Table 4.1.

Experimental modal analysis of the beam was performed to measure its modal properties.

Six equally spaced translational degrees of freedom shown in Figure 4.9 were selected as

measurement locations. The modal properties of the first three modes of vibration were

determined using frequency domain techniques and single degree of freedom curve fitting

algorithms. The excitation source used in the testing was an impact hammer and the driving

point measurement was an accelerometer mounted at the free end of the beam. Impact and

exponential windows were utilized to improve frequency response calculations. At each

measured degree of freedom, five frequency response measurements were made and

averaged to reduce the effects of measurement noise. Natural frequencies, damping ratios

and mode shapes of the beam's first three modes of vibration were identified and are reported

in Tables 4.2 and 4.3.


6 5


4 3 2 1


Figure 4.9 Experimental Cantilever Beam.








Table 4.1 Structural Properties of the Cantilever Beam.


Table 4.2 Measured Natural Frequencies and Damping Ratios of the
Cantilever Beam.

Mode # Natural Frequency (Hz) Damping Ratio
(%)
1 7.25 4.41
2 45.55 0.68
3 127.01 0.33


Table 4.3 Measured Mode Shapes of the Cantilever Beam.

Mode 1 2 3
DOF
1 1.00 1.00 1.00
2 0.95 0.16 -0.30
3 0.65 -0.53 -0.61
4 0.36 -0.72 0.20
5 0.15 -0.52 0.75
6 0.03 -0.12 0.28


4.3.2.2 Finite Element Model Description


A twelve DOF undamped finite element model (FEM) of the beam was generated using

six equal length beam elements as shown in Figure 4.9. The beam element has two degrees of

freedom (DOFs) at each node; bending and rotation. This model was then reduced using


Length 0.84 m
Mass/Length 2.364 kg/m
Moment of Inertia 3.02e-9 m4
Youngs Modulus 70 GPa








Guyan reduction (1965) eliminating the rotational degrees of freedom. There are several

possible errors affecting the accuracy of this FEM. The most obvious is the fact that a perfect

cantilever condition is assumed. In addition, an artificial error was purposely introduced by

selecting the Young's Modulus higher than that often assumed for aluminum.

4.3.2.3 Application of the SEAMRA.

Because the "true" finite element model is unknown, a comparison between the "true"

and updated structural matrices is not possible. Besides such comparison, a fair and useful

judgement on the quality of SEAMRA updating capability can be obtained by comparing

actual experimental frequency response functions with those predicted by the initial and

updated FEMs. Figure 4.10 shows a comparison of frequency response functions measured

between degrees of freedom 1 and 3 (i.e. sensor measurement at DOF 1 and impact excitation

at DOF 3). The solid curve corresponds to the experimental data, while the dotted line

corresponds to that predicted by the original analytical FEM. It is apparent that the

discrepancies between the frequency response function increases as the frequency of

excitation increases. This is in part due to the fact that the assumption of a perfect cantilever

condition affects the higher modes of vibration to a greater extent. The dashed lines in this

figure corresponds to the SEA-SR updated finite element model. It is clear from this

comparison that the SEA-SR provided a great deal of improvement to the original analytical

FEM. Inspection of the updated stiffness matrix indicates that changes occur throughout the

matrix, indicating that the discrepancy between the original and refined FEMs was due to

degradation of some global structural property (Youngs Modulus), as opposed to some form

of local damage, as seen in the previous problem.


4.4 Discussion of the SEAMRA's Formulation

In some problems SEAMRA in conjunction with either the subspace rotation or the

modal selection method, failed to find a symmetric updated FEM (symmetric Da and Ka).

This shortcoming was especially encountered in practical situations when the experimental









SEAMRA
10-2 ,
Solid Experimental Measurement
Dash Modified Analytical Model
10-3 Dotted Original Analytical Model


10-4


10.-5 ,


10-6 .,


10-7


10-8 -----
0 20 40 60 80 100 120 140 160 180 200

FREQUENCY (Hz)


Figure 4.10 Experimental and Analytical Frequency Response
Functions of the Cantilever Beam.



modal properties were corrupted by measurement errors. This can be attributed to the fact

that there is no symmetric updated FEM that is consistent with the present SEAMRA's

formulation. Recall that SEAMRA's modifications to the initial stiffness and damping

matrices in its updating process are given by

AK = BoHoBT with Ho = FGo
(4.5.1)
AD = BoHIBo with H, = FG1

Clearly, from Eq. (4.5.1), the perturbations (i.e. modifications) to the initial stiffness and

damping matrices are constrained by the relationship


range(AK) = range(AKT) = range(AD) = range(ADT)


(4.5.2)








This relationship can always be satisfied for the cases when the experimental modal

properties are consistent with an update in which (i) either AK or AD is zero or (ii) AK is

proportional to AD (AK = P AD, P3 is a scalar). For all other cases, SEAMRA might fail to

produce symmetric AK and AD. A more flexible and general formulation that accounts for

such shortcoming is proposed in Section 6.5 of Chapter 6. The formulation as presented in

Chapter 6 is more elegant, efficient and guarantees, for all situations, a symmetric updated

FEM.


4.5 Summary

A previously developed model refinement algorithm based on the general mathematical

framework of eigenstructure assignment theory (Zimmerman and Widengren, 1989, 1990)

has been extended and improved. A technique to enhance eigenvector assignability of the

algorithm has been developed. The method consists of rotating the achievable eigenvector

subspaces into the experimentally measured eigenvectors. The subspace rotation method,

used in conjunction with one of eigenvector expansion techniques discussed in Chapter 2,

results in both a decrease in the computational burden as well as an increase in the accuracy of

the assigned eigenvectors. Finally, the improved algorithm (SEA-SR) was tested for its

suitability for model refinement and structural damage assessment. The performances of

SEA-SR in a damage assessment problem on a challenging simulated structure was

presented and compared to other algorithms. The results acquired using the SEA-SR were

superior.















CHAPTER 5
DAMAGE LOCATION: THE SUBSPACE ROTATION ALGORITHM



5.1 Introduction

In this chapter, a computationally attractive algorithm is proposed to provide an insight

to the location of structural damage. The proposed algorithm is similar to the Modal Force

Error Criteria proposed by several researchers ( Ojalvo and Pilon 1988; Gysin, 1990).

However, a greater insight of the Modal Force Error criteria is provided. Further, a new

viewpoint which allows for the reduction of the effects of measurement errors in the

experimental modal parameters for a certain class of structures is also discussed. As will be

shown in the next sections, the proposed damage location algorithm requires only

matrix-scalar and matrix-vector multiplication.


5.2 The Subspace Rotation Algorithm: The Direct Method

Assume that an n-DOF finite element model of the "healthy" (undamaged) structure

exists. As seen in the earlier chapters, the standard differential equation governing the

dynamic motion of such structures is given by

Mw + Dw + Kw = 0 (5.2.1)

where M, D, and K are the n x n analytical mass, damping, and stiffness matrices, w is a n

x 1 vector of positions and the overdots represent differentiation with respect to time. The

eigenvalue problem associated with Eq. (5.2.1) in second order (lambda) form is given as

(2M + XhiD + K)vhi = 0 (5.2.2)








where kh and vh denote the ith eigenvalue and eigenvector, respectively, of the pre-damaged

"healthy" structure. It is assumed that Eq. (5.2.2) is satisfied for all measured "healthy"

eigenvalues/eigenvectors. This can be enforced by correlating the original FEM (M,D, and

K), possibly through the use of a model refinement procedure.

Next, consider that the p eigenvalues and eigenvectors, 4i and vvi, of a post-damage

modal survey of the structure are available, in which Xd e X i, Vdi VYhi. In the present

formulation, it is assumed that the dimension of the measured eigenvector is the same as the

analytical eigenvector. As discussed in Chapter 2, this is true (i) when all FEM DOFs are

measured (ii) after the application of an eigenvector expansion algorithm, or (iii) after the

application of a finite element model reduction algorithm. The ideal situation would be to

measure all FEM DOFs since the eigenvector expansion process would introduce additional

errors in the "expanded" eigenvectors and the model reduction process would introduce

errors in the FEM. It should be noted that in both cases the additional errors may become

significant as the ratio of measured to unmeasured DOFs become smaller.

Let AMd, ADd, and AKd be the exact perturbation matrices that reflect the nature of the

structural damage. For localized damage, the exact perturbation matrices are sparse matrices

with the nonzero elements reflecting the state of damage. The lambda equation for the

damaged structure is, defined by

(12(M AMd) + d(D ADd) + (K AKd))Vd = 0 (5.2.3)

Although only p of the n eigenvalues/eigenvectors are assumed measured, p << n, Eq. (5.2.3)

holds for any particular eigenvalue and eigenvector of the damaged structure because the

perturbation matrices are assumed to be exact. Grouping all perturbation matrices on the

right-hand side defines a damage vector di,

d. m Zdd (5.2.4a)

= (X2AMd + XdADd + AKd)Vd (5.2.4b)

where Zd = M + d dD + K








Although Eqs. (5.2.4a, b) yield the same damage vector, it should be noted that Zdj and

the coefficient matrix of Eq. (5.2.4b) are not equal. When the measured eigendata are not

corrupted by noise, an inspection ofdi in terms of the Eq. (5.2.4b) reveals that the jth element

ofdi will be zero when thejth rows of the perturbation matrices are zero, i.e. the finite element

model for thejj* degree of freedom is not directly affected by damage. Conversely, a degree

of freedom whose finite element model has been affected by damage will result in a nonzero

entry in di. Thus, the degrees of freedom which have been affected by damage can be

determined by inspecting the elements of di. Vector di as defined in Eq. (5.2.4b) also reveals

that only a single mode of vibration needs to be measured exactly to determine the damage

locations exactly. This is true in even multiple member damage situations. More

importantly, the vector di can be determined from the original finite element model (M,D,K)

and the measured eigenvalues and eigenvectors, Xd and vdi, using Eq. (5.2.4a). Thus, there is

no need to use a model refinement algorithm to attempt to estimate the exact perturbation

matrices in order to locate the damage. If the damping term is ignored, Eq. (5.2.4a) is

essentially the Modal Force Error criteria as proposed by Ojalvo (1988) for use as a

diagnostic "tool" to locate modelling errors in FEMs. A physical interpretation of Eq.

(5.2.4a) provided by Ojalvo was "... di is the applied harmonic force error distribution,

applied at frequency Xdi, which is necessary to cause the analytical model to vibrate with

mode shape vd ...". However, Eq. (5.2.4b) provides a much clearer interpretation of the

damage vector di for the damage location problem in which the perturbation matrices are

sparse.

In practice, the perfect zero/nonzero pattern of the damage vector di rarely occurs due to

errors present in the experimentally measured eigenvalues and eigenvectors. This scenario

was studied and discussed for the undamped case by Gysin (1990) in the context of

eigenvector expansion techniques. Gysin (1990) observed that in certain specific cases of

eigenvector expansion errors, the damage vector defined by Eq. (5.2.1a) may lead to

incorrect conclusions concerning the location of damage. The next section proposes a new








viewpoint which allows for the reduction of the effects of measurement errors for certain

classes of structures.


5.3 The Subspace Rotation Algorithm: The Angle Perturbation Method

In order to provide an alternative view of the state of damage, Eq. (5.2.4a) is rewritten as

d 4 = 1 IId II dcos(W) (5.3.1)


where d! is the jth component (or j* DOF) of the ith damage vector, zJ is the j* row of the

matrix Zd and 01 is the angle between the vectors zJ and vd*

In the case when the measurements are free of error, a zero dJ corresponds to a 06 of
I I

ninety degrees, whereas a nonzero d. corresponds to a OQ different from ninety degrees.
1 I

Errors in the experimental measurements of modal parameters will cause slight

perturbations in the angles 06 that destroy the zero/nonzero pattern of the damage vector.
I

One would initially expect that the components of di corresponding to the damaged DOFs

would be substantially larger than the other elements. However, by inspecting Eq. (5.3.1), a

large di component could be due to a zJ row norm substantially larger than other rows of Zd.,
di ,

coupled with a slight deviation of O9 from ninety degrees due to measurement noise. Hence,

when dealing with a structure whose FEM results in zJ row norms of different order of
-di

magnitude, it is more reasonable to use the deviation of the angles, W1, from ninety degrees

for damage location,

ai = 18 900 (5.3.2)


d!
where Oi = cos 1 -I d



The angle 01 is determined from Eq. (5.3.2) and a1 is the jth component of ai.
I I








5.4 Practical Issues


5.4.1 Cumulative Damage Location Vectors

The discussion in the previous section suggests that for a given mode, the damage is

locatable if the perturbation to the angle, OQ due to the measurement error is less than the
I
angle perturbation due to the damage. Hence, modes that are highly affected by the damage

are expected to provide better assessment to the location of the damage when errors are

present in the measured eigendata. As will be seen in example problems (Chapter 7), certain

modes are more susceptible to a given state of damage than others. This is mainly due to the

fact that different elements of the structure have different levels of contribution to the total

strain energy of a given mode (Kashangaki 1992). Furthermore, a particular part of the

structure usually has different orders of strain energy contribution for different modes.

Usually, if the damage occurs in a region of high strain energy for a given mode, that mode

would be highly susceptible to the damage and, hence would reflect the state of damage. To

accommodate this type of problem, when the number of measured modes p is greater than

one, two different composite damage vectors may be defined as


d i= 1 di


a 1 i (5.4.2)
-p
i=l

In Eq. (5.4.1), the damage vectors, di, are normalized with respect to their corresponding

eigenvectors, Vdi. The reason for this normalization is such that the composite vector ignores

the inherent "weighting" of I| yd 1i, which is usually of different orders of magnitudes for

different measured modes. It should be noted that in the multi-mode measurement case, Eq.

(5.4.2) is preferable when the values of 1| zJ || are of different orders of magnitude for
different measured modes. Again, in practice,
different measured modes. Again, in practice, the DOFs affected by the damage are expected








to have substantially larger d or _q. Finally, the damaged areas of the structure can then be

located using the knowledge of the "damaged" DOFs and the connectivity of the FEM.

It is interesting to note that Eqs. (5.2.4a, b) reveal an interesting relationship between

various model refinement algorithms. Model refinement techniques attempt to approximate

the exact perturbation matrices by using limited modal data, but do so in different manners.

However, Eq. (5.2.4a, b) indicates that if the model refinement technique has satisfaction of

the eigenproblem as an equality constraint, the calculated perturbation matrices AM, AD,

and AK are constrained to be related to the original finite element model M, D and K and the

measured eigendata by Eqs. (5.2.4a, b).

5.4.2 Eigenvector Filtering Algorithm

In a modal survey, the errors associated with the measured eigenvectors are typically

greater than the error associated with the measured eigenvalues. In addition, in the case of

incomplete eigenvector component measurements, these measurement errors are often

compounded with eigenvector expansion induced errors. A simple eigenvector noise

filtering algorithm is proposed assuming the measured eigenvalues to be correct. From the

cumulative damage vector defined in Eqs. (5.2.4a) or (5.2.4b), and the original FEM

connectivity, the engineer can deduce which DOFs have been damaged. It is reasonable to

assume that nonzero elements in each damage vector di associated with "undamaged DOFs"

are due to eigenvector errors. These elements can then be set to zero. In addition, the

magnitude of the elements ofdi at the "damaged" DOFs can be adjusted by using knowledge

of DOF connectivity and the properties of the element property matrices connecting the

"damaged DOFs." The element property matrices provide constraints relating the effect of

damage on each element DOF. The noise filtering algorithm consists simply of replacing the

di vectors by df, where df is obtained from di as described above. The ith filtered

eigenvector, vdf, can then be obtained from solving

(dXM + ,dD + K)vdf = df (5.4.3)








using Gaussian elimination. In this calculation, the bandedness of typical FEM matrices

should be exploited. Essentially, the filtered eigenvector is just the eigenvector that if

measured would have produced the damage vector df. Experience gained in using the

eigenvector filtering algorithm indicates that it is best to use structural matrix properties (M,

D, K) that (i) are finite element consistent, and (ii) have not been "corrupted" by

measurement noise. By finite element consistent, it is meant that the property matrices can

be achieved by a finite element program. Note that measurement noise can be introduced

in the property matrix through a FEM refinement algorithm. Hence, the property matrices

that should be used are the original property matrices (unrefined). The present eigenvector

filtering algorithm can be useful in improving the damage extent assessment. This concept

is discussed in Chapters 6 and 7.


5.5 Summary

A computationally attractive algorithm to determine the location of damage in

structures was developed. The algorithm completely bypasses the general framework of the

model refinement problem and involves only matrix-scalar and matrix-vector

multiplications. The effect of measurement error in the eigendata was discussed and

techniques to reduce these effect were presented. Furthermore, a simple eigenvector

filtering algorithm was developed. Practical example problems to illustrate and evaluate the

performance of the developed algorithm will be presented in Chapter 7.














CHAPTER 6
THE MINIMUM RANK PERTURBATION THEORY



6.1 Background

The theory developed in Chapter 5 is limited to determining the location of structural

damage. In a practical situation, it is essential to determine the extent of the damage to get a

good estimate about the overall integrity of the structure. In general, the extent problem, as

discussed in Chapter 1, has been approached by several researchers using existing model

refinement algorithms. The formulations of these algorithms were obviously based on the

"model refinement philosophy": minimum change made to the original FEM. The minimum

change constraint has a clear tendency to smear the changes throughout the entire FEM.

However, in most cases, this philosophy is not consistent with the effect of structural damage

on FEMs. In fact, the effects of structural damage on FEMs are usually "non-minimal"

localized perturbations. Structural damage often occurs at discrete locations. The effect of

damage on the analytical model is often restricted to just a few elements of the finite element

model. The rank of each element mass, damping or stiffness matrix is dependent on the

number of degrees of freedom defined by the element and the shape functions utilized.

However, it should be noted that in general the element matrices are not of full rank. For

example, the rank of the 6x6 element stiffness matrix of a three dimensional truss element is

just one. Thus, instead of using the matrix Frobenius norm minimization formulation to

arrive at unique perturbation matrices, minimum rank perturbation constraints are enforced.

In this Chapter, a computationally attractive damage extent algorithm is proposed. The

proposed damage extent algorithm is a minimum rank perturbation, which is consistent with

the effects of many classes of structural damage on a FEM.








Assume that "p damaged" eigenvalues and eigenvectors have been measured and that

the original FEM has been corrected such that its modal properties match the measured

modal properties of the healthy model. The eigenvalue problem of a damaged structure

shown in Eq. (5.2.3), for all p measured modes, can be written in matrix form, as

MVdAd + DVdAd + KVd = AMdVdAd + ADdVdAd + AKdVd B (6.1.1)

where Ad = diag(Xd,. d2' ... dp)

Vd= [d,,Y.d2, ..., Y-d

B = [dl,d2 ... dp]


where all variables have the same definitions as in the previous chapter. Note that matrix

B can be determined from the FEM (M, D, K) and the "p" measured eigenvalues and

eigenvectors. As discussed earlier, the damage extent problem consist of finding the

perturbation matrices, AMd, ADd, and AKd, such that Eq. (6.1.1) is satisfied. As already

discussed in Chapter 2, structures can be modelled using either undamped, proportionally

damped or nonproportionally damped finite element models. The proposed extent algorithm

is formulated to accommodate all three types of structural models. For each type of model,

several scenarios of damage effects are considered. Practical issues that can be used to

improve the damage extent estimate are also presented.


6.2 The Minimum Rank Perturbation Theory: Theoretical Background

In this section, the theoretical foundation of the Minimum Rank Perturbation Theory

(MRPT) is derived. This theory will be extensively used throughout the remainder of this

chapter.

PROPOSITION 6.1 Suppose that X, Y E RnxP are given where p < n and

rank(X)=rank(Y)=p.

Define % to be the set of matrices A in R"' that satisfy,








AX = Y with AT = A


Then,

(1.a) If the set % is nonempty, the minimum rank of any matrix, A, in % is p.

Next, define %P to be a subset of % comprised of all A such that rank(A) =p.
Then

(1.b) If the matrix YTX is symmetric, then one member of %P is given by


AP = YHYT


(6.2.2)


with H = (yTX)


and


(1.c) The matrix defined by Eq. (6.2.2) is the unique member of %9P.



Proof:
To prove Proposition (6.1.a), note that Eq. (6.2.1) is exactly satisfied if and only if
range(Y) is included in range(A), which is also the range(AT) by symmetry. This implies that
rank(Y) = p : rank(A).

///

To investigate Proposition (6.1.b), assume that the expanded singular value

decomposition of one member, AN, of 9%P to be of the form


(6.2.3)


where


Uj = [u{, u ... ..]

XJ = diag(oJ{, oJ, ... i)


(6.2.1)


APJ = Uj:jUjw








where the superscript j indicates the jth family member of 9%P, the uj are the left and right

singular vectors and the ao are the non-zero singular values of Ap. In the expanded singular

value decomposition, the (p+l) to n singular vectors are not shown in the factorization

because of their corresponding zero singular values. Note that the left and right singular

vectors are the same because AN is restricted to be symmetric. For Eq. (6.2.1) to be satisfied,

the range of Y, AP'N, ApJT and Ui must be equal. Thus, any column of Y can be written as

a linear combination of the u.'s. The matrices Y and Ui are then related by a unique pxp

invertible matrix QJ,

Y = UJ Q (6.2.4)

Substituting Eq. (6.2.4) into Eq. (6.2.3) gives

AP = Y(Qj-12jQJ-T)YT = yHJyT (6.2.5)


Thus, each family member is uniquely defined by the factorization of Eq. (6.2.3). From Eq.

(6.2.5), it is evident that HJ is of full rank because its inverse exists (HI-' = QjTFji~Q).
Inspection of Eq. (6.2.5) reveals that the only unknown term in the factorization is Hi.

By using the factorization of AN as defined by Eq. (6.2.5), Eq. (6.2.5) can be rewritten as

Y = A-PX = (YHjiYX = Y(HjYTX) (6.2.6)


Equation (6.2.6) is satisfied if and only if HJYTX = Ipxp, where Ipxp is the pxp identity

matrix. This is true because Y and X are of full column rank. Thus, Hi is uniquely calculated
to be

Hi = (yTX)1 (6.2.7)

///
Proposition (6.1.c) follows immediately by inspecting the right hand side of Eq. (6.2.7).
Inspection reveals that HJ is the same for all members of %P. This fact, in conjunction with








Eq. (6.2.5) leads to the conclusion that AN'j is the unique member of the set %P. This member

is given by Eq. (6.2.2).





At this point, the MRPT as defined in Proposition 6.1 assumes that the matrices X and Y

are of full rank. In practical uses, as will be seen later, matrix X is usually of full rank. The

rank requirement on matrix Y can be of some concern since it is directly related to the rank of

matrix A. The next proposition addresses the case in which matrix Y is rank deficient.


PROPOSITION 6.2 Suppose that X, Y E Rnxm are given and rank(X)=m and rank(Y)=p,

where p < m < n. Further, suppose that the matrix YTX is symmetric.

Define cl, to be the set of matrices A in Rnxn that satisfies the problem,

APXP = YP with (AP)T = AP (6.2.8)

where the superscript p indicates a rank p matrix. In Eq. (6.2.8), XP, YP E Rnxp are

corresponding full rank submatrices of X and Y.

Then

The set U contains a single member, AP, that can be calculated from Eq. (6.2.2) using

any corresponding XP and YP.

Proof:


The jth member of the set cM is given by

AP = YPHjYPj (6.2.9)

with HJ = (YPJTXPj)

where the additional superscript ( )-J indicates the jth member of c. Note that Hi is

symmetric since YTX is symmetric. The range of any YP is equal to the range of Y, thus

the YP'1 and YP'J are related by








YP- = YPj Qi' (6.2.10)

where Qid C Rpxp and rank(QiJ) = p. By utilizing Eq. (6.2.10), Eq. (6.2.9) can be written

for the ith member of cU as

Ap, = YPJ(Qi'HiQiT)YPJT (6.2.11)

with H' = (QijTYPTp,i) 1

Again, Hi is symmetric because YTX is symmetric. In comparing Eqs. (6.2.9) and (6.2.11),

it is seen that AN = Ap' if

HJ = Q'J H' QjiT (6.2.12)

or equivalently,

Hij- = Qi'-THi-TQ'ji' (6.2.13)

where Eq. (6.2.13) makes use of the symmetry of Hi. By using the definitions of Hi and H9,

Eq. (6.2.13) can be written as

YPjTXP' = Qi,-TXPiYPJ (6.2.14)

Pre-multiplying Eq. (6.2.14) by Qi'j and utilizing the relation in Eq. (6.2.10) gives the

condition such that AN = AP", namely

YPirXPj = XpirYPj (6.2.15)

This condition is clearly satisfied since YTX is symmetric.



The actual uses of Proposition 6.1 and Proposition 6.2 will be clearly seen in the next

sections. The practical implication of Proposition 6.2 is discussed in detail in Section 6.6.


6.3 Damage Extent: Undamped Structures

In some cases, the damping of the system under consideration is assumed to be

negligible. For this type of system, MRPT based algorithms will be developed assuming that








the structural damage affects (i) only the mass properties, or (ii) only the stiffness properties,

or (iii) simultaneously the mass and stiffness properties.

6.3.1 Damage Extent: Mass Properties

In this case, it is assumed that the effect of damage on the stiffness properties of the

structure is negligible. With this assumption, Eq. (6.1.1) can be rewritten as
MVdA2 + KVd = AMVdA2 = B (6.3.1)


Note that the eigenvectors are real and the eigenvalues are purely imaginary. Further, the

eigenvectors are linearly independent, which implies that the matrix product VdA2 is of full

column rank if rigid body modes are not included. Assume, for the moment, that B is of full

rank (rank(B) = p). Then, Proposition 6.1 can be applied to determine the perturbation

matrix, AMd, as

AMd = B(BTVdA) BT (6.3.2)

by letting Y=B and X=VdA Note that the required inversion is that of a pxp matrix, where

"p" is the number of measured modes. As discussed in Proposition 6.1, this inversion is

feasible if matrix B is of full rank and the rigid body modes of the system are omitted in the

computations. When matrix B is rank deficient, Proposition 6.2 should be used to render

the computation possible.

The properties associated with AMd as computed in Eq. (6.3.2) are as follows:

PROPERTY 6.3.1 The perturbation matrix, AMd, defined in Eq. (6.3.2) will be symmetric if

the eigenvectors, Vd, are stiffness orthogonal, i.e., the eigenvectors are orthogonal with

respect to the original stiffness matrix, K.

Proof:

Proposition (6.1.c) in conjunction with Proposition (6.1.b) implies that the existence of

the unique symmetric rank p AMd requires the symmetry of the matrix product BTVdA 2

The symmetric equivalence associated with this matrix product is








BTVdA -= A VB (6.3.3)

Substituting the expression for B from Eq. (6.3.1) into Eq. (6.3.3) gives

AdV MVdAd + V KVdA2 AdVdMVdAd + A2VTKVd (6.3.4)

where the symmetry of M, K and A2 has been used in writing Eq. (6.3.4). From Eq. (6.3.4),

it is clear that the equivalence is true if

(VKVd)A == Ad(V KVd) (6.3.5)

Equation (6.3.5) will obviously be satisfied if the measured eigenvectors are stiffness

orthogonal. Baruch (1978) treated one approach to mass orthogonalize the measured

eigenvectors. A similar approach can be used to orthogonalize the measured eigenvectors

with respect to the stiffness matrix.




PROPERTY 6.3.2 The updated finite element model (FEM) defined by the original mass

and stiffness matrix along with the perturbation mass matrix computed using Eq. (6.3.2)

preserve the rigid body characteristics of the original FEM.

Proof:

This is apparent in that the original stiffness matrix is unchanged and that the rigid body

modes are defined as modes whose eigenvectors lie in the null space of the stiffness matrix.

6.3.2 Damage Extent: Stiffness Properties

Here, it is assumed that the effect of damage on the mass properties of the structure is
negligible. With this assumption, Eq. (6.1.1) can be rewritten as

MVdA2 + KVd = AKdVd -= B (6.3.6)

For this problem, the eigenvectors are real and the eigenvalues are purely imaginary. The
eigenvectors are also linearly independent, which implies that matrix Vd is of full rank. If








matrix B is assumed to be of full rank (rank(B)=p), Proposition 6.1 can be used to determine

the perturbation to the original stiffness matrix,

AKd = B(BTVd) IB (6.3.7)

This expression for AKd is determined by setting Y=B and X=Vd in Eq. (6.1.3).

The properties associated with AKd as computed by Eq. (6.3.7) are as follow.

PROPERTY 6.3.3 The matrix AKd will be symmetric if the eigenvectors are mass

orthogonal, i.e., the eigenvectors are orthogonal with respect to the original mass matrix.

The proof of Property 6.3.3 follows very much the same pattern as the one presented for AMd

(Property 6.3.1).

PROPERTY 6.3.4 The updated FEM defined by the original mass and stiffness matrices and

the perturbation stiffness matrix, AKd, preserves the rigid body mode characteristics if the

measured eigenvectors and the rigid body modes are mass orthogonal.



The original rigid body modes of an undamped system are defined by the eigenvalue

problem,

KVr = XrMVr = 0 (6.3.8)

where the subscript r denotes the rigid body mode(s) and Xr is equal to zero. Thus, the rigid

body modes lie in the null space of the original stiffness matrix. The rigid body modes of

the system will be preserved in the updated model if the original rigid body modes lie in the

null space of the updated stiffness matrix,

e = (K AKd)Vr (6.3.9a)
= AKdYr (6.3.9b)

where vector e is zero if the the rigid body modes are preserved. Equation (6.3.8) has been

used to arrive at the expression shown in Eq. (6.3.9b). Substituting Eq. (6.3.7) into (6.3.9b)

gives








e = B(BTVd) BTr (6.3.10)

By utilizing the symmetry of the original mass and stiffness matrices, along with Eq. (6.3.6),
Eq. (6.3.10) can be expanded as

e = B(BTVd)- [VTKKv + AdVdMv,] (6.3.11)

The first term in the parenthesis is zero because the matrix-vector product Kvr is zero by
definition. The second term will be zero if the rigid body modes and the measured mode
shapes are mass orthogonal.



6.3.3 Damage Extent: Mass and Stiffness Properties

In this case, it is assumed that the structural damage affects simultaneously the mass and
stiffness properties of the structure. With this assumption, Eq. (6.1.1) can be rewritten as

MVdAd + KVd = AMdVdA2 + AKdVd =- B (6.3.12)

6.3.3.1 Application of The MRPT

Assume that Eq. (6.1.12) can be decoupled as follows,

AMdVd = Bm (6.3.13a)
AKdVd = Bk (6.3.13b)

Then, the MRPT, as formulated in Proposition 6.1, can be applied to determine the
perturbation matrices AMd and AKd, as

AMd=Bm (BLVd) -1 B (6.3.14a)

AKd = Bk (B Vk) Bk (6.3.14b)


Note that the matrices BLVd and B Vd are invertible if Bm and Bk are of full rank. When

these rank requirements are not met, Proposition 6.2 can be used to make the computations
possible.








6.3.3.2 Decomposition of Matrix B

The decomposition problem as illustrated in the previous section is equivalent to the

problem of solving for the matrices Bm and Bk. So far, the only constraint that these

unknown matrices must satisfy is given by the expression,

B = Bm A2 + Bk (6.3.15)

which results from Eqs. (6.3.12) and (6.3.13). Naturally, there is an infinite set of solutions

(Bm, Bk) that satisfy Eq. (6.3.15). To arrive at a unique solution, additional physically

meaningful constraints can be enforced. The decomposition proposed herein exploits the

cross-orthogonality relations that arise from the symmetric nature of the property matrices

and the undamped assumption. By measuring mass normalized "damaged" eigenvectors

(which is possible if a driving point measurement is made), the cross-orthogonality relations

associated with the damaged structure can be written as

V(M AMd)Vd = Ipxp (6.3.16a)

VTK AKd)Vd = diag( d,2, 2) = 2 d (6.3.16b)

in which Odi is the natural frequency of the ith mode of the "damaged" structure. Matrix

Ipxp is the pxp identity matrix. A rearrangement of Eq. (6.3.16) yields

VT AMd Vd = VT M Vd Ipxp VT Bm (6.3.17a)

VdT AKd Vd = V K Vd d = V Bk (6.3.17b)

Clearly, the matrices Bm and Bk can be computed from Eqs. (6.3.17). In the rare situation

that the number of measured modes is equal to the number of DOFs in the FEM (p = n), these
can be computed by simply inverting matrix Vd. Unfortunately, as discussed earlier, the

number of measured modes is usually much less than the number of FEM DOFs (p << n).
In this case, the solution that naturally comes to mind is to use the pseudo-inverse of matrix

VT. The inconvenience of this approach is that the sparsity pattern of matrix B will not be








reflected in the computed matrices Bm and Bk. Remember that the sparsity pattern of B, as

discussed in Chapter 5, indicates the location of the damage affecting the structure. A more

physically intuitive approach is to constrain Bm and Bk to exhibit the same sparsity pattern

as matrix B. This is done by casting B in an equation similar to the expressions of Eqs.

(6.3.17). The problem in question is then to find an nxp matrix P that satisfies

P(VTB) = B (6.3.18)

Matrix P can be computed as


P = B (VTB) (6.3.19)

The inverse involved in this computation is that of a pxp matrix which is invertible if matrix

B is of full rank. Now that P is computed, Bm and Bk can be computed using Eq. (6.3.19)

as


B = P(V M Vd Ipxp) (6.3.20a)

Bk = P K Vd Qd) (6.3.20b)

It is clear from Eq. (6.3.19) that P will have the same sparsity pattern as matrix B. Hence

Bm and Bk will also reflect the important sparsity pattern of B. The computed matrices Bm

and Bk can also be used to determine the effect of the damage, respectively, on the mass, and

stiffness properties. As in Chapter 5, cumulative vectors associated to Bm and Bk can also be

defined when more than one measured mode is available.

d 1 IdrI
Am = I --i (6.3.21a)


dk Idl (6.3.21b)
i=111 di


where dm. and dki are, respectively, the ith column of matrix Bm and Bk.








PROPERTY 6.3.5 The perturbation matrices (AMd, AKd) computed from the MRPT using
the Bm and Bk resulting from the decomposition discussed above will be symmetric.

Proof:
The perturbation matrices AMd and AKd will be symmetric because they satisfy the
relationships in Eqs. (6.3.17) and the right hand sides of these equations are symmetric.


6.4 Damage Extent: Proportionally Damped Structures

Since many structures have non-negligible damping, it is of practical interest to extend
the MRPT to address damped structures. In this analysis, the structure under consideration is
assumed to exhibit proportional damping.

6.4.1 Damage Extent: Stiffness and Damping Properties

It is assumed that the effect of the structural damage on the mass properties is negligible.
In this context, Eq. (6.1.1) is rewritten as

MVdA2 + DVdAd + KVd = ADdVdAd + AKdVd B (6.4.1)

The complex conjugate of Eq. (6.4.1) is

ADdVdAd + AKdVd = B (6.4.2)

where the overbar indicates the complex conjugate operator, and the fact that ADd, AKd and
Vd are real has been used in writing Eq. (6.4.2). Subtracting Eq. (6.4.2) from Eq. (6.4.1)
gives

ADdVd(Ad Ad) = (B B) (6.4.3)

If (B B) is assumed to be of full rank, Proposition 6.1 can be applied to determine the

perturbation matrix, ADd, as

ADd = (B B)Hd(B BT

with Hd = [(B TVd(Ad d)-1 (6.4.4)







Note that ADd as defined by Eq. (6.4.4) is real. Post-multiplying Eq. (6.4.1) by Ad and
Eq. (6.4.2) by Ad and subtracting the two equations leads to

AKdVd(Ad Ad) = (BAd BAd) (6.4.5)

where the fact that Ad and Ad are diagonal matrices has been used in writing Eq. (6.4.5).

If (BAd BAd) is assumed to be of full rank, Proposition 6.1 can also be applied to
determine the perturbation matrix, AKd, as

AKd = (BAd BAd)Hk(BAd BAd)
-1 (6.4.6)
with Hk = [(BAd BAd)TVd(Ad Ad)] (6.4.6)

Note that AKd as defined by Eq. (33) is also real.

PROPERTY 6.4.1 The perturbation matrices ADd and AKd, as computed above, will be
symmetric if the measured eigenvectors, Vd, are mass orthogonal; i.e., the eigenvectors are
orthogonal with respect to the original unperturbed mass matrix.


Proof:
Matrix ADd is symmetric if Hd is symmetric or, equivalently, if Hd 1 is symmetric.
Hence, to get a symmetric ADd, the following equivalence must be satisfied.

(B B)TVd(Ad Ad) = (Ad AVT(B B) (6.4.7)


Substituting the expressions for B and B, from Eqs. (6.4.1) and (6.4.2) respectively, into
Eq. (6.4.7) yields

(A2VTM + AdVTD AdV2TM AdVD Vd Ad)
d d d (6.4.8)
(Ad Ad)V (MVdA2 + DVdAd MVdd + DVdAd







Note that in Eq. (6.4.8) the terms involving matrix AKd canceled out. A further expansion
and simplification of Eq. (6.4.8) yields

(Ad Ad) VMMV (Ad d) (Ad d) VMVd (A Ad) (6.4.9)


which is clearly satisfied if the measured "damaged" eigenvectors, Vd, are mass orthogonal.


Likewise, the perturbation matrix AKd as computed in Eq. (6.4.6) is symmetric if Hk is

symmetric or, equivalently, if Hk 1 is symmetric. This symmetry requirement yields the
following equivalence.

(BAd BAd)TVd(Ad Ad) (Ad Ad)VT(BAd BAd) (6.4.10)

Substitution of the expressions for B and B into Eq. (6.4.10) yields

(AdA2VM + AdVK AdAdV M AdVK) Vd(Ad Ad)
T 2 2 (6.4.11)
S(Ad Ad)Vd (MVdA dAd + KVdAd dAd KVdAd)

in which the terms involving matrix ADd cancel. Manipulating and simplifying Eq. (6.4.11)
yields

(?dAdA A dAd)VMVd(Ad Ad) = (Ad Ad)V dMVd(XdAd AdAd) (6.4.12)

This equivalence is obviously satisfied if the eigenvectors are mass orthogonal.
///




PROPERTY 6.4.2 The updated FEM, defined by the original FEM and the perturbation
matrices ADd and AKd computed from Eqs. (6.4.4) and (6.4.6), preserves the original rigid
body modes if the measured eigenvectors and the rigid body modes are mass orthogonal.








Proof:
As discussed earlier, a rigid body mode is defined as a mode whose eigenvalue is equal to
zero and whose eigenvector lies in the null space of the FEM stiffness matrix. Hence, the
rigid body modes of the original system are preserved in the updated FEM if they lie in the
null space of the perturbed stiffness matrix. Consider the relationship

e = (K AKd)Vr (6.4.13)

where Yr is a rigid body mode eigenvector. Clearly, the rigid body mode associated to
eigenvector Yr is preserved if e = Q. By definition, _r is a rigid body eigenvector of the
original system, hence Eq. (6.4.13) can be simplified as

e = AKdYr (6.4.14)

Substituting the expression for AKd as defined in Eq. (6.4.6), into Eq. (6.4.14) gives

e = (BAd BAd)Hk(BAd BAd) r (6.4.15)

Substitution of the expressions for B and B into this equation yields

e = (BAd Ad)Hk[ AdAVM + A K AdAdVdM AdVK ]vr (6.4.16)

By using the fact that vr is a rigid body eigenvector of the original system (i.e. Kvr = 0), Eq.
(6.3.16) can be simplified as

e = (BAd BAd)Hk( AdA2 Ad )VMvr (6.4.17)

It is clear from Eq. (6.4.17) that e = 0 if the rigid body mode Yr and the measured

eigenvectors Vd are mass orthogonal (i.e. VTMvr = 0).



6.4.2 Damage Extent: Mass and Damping Properties

In this case it is assumed that the effect of the structural damage on the stiffness
properties is negligible. In this context, Eq. (6.1.1) is rewritten as







MVdAd + DVdAd + KVd = AMdVdA2 + ADdVdAd = B (6.4.18)

By using an approach similar to one used in the preceding section, Eq. (6.4.18) and its
complex conjugate can be manipulated to yield the following decomposition

AMdVd(Ad Ad) (BAd BA) (6.4.19)

ADdVd(AdAd Ad) = (BAd dBA) (6.4.20)

Again by applying the MRPT to the preceding equations, AMd and ADd are determined to
be

AMd = (BAd Ad)Hm(BAd BAd)T
S-I (6.4.21)
with Hm = (BAd- 2BA) V d Ad


ADd = (BAd RAd)Hd(BAd TA2

with Hd ( BA) Vd(Add dA(6.4.22)

Clearly, the perturbation matrices AMd and ADd as defined by Eqs. (6.4.21) and (6.4.22) are
real.

PROPERTY 6.4.3 The perturbation matrices AMd and ADd, as computed above, will be
symmetric if the measured eigenvectors, Vd, are stiffness orthogonal; i.e., the eigenvectors
are orthogonal with respect to the original unperturbed stiffness matrix.

PROPERTY 6.4.4 The updated FEM, defined by the original FEM and the perturbation
matrices, AMd and ADd, preserves the original rigid body modes.
The proof of Property 6.4.4 is straightforward since the original stiffness matrix is
unchanged (see Property 6.3.2). The proof of Property 6.4.4 follows very much the same
pattern as the proof of Property 6.4.2.







6.4.3 Damage Extent: Mass and Stiffness Properties

In this problem, it is assumed that the effect of the structural damage on the damping
properties is negligible. For this situation, the general eigenvalue problem defined in Eq.
(6.1.1) associated to this case can be simplified as

MVdA2 + DVdAd + KVd = AMdVdA2 + AKdVd B (6.4.23)

Algebraic manipulations of Eq. (6.4.23) and its complex conjugate yield the following
decomposition

AMdVd(A Ad) = (B B) (6.4.24)

AKdVd(A A) = (BAd dBA) (6.4.25)

The perturbation matrices AMd and AKd can then be computed using the MRPT.

AMd = (B )Hm(B B)T

with Hm = (B BTVdA2 1 (6.4.26)

T
AKd = (BAd BA)HA BAS d
T (6.4.27)
with Hk (BA A) Vd( (6.4.27)

Note that AMd and AKd as defined by Eqs. (6.4.26) and (6.4.27) are real.

PROPERTY 6.4.5 The perturbation matrices AMd and AKd, as computed above, will be
symmetric if the measured eigenvectors, Vd, are damping orthogonal; i.e., the eigenvectors
are orthogonal with respect to the original unperturbed damping matrix.


PROPERTY 6.4.6 The updated FEM, defined by the original FEM and the perturbation
matrices, AMd and AKd, preserves the original rigid body modes if the measured
eigenvectors and the rigid body modes are damping orthogonal.







These proofs of the above two properties are not reported here. They follow very much the
same pattern as the proofs in Section 6.4.2.

6.4.4 Damage Extent: Mass. Damping and Stiffness Properties

The eigenvalue problem of a proportionally damped system with all property matrices
simultaneously affected by damage can be rearranged into the form

MVdAd + DVdAd + KVd = AMdVdA + ADdVdAd + AKdVd = B (6.4.28)

The theory developed in Section 6.3.3 can be expanded to address this particular problem.
The cross-orthogonality relationships associated with this type of structures are

Vj(M AMd)Vd = Ipxp (6.4.29a)

VJ(D ADd)Vd = diag(2Odd1, ,2E ddp) = (6.4.29b)

VT(K AKd)Vd = diag(wd2, .. ap2) = d (6.4.29c)

Notice that the cross-orthogonality relationships in Eqs. (6.4.29a) and (6.4.29c) are exactly
the same as the ones associated with undamped systems reported in Eqs. (6.3.16a) and
(6.3.16b). As before, these cross-orthogonality conditions can also be rearranged as

V AMd Vd = V M Vd pxp = V Bm (6.4.30a)

VT ADd Vd = V D Vd d VT Bd (6.4.30b)

VT AKd Vd = V K Vd d = V Bk (6.4.30c)

Following the exact same argument discussed for undamped systems in Section 6.3.3, an nxp
matrix P that satisfies the relation,

P(V dB) = B (6.4.31)

is sought, where B is computed using Eq. (6.4.18) and (vdTB) is a pxp matrix. Although B

is a complex matrix, the nxp matrix P is real, since Vd is real. Hence, for computational
efficiency, matrix P can computed from








P = Br(V Br) (6.4.32)


where Br is the real part of B. In Eq. (6.4.32), it is assumed that matrix (VTB) is invertible.

With P computed, the next step is to determine the decomposed damage vectors that indicate
the effects of the damage on the mass, damping and stiffness matrices,

Bm = AMVd = P (V M Vd pxp) (6.4.33a)

Bd = ADVd = P (V D Vd Yd) (6.4.33b)

Bk = AKVd = P (V' K Vd d) (6.4.33c)

The minimum rank perturbation theory (MRPT), as formulated in Proposition 6.1, can again
be applied to determine the perturbation matrices, AMd, ADd and AKd, as

AMd = Bm (BTVd)-1 Bm (6.4.34a)

ADd =Bd (B Vd)-1 BT (6.4.34b)

AKd = k (BVk) -1 Bk (6.4.34c)


Note that the matrices BTVd, BJVd and B Vd are pxp matrices that are invertible if Bm,

Bd and Bk are of full rank. As in all other cases already studied, Proposition 6.2 can be used
to deal with the situation when any one of these matrices are rank deficient. The cumulative
damage location vector associated to Bm and Bk, defined in Eqs. 6.3.21, are also applicable
to this problem. An additional cumulative damage vector associated to the perturbations in
the damping properties can be similarly defined as


dd = I (6.4.35)
pi=l 1 d


where dd is the ith column of matrix Bd.




Full Text
xml version 1.0 encoding UTF-8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID E6Q7S0BRS_TJDVUO INGEST_TIME 2017-07-13T21:38:36Z PACKAGE AA00003663_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES


STRUCTURAL DAMAGE ASSESSMENT AND FINITE ELEMENT
MODEL REFINEMENT USING MEASURED MODAL DATA
By
MOHAMED KAOUK
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1993

A Mes Parents
Tout le mérite de ce travail, s’il en est, vous revient.

ACKNOWLEDGEMENTS
I would like to express my sincere gratitude toward my advisor, Dr. David Zimmerman,
for his expert guidance, friendship, endless encouragement and support. I will forever be
indebted to him for inspiring me in my research and for the priceless education I acquired
from him. I am also grateful for the financial support he provided me during the course of my
graduate studies.
Words are not enough to express my deepest gratitude to my parents for their love, moral
and financial support; it is these things that have made this work possible. I also wish to
acknowledge my sisters and brothers for their continuous encouragement.
I would like to thank the members of my supervisory committee, Drs. Norman Fitz-Coy,
Lawrence Malvern, Bhavani Sankar, and Kermit Sigmon, for their helpful advice.
I wish to thank my colleagues of the Dynamic Systems and Control Laboratory for their
consideration and entertaining discussions. In particular, William Leath and Cinnamon
Larson for their friendship and encouragement.
I would like to thank the staff of the Aerospace Engineering, Mechanics, and
Engineering Science department for their assistance throughout the years, especially Shirley
Robinson for making my life easier during registration, John Young for his prompt responses
in fixing my computer problems, and Ronald Brown for his assistance in the machine shop
and for many stimulating discussions.
I would like to acknowledge the financial support received from Harris Corporation,
NASA/Florida Space Grant Consortium and Florida High Technology Council. I would like
to thank Dr. T. A. Kashangaki of the NASA Langley Research Center and Dr. S. W. Smith of
the University of Kentucky for providing the data of the Eight-Bay Truss used in this study.
Last, but not least, I am grateful to my good friends Joel Payabyab and Fadel Abdallah
iii
for their continuous support.

TABLE OF CONTENTS
page
ACKNOWLEDGEMENTS iii
LIST OF TABLES ix
LIST OF FIGURES x
ABSTRACT xiii
CHAPTERS
1 INTRODUCTION 1
1.1 Finite Element Model Refinement 1
1.1.1 Overview 1
1.1.2 Literature Survey 3
1.2 Structural Damage Assessment 6
1.3 Objective of the Present Study 8
2 MATHEMATICAL PRELIMINARIES AND PRACTICAL
ISSUES TO THE PROBLEMS OF MODEL
REFINEMENT AND DAMAGE DETECTION 11
2.1 Introduction 11
2.2 The Eigenvalue Problem of Discrete Systems 11
2.2.1 Undamped Models 15
2.2.2 Proportional Damped Models 16
2.3 Experimental Modal Analysis 16
2.4 Analytical/Experimental Model Dimensions Correlation 19
2.4.1 Model Reduction Methods 19
2.4.1.1 Static Reduction 21
2.4.1.2 IRS Reduction 22
2.4.1.3 Dynamic Reduction 22
iv

2.4.2Eigenvector Expansion Methods 23
2.4.2.1 Dynamic Expansion 23
2.4.2.2 Orthogonal Procrustes Expansion 24
2.5 Eigenvector Orthogonalization 26
2.5.1 Optimal Weighted Orthogonalization 27
2.5.2 Selective Optimal Orthogonalization 27
2.6 Load Path Preservation 28
3 INVERSE / HYBRID PROBLEM APPROACH FOR FINITE
ELEMENT MODEL REFINEMENT 31
3.1 Introduction 31
3.2 Theoretical Formulation 31
3.3 Numerical Illustrations 35
3.3 Summary 37
4 SYMMETRIC EIGENSTRUCTURES ASSIGNMENT
MODEL REFINEMENT ALGORITHM 39
4.1 Introduction 39
4.2 Problem Formulation 39
4.2.1 Standard Eigenstructure Assignment Formulation 40
4.2.2 Symmetric Eigenstructure Assignment Formulation 42
4.2.3 Best Achievable Eigenvectors 44
4.2.4 Selection of Bo : The Subspace Rotation Method 46
4.3 Numerical Illustrations 47
4.3.1 Damage Detection: Kabe’s Problem 48
4.3.1.1 Local to Global Mode Change 49
4.3.1.2 Consistent Modes 52
4.3.2 Model Refinement of a Cantilever Beam:
Experimental Study 55
4.3.2.1 Modal Test Description 55
4.3.2.2 Finite Element Model Description 56
4.3.2.3 Application of SEAMRA 57
4.4 Discussion of the SEAMRA’s Formulation 57
4.5 Summary 59
v

5 DAMAGE LOCATION: THE SUBSPACE ROTATION
ALGORITHM 60
5.1 Introduction 60
5.2 The Subspace Rotation Algorithm: The Direct Method 60
5.3 The Subspace Rotation Algorithm: The Angle
Perturbation Method 63
5.4 Practical Issues 64
5.4.1 Cumulative Damage Vectors 64
5.4.2 Eigenvector Filtering Algorithm 65
5.5 Summary 66
6 THE MINIMUM RANK PERTURBATION THEORY 67
6.1 Background 67
6.2 The Minimum Rank Perturbation Theory:
Theoretical Background 68
6.3 Damage Extent: Undamped Structures 72
6.3.1 Damage Extent: Mass Properties 73
6.3.2 Damage Extent: Stiffness Properties 74
6.3.3 Damage Extent: Mass and Stiffness Properties 76
6.3.3.1 Application of the MRPT 76
6.3.3.2 Decomposition of Matrix B 77
6.4 Damage Extent: Proportionally Damped Structures 79
6.4.1 Damage Extent: Stiffness and Damping Properties 79
6.4.2 Damage Extent: Mass and Damping Properties 82
6.4.3 Damage Extent: Mass and Stiffness Properties 84
6.4.4 Damage Extent: Mass, Damping and Stiffness Properties .... 85
6.5 Damage Extent: Nonproportionally Damped Structures 87
6.5.1 Damage Extent: Damping and Stiffness Properties 89
6.5.2 Damage Extent: Mass and Damping Properties 90
6.5.3 Damage Extent: Mass and Stiffness Properties 92
6.6 Practical Issues 94
6.6.1 The Concept of “Best” Modes 95
6.6.2 Application of the Eigenvector Filtering Algorithm 97
6.7 Summary 97
vi

7 VALIDATION AND ASSESSMENT OF THE SUBSPACE
ROTATION ALGORITHM AND THE MINIMUM
RANK PERTURBATION THEORY 98
7.1 Introduction 98
7.2 Kabe’s Problem 98
7.2.1 Problem Description 98
7.2.2 Damage Location 100
7.2.3 Damage Extent 103
7.3 Damage Detection: Fifty-Bay Two-Dimensional
Truss: Undamped FEM 105
7.3.1 Problem Description 105
7.3.2 Damage Location 106
7.3.3 Damage Extent 107
7.4 Experimental Study: The NASA 8-Bay Truss 112
7.4.1 Problem Description 112
7.4.2 Refinement of the Original FEM 115
7.4.3 Damage Location 118
7.4.4 Damage Extent 120
7.4.4.1 The Brute Force Method 121
7.4.4.2 The Damage Consistent Method 121
7.4.4.3 Application of the Eigenvector Filtering
Algorithm 122
7.5 Experimental Study: Mass Loaded Cantilevered Beam 140
7.5.1 Problem Description 140
7.5.2 Analytical and Experimental Models Dimension
Correlation 141
7.5.3 Refinement of the Original FEM 141
7.5.4 Damage Location 142
7.5.5 Damage Extent 143
7.6 Fifty-Bay Two-Dimensional Truss: Proportionally
Damped FEM 144
7.6.1 Problem Description 144
7.6.2 Damage Location 145
7.6.3 Damage Extent 146
vii

7.7 Eight-Bay Two-Dimensional Mass-Loaded
Cantilevered Truss 148
7.7.1 Problem Description 148
7.7.2 Proportionally Damped Configuration: Damage of
Small Order of Magnitude 149
7.7.2.1 Damage Location 150
1.1.22 Decomposition of Matrix B 150
1.1.2.3 Damage Extent 152
7.7.3 Undamped Configuration: Damage of Large
Order of Magnitude 153
7.7.3.1 Noise Free Eigendata 153
7.7.3.2 Noisy Eigendata 154
7.8 Summary 156
8 CONCLUSION AND SUGGESTIONS FOR FUTURE WORK 159
REFERENCES 162
BIOGRAPHICAL SKETCH 167

LIST OF TABLES
Table Page
3.1 Kabe’s Problem: Elemental Stiffness Components 38
4.1 Properties of the Cantilever Beam 56
4.2 Measured Natural Frequencies and Damping Ratios
of the Cantilever Beam 56
4.3 Measured Mode Shapes of the Cantilevered Beam 56
7.1 Fifty-Bay Truss: Summary of Damage Detection
Results using the MRPT Ill
7.2 Mass Properties of the Eight Bay Truss 113
7.3 Strut Properties of the Eight Bay Truss 114
7.4 NASA 8-Bay: Truss Damage Case Definitions 115
7.5 Comparison of Analytical and Experimental Frequencies 116
7.6 Summary of the Filtering Process for Single Member Damage Cases 123
7.7 NASA 8-Bay Truss: Summary of the Damage Assessment Results 139
7.8 Mass Loaded Cantilevered Beam Properties 140
7.9 Analytical and Experimental Frequencies of the “Healthy” Structure 142
7.10 50-Bay 2-Dimensional Truss: Summary of the Percentage
Error with Respect to the Exact Damage 147
7.11 Problem 7.7: Percentage Error of Damage Estimate with
Respect to Exact Damage 156
IX

LIST OF FIGURES
Figure Page
1.1 Overview of Finite Element Model Refinement 2
1.2 Overview of FEM Model Refinement Process Used for
Damage Assessment 7
2.1 Components of a Vibration Measurement System for
Modal Analysis 17
2.2 A simple Experimental Modal Analysis Setup 18
2.3 Flow Chart of the Iterative Load Preservation Path Algorithm 29
3.1 Kabe’s Problem: Analytical Test Structure 36
4.1 Best Achievable Eigenvector Projection 45
4.2 Rotation of the Achievable Subspace 47
4.3 Kabe’s Problem 48
4.4 Results for Kabe’s Problem using the 1st Mode,
Full Eigenvector 50
4.5 Results for Kabe’s Problem Modes 1,2,3 and
Eigenvectors Components 1, 2, 3 51
4.6 Results for Kabe’s Problem using Load Path Preservation,
Modes 1, 2, 3 and Eigenvectors Components 1, 2, 3 52
4.7 Results for Kabe’s Problem using the 1 st Mode,
Full Eigenvector 53
4.8 Results for Kabe’s Problem Modes 1,2,3 and
Eigenvectors Components 1,2,6 54
4.9 Experimental Cantilever Beam 55
4.10 Experimental and Analytical Frequency Response Function
of the Cantilever Beam 58
x

7.1 Kabe’s Problem 99
7.2 Kabe’s Problem: Damage Location Results using the Subspace
Rotation Direct Method with the Eigendata of the 1st Mode 101
7.3 Kabe’s Problem: Damage Location Results using the Angle
Perturbation Method with the Eigendata of the 1st Mode 101
7.4 Kabe’s Problem: Damage Location Results using Lin’s
Algorithm with the Eigendata of the 1st Mode 102
7.5 Kabe’s Problem: Damage Location Results using the Angle
Perturbation Method with the Eigendata of the 1st and 2nd Modes 102
7.6 Kabe’s Problem: Damage Location Results using Lin’s
Algorithm with the Eigendata of the 1st and 2nd Modes 103
7.7 Kabe’s Problem: Damage Extent Results using the MRPT
with the Eigendata of Mode 2 104
7.8 Kabe’s Problem: Damage Extent Results using Baruch’s Method 105
7.9 Fifty-Bay Two-Dimensional Truss 106
7.10 Fifty-Bay Truss: Damage Location Results using the Subspace
Rotation Algorithm with the Eigendata of the First Ten Modes 107
7.11 Fifty-Bay Truss: Damage Extent Results using the
MRPT with the Eigendata of Modes 8 and 9 109
7.12 Fifty-Bay Truss: Damage Extent Results using the
MRPT with the Eigendata of the First Ten Modes 110
7.13 Fifty-Bay Truss: Damage Extent Results using Baruch’s Algorithm 110
7.14 The NASA Eight-Bay Hybrid-Scaled Truss: Damage Cases 112
7.15 The NASA 8-Bay Truss: Lacing Pattern 113
7.16 NASA 8-Bay Truss: Typical Frequency Response Comparison 117
7.17 NASA 8-Bay Truss: Perturbation to the Original Stiffness
Matrix that Resulted From the Refinement Process 118
7.18 NASA 8-Bay Truss: Cumulative Damage Vector Associated
with Case F 124
7.19 NASA 8-Bay Truss: Damage Assessment of Case A 125
7.20 NASA 8-Bay Truss: Damage Assessment of Case C 126
xi

7.21 NASA 8-Bay Truss: Damage Assessment of Case D 127
7.22 NASA 8-Bay Truss: Damage Assessment of Case E 128
7.23 NASA 8-Bay Truss: Damage Assessment of Case G 129
7.24 NASA 8-Bay Truss: Damage Assessment of Case H 130
7.25 NASA 8-Bay Truss: Damage Assessment of Case I 131
7.26 NASA 8-Bay Truss: Damage Assessment of Case J 132
7.27 NASA 8-Bay Truss: Damage Assessment of Case K 133
7.28 NASA 8-Bay Truss: Damage Assessment of Case L 134
7.29 NASA 8-Bay Truss: Damage Assessment of Case M 135
7.30 NASA 8-Bay Truss: Damage Assessment of Case N 136
7.31 NASA 8-Bay Truss: Damage Assessment of Case O 137
7.32 NASA 8-Bay Truss: Damage Assessment of Case P 138
7.33 The Mass Loaded Cantilevered Beam 140
7.34 Mass Loaded Cantilevered Beam: Damage Assessment 143
7.35 50-Bay 2-Dimensional Truss 145
7.36 50-Bay 2-Dimensional Truss: Damage Location 146
7.37 50-Bay 2-Dimensional Truss: Damage Extent 148
7.38 The Eight-Bay Two-Dimensional Mass-Loaded Cantilevered Truss 149
7.39 Problem 7.7: Cumulative Damage Location Vector: First Four Modes 150
7.40 Problem 7.7: Cumulative Vectors Associated with the Exact
and Computed Bm, Bd, B^: First Three Modes 151
7.41 Problem 7.7: Exact and Computed AMd, ADd, AKd 152
7.42 Problem 7.7: Cumulative Vector Associated with B, Bm, Bj, B^
B Computed using Modes 1-4
Bm, Bj, Bk Computed using Modes 3,4 & 5 155
7.43 Problem 7.7: Exact and Computed AMd, ADd, AKd 157
xii

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirement for the Degree of Doctor of Philosophy
STRUCTURAL DAMAGE ASSESSMENT AND FINITE ELEMENT
MODEL REFINEMENT USING MEASURED MODAL DATA
By
Mohamed Kaouk
August 1993
Chairperson: Dr. David C. Zimmerman
Major Department: Aerospace Engineering, Mechanics and Engineering Science
This study investigates the problems of model refinement and structural damage
assessment. The essence of the model refinement problem is to adjust finite element models
(FEMs) of structures with the intent of producing a correlation between experimental and
analytical modal properties. Recently, the framework of model refinement has been adopted
to determine the location and extent of structural damage. Damage will result in changes to
the modal properties of the healthy structure. A further refinement of an accurate FEM of the
structure using damaged modal parameters is expected to generate adjustments to the FEM at
locations associated with damage. Analysis of these adjustments can then be used to assess
damage. In this investigation, four algorithms relevant to the subjects of model refinement
and damage detection are presented. The development of a model refinement algorithm with
its basis in inverse theory is first presented. The algorithm, termed the inverse/hybrid
approach, is illustrated in a comparative computer simulated study. Next, an existing
eigenstructure assignment model refinement algorithm is improved to better approach the
damage assessment problem. The enhanced algorithm is evaluated and compared to other
techniques using simulated and experimental data. The algorithm is shown to perform well

in assessing damage and refining FEMs. A damage location algorithm that bypasses the
general framework of model refinement is discussed. The damage location algorithm,
termed the subspace rotation, is similar to the modal force error criteria proposed by several
researchers. Greater insight to the modal force error criteria, along with a new viewpoint that
reduces the effects of measurement noise, is discussed. Furthermore, an efficient damage
extent algorithm based on a minimum rank perturbation theory (MRPT) is developed. The
formulation of the MRPT is consistent with the effect of most structural damage on FEMs.
The characteristics of the subspace rotation algorithm and the minimum rank perturbation
theory are illustrated using simulated and experimental testbeds. The decomposition of the
damage assessment problem into location and extent subproblems is shown to be
advantageous from both for computational efficiency and for engineering insight.

CHAPTER 1
INTRODUCTION
1.1 Finite Element Model Refinement
1.1.1 Overview
An important aspect in the design process of a structure is the evaluation of its
performance under expected dynamic loading conditions. Dynamic performance can be
analyzed by using either analytical or experimental techniques. Experimental analyses are
generally very costly, time consuming and can encounter technical difficulties. One example
of a commonly encountered technical barrier is the ground base reproduction of a weightless
environment for the testing of space structures. This shortcoming and the advent of
computers have sparked a growing interest in the uses of analytical techniques. This type of
analysis utilizes a mathematical model of the actual structure for computer simulated
evaluation of structural performance. Presently, most analytical models used in industry are
finite element models generated by using Finite Element Methods. The accuracy of a finite
element model (FEM) can be improved to some extent by increasing the number of degrees
of freedom (DOFs) included in the model. However, accuracy of the FEM may still be
lacking due to uncertainties in material properties, geometric layout and fabrication induced
errors. Thus, it is essential to “validate” these FEMs prior to their acceptance as a basis for
analysis. One way to validate them is to compare their modal properties (eigenvalues/
eigenvectors) with the measured modal parameters of the actual structure as obtained from
experimental modal analysis (EMA). A FEM is acceptable when these two sets of modal
properties are in agreement. Unfortunately, this agreement rarely occurs. As a result, the
1

2
FEM must be adjusted to produce a correlation of analytical and experimental modal
properties. An overview of this procedure is schematically illustrated in Figure 1.1
(Zimmerman and Smith, 1992).
Experimental
Figure 1.1 Overview of Finite Element Model Refinement
(Zimmerman and Smith, 1992)
In the past, the process of adjusting FEMs was performed on an ad-hoc basis aided by
engineering experience. This practice was naturally time consuming and in most cases
inadequate for large-order, complex FEMs that are typically needed for accurate dynamic
modelling. In the past years, growing interest has been focused on developing systematic
procedures to produce correlated FEMs. These efforts have resulted in the development of a
large number of algorithms. Finite element model adjustment procedures have been

3
commonly referred to in the literature as, FEM refinement, FEM adjustment, FEM
correction, FEM correlation, and FEM identification.
1.1.2 Literature Survey
Comprehensive literature surveys covering a large portion of the work that addresses the
model refinement problem can be found in the book chapter by Zimmerman and Smith
(1992), and in the papers by Ibrahim and Saafan (1987) and Heylen and Saas (1987). For
completeness and to properly underline the objective of the current study, it is appropriate to
include a brief survey of the development made in this area.
The concept of using experimental modal data in analytical studies was initiated by an
early work presented by Rodden (1967). In his work, Rodden explored the possibilities of
generating mass and stiffness property matrices by using experimentally measured modal
data. The experimentally generated mass and stiffness matrices were nonsymmetric. Brock
(1968) improved the work of Rodden by proposing a strategy to insure symmetry of property
matrices.
The essence of the model refinement concept, as adopted by most researchers, is to
modify the finite element model (FEM) of systems with the intention of producing a
correlation of test and analytical modal parameters. In general, this task has been approached
through two different philosophies. The first amounts to modifying globally the finite
element property matrices of the system. This approach is commonly referred to as the
property matrix update. The alternative approach is to individually correct parameters of
each element of the finite element model. This will be addressed here as the physical
parameter update.
Property matrix update algorithms can be classified into two subclasses of
methodologies: optimal-matrix update and control-based eigenstructure assignment
techniques.

4
The basic philosophy of the optimal-matrix update is to minimize the correction to the
FEM property matrices to accomplish the analytical/test modal correlation. The pioneering
work in this area can be credited to Baruch and Bar Itzhack (1978). In their formulation, by
assuming that the mass matrix is correct, the refinement of the stiffness matrix of an
undamped FEM is cast as a constrained minimization problem. The objective of their
formulation is to determine the minimal Frobenius-norm symmetric stiffness adjustment
that satisfies the eigenvalue problem in terms of the experimentally measured eigenvalues
and eigenvectors. A computationally efficient closed form solution was developed for the
updated stiffness matrix. Berman and Nagy (1983) extended the Baruch and Bar Itzhack
approach to the refinement of both the mass and stiffness matrices. This same methodology
was further extended by Fuh et al. (1984) to update the mass, damping and stiffness matrices
of damped FEMs. Fuh and his colleagues used cross-orthogonality relationships to correct
the mass and damping matrices and a constrained minimization problem similar to the one
proposed by Baruch and Bar Itzhack to determine the refined stiffness matrix. The problem
of adjusting the mass, damping and stiffness matrices was also attempted by Hanagud et al.
(1984). In their approach, all three property matrices of a nonproportionally damped FEM
are incorporated in the constrained minimization problem.
The previously stated algorithms do not preserve the load path (sparsity pattern) of the
original analytical property matrices. Kabe (1985) proposed a reformulation of the Baruch
and Bar Itzhack algorithm that constrains the updated stiffness matrix to preserve the load
path of the original stiffness matrix. In addition, in his reformulation, he adopted a percent
weighting on the stiffness changes instead of mass matrix weighting as used in the Baruch
and Bar Itzhack algorithm. Kammer (1987) proposed an alternative solution approach to the
problem defined by Kabe that uses projection matrix theory and the Moore-Penrose
generalized inverse. The algorithms formulated by Kabe and Kammer involve an excessive
amount of computational effort. Another alternative and more efficient formulation of the

5
Kabe problem that utilized a generalization of Marwill-Toint updates was developed by
Smith and Beattie (1991).
The other subclass of property matrix updates is based on the general framework of
control-based eigenstructure assignment algorithms. The essence of this approach is to
determine pseudo-controllers that would assign the experimentally measured modal
parameters to the original analytical FEM. The pseudo-controllers are then translated into
matrix adjustment to the original property matrices. This approach was initially formulated
by Inman and Minas (1990) to adjust the damping and stiffness matrices of the original FEM.
In their formulation, the symmetry of the adjusted property matrices was enforced through
an iterative process that involves a numerical nonlinear optimization process. Zimmerman
and Widengren (1989, 1990) proposed a formulation that replaces the iterative process of
Inman and Minas with a generalized algebraic Riccati equation. Further details about this
approach are treated in Chapter 4.
The alternative philosophy to the model refinement problem is the physical parameter
update. The basic idea of this alternative approach consists of adjusting some or all structural
physical parameters to produce test/analysis modal correlation. Structural parameters are
design variables such as material densities, modulus of elasticity, cross-sectional areas,
element lengths and others. This type of procedure results in corrected FEMs that are
consistent with the framework of the original FEM.
Most methodologies that have adopted this philosophy have used the sensitivity
derivatives of the system eigenvalues and/or eigenvectors with respect to the structural
parameters. Generally, the refinement process amounts to solving for the corrected
structural parameters through an iterative optimization problem directed by the sensitivity
derivatives. Some notable work in this area was accomplished by Collins et al. (1974), Chen
and Garba (1980), Adelman and Haftka (1986), Creamer and Hendricks (1987), Flanigan
(1991), Martinez et al. (1991), to name only a few. In these studies, different sensitivity
formulations and iterative optimization strategies are proposed. A direct approach that

6
bypasses the use of closed-form sensitivity derivatives was presented in the papers of Hajela
and Soeiro (1990) and Soeiro (1990).
White and May turn (1976) set forth another alternative methodology that can be
classified in the physical parameter update group. In their approach, adjustments to the
stiffness matrix are viewed as a collection of known submatrices, elements or group of
elements. Each submatrix is multiplied by an unknown scaling factor. Correction of the
original stiffness matrix amounts to determining the scaling factors that would improve the
correlation of analytical and experimental modal properties. The scaling factors can be
physically viewed as functions of the stiffness structural parameters. A large
underdetermined system of equations in terms of the scaling factor is generated. The scaling
factors are then computed by using a pseudo-inverse. An improvement to the White and
Maytum approach is presented in the paper by Lim (1990).
1.2 Structural Damage Assessment
Structures, in general, are prone to structural damage during their service lives, that
could affect their overall performance and could result in catastrophic failures. This is of
critical concern when expensive systems and/or human lives are at stake. On this basis, it is
highly desirable to monitor the structural health of systems such as airplanes, space systems,
bridges, buildings, oil platforms, among others, to prevent such catastrophic events.
Currently, the most common structural health monitoring methods involve visual
inspections supplemented with X-rays, acoustic emission, magnetic resonance and
ultrasonic testing. These approaches can be time consuming, costly and difficult to perform
on inaccessible structural components. Additionally, none of these approaches provide a
quantitative assessment of the magnitude of the damage.
In response to these shortcomings, structural health monitoring approaches based on
the framework of existing model refinement techniques have been recently proposed by
several researchers (Smith and Hendricks, 1987; Chen and Garba, 1988; Rides and

7
Kosmatka, 1992; Zimmerman and Kaouk, 1992a,b; Kaouk and Zimmerman, 1993a,b).
These techniques utilize the vibration signature of the pre- and post- damaged structure, in
conjunction with an analytical model of the original structure, to assess both the location and
extent of the structural damage. The pre-damage modal parameters are used to correct
(refine) the original finite element model (FEM) to determine an “accurate” reference
baseline. Once damaged, the post-damage modal properties of the structure are used to
further “refine” the refined analytical model. This results in perturbations to the refined
analytical model. Analysis of the perturbations could indicate the damage location and
extent. An overview of the application of model refinement algorithms in assessing
structural damage is summarized in Figure 1.2.
Figure 1.2 Overview of Finite Element Model Refinement
Process Used for Damage Assessment

8
Notable exceptions to the direct use of FEM refinement algorithms to the damage
detection problem are the work of Lin (1990), Ojalvo and Pilón (1988), and Gysin (1990). In
the work of Lin, a flexibility matrix is determined using experimental data. This matrix is
then multiplied by the original stiffness matrix, with those rows and/or columns that differ
significantly from a row and/or column of the identity matrix indicating which degrees of
freedom have been most affected by the damage. It is then assumed that damage has occurred
in structural elements connecting those degrees of freedom. An overview of the work by
Ojalvo and Pilón (1988), and Gysin (1990) is given in Chapter 5.
Although the problems of damage assessment and model refinement are related to one
another, they have quite different characteristics. In essence, the model refinement concept
is based on the philosophy of the minimum change to the original FEM. Naturally, this
minimum change constraint has a tendency of smearing the changes throughout the entire
FEM. This aspect is inconsistent with the effects of structural damage on FEMs which are
usually localized perturbations of possibly large magnitude. In fact, structural damage often
occurs at discrete locations and only affects a few elements of the FEM.
1.3 Objective of the Present Study
The present study investigates the development of new and promising model refinement
and damage detection methodologies. Although considerable research has been done in
these areas, no methodology has been fully successful in dealing with the refinement or the
damage detection problem of “real life” systems. The main objective of this investigation is
to formulate efficient model refinement algorithms that are consistent with the effect of
structural damage.
In chapter 2, practical concepts and issues related to the general areas of model
refinement are presented. First, the concept of the eigenvalue problem of discrete structural
models is reviewed with emphasis on the associated cross-orthogonality conditions. A brief
discussion of modal analysis follows. The problem associated with incomplete eigenvector

9
measurements is then investigated and already existing techniques to deal with this problem
are presented. Furthermore, eigenvector orthogonalization techniques useful to a large
number of model refinement algorithms are reported. Finally, an iterative approach to the
problem of load path preservation encountered in a large number of property matrix update
algorithms is discussed.
In chapter 3, the author proposes and formulates a new model refinement algorithm.
The newly developed algorithm, named the inverse/hybrid method, is naturally based on the
inverse problem. The model type under consideration is proportionally damped and the
inaccuracies of the analytical model is assumed to be due to errors in the modelling of the
damping and stiffness properties. A complete hybrid set of modal data is generated by
approximating the unmeasured modal information by the corresponding analytical modes.
Symmetry of the corrected stiffness and damping matrices is enforced by mass
orthogonalization the complete hybrid eigenvector matrix. An orthogonalization procedure
that assigns more confidence on the measured eigenvectors is proposed. A comparative
study of the inverse/hybrid method and the algorithm proposed by Baruch and Bar Itzhack
(1978) shows that both algorithms give similar results. However, it is shown that the
inverse/hybrid approach is less computationally efficient.
Chapter 4 proposes improvements to the symmetric eigenstructure assignment model
refinement algorithm (SEAMRA) formulated by Zimmerman and Widengren (1989). The
author develops a technique to enhance eigenvector assignability. The technique, termed the
subspace rotation method, is based on rotating the achievable eigenvector subspaces into the
experimental eigenvectors. The subspace rotation method results in both a decrease in the
computational burden and an increase in the accuracy of the assigned eigenvector. The
enhanced SEAMRA is then evaluated and compared to other algorithms using both
computer simulated and experimental testbeds. It is shown that the enhanced algorithm is
suitable for damage detection applications.

10
In Chapter 5, an efficient damage location algorithm that utilizes modal data
information but bypasses the general framework of the model refinement problem is
presented. This location algorithm is an outgrowth of the subspace rotation method used to
enhance eigenvector assignability of the SEAMRA. The proposed location algorithm is
similar to the modal force error criteria presented by several researchers (Ojalvo and Pilón,
1988;Gysin, 1990). Further interpretation of the algorithm operation is given. Additionally,
the author proposes and formulates a new viewpoint that reduces the effect of measurement
noise for certain types of structures. Based on this formulation, an eigenvector filtering
algorithm is also developed.
Chapter 6 presents the formulation of computationally attractive damage extent
algorithms that are based on a minimum rank perturbation theory (MRPT) developed by the
author. The formulation of the MRPT is consistent with the effect of many classes of
structural damage on FEMs. Several MRPT based algorithms are formulated to
accommodate structures with undamped, proportionally damped, and nonproportionally
damped FEMs. For each type of FEM, several damage scenarios are considered.
Discussions of the characteristics and properties of the developed algorithms are presented
along with practical issues that can be used to improve their performance.
In Chapter 7, the algorithms developed in Chapters 5 and 6 are demonstrated and
evaluated using both computer simulated and actual experimental data. The main objective
of these example problems is to illustrate the potential of these algorithms in assessing
structural damage. In all examples, the location of damage is first determined by using the
algorithm presented in Chapter 5. An MRPT based algorithm is then utilized to assess the
extent of the damage. Several key points made throughout the formulation in Chapters 5 and
6 are emphasized. In particular, it is shown that the damage extent calculations can be greatly
enhanced by making use of the damage location algorithm.
In Chapter 8, summaries and conclusions of the issues discussed in this study are
presented along with some suggestions for future work.

CHAPTER 2
MATHEMATICAL PRELIMINARIES AND PRACTICAL ISSUES RELATED TO
THE PROBLEMS OF MODEL REFINEMENT AND DAMAGE DETECTION
2.1 Introduction
In this chapter, general mathematical preliminaries and practical issues relevant to the
areas of model refinement and damage detection are discussed. In Section 2.2, an overview
of the general eigenvalue problem for discrete models is given. Further, the concept of modal
analysis is introduced in Section 2.3. Section 2.4 deals with the concept of incomplete
degrees of freedom measurement. Two alternative approaches are discussed as possible
solutions to the incomplete measurement problem. Two eigenvector orthogonalization
techniques are discussed in Section 2.5. Finally, in Section 2.6, an algorithm to preserve
sparsity in updated property matrices is presented. The concepts discussed in this chapter
will be frequently referred to in the course of the forthcoming chapters.
2.2 The Eigenvalue Problem of Discrete Systems
In practice, most engineering structures are continuous systems with spatially
distributed material properties. The vibration motion of these structures in terms of their
distributed properties is usually governed by one or several partial differential equations.
With complex systems, both the development and the analysis of partial differential
equations of motion are tedious and in many cases impossible. These shortcomings and the
advent of digital computers have motivated the development of approximate modelling of
continuous systems in order to simplify the equations of motion. The general idea behind
these approximations is to represent the exact distributed model of a system by a discrete one.
11

12
This concept is known as spatial discretization which eliminates the continuous spatial
dependence of the distributed properties. The discrete model is of finite order and is
described by a finite number of variables known as degrees of freedom (DOFs). The number
of DOFs used in the discrete model depends on the desired accuracy in representing the
continuous model. Commonly, the vibration motion of systems in terms of their discrete
models is described by a set of simultaneous ordinary differential equations that are usually
simpler to develop and analyze than the partial differential equations of the continuous
system. One of the most commonly used approximate discrete modelling techniques is the
finite element method. The model generated by a finite element method is called a finite
element model (FEM). More elaborate discussions of the concepts of continuous and
discrete models as well as finite element methods are covered in detail in the books of
Meirovitch (1986, 1980), Inman (1989), and Hughes (1987).
Commonly, the free vibration motion of a structure in terms of an nth order discrete
model is represented by the following set of simultaneous ordinary differential equations
Mw(t) + Dw(t) + Kw(t) = 0 (2.2.1)
where M, D and K are termed, respectively, the mass, damping and stiffness matrices. They
are models of the mass, damping and stiffness properties of the structure. Since the system
model is order n, these matrices are of dimension nxn and are generally real. The variable
w(t) represents the n displacements of the n-DOF model of the structure. The overdots
represent differentiation with respect to time. The mass matrix, M, is always symmetric
positive definite. The stiffness (K) and damping (D) matrices of nongyroscopic and
noncirculatory systems are symmetric. In general, the modelling of the mass and stiffness
properties of the structure is simpler and more accurate than the modelling of the damping
properties. In the forthcoming discussion, it is assumed that the system under consideration
is nongyroscopic and noncirculatory. The standard solution to Eq. (2.2.1) is of the form
w(t) = veXt
(2.2.2)

13
where v is a constant nxl vector and X is a constant scalar. Substituting Eq. (2.2.2) into Eq.
(2.2.1) and dividing the resultant equation by e^1 yields the condition
My X2 + Dv X + Kv = 0 (2.2.3)
There are n sets of nontrivial complex conjugate solutions (X^, v¡) to Eq. (2.2.3). Note that
since the property matrices (M, D, K) are real, if (X¡, v¡) is a solution set to Eq. (2.2.3), the
complex conjugate of that set is also a solution. The problem of solving for these solutions
is commonly known as the eigenvalue problem and is sometimes referred to as the
characteristic value problem. The scalar X¡ and the vector v¡ are known, respectively, as the
eigenvalue and eigenvector of the ith mode of vibration of the structure. For a general
damped system, eigenvalues and eigenvectors are both complex. Note that Eq. (2.2.3) can
be easily rearranged in the more general mathematical eigenvalue problem format as
0
- M~'K
Inxn
- M“'D
(2.2.4)
where Inxn is the nxn identity matrix. Equation (2.2.4) is called a state space representation
of Eq. (2.2.3).
The eigenvalue and eigenvector can be related to some physical characteristic properties
of structures. For that matter, the ith eigenvalue is written as
\ = - CjW¡ + j ü)¡ y 1 - {22.5)
where j = / — 1 . Note that in writing this equation it is assumed that the system is
underdamped. The real scalar 0)¡ and are the natural frequency and damping ratio (or
damping factor), respectively, of the ith mode of the structure. The eigenvector v¡ indicates
the “shape” of vibration of the ith mode of the structure. The sets of frequencies, damping
ratios and mode shapes are sometimes referred to as modal parameters.
The symmetric nature of the property matrices (M, D, K) constrains the eigenvectors to
satisfy some cross-orthogonality relationships. For the purpose of discussing these

14
cross-orthogonality relationships, consider the following alternative state space
representation of the eigenvalue problem in Eq. (2.2.3)
[M ol
VA VA
[a ol
[D K1
VA VA
r°i
+
—
0 - K
V V
o
>
K 0
V V
0
where V = [ Vj, v2, . . . , vn ]
A = diag(Xj, X2, . . . , Xn)
where the overbar denotes the matrix complex conjugate operator. Based on this particular
representation and the fact that the state matrices of Eq. (2.2.6) are symmetric, by proper
normalization of the eigenvectors, the cross-orthogonality relationship associated with the
system are given by
n -,T_
M 0
0 - K
VA VA
V V
VA VA
V V
= I
2nx2n
(2.2.7)
T
VA VA
D K
VA VA
A 0
V V
K 0
V V
0 A
(2.2.8)
Equations (2.2.7) and (2.2.8), respectively, clearly imply the following relations
*
"VA'
"M
o'
"VA"
'o'
V
0
- K
V
0
*
'VA'
"D
K"
'VA'
'o'
V
K
0
V
0
(2.2.10)
where [ ] denotes the complex conjugate transpose operator ( [ ] = ^ J ). Note that,
contrary to Eqs. (2.2.7-8), no particular normalization of the eigenvector is needed in order
for Eqs. (2.2.9-10) to be satisfied.
Another state space representation of Eq. (2.2.3) is given by

15
O
M
M
D
VA VA
V V
A
O
O
A
+
O
K
VA VA
V V
(2.2.11)
Based on the same argument discussed earlier, the cross-orthogonality conditions that arise
from this representation are
T
VA VA
V V
0
M
M
D
VA VA
V V
^2nx2n
r
T
_
VA
VA
- M
0
VA
VA
A
0
V
V
0
K
V
V
0
A
Again, the following two relationships follow,
*
"VA"
'o
m'
'VA'
'o'
V
M
D
V
0
*
'VA'
'- M
o'
'VA'
'o'
V
0
K
V
0
(2.2.12)
(2.2.13)
(2.2.14)
(2.2.15)
2.2.1 Undamped Models
In the modelling of structures, it is often assumed that the damping is negligible and
hence is set to zero. For these type of system models, the eigenvalue problem in matrix form
is given by
MV A2 + KV = [0] (2.2.16)
The matrices V and A are as defined earlier. For undamped systems, the eigenvalues are
pure imaginary and the eigenvectors y¡ are real. Note that the eigenvalues are related to the
system natural frequency by
X2 = - co2 (2.2.17)
Furthermore, by proper normalization of the eigenvectors, the cross-orthogonality relations
associated with this type of systems are

16
VtMV = Inxn (2.2.18)
VTKV = diag(wj, u)^, . . . , O0n) (2.2.19)
2.2.2 Proportionally Damped Models
When the damping of the structure is accounted for, it is sometimes modelled to be
proportional,
D = aM + pK (2.2.20)
where a and (3 are real scalars. The eigenvectors of a proportionally damped system are the
same as the eigenvectors associated with the corresponding undamped system. The
cross-orthogonality relationships associated with proportionally damped models are
VtMV = Inxn
(2.2.21)
VtDV = diag^jCDj, 2£2C02, .
• ■ > 2^no)n)
(2.2.22)
VtKV = diag(o)j, oc>2, . .
• , o>2)
(2.2.23)
where all variables have the same definitions as in the previous discussion.
A more detailed development of the eigenvalue problem can be found in the books of
Meirovitch (1986, 1980) and Inman (1989).
2.3 Experimental Modal Analysis
The vibration characteristics of structures can also be measured experimentally. Recall
that the structural vibration characteristics are given by natural frequencies, damping ratios
and mode shapes (eigenvectors). The process of measuring the modal parameters is known
as modal testing or experimental modal analysis. An introductory treatment of the concept
of experimental modal analysis can be found in the book of Inman (1989); a more rigorous
coverage is treated in the book of Ewins (1986).
The hardware components needed in a modal analysis experiment are identified in
Figure 2.1 (Inman, 1989). A schematic of a simple modal vibration measurement test setup
6

17
is shown in Figure 2.2. Brief descriptions of some of the components and their functions are
given below.
T := Transducer
SC := Signal Conditioner
Figure 2.1 Components of a Vibration Measurement System
For Modal Analysis (Inman, 1989).
The exciter is used to deliver the driving force that puts the structure in motion. The
two most commonly used exciters are the shaker and the impulse hammer.
Transducers are devices that measure the driving force as well as the response of the
structure. They sense mechanical forces or motions and, then, convert them into
electrical signals. Mechanical forces are usually recorded by a force transducers.
Commonly, the vibration response of structures is measured by accelerometers.
Accelerometers are mounted directly on the structure and, naturally, record the
local accelerations.
Signal conditioners are used to match the signal, received from the transducers, to
the specification of the analyzer. These devices can also be used to amplify the
signals.

18
Signal Analyzer
Figure 2.2 A Simple Experimental Modal Analysis Setup.
The signal analyzer processes the electrical signal received from the signal
conditioners. The standard type of analyzer allows time domain signals to be
viewed in the frequency domain via a Fast Fourier Transform (FFT) algorithm. In
an FFT, the signals are first filtered, digitized and then transformed into discrete
frequency spectra. The frequency spectra can then be manipulated to compute the
modal properties of the structure.
It is important to note that the experimentally identified modal parameters are usually
affected by unpredictable measurement noise. Typically, natural frequencies are identified
to within 1 to 2% repeatability; damping ratios to within 5 to 15% repeatability, and mode
shapes to within 5 to 10% repeatability. In practice, due to testing limitations, the set of
£

19
structural modal parameters identified experimentally is incomplete with respect to the
analytical model. Experimental incompleteness is manifested in two forms, (i) a limited
number of measured modes of vibration, (ii) a limited number of eigenvector DOF
measurements.
2.4 Analytical and Experimental Model Dimensions Correlation
One major prerequisite common to most model refinement algorithms is to match the
number of degrees of freedom (DOFs) in the experimentally measured eigenvector and in
the discrete analytical model. Two different approaches have been commonly used to
resolve this matching problem when an incomplete set of eigenvector DOFs have been
measured. The first approach consists of reducing the dimension of the discrete analytical
model to the number of the measured DOFs. The other approach is to expand the measured
eigenvector to the size of the analytical model. A good evaluative survey of a number of
analytical model reduction techniques has been compiled by McGowan (1991). The subject
of eigenvector expansion is discussed in fair detail in the papers by Gysin (1990), and
Zimmerman and Kaouk (1992a). In the next two sections, some commonly used model
reduction and eigenvector expansion algorithms are reported and described.
2.4.1 Model Reduction Methods
In this section the general framework of the model reduction concept is first presented.
Then formulations of three commonly used model reduction techniques are summarized.
Mostly, the concept of model reduction has only been studied for undamped models. In this
presentation no attempt has been made to generalize these concepts to damped models.
For the purpose of reporting the general concept of model reduction, consider the
eigenvalue problem associated with an undamped model,
MV A2 + KV = [0]
(2.4.1)

20
where, as defined earlier, M and K are, respectively, the mass and stiffness matrix; V is the
eigenvector matrix and A is the diagonal eigenvalue matrix. Assume that only a subset of
the eigenvector DOFs has been experimentally measured.
Equation (2.4.1) can be reordered such that the DOFs associated with the measured
DOFs are in the upper rows of the equation,
M0V0A2 + K0V0 =
= [0]
"Vm"
K-mm Kmu
Mmm Mmu
where V0 =
Vu
K0 =
Kum Kuu
M0 =
Mum Muu
The matrices M0, K0, and V0 are, respectively, the reordered mass, stiffness and
eigenvector matrices. In the above equation, the subscripts “m” and “u” denote, respectively,
the components associated with the measured and unmeasured DOFs. A transformation
matrix P, that relates matrices Vu and Vm can be defined as
Vu = PVm
(2.4.3)
A substitution of this relationship in the reordered eigenvector matrix V0 results in
V0
TVm
(2.4.4)
Substituting Eq. (2.4.4) into Eq. (2.4.2) and premultiplying by TT yields the eigenvalue
problem of the reduced model,
MrVmA2 + KrVm = [0] (2.4.5)
where Mr = TtM0T
Kr = TtK0T
where Mr and Kr are the reduced mass and stiffness matrices, respectively. In terms of the
partitioned matrices, the reduced matrices are defined as

21
Mr
~ Mmm
+
PTMum
+
MmuP
+
ptmuup
(2.4.6)
Kr
= Kmrn
+
PTKum
+
KmuP
+
PTKuuP
(2.4.7)
At this point, the only condition placed on P is the relationship of Eq. (2.4.3). Naturally,
matrix P can be computed directly from Eq. (2.4.3) if the eigenvectors of the system are
available. This approach is known as the exact reduction method and has been discussed in
the papers by Kammer (1987) and O’Callahan et al. (1989). The exact reduction method
requires solving for a large number of eigenvectors, which can be computationally
expensive. The three reduction methods that are presented in the forthcoming discussions
propose alternative techniques to compute matrix P. The first two do not require the
computation of the system eigenvalue problem. The last one requires the knowledge of one
eigenvalue which is computationally admissible.
2.4.1.1 Static Reduction
This reduction method is often referred to as Guyan (1965) reduction. In the static
reduction, the mass properties associated to the unmeasured DOFs are assumed negligible.
With that assumption, Eq. (2.4.2) can be written as
Mmm
0
0
0
A2 +
Kmm
K-um
Kmu
1
E
>
i
c
c
Vu
The second row of this matrix equation can then be manipulated as
Vu = - KjKumVm
(2.4.8)
(2.4.9)
From comparing Eq. (2.4.9) to Eq. (2.4.3), it can deduced that the transformation
matrix, P, computed using the Guyan approach is given by
Pg = - Kj Kum (2.4.10)
The reduced mass and stiffness matrices can then be computed by substituting matrix Pg
for matrix P in Eqs. (2.4.6) and (2.4.7). Naturally, the Guyan assumption (Eq. (2.4.8))

22
suggests that if the mass properties of the omitted DOFs are not small, the accuracy of the
Guyan reduced model could be lacking.
2.4.1.2 IRS Reduction
The improved reduction method (IRS) was formulated by O’Callahan (1989). It is an
improvement over the Guyan reduction in that it accounts for the mass properties of the
unmeasured DOFs. In the formulation of the IRS method, the Guyan reduced model is
corrected to include the mass influence of the unmeasured DOFs. This formulation is
somewhat lengthy and the interested reader is referred back to the paper of O’Callahan
(1989) or the thesis of McGowan (1991). The transformation matrix P computed using the
IRS reduction is
PIRS = Pg + Kjf Mum + MuuPg] M-'Krg (2.4.11)
The reduced IRS model is then computed by substituting matrix Pirs in Eqs. (2.4.6-7).
2.4.1.3 Dynamic Reduction
The dynamic reduction was proposed as another improvement to the Guyan reduction
(Kidder, 1973; Miller, 1980; and Paz, 1984). This reduction utilizes the dynamic equation
associated with a single mode of vibration to compute the transformation matrix P. In this
technique, the transformation matrix P is arrived at by considering the reordered dynamic
equation associated with the ith mode,
^•j Mmrn + Kmm
Mmu + Kmu
—m¡
'O'
+ Kum
^fMuu + Kuu
0
where is the ith eigenvalue; vm and vu are, respectively, the measured and unmeasured
eigenvector DOFs associated with the ith mode of vibration. Based on this partition, two
equations can be generated. By using the rows that correspond to the unmeasured DOFs
(lower rows), the following relationship is obtained,

23
vu¡ = -
-i
MUUX? + K,„, M„m^? + K,
—m¡
(2.4.13)
Hence, the transformation matrix associated with the ith mode is defined by
Pd; = -
Mn,A? + K„„ MlimX? + K,
(2.4.14)
The reduced mass and stiffness matrices are then computed by using Eqs. (2.4.6) and (2.4.7).
Note that different modes will result in different transformation matrices Pd and, hence,
different reduced mass and stiffness matrices.
2.4.2 Eigenvector Expansion Methods
Alternatively, the dimension of the measured eigenvectors can be correlated to the
dimension of the analytical model by using eigenvector expansion algorithms. The common
basis of these algorithms is the interpolation of the unmeasured eigenvector components. In
the forthcoming sections, two eigenvector expansion algorithms are reviewed.
2.4.2.1 Dynamic Expansion
The dynamic expansion technique (Berman and Nagy, 1983) is one of the most
commonly used eigenvector expansion algorithms. A slight modification of the Berman and
Nagy formulation is presented here to accommodate damped systems (Fuh et al., 1984). In
the formulation of the dynamic expansion, it is assumed that the measured modes satisfy the
eigenvalue problem involving the property matrices of the original model (M, D, K). For the
ith measured mode, this assumption takes the form
(X|.M + KP + K) ve. = 0 (2.4.15)
where Xe. and ve are, respectively, the ith experimental eigenvalue and eigenvector. The
matrices M, D and K have the same definitions as in the earlier sections. Assume that only
a subset of the DOFs of eigenvector ve has been measured. Equation (2.4.15) can be
reordered, as in Section 2.4.1, such that the measured eigenvector DOFs reside in the upper
half of the equation,

24
Mmrn
Mmu
X-e. +
Dmm
Dmu
X*p "I"
Kmm
Kmu
>
>
Xem,
MUm
Muu
Dum
Duu
ei
KUm
Kuu
Xeu¡
*
• -
(2.4.16)
where vem and veu are, respectively, the measured and unmeasured DOFs of ve.. The
subscripts “m” and “u” denote measured and unmeasured components. A rearrangement of
Eq. (2.4.16) yields
^e^mm 3" ^e¡Dmm "b Kmrn
^-eiMmu + X,ejDmu "b Kmu
rv i
—em¡
'O'
X-^Mum "b ^-ejDum "b KUm
X.g.Muu "b Xe.Duu "b Kuu
1
s'
>i
i
0
From Eq. (2.4.17), two matrix equations in function of vem. and veu. can be generated.
By using the equation associated with the second row of the partition, the unmeasured
components of the ith experimental eigenvector are computed to be
Xeu,
-1
MUUX¡. + DIluXe. + Kuu M umll + D um^e: + K,
Lum
Vemi (2-4.18)
Note that this expansion works on a single mode at a time. Also, notice that it involves
the original analytical model (M, D, K). This implies that the accuracy of the expansion
depends on the validity of the original analytical model.
2.4.2.2 Orthogonal Procrustes Expansion
Another expansion process that has shown great promise is the orthogonal Procrustes
(OP) expansion method presented in the papers of Smith and Beattie (1990) and Zimmerman
and Kaouk (1992a). The technique uses the general mathematical framework of the
orthogonal Procrustes problem (Golub and Van Loan, 1983).
Let Vem be the experimentally measured eigenvector component matrix and Vam be the
corresponding analytical eigenvector component matrix. The essence of the OP expansion is
to find a unitary transformation matrix Pop that closely rotates Vam into Vem. This is
attempted by solving the following problem,

25
Minimize || Vem - VamPop ||F
subject to PjpPop = I
(2.4.19)
The solution to this general problem is discussed in the book of Golub and Van Loan and
is given by
Pop = YZt (2.4.20)
where Y and Z are, respectively the left and right singular matrices of matrix O defined by
* = VimVem (2.4.21)
Let Vau be the eigenvector matrix associated with the unmeasured DOFs. In the
orthogonal Procrustes expansion, it is assumed that the transformation matrix computed
above also rotates Vau into the unmeasured “experimental” eigenvector component matrix,
Veu, as
Yeu — VauP0p
(2.4.22)
Two different approaches have been defined to generate the expanded experimental
eigenvector matrix. The first is suggested in the paper by Smith and Beattie (1990),
Ve
op
(2.4.23)
In this definition, the expanded “experimental” eigenvector matrix is the rotated analytical
eigenvector matrix. The actual experimental measured eigenvector components Vem are
replaced by matrix VamPop. The advantage of the approach is that the resulting
“experimental” eigenvector satisfies the cross-orthogonality conditions (see Section 2.4).
The second viewpoint was proposed by Zimmerman and Kaouk (1992a),
Ve
' vem '
VauPop
(2.4.24)
Here, the unaltered eigenvector components measured experimentally are inserted in Ve. In
this viewpoint, if cross-orthogonalization of the expanded experimental eigenvectors is

26
required, a separated orthogonality algorithm can be used (Section 2.5). In the paper by
Smith at al. (1993), it is shown that for actual model refinement problems, both viewpoints
give equivalent eigenvector expansion results. However, for damage detection problems,
a preliminary study indicates that eigenvectors expanded by using the viewpoint defined in
Eq. (2.4.24) give better assessment of the damage.
2.5 Eigenvector Orthogonalization
Most matrix update algorithms require the measured eigenvectors to satisfy a
cross-orthogonality condition. This is especially true in matrix update algorithms in which
(i) the model of the structure is assumed undamped and the modelling errors are assumed to
be in only one of the two property matrices (M or K is assumed correct) or (ii) the system is
modeled by a proportionally damped model with errors in only two of the three property
matrices (M, D or K). In these situations, in order to insure symmetry of the updated property
matrices, it is required that the expanded experimental eigenvectors be orthogonal with
respect to the property matrix assumed accurate. This situation is encountered in the optimal
update algorithms developed by Baruch and Bar Itzhack (1978), Kabe (1985), Kammer
(1985), and Smith and Beattie (1991), Zimmerman and Kaouk (1992b), Kaouk and
Zimmerman (1993b) among others. In most of these algorithms, it is assumed that the mass
matrix is correct. This assumption is used in a number of the model refinement algorithms
since the inertial properties of structures are known to a good extent. In these cases, one
would expect the expanded experimental eigenvectors to be mass orthogonal. However,
because of measurements errors, this condition rarely occurs. For this reason, a great deal of
effort was focussed on the development of mass orthogonalization techniques. Some of the
most notable work in that area was performed by Targoff (1976), Baruch and Bar Itzhack
(1978), and Baruch (1979). In the next two sections, two orthogonalization techniques
(Baruch and Bar Itzhack, 1978; Baruch, 1979) are discussed. Both techniques are mass
orthogonalization techniques; however, with obvious modifications, these techniques can be

27
adopted to solve the orthogonalization problem of the eigenvectors with respect to the
stiffness or the damping matrices.
2.5.1 Optimal Weighted Orthogonalization
The essence of the standard mass orthogonalization technique is to modify the measured
eigenvectors such that the mass cross-orthogonality condition is satisfied. Baruch and Bar
Itzhack (1978) proposed an elegant solution to that problem. An overview of their problem
statement and solution is given below.
Assuming that Ve is a matrix of expanded experimental eigenvectors that need to be
mass orthogonalized. The present formulation searches for the optimal mass weighted
change of matrix Ve such that the mass cross-orthogonality condition is satisfied. This
problem is cast as
Minimize
|| N (Veo - Ve) ||p
(2.5.1)
subject to
Vio M Veo = I
(2.5.2)
where N=M1/2 and M is the mass matrix. By means of a Lagrange multiplier, Eq. (2.5.2)
can be incorporated into Eq. (2.5.1); then the application of the optimality conditions yields
the following expression for Ve0,
Veo = Ve(V¡F M Ve)'1/2 (2-5.3)
Before being incorporated into the orthogonalization process, the measured
eigenvectors have to be unit mass normalized, i.e.,
(2.5.4)
where ve is the ith expanded experimental eigenvector (ith column of Ve).
2.5.2 Selective Optimal Orthogonalization
Some structures exhibit rigid body modes (modes with zero eigenvalues). It is desirable
to preserve these rigid body modes in the refinement process. However, some matrix update

28
algorithms require the rigid body modes and the experimental eigenvectors to be mass
orthogonal to insure symmetry of the updated property matrices (see Chapter 6). Naturally,
the rigid body modes will be corrupted if they are incorporated along with the expanded
experimental eigenvectors in the above orthogonalization process. Thus, Baruch (1979)
presented a modification of the procedure in Section 2.5.1 to deal with such a problem. The
resulting problem is a selective orthogonalization and is formulated as follows,
minimize
|| N (Veo - Ve) ||F
(2.5.5)
Subject to
Vio M Veo = I
(2.5.6)
and
V¡T0 M Vr = [0]
(2.5.7)
In the above equations, Ve and Vr are, respectively, the expanded experimental eigenvector
matrix and the rigid body mode matrix. Again, the Lagrange multiplier is used, and the
orthogonalized experimental eigenvector matrix that satisfies the conditions in Eqs. (2.5.6)
and (2.5.7) is found to be
Veo = Q(QT M Q)-1/2
where Q = Ve - VrV?MVe
(2.5.8)
Note that, as in previous process, the expanded experimental eigenvectors have to be
unit mass normalized.
2.6 Load Path Preservation
Many matrix update algorithms introduce additional load paths in their updated models,
i.e., elements of the mass, damping or stiffness matrices that were originally zero may
become nonzero. Whether or not preserving the original load path is a practical problem is
still a matter of current debate. It seems that for damage assessment of truss structures it is
desired to maintain load paths. In the paper of Zimmerman and Kaouk (1992a), an iterative
approach to preserve the load path of the original property matrices was developed. The

29
approach was presented in the context of the symmetric eigenstructure assignment model
refinement algorithm (discussed in Chapter 4); however, its application can also be extended
to other model refinement algorithms. In Figure 2.3, a flow chart of the iterative load
preservation algorithm is presented. The procedure is illustrated for a general model
refinement scenario in which all three property matrices (M, D, K) are being updated.
However, it can be easily modified to accommodate other refinement problems.
Figure 2.3 Flow Chart of the Iterative Load Path Preservation Algorithm
In the flow chart, the matrices Ma m, Da m, and Ka m are respectively the adjusted
masked mass, damping and stiffness matrices defined by

30
Ma,m — Ma O Mm
Da,m = Da O Dm (2.6.2)
Ka,m = Ka O Km
where Ma, Da, and Ka are the adjusted mass damping and stiffness matrices. The matrices
Mm, Dm, and Km are the masking matrices associated with the original mass, damping and
stiffness matrix. By definition, the masking matrix, Am, associated with matrix A is given
by
Am(i,j) = 1 if A(i,j) * 0
(2.6.1)
Am(i,j) = 0 if A(i,j) = 0
In Eqs. (2.6.2), the operator O is the element-by-element (scalar) matrix multiplication. Let
B and C be two nxn matrices, then the element-by-element multiplication of B and C is given
by
S = BOC => S(i,j) = B(i,j) * C(i,j) i, j = l,...,n (2.6.3)
At every iteration, the norms of the matrix differences between corresponding adjusted
and adjusted masked property matrices are computed. At a given iteration, if the three
computed norms are equal to zero or within user set limits, then the load paths of the original
three property matrices have been exactly achieved or achieved within user state guidelines.
Thus, the procedure is halted, and the refined model consists of the adjusted property
matrices computed at that particular iteration. It should be noted that there is no formal
guarantee of convergence in using this iterative procedure. Experience gained in using the
present algorithm indicates that if the experimental modal data are consistent with the
sparsity pattern, the procedure will converge. Consistent data means that there exist mass,
damping and stiffness matrices that have the same sparsity pattern as the original matrices
and also exhibit the measured test data. Otherwise, if the data are inconsistent, the original
sparsity patterns will not be exactly preserved. In this case, the added load path terms of Ma,
Da and Ka which should be zero will be closer to zero after application of the algorithm.

CHAPTER 3
INVERSE / HYBRID APPROACH FOR FINITE
ELEMENT MODEL REFINEMENT
3.1 Introduction
The inverse eigenvalue problem is concerned with the construction of the property
matrices (mass, damping or stiffness) of a dynamic model using experimentally measured
modal data. These techniques require complete modal properties. Thus, for an n-DOF
model, n natural frequencies, damping ratios and mode shapes (eigenvectors) must be
measured, and the identified mode shapes must be of dimension n. Due to practical testing
limitations, this is rarely accomplished for typical large structures. In this chapter, the
application of the inverse problem is extended to model updating by combining
experimental measurements and original analytical FEM modal information. Again,
refinement implies correlating the measured and analytical modal properties.
3.2 Theoretical Formulation
The dynamic structure under consideration is assumed to be successfully modelled by
an n-DOF proportionally damped nongyroscopic and noncirculatory (symmetric property
matrices) FEM. The free vibration motion of such a dynamic structure can be analytically
represented by a differential equation of the form
Mw(t) + Dw(t) + Kw(t) = 0 (3.2.1)
where the variables M, D, and K are nxn real symmetric matrix models of the mass, damping
and stiffness properties of the structure. The nxl time varying vector w(t) represents the n
31

32
displacements of the n-DOF model of the system. The overdots represent differentiation
with respect to time. The eigenvalue problem associated with the differential equation
shown in Eq. (3.2.1) is of the form
Mv¡ X? + Dyj X¡ + Kvj = 0 (3.2.2)
where Xj and v¡ are, respectively, the eigenvalue and eigenvector of the ith mode of vibration.
In this problem, it is assumed that the accuracy of the original FEM is lacking and, hence,
needs improvement. Furthermore, it is assumed that the inaccuracy of the original FEM is
solely due to modeling errors in the stiffness and damping properties.
The model refinement, proposed herein, exploits the cross-orthogonality relations that
arise from the symmetric nature of the property matrices and the proportional damping
assumption. As discussed in Chapter 2, by proper normalization of the eigenvectors these
cross-orthogonality relations have the form
VT M V =
Inxn
(3.2.3a)
VT D V = diag(2^1co1,
• • • »2^nwn) = ^
(3.2.3b)
VtKV = diag(o),2, .
. . ,wn2) = Q
(3.2.3c)
>T
ii
>
- vn ]
where tOj and £¡ are the natural frequency and damping ratio, respectively, of the ith mode
of the structure. The matrix Inxn is the nxn identity matrix. It is important to recognize that
Eq. (3.2.3a) represents necessary and sufficient conditions for conserving symmetry and
damping proportionality when updating the stiffness and damping properties of
proportionally damped systems.
Suppose that p (p « n) modes of an existing structure have been experimentally
identified (mode shapes or eigenvectors, frequencies and damping ratios). Assume that the
dimension of measured eigenvectors is equal to the dimension of the FEM, i.e., all n
components of the measured eigenvectors are available. It is widely accepted that in the

33
absence of specific experimental measurements a good approximation to the unmeasured
modes is their corresponding analytical modal information. With that in mind, a complete
hybrid set of modal data is generated by combining experimental and analytical information
Vea = [Ve VJ
Qe 0
0
2P, =
Ze 0
0
(3.2.4)
where V is the eigenvector matrix; £2 and X are diagonal matrices of frequencies squared and
damping ratios, respectively. The subscripts e and a denote, respectively, experimental and
analytical sets. At this point, the complete “hybrid” set of eigenvectors, Vea, does not satisfy
the cross-orthogonality conditions defined in Eq. (3.2.3a); thus the conditions in Eqs.
(3.2.3b) and (3.2.3c) are not met. One possible solution to this problem is to modify all of
Vea in an optimal way to comply with the orthogonality requirement. This approach treats
all parts of Vea equally, and thus overlooks the fact that the experimental modes are known
with a higher confidence. Naturally, a technique that assigns a higher credibility to the
experimental eigenvectors, Ve, is preferable. This can be achieved by incorporating Vea into
the orthogonalization process group by group in the order of their descending credibility
(experimental then analytical). If the problem is set such that the experimental modes, Ve,
are corrected first, it is clear that the analytical modes, Va, will be subject to larger correction
when incorporated into the orthogonalization process since they will be subject to more
constraints.
The experimental modes, Ve, are orthogonalized by using the orthogonalization
technique formulated by Baruch and Bar Itzhack (1978). The general formulation of this
orthogonalization technique is discussed in Chapter 2. For this particular application, the
problem consists of finding the matrix ,Veo, that satisfies the cross-orthogonality condition,

34
VT M V = T
v eo iVA v eo Apxp
and that minimizes the weighted Euclidean norm,
<*» =|| N (Veo - Ve) ||F
where N = M1/2
The solution to this problem, as reported in Chapter 2, is
Veo = VefvJ M Ve)“1/2
(3.2.5)
(3.2.6)
(3.2.7)
The next step is to invoke the orthogonality requirement on the analytical eigenvector
matrix, Va, by searching for a matrix Vao that satisfies the following two conditions,
Vio M Vao = I(n_p)x(n_p) (3.2.8)
and Vüo M Veo = [0]
while minimizing the objective function,
r =11 N (Vao - Va) ||p
where N = M1/2
(3.2.9)
(3.2.10)
A similar problem was also treated by Baruch (1980) in a different context. A brief
discussion of the solution approach can be found in Chapter 2. The set of eigenvectors, Vao
that satisfies this problem is given by
vao = q(qt m q) 1/2
Q = Va - VeoVjoMVa
(3.2.11)
Clearly, the resultant matrix, Veao = [ Veo , Vao ], satisfies Eq. (3.2.3a). The corrected
stiffness and damping matrices are then computed using Eqs. (3.2.2b) and (3.2.2c).

35
Ka = MVeao Q eaVjaoM (3.2.12a)
Da = MVeao © (3.2.12a)
where 0ea = 2TleaQea
Note that the matrices Ka and Da computed from Eqs. (3.2.12a,b) will be symmetric.
The above formulation suggests that the system modelled by the original mass matrix (M)
and the stiffness (Ka) and damping (Da) matrices computed from Eqs. (3.2.12a,b) will have
eigenvectors Veao, frequencies £^a, and damping ratios nea. Some structures exhibit rigid
body modes of vibration. Commonly, it is desirable to preserve these rigid body modes in the
updated model. The above formulation also suggest that the updated model will preserve the
original rigid body modes.
The procedure developed above can be easily contracted to address the case when the
system model does not account for the effects of damping (undamped model). The
contraction can be obtained by setting to zeros matrices D, 0ea, and nea in Eqs. (3.2.2-4).
The computational burden limits the size of the FEM which can be updated (order of
200). Essentially, the limiting factor is that all mode shapes of the structure that are not
available from experimental measurements must be calculated analytically by solving the
eigenvalue / eigenvector problem.
3.3 Numerical Illustration
The system addressed in this investigation is the commonly used eight degrees of
freedom model shown in Figure 3.1. This model was developed by Kabe (1985) to give a
common testbed for the evaluation of the performance of model refinement algorithms. An
original undamped analytical model of the system was generated by using the mass and
stiffness properties shown in Figure 3.1. The elements of the original analytical stiffness
matrix are displayed in the second column of Table 3.1. In this problem, it is assumed that the
original stiffness matrix of the model is incorrect. These inaccuracies were simulated by

36
mi = 0.001 nift = 0.002 mj = 1.0 j = 2,..., 7
kj = 1000 k2=10 k3 = 900 k4=100 k5 = 1.5 ^ = 2.0
Figure 3.1 Kabe’s Problem: Analytical Test Structure.
using incorrect stiffness constants for most of the springs. The elements of the actual correct
stiffness matrix are given in the last column of Table 3.1. Note that the present model
refinement problem is very challenging and because of the large difference between the
stiffness matrices of the original and exact model. In this study, two cases will be considered.
In the first case, it is assumed that only the modal parameters (eigenvalue and eigenvector) of
the first mode were measured. In the other, the modal parameters of first three modes are
assumed to be available. In both cases, the measured eigenvectors are supposed to be full (all
degrees of freedom of the eigenvector(s) are measured).
The main objective of this investigation is to compare the performance of the
inverse/hybrid method to the algorithm proposed by Baruch and Bar Itzhack (1978). The
Baruch and Bar Itzhack model update technique is one of the most commonly used model
refinement algorithm. One of the main reasons for its common use is because the algorithm
is computationally efficient. The updated stiffness matrices generated by using the Baruch
and Bar Itzhack approach for the one mode and three mode cases are shown in the third and

37
fourth column of Table 3.1, respectively. The fifth and sixth columns of Table 3.1 display the
results of using the inverse/hybrid algorithm for the one mode and three modes cases. For
both cases, it is clear that the performances of both algorithms is lacking in predicting the
exact stiffness matrix. As expected, It can also be seen that both algorithms generate better
results when using three measured modes. A comparison of the results generated using the
Baruch and Bar Itzhack approach and the inverse/hybrid method shows that both algorithms
give the same type of results. This similarity of results was also encountered in other
example problems.
3.4 Summary
A model refinement approach that uses a hybrid set of experimental and analytical
modal properties was formulated. The developed approach, termed the inverse/hybrid
algorithm, was illustrated by using a computer simulated example. Part of the evaluation of
the proposed algorithm was the comparison of its performance with the performance of the
Baruch and Bar Itzhack algorithm. It was found that both algorithms give the same type of
results. However, the computations involved in the inverse/hybrid approach exceed those
involved in the Baruch and Bar Itzhack algorithm. Essentially, the inverse/hybrid approach
requires the computation of all eigenvalues and eigenvectors of the structure that are not
available from experimental measurements. As will be illustrated in the forthcoming
chapter, the Baruch type approaches are not suited for damage assessment applications. For
these reasons, further development of the Inverse/Hybrid algorithm was not investigated and
new formulations (Chapters 4 & 6) were considered.

38
Table 3.1 Kabe’s Problem: Elemental Stiffness Components.
Baruch
Inverse/Hybrid
Element #
Original
Mode 1
Modes 1-3
Mode 1
Modes 1-3
Exact
(1,1)
2.0
2.0
2.0
2.0
2.0
1.5
(1,2)
-2.0
-2.0
-3.0
-2.0
-3.0
-1.5
(1,3)
0.0
0.0
-0.1
0.0
-0.1
0.0
(1,4)
0.0
0.0
-0.1
0.0
-0.1
0.0
(1,5)
0.0
0.0
-0.1
0.0
0.1
0.0
(1,6)
0.0
0.0
0.1
0.0
0.1
0.0
(1,7)
0.0
0.0
0.1
0.0
0.0
0.0
(1,8)
0.0
0.0
0.0
0.0
0.0
0.0
(2,2)
1512.0
1508.6
1024.2
1510.6
1024.3
1011.5
(2,3)
-10.0
-31.5
-68.5
-21.5
-71.4
-10.0
(2,4)
0.0
-8.9
-9.0
-15.0
-8.4
0.0
(2,5)
0.0
-8.9
-20.9
-15.0
23.8
0.0
(2,6)
0.0
-21.6
38.5
-11.5
35.2
0.0
(2,7)
0.0
-3.9
-9.1
-1.6
-8.7
0.0
(2,8)
0.0
-0.1
0.1
-0.1
0.1
0.0
(3,3)
1710.0
1574.6
1560.8
1624.1
1612.2
1110.0
(3,4)
0.0
-44.9
^19.8
-75.6
-76.2
0.0
(3,5)
-200.0
-244.8
-244.1
-275.6
-276.9
-100.0
(3,6)
0.0
-136.5
-123.4
-86.2
-74.5
0.0
(3,7)
0.0
-24.6
50.8
-13.1
46.8
0.0
(3,8)
0.0
-0.4
-0.1
-0.4
0.0
0.0
(4,4)
850.0
1083.1
1087.9
1102.1
1099.9
1100.0
(4,5)
-200.0
3.27
25.1
51.8
48.0
-100.0
(4,6)
-200.0
-254.3
-242.2
-276.5
-274.9
-100.0
(4,7)
0.0
-10.1
27.0
-17.2
30.8
0.0
(4,8)
0.0
-0.2
-0.1
-0.2
-0.2
0.0
(5,5)
850.0
1082.2
1089.5
1101.4
1101.1
1100.0
(5,6)
0.0
-45.2
-49.5
-76.5
-75.1
0.0
(5,7)
0.0
-10.1
-12.4
-17.2
-11.7
0.0
(5,8)
0.0
-0.2
-0.1
-0.2
-0.2
0.0
(6,6)
1714.0
1576.4
1565.0
1617.5
1610.8
1112.0
(6,7)
-10.0
-34.8
-82.6
-23.1
-76.2
-10.0
(6,8)
-4.0
-4.4
-4.1
-4.4
-4.5
-2.0
(7,7)
1512.0
1507.5
1027.9
1510.1
1028.0
1011.5
(7,8)
-2.0
2.1
-4.1
-2.1
-4.2
-1.5
(8,8)
6.0
6.0
6.0
6.0
6.0
3.5

CHAPTER 4
SYMMETRIC EIGENSTRUCTURE ASSIGNMENT MODEL
REFINEMENT ALGORITHM
4.1 Introduction
Eigenstructure assignment is a control concept used to alter the transient response of
linear systems. This is done by forcing the system to have some predetermined eigenvalues
and eigenvectors. A detailed overview of eigenstructure assignment theories can be found in
the paper by Andry et al. (1983). Inman and Minas (1990), Zimmerman and Widengren
(1989,1990), and Widengren (1989) have developed model refinement algorithms based on
the mathematical framework of eigenstructure assignment. The basic idea of these model
refinement techniques is to design the pseudo-controller which is required to produce the
measured modal properties (natural frequencies, damping ratios and mode shapes) with the
original finite element model (FEM) of the structure. The pseudo-controller is then
translated into matrix adjustments applied to the initial FEM.
In this work, the eigenstructure assignment based model refinement algorithm proposed
by Zimmerman and Widengren (1989, 1990) is extended to better approach the damage
assessment problem. A subspace rotation algorithm is developed to enhance eigenvector
assignability. Finally, the enhanced algorithm is tested and compared to other techniques on
both “simulated” and actual experimental data.
4.2 Problem Formulation
In this section, a review of the Zimmerman and Widengren (1989, 1990) refinement
technique, which is termed the symmetric eigenstructure assignment model refinement
39

40
algorithm (SEAMRA), is presented. This review is essential in order to properly introduce
and discuss the extension and improvement proposed in this work.
4.2.1 Standard Eigenstructure Assignment Formulation
Consider the standard differential equation of motion of an n degrees of freedom
damped, nongyroscopic and noncirculatory structure with control feedback,
Mw(t) + Dw(t) + Kw(t) = Bgu(t) (4.2.1)
Again, M, D, and K are n-by-n real symmetric matrix models of the mass, damping and
stiffness properties of the structure. Assume that these matrices were generated using the
finite element method. The nx 1 time varying vector, w(t), represents the n displacements
of the n-DOF FEM of the system. The overdots represent differentiation with respect to
time. In control terminology, Bo is the nxm (m « n) control influence matrix describing
the actuator force distributions and u(t) is the mxl vector of output feedback control forces
defined by
u(t) = Fy(t) (4.2.2)
In Eq. (4.2.2), F is the mxr feedback gain matrix and y(t) is the rxl output of sensor
measurements defined by
y(t) = CqW + C,w (4.2.3)
in which Co and C\ are the rxn output influence matrices corresponding to position and
velocity, respectively. A substitution of Eqs. (4.2.2) and (4.2.3) into Eq. (4.2.1) yields
Mw(t) + (D - B0FC|)w(t) + (K - B0FC0)w(t) = 0 (4.2.4)
It is clear, from Eq. (4.2.4), that the feedback controller results in residual changes, BoFCo
and BoFCj, to the stiffness and damping matrices, respectively. These changes can be
viewed as perturbations to the initial finite element model (FEM) such that the adjusted FEM
matches closely the experimentally measured modal properties. The adjusted FEM consists
of the original mass matrix and the adjusted stiffness and damping matrices given by

41
Ka = K — B0FC0
Da = D — BqFCj
(4.2.5)
Assume that modal analysis of the structure under consideration has been performed and that
p modes (p eigenvalues X,e., and p eigenvectors ve.) have been identified. As discussed earlier
in Chapter 2, in practice p is typically much less than n. The feedback gain matrix F, such
that the adjusted FEM eigendata matches the experimental modal parameters, is computed
using standard eigenstructure assignment theories (Andry et al. 1983):
F = [Z - AUV]<
[Co C,
„ -1
w w
WA WA
(4.2.6)
where
A =
0 Inxn
M_1K -m_id
B =
0
M_1K
T = [B P]
Á = T-1AT =
'Au
Imxm
Al
B = T_1B =
0
— x -1
V = T
W W
WA WA
A = diag(Xe1,Xe2,.-,Xep) W = [veai,vea2,...,veap]
— C - 1
z = s
r-
's,'
w
w
T — 1
WA
A
A
1
c
WA
The overbar in the above equations indicates the complex conjugate operator. The vectors
vea in matrix W are the expanded “best achievable” eigenvectors associated with the
experimentally measured eigenvectors ve.. An explanation of the concept of “best

42
achievable” eigenvectors is discussed in Section 4.2.3. The submatrix P of matrix T is
arbitrary as long as T is invertible.
At this point, the variables Bo, Co and Cj are still arbitrary. A random selection of these
variables will usually result in nonsymmetric perturbation matrices and, consequently
nonsymmetric adjusted stiffness and damping matrices. This clearly conflicts with the
fundamental symmetry requirement of most structures’ FEM. In the formulation of Inman
and Minas (1990), the resulting perturbation matrices from the pseudo-controller are forced
to be symmetric through a nonlinear unconstrained optimization problem. Zimmerman and
Widengren (1989, 1990) proposed a non-iterative and computationally more efficient
approach to satisfy the symmetry requirement. This approach in discussed in the following
section.
4.2.2 Symmetric Eigenstructure Assignment Formulation
The perturbation matrices are symmetric if the following conditions are met,
B„FC0 = CjFTBj (a)
B0FC, = C]FTBj (b)
(4.2.7)
At this point, two additional assumptions are made. As a prerequisite to the existence of the
inverse of some matrices used in the computations, it is assumed that the number of pseudo
sensors and actuators is equal to twice the number of measured modes (m=r=2p). The other
assumption consists of restricting the matrices Co and Q by the conditions
(a)
(b)
c0 - g0bJ
(4.2.8)
C, = G,Bj
where Go and Gj are mxm invertible matrices. A substitution of Eq. (4.2.8) into Eq. (4.2.7)
simplifies the symmetry conditions to the following relationships,
(a)
(b)
FG0 = GjFT
(4.2.9)
FG, = G}Ft

43
By using the conditions in Eq. (4.2.9), along with the expression for the feedback gain matrix
(Eq. (4.2.6)), a necessary but not sufficient condition on Go and Gj, for symmetric
perturbation matrices, is expressed in the form of a generalized algebraic Riccati equation,
A,X + XA2 + XA3X + A4 = [0] (4.2.10)
where
X = Gj-'Go
A, =
* _ 1 _ 1 / * * \ *
a a ‘a Mat — xa) - x
a
. * _ i _ i *
A2 = t a a ‘a
A3 = x a ‘a Mat - tala
A4 = a a 'a 'a - Imxm
[W*B01
AW*B0"
X =
*
a =
*
WBo
AW B0
a = Z - A,V
The matrices A\, W, Z, and V are defined in Eqs. (4.2.6). The superscript ()-* indicates the
inverse of the complex conjugate transpose matrix. Equation (4.2.10) can be solved for X
by using the techniques described in the papers of Potter (1966) or Martensson (1971). In
general, there exist multiple solutions (X’s) to this generalized algebraic Riccati equation.
With all solutions computed, the next step is to decompose these solutions into Go’s and Gj’s.
It is shown in the paper by Zimmerman and Widengren (1989) that for a given solution X,
any selection of Go and G] satisfying X = G,- 'G0 results in the same adjusted damping
(Da) and stiffness (Ka) matrices. Hence, either Gi (or Go) can be chosen arbitrarily, as long
as its inverse exists. Then, Go (or Gj) is calculated from the relationship X = Gj“ 'Gq.
For each set (Go, G)), a feedback gain matrix F is calculated from Eq. (4.2.6), and the
corresponding adjusted damping (Da) and stiffness (Ka) matrices are computed using Eqs.

44
(4.2.5). At this point, a rationale is proposed to choose the most meaningful adjusted
damping and stiffness matrices. Among all computed sets (Da, Ka), it is apparent that only
the ones that are real and symmetric are acceptable. When dealing with a model refinement
problem, among all acceptable solutions, the final selection could be made by choosing the
set (Da, Ka) that minimizes the cost function,
J = q || K - Ka ||F + || D-Da ||F (4.2.11)
Clearly, this process selects the set (Da, Ka) that results in a minimum change from the
original set (D, K). The scale factor q in Eq. (4.2.11) is used to give equal weight to the
changes in D and K. For the damage detection problem, there is no unique rationale to
choose the “best” set (Da, Ka). A physically intuitive approach is to use engineering
judgement in selecting the “best” updated model. Thus, all acceptable “adjusted” sets of
solutions should be inspected to determine which best provides information concerning the
state of damage.
4.2.3 Best Achievable Eigenvectors
From standard eigenstructure assignment theory (Andry, et al. 1983), it is shown that the
measured eigenvectors are not always exactly assignable to the adjusted finite element
model. In fact, it can be shown that the measured eigenvectors are assigned exactly if and
only if they lie in their respective achievable subspace. The achievable subspace associated
to the ith mode is defined by
Lj = (MX? + DX,j + K)-1B0 (4.2.12)
where h{ is the measured eigenvalue of the ith mode. When all n components of the
experimental eigenvectors are available, the ith best achievable eigenvectors is defined as the

45
least square projection of the ith experimental eigenvector ve on the ith achievable subspace
Lj. This projection is schematically illustrated in Figure 4.1.
Figure 4.1 Best Achievable Eigenvector Projection.
This best achievable eigenvector is given by
V
—ea¡
= L:
Li Lj
-l
LiXei
(4.2.13)
When only a subset s of the eigenvector components are measured, s < n, the least square
projection discussed above can be used to simultaneously expand and project the measured
eigenvectors. In this case, the ith expanded best achievable experimental eigenvector is
given by
—ea. Lj
-1
L; Lj
LjYe,
(4.2.14)
where L¡ are the rows of L¡ which correspond to the measured eigenvector components.
Notice that the calculation of the p achievable subspace using Eq. (4.2.13) requires p
inversions of an nxn matrix. Although the matrix to be inverted is typically banded, this
may present a practical computational burden when dealing with large FEMs. The next
section discusses an approach that does not require the actual computation of the achievable
subspaces and hence avoids this computational burden.

46
4.2.4 Selection of Bn : The Subspace Rotation Method
So far, the control influence matrix has not yet been completely defined. The preceding
formulation suggest that different Bo may possibly result in different adjusted FEM. Hence,
it is essential to develop a physically meaningful rationale to select Bo-
Zimmerman and Widengren (1989, 1990) proposed an approach, termed the mode
selection method, that consists in selecting Bo such that the unmeasured modes of the
structure are nearly unchanged. In other word, Bo is selected such that only the measured
modes of the structure are corrected. This selection technique fixes the achievable subspaces
in which the eigenvectors must lie, and hence places a limitation on the assignment process.
In most studied cases, the experimental eigenvectors were not assigned exactly since their
assignment “success” depends on the locations vis-a-vis the achievable subspaces set by the
selection of Bo-
In this work, a new method of selecting Bq, termed the subspace rotation method, is
proposed. The subspace rotation method is based on selecting Bo such that the measured
eigenvectors lie exactly in the achievable eigenvectors subspaces. This procedure is
illustrated in Figure 4.2 and is accomplished by setting Bq as
B0 =
br,, bf2, ... ,brp I biit bj2, ... , b
(4.2.15)
where br. = real
(mX2 + Dkj + K)vej
br = imaginary (Mk2 + Dkj + K)vej
j = 1,... p
where ve. is the eigenvector associated with the jth experimentally measured mode, and it
is assumed that all n components of the experimental eigenvectors are available. This could
be accomplished by any of the procedures discussed in Chapter 2. Clearly, when Bq is
selected as shown in Eq. (4.2.15), the measured expanded eigenvectors lie exactly in the
achievable subspaces defined in Eq. (4.2.12). Hence, there is no need for the projection
operations defined in Eq. (4.2.13). This eliminates the required p inverses of nxn matrices

47
involved in computing the achievable subspaces. As will be seen in Chapters 5 and 6, the
elements of Bq, as defined by Eq. (4.2.15), give an indication to the pseudo-controller about
the extent of modification of each DOF in order for the structure to exhibit the jth measured
eigenvalue and eigenvector.
Figure 4.2 Rotation of the Achievable Subspace.
4.3 Numerical Illustrations
In this section, the characteristics of the proposed enhancement to the symmetric
eigenstructure assignment model refinement algorithm (SEAMRA) are evaluated and
compared to other refinement techniques for two example problems. The first problem is a
widely-used spring-mass computer simulated example (Kabe, 1985). It is used here for the
purposes of illustrating model refinement for a large local discrepancy, analogous to a
damage detection situation. The phenomena of global/local mode switching and load path
preservation are examined in this problem. The second problem is used to illustrate the
characteristics of the enhanced SEAMRA in updating the finite element model (FEM) of a
laboratory cantilever beam using actual measured modal parameters.

48
4.3.1 Damage Detection: Kabe’s Problem
Kabe’s eight degree of freedom spring-mass system is shown in Figure 4.3. The mass
and stiffness properties of the system are included in the figure. This problem presents a
challenging situation for damage detection in that stiffness values of various magnitudes are
included. The model exhibits closely-spaced frequencies and both local and global modes of
vibration.
mi
kj= 1000
0.001 mg = 0.002 mj = 1.0 j = 2,...
k2 = 10 k3 = 900 k4=100 k5 = 1.5
7
k6 = 2.0
Figure 4.3 Kabe’s Problem.
A variation of Kabe’s original problem is used here. Rather than the standard initial
model commonly used, which has incorrect values for all of the connecting springs, only a
single spring constant is changed. This is reflective of the fact that damage may occur as a
large local change in the stiffness of a structural member.

49
4.3.1.1 Local to Global Mode Change
In the first problem, Kabe’s initial model is only incorrect for the spring between
masses 3 and 5. A value of500, five times that of the exact spring, is assumed in this problem.
Changing the spring value from 500 to 100 also causes a local mode of vibration to be
replaced by a global mode, thus presenting a difficult challenge for damage detection.
Figure 4.4 presents element-by-element stiffness matrix results for applying the Baruch
and Bar Itzhack update (1978) and the symmetric eigenstructure assignment model
refinement algorithm. Baruch Damage indicates that the update was made using Baruch and
Bar Itzhack’s algorithm. SEA-M Damage indicates that the update was made using the
SEAMRA with Bo selected by using the modal (M) selection method. SEA-SR Damage
indicates that the update was made by using the SEAMRA with Bo selected using Subspace
Rotation (SR) method. The x-coordinate on all plots are the indices of a column vector
constructed by storing the upper triangular portion of the stiffness matrix in a column vector.
The y-coordinate on each plots consists of the difference between the updated stiffness
matrix elements and the original stiffness matrix.
In the first case, as shown in Figure 4.4, it is assumed that only the fundamental mode of
vibration is measured, but all eigenvector components have been measured. Thus, no
expansion of eigenvectors is required. It is evident from Figure 4.4 that the Baruch update is
unable to discern the damage, but that both the SEA-M and SEA-SR are able to clearly locate
the damage. In fact, the SEA-SR was able to exactly reproduce the correct stiffness matrix.
This was true independent of which mode was used in the update. Also, it should be noted
that the Baruch update tends to focus elemental changes in the third and fifth row of the
stiffness matrix, indicating the possibility of damage between these degrees of freedom, but
certainly giving no clear indication to the extent of damage. As is evident from the plot, the
Baruch update has spread errors over several elements. Using the algorithm of Lin (1990),
the damage vector is given as a = [ 1.0 0.93 0.72 0.83 0.70 0.90 0.97 1.0]T, where the
element number corresponds to the structural DOFs and a number less than 1 indicates the

50
500
Actual Damage
a
E
'5b
â– c
o
C3
x
W
-500
500
d
E
'5b
•c
O
<
w
on
-500
10 20 30
Indices
SEA-M Damage
10 20 30
Indices
500
n
c
'5b
•c
O
i
fsá
on
i
<
W
on
-500
SEA-SR Damage
10 20 30
Indices
Figure 4.4 Results for Kabe’s Problem using the 1st Mode, Full Eigenvector.
possibility of damage affecting that DOF. It is obvious that DOFs 3 and 5 are affected by
damage, but the results also indicate strong damage of DOF 4.
In the second case, as shown in Figure 4.5, it is assumed that the first three modes of
vibration have been measured, but only the first three components of the eigenvectors have
been measured. The eigenvectors components were expanded for the Baruch update using
dynamic expansion (Berman and Nagy, 1983) with subsequent orthogonalization (Baruch
and Bar Itzhack, 1978). The least squares expansion was used for the SEA-M update. The
SEA-SR update utilized the orthogonal Procrustes expansion (Chapter 2). In comparing
Figure 4.5 to Figure 4.4, it is clear that the damage detection capabilities of all three
algorithms have been degraded when using expanded mode shapes, even though more
modes have been measured. However, both the SEA-M and SEA-SR updates give a clear

51
500
Actual Damage
C3
c
'3b
•G
O
o
cd
x
W
-500
500
500
Baruch Damage
cd
c
'3)
•G
O
<
W
on
-500
10 20 30
Indices
SEA-M Damage
10 20 30
Indices
cd
C
'3d
'C
O
o
CQ
-500
500
cd
C
'3d
•c
O
i
oí
on
I
C
W
on
-500
10 20 30
Indices
SEA-SR Damage
10 20 30
Indices
Figure 4.5 Results for Kabe’s Problem using Modes 1, 2, 3,
and Eigenvectors Components 1, 2, 3.
indication to both the location and extent of damage. Using Lin’s algorithm, the damage
vector is given as a =[ 1.0 0.81 0.75 0.83 0.82 0.79 0.85 1.0]T. It is difficult from
inspection of a to determine the location of damage.
The effect of applying the iterative load path algorithm described in Chapter 2 in the
update procedure is shown in Figure 4.6. For the Baruch update, 100 iterations were
performed. For the SEA-M and SEA-SR updates 2 and 3 iterations respectively, were
performed. The iterations were halted early for both SEA updates because the discrepancy
between the eigenstructure before and after masking was within the numerical precision of
the symmetric eigenstructure assignment software. It is seen that the load path enforcement
further enhances the damage detection capability of both SEA updates.

52
500
r3
C
’5b
•C
O
Q
cd
x
W
-500
500
Actual Damage
n
c
'5b
‘C
O
<
w
on
-500
10 20 30
Indices
0 10 20 30
Indices
SEA-M Damage
1
500
n
C
'5b
â– C
O
â– 
j=
o
g
CQ
-500
500
Baruch Damage
C
‘5b
*C
o
I
C/3
i
<
w
on
-500
1 1 1
0
10 20 30
Indices
SEA-SR Damage
10 20 30
Indices
Figure 4.6 Results for Kabe’s Problem using Load Path Preservation,
Modes 1, 2, 3, and Eigenvectors Components 1, 2, 3.
4.3.1.2 Consistent Modes
In the second problem, the initial model is only incorrect for the spring between masses 4
and 6. A value of 200, two times that of the exact spring, is assumed in this problem. In this
problem, all global and local modes remain global and local modes respectively after
damage. It should be noted that finding a problem with this feature was difficult.
In the first case, as shown in Figure 4.7, it is assumed that only the fundamental mode of
vibration is measured, but all eigenvector components have been measured. It is evident
from Figure 4.7 that the Baruch and SEA-M update are unable to discern the damage, but
that the SEA-SR is able to clearly locate damage. In fact, the SEA-SR was able to exactly
reproduce the correct stiffness matrix. Again, this was true independent of which mode was

53
Indices
SEA-M Damage
73 100-
c
'5b
<
W
^ -100
0 10 20 30
Indices
O 10 20 30
Indices
Figure 4.7 Results for Kabe’s Problem using the 1st Mode, Full Eigenvectors.
used in the update. It should be noted that the Baruch and SEA-M update tends to focus
elemental changes in the fourth and sixth rows of the stiffness matrix, indicating the
possibility of damage between these degrees of freedom, but certainly giving no clear
indication to the extent of damage. Using Lin’s algorithm, the damage vector is given as a =
[1.0 0.98 0.92 0.85 0.87 0.84 0.95 1.0]T. This algorithm does not clearly identify the
damage location.
In the second case, as shown in Figure 4.8, it is assumed that the first three modes of
vibration have been measured, but only DOFs 1,3, and 6 of the eigenvectors have been
measured. In comparing Figure 4.8 to Figure 4.7, it is clear that the damage detection
capability of all three algorithms has again been degraded when using expanded mode
shapes. Only the SEA-SR update gives a clear indication to the location of damage, but is

SEA-M - Original Exact - Original
54
unable to predict the exact extent. Using Lin’s algorithm, the damage vector is given as a_=
[0.99 0.18 0.55 0.34 0.52 0.33 0.41 1.0]T. Again, it is difficult from inspection of a to
determine the location of damage. In fact, inspection of a indicates that DOF 2 is the most
likely damaged DOF.
Indices
m -100
Baruch Damage
0 10 20 30
Indices
Figure 4.8 Results for Kabe’s Problem using Modes 1, 2, 3,
and Eigenvectors Components 1, 2, 6.
It should be noted that in this problem, it was critical to have the proper DOFs measured.
When the second test case was run with the first three DOFs measured, no algorithm was able
to locate damage. In this case, the eigenvectors components were relatively unaffected by
damage, thus causing substantial error in the eigenvector expansion process.

55
4.3.2 Model Refinement of a Cantilever Beam: Experimental Study
4.3.2.1 Modal Test Description
The structure used in this investigation is the aluminum cantilevered beam shown in
Figure 4.9. The dimensions and material properties of the beam are given in Table 4.1.
Experimental modal analysis of the beam was performed to measure its modal properties.
Six equally spaced translational degrees of freedom shown in Figure 4.9 were selected as
measurement locations. The modal properties of the first three modes of vibration were
determined using frequency domain techniques and single degree of freedom curve fitting
algorithms. The excitation source used in the testing was an impact hammer and the driving
point measurement was an accelerometer mounted at the free end of the beam. Impact and
exponential windows were utilized to improve frequency response calculations. At each
measured degree of freedom, five frequency response measurements were made and
averaged to reduce the effects of measurement noise. Natural frequencies, damping ratios
and mode shapes of the beam’s first three modes of vibration were identified and are reported
in Tables 4.2 and 4.3.
Figure 4.9 Experimental Cantilever Beam.

56
Table 4.1 Structural Properties of the Cantilever Beam.
Length - 0.84 m
Mass/Length - 2.364 kg/m
Moment of Inertia - 3.02e-9 m4
Youngs Modulus - 70 GPa
Table 4.2 Measured Natural Frequencies and Damping Ratios of the
Cantilever Beam.
Mode #
Natural Frequency (Hz)
Damping Ratio
(%)
1
7.25
4.41
2
45.55
0.68
3
127.01
0.33
Table 4.3 Measured Mode Shapes of the Cantilever Beam.
Mode
1
2
3
DOF
1
1.00
1.00
1.00
2
0.95
0.16
-B.30
3
0.65
-0.53
-0.61
4
0.36
-0.72
0.20
i 5
0.15
-0.52
0.75
6
0.03
-0.12
0.28
4.3.2.2 Finite Element Model Description
A twelve DOF undamped finite element model (FEM) of the beam was generated using
six equal length beam elements as shown in Figure 4.9. The beam element has two degrees of
freedom (DOFs) at each node; bending and rotation. This model was then reduced using

57
Guyan reduction (1965) eliminating the rotational degrees of freedom. There are several
possible errors affecting the accuracy of this FEM. The most obvious is the fact that a perfect
cantilever condition is assumed. In addition, an artificial error was purposely introduced by
selecting the Young’s Modulus higher than that often assumed for aluminum.
4.3.2.3 Application of the SEAMRA.
Because the “true” finite element model is unknown, a comparison between the “true”
and updated structural matrices is not possible. Besides such comparison, a fair and useful
judgement on the quality of SEAMRA updating capability can be obtained by comparing
actual experimental frequency response functions with those predicted by the initial and
updated FEMs. Figure 4.10 shows a comparison of frequency response functions measured
between degrees of freedom 1 and 3 (i.e. sensor measurement at DOF 1 and impact excitation
at DOF 3). The solid curve corresponds to the experimental data, while the dotted line
corresponds to that predicted by the original analytical FEM. It is apparent that the
discrepancies between the frequency response function increases as the frequency of
excitation increases. This is in part due to the fact that the assumption of a perfect cantilever
condition affects the higher modes of vibration to a greater extent. The dashed lines in this
figure corresponds to the SEA-SR updated finite element model. It is clear from this
comparison that the SEA-SR provided a great deal of improvement to the original analytical
FEM. Inspection of the updated stiffness matrix indicates that changes occur throughout the
matrix, indicating that the discrepancy between the original and refined FEMs was due to
degradation of some global structural property (Youngs Modulus), as opposed to some form
of local damage, as seen in the previous problem.
4.4 Discussion of the SEAMRA’s Formulation
In some problems SEAMRA in conjunction with either the subspace rotation or the
modal selection method, failed to find a symmetric updated FEM (symmetric Da and Ka).
This shortcoming was especially encountered in practical situations when the experimental

58
SEAMRA
Figure 4.10 Experimental and Analytical Frequency Response
Functions of the Cantilever Beam.
modal properties were corrupted by measurement errors. This can be attributed to the fact
that there is no symmetric updated FEM that is consistent with the present SEAMRA’s
formulation. Recall that SEAMRA’s modifications to the initial stiffness and damping
matrices in its updating process are given by
AK = BqHqBJ with H0 = FG0
AD = B0H]Bj
(4.5.1)
with H, = FG]
Clearly, from Eq. (4.5.1), the perturbations (i.e. modifications) to the initial stiffness and
damping matrices are constrained by the relationship
range(AK) = range(AKT) = range(AD) = range(ADT) (4.5.2)

59
This relationship can always be satisfied for the cases when the experimental modal
properties are consistent with an update in which (i) either AK or AD is zero or (ii) AK is
proportional to AD (AK = (3 AD, (3 is a scalar). For all other cases, SEAMRA might fail to
produce symmetric AK and AD. A more flexible and general formulation that accounts for
such shortcoming is proposed in Section 6.5 of Chapter 6. The formulation as presented in
Chapter 6 is more elegant, efficient and guarantees, for all situations, a symmetric updated
FEM.
4.5 Summary
A previously developed model refinement algorithm based on the general mathematical
framework of eigenstructure assignment theory (Zimmerman and Widengren, 1989, 1990)
has been extended and improved. A technique to enhance eigenvector assignability of the
algorithm has been developed. The method consists of rotating the achievable eigenvector
subspaces into the experimentally measured eigenvectors. The subspace rotation method,
used in conjunction with one of eigenvector expansion techniques discussed in Chapter 2,
results in both a decrease in the computational burden as well as an increase in the accuracy of
the assigned eigenvectors. Finally, the improved algorithm (SEA-SR) was tested for its
suitability for model refinement and structural damage assessment. The performances of
SEA-SR in a damage assessment problem on a challenging simulated structure was
presented and compared to other algorithms. The results acquired using the SEA-SR were
superior.

CHAPTER 5
DAMAGE LOCATION: THE SUBSPACE ROTATION ALGORITHM
5.1 Introduction
In this chapter, a computationally attractive algorithm is proposed to provide an insight
to the location of structural damage. The proposed algorithm is similar to the Modal Force
Error Criteria proposed by several researchers ( Ojalvo and Pilón 1988; Gysin, 1990).
However, a greater insight of the Modal Force Error criteria is provided. Further, a new
viewpoint which allows for the reduction of the effects of measurement errors in the
experimental modal parameters for a certain class of structures is also discussed. As will be
shown in the next sections, the proposed damage location algorithm requires only
matrix-scalar and matrix-vector multiplication.
5.2 The Subspace Rotation Algorithm: The Direct Method
Assume that an n-DOF finite element model of the “healthy” (undamaged) structure
exists. As seen in the earlier chapters, the standard differential equation governing the
dynamic motion of such structures is given by
(5.2.1)
Mw + Dw + Kw = 0
where M, D, and K are the n x n analytical mass, damping, and stiffness matrices, w is a n
x 1 vector of positions and the overdots represent differentiation with respect to time. The
eigenvalue problem associated with Eq. (5.2.1) in second order (lambda) form is given as
(5.2.2)
60

61
where Xh and vh denote the ith eigenvalue and eigenvector, respectively, of the pre-damaged
“healthy” structure. It is assumed that Eq. (5.2.2) is satisfied for all measured “healthy”
eigenvalues/eigenvectors. This can be enforced by correlating the original FEM (M,D, and
K), possibly through the use of a model refinement procedure.
Next, consider that the p eigenvalues and eigenvectors, Xj; and v^j, of a post-damage
modal survey of the structure are available, in which Xd ^ X,h, vd ^ yh. In the present
formulation, it is assumed that the dimension of the measured eigenvector is the same as the
analytical eigenvector. As discussed in Chapter 2, this is true (i) when all FEM DOFs are
measured (ii) after the application of an eigenvector expansion algorithm, or (iii) after the
application of a finite element model reduction algorithm. The ideal situation would be to
measure all FEM DOFs since the eigenvector expansion process would introduce additional
errors in the “expanded” eigenvectors and the model reduction process would introduce
errors in the FEM. It should be noted that in both cases the additional errors may become
significant as the ratio of measured to unmeasured DOFs become smaller.
Let AM structural damage. For localized damage, the exact perturbation matrices are sparse matrices
with the nonzero elements reflecting the state of damage. The lambda equation for the
damaged structure is, defined by
(*d,(M - AMd) + Xd(D - ADd) + (K - AKd))vd¡ = 0 (5.2.3)
Although only p of the n eigenvalues/eigenvectors are assumed measured, p « n, Eq. (5.2.3)
holds for any particular eigenvalue and eigenvector of the damaged structure because the
perturbation matrices are assumed to be exact. Grouping all perturbation matrices on the
right-hand side defines a damage vector d¡,
(5.2.4a)
(5.2.4b)
where Zd = XdM + XdD + K

62
Although Eqs. (5.2.4a, b) yield the same damage vector, it should be noted that Zd and
the coefficient matrix of Eq. (5.2.4b) are not equal. When the measured eigendata are not
corrupted by noise, an inspection of d¡ in terms of the Eq. (5.2.4b) reveals that the jth element
of dj will be zero when the jth rows of the perturbation matrices are zero, i.e. the finite element
model for the jth degree of freedom is not directly affected by damage. Conversely, a degree
of freedom whose finite element model has been affected by damage will result in a nonzero
entry in d¡. Thus, the degrees of freedom which have been affected by damage can be
determined by inspecting the elements of d¡. Vector d¡ as defined in Eq. (5.2.4b) also reveals
that only a single mode of vibration needs to be measured exactly to determine the damage
locations exactly. This is true in even multiple member damage situations. More
importantly, the vector d¡ can be determined from the original finite element model (M,D,K)
and the measured eigenvalues and eigenvectors, Xd and yd, using Eq. (5.2.4a). Thus, there is
no need to use a model refinement algorithm to attempt to estimate the exact perturbation
matrices in order to locate the damage. If the damping term is ignored, Eq. (5.2.4a) is
essentially the Modal Force Error criteria as proposed by Ojalvo (1988) for use as a
diagnostic “tool” to locate modelling errors in FEMs. A physical interpretation of Eq.
(5.2.4a) provided by Ojalvo was “... d¡ is the applied harmonic force error distribution,
applied at frequency Xd, which is necessary to cause the analytical model to vibrate with
mode shape vd ...”. However, Eq. (5.2.4b) provides a much clearer interpretation of the
damage vector d¡ for the damage location problem in which the perturbation matrices are
sparse.
In practice, the perfect zero/nonzero pattern of the damage vector d¡ rarely occurs due to
errors present in the experimentally measured eigenvalues and eigenvectors. This scenario
was studied and discussed for the undamped case by Gysin (1990) in the context of
eigenvector expansion techniques. Gysin (1990) observed that in certain specific cases of
eigenvector expansion errors, the damage vector defined by Eq. (5.2.1a) may lead to
incorrect conclusions concerning the location of damage. The next section proposes a new

63
viewpoint which allows for the reduction of the effects of measurement errors for certain
classes of structures.
5.3 The Subspace Rotation Algorithm: The Angle Perturbation Method
In order to provide an alternative view of the state of damage, Eq. (5.2.4a) is rewritten as
dj = zJjYd, = || 4( || || Yd, II cos(0J.) (5.3.1)
where dl is the jth component (or jth DOF) of the ith damage vector, zJd is the jth row of the
matrix Z . and 0Í is the angle between the vectors z\ and vrI.
°i i ° —dj u¡
In the case when the measurements are free of error, a zero d| corresponds to a 0j of
ninety degrees, whereas a nonzero dj corresponds to a 0j different from ninety degrees.
Errors in the experimental measurements of modal parameters will cause slight
perturbations in the angles 0Í that destroy the zero/nonzero pattern of the damage vector.
One would initially expect that the components of d, corresponding to the damaged DOFs
would be substantially larger than the other elements. However, by inspecting Eq. (5.3.1), a
large d¡ component could be due to a zJd row norm substantially larger than other rows of Zd,
coupled with a slight deviation of 01 from ninety degrees due to measurement noise. Hence,
when dealing with a structure whose FEM results in z\ row norms of different order of
di
magnitude, it is more reasonable to use the deviation of the angles, 0j, from ninety degrees
for damage location,
aj = 0j - 90°
where 0)
1
COS
(5.3.2)
The angle 0Í is determined from Eq. (5.3.2) and aj is the jth component of a¡.

64
5.4 Practical Issues
5.4.1 Cumulative Damage Location Vectors
The discussion in the previous section suggests that for a given mode, the damage is
locatable if the perturbation to the angle, 0J. due to the measurement error is less than the
angle perturbation due to the damage. Hence, modes that are highly affected by the damage
are expected to provide better assessment to the location of the damage when errors are
present in the measured eigendata. As will be seen in example problems (Chapter 7), certain
modes are more susceptible to a given state of damage than others. This is mainly due to the
fact that different elements of the structure have different levels of contribution to the total
strain energy of a given mode (Kashangaki 1992). Furthermore, a particular part of the
structure usually has different orders of strain energy contribution for different modes.
Usually, if the damage occurs in a region of high strain energy for a given mode, that mode
would be highly susceptible to the damage and, hence would reflect the state of damage. To
accommodate this type of problem, when the number of measured modes p is greater than
one, two different composite damage vectors may be defined as
P
o = p ¿L I —i I (5.4.2)
1=1
In Eq. (5.4.1), the damage vectors, d¡, are normalized with respect to their corresponding
eigenvectors, vd . The reason for this normalization is such that the composite vector ignores
the inherent “weighting” of || yd ||, which is usually of different orders of magnitudes for
different measured modes. It should be noted that in the multi-mode measurement case, Eq.
(5.4.2) is preferable when the values of || zJd || are of different orders of magnitude for
different measured modes. Again, in practice, the DOFs affected by the damage are expected

65
to have substantially larger d or a. Finally, the damaged areas of the structure can then be
located using the knowledge of the “damaged” DOFs and the connectivity of the FEM.
It is interesting to note that Eqs. (5.2.4a, b) reveal an interesting relationship between
various model refinement algorithms. Model refinement techniques attempt to approximate
the exact perturbation matrices by using limited modal data, but do so in different manners.
However, Eq. (5.2.4a, b) indicates that if the model refinement technique has satisfaction of
the eigenproblem as an equality constraint, the calculated perturbation matrices AM, AD,
and AK are constrained to be related to the original finite element model M, D and K and the
measured eigendata by Eqs. (5.2.4a, b).
5.4.2 Eigenvector Filtering Algorithm
In a modal survey, the errors associated with the measured eigenvectors are typically
greater than the error associated with the measured eigenvalues. In addition, in the case of
incomplete eigenvector component measurements, these measurement errors are often
compounded with eigenvector expansion induced errors. A simple eigenvector noise
filtering algorithm is proposed assuming the measured eigenvalues to be correct. From the
cumulative damage vector defined in Eqs. (5.2.4a) or (5.2.4b), and the original FEM
connectivity, the engineer can deduce which DOFs have been damaged. It is reasonable to
assume that nonzero elements in each damage vector d¡ associated with “undamaged DOFs”
are due to eigenvector errors. These elements can then be set to zero. In addition, the
magnitude of the elements of d¡ at the “damaged” DOFs can be adjusted by using knowledge
of DOF connectivity and the properties of the element property matrices connecting the
“damaged DOFs.” The element property matrices provide constraints relating the effect of
damage on each element DOF. The noise filtering algorithm consists simply of replacing the
d¡ vectors by df, where df is obtained from d¡ as described above. The ith filtered
eigenvector, ydf, can then be obtained from solving
(5.4.3)

66
using Gaussian elimination. In this calculation, the bandedness of typical FEM matrices
should be exploited. Essentially, the filtered eigenvector is just the eigenvector that if
measured would have produced the damage vector df. Experience gained in using the
eigenvector filtering algorithm indicates that it is best to use structural matrix properties (M,
D, K) that (i) are finite element consistent, and (ii) have not been “corrupted” by
measurement noise. By finite element consistent, it is meant that the property matrices can
be achieved by a finite element program. Note that measurement noise can be introduced
in the property matrix through a FEM refinement algorithm. Hence, the property matrices
that should be used are the original property matrices (unrefined). The present eigenvector
filtering algorithm can be useful in improving the damage extent assessment. This concept
is discussed in Chapters 6 and 7.
5.5 Summary
A computationally attractive algorithm to determine the location of damage in
structures was developed. The algorithm completely bypasses the general framework of the
model refinement problem and involves only matrix-scalar and matrix-vector
multiplications. The effect of measurement error in the eigendata was discussed and
techniques to reduce these effect were presented. Furthermore, a simple eigenvector
filtering algorithm was developed. Practical example problems to illustrate and evaluate the
performance of the developed algorithm will be presented in Chapter 7.

CHAPTER 6
THE MINIMUM RANK PERTURBATION THEORY
6.1 Background
The theory developed in Chapter 5 is limited to determining the location of structural
damage. In a practical situation, it is essential to determine the extent of the damage to get a
good estimate about the overall integrity of the structure. In general, the extent problem, as
discussed in Chapter 1, has been approached by several researchers using existing model
refinement algorithms. The formulations of these algorithms were obviously based on the
“model refinement philosophy”: minimum change made to the original FEM. The minimum
change constraint has a clear tendency to smear the changes throughout the entire FEM.
However, in most cases, this philosophy is not consistent with the effect of structural damage
on FEMs. In fact, the effects of structural damage on FEMs are usually “non-minimal”
localized perturbations. Structural damage often occurs at discrete locations. The effect of
damage on the analytical model is often restricted to just a few elements of the finite element
model. The rank of each element mass, damping or stiffness matrix is dependent on the
number of degrees of freedom defined by the element and the shape functions utilized.
However, it should be noted that in general the element matrices are not of full rank. For
example, the rank of the 6x6 element stiffness matrix of a three dimensional truss element is
just one. Thus, instead of using the matrix Frobenius norm minimization formulation to
arrive at unique perturbation matrices, minimum rank perturbation constraints are enforced.
In this Chapter, a computationally attractive damage extent algorithm is proposed. The
proposed damage extent algorithm is a minimum rank perturbation, which is consistent with
the effects of many classes of structural damage on a FEM.
67

68
Assume that “p damaged” eigenvalues and eigenvectors have been measured and that
the original FEM has been corrected such that its modal properties match the measured
modal properties of the healthy model. The eigenvalue problem of a damaged structure
shown in Eq. (5.2.3), for all p measured modes, can be written in matrix form, as
MVdA2 + DVdAd + KVd = AMdVdA2 + ADdVdAd + AKdVd ee B
where Ad = diag^d|,Xd2,...,XdJ
(6.1.1)
vd =
B =
dj, d2, •••> dp
where all variables have the same definitions as in the previous chapter. Note that matrix
B can be determined from the FEM (M, D, K) and the “p” measured eigenvalues and
eigenvectors. As discussed earlier, the damage extent problem consist of finding the
perturbation matrices, AMd, ADd, and AK<¡, such that Eq. (6.1.1) is satisfied. As already
discussed in Chapter 2, structures can be modelled using either undamped, proportionally
damped or nonproportionally damped finite element models. The proposed extent algorithm
is formulated to accommodate all three types of structural models. For each type of model,
several scenarios of damage effects are considered. Practical issues that can be used to
improve the damage extent estimate are also presented.
6.2 The Minimum Rank Perturbation Theory: Theoretical Background
In this section, the theoretical foundation of the Minimum Rank Perturbation Theory
(MRPT) is derived. This theory will be extensively used throughout the remainder of this
chapter.
PROPOSITION 6.1 Suppose that X,Y £ Rnxp are given where p < n and
rank(X)=rank(Y)=p.
Define % to be the set of matrices A in R1â„¢1 that satisfy,

69
AX = Y with AT = A (6.2.1)
Then,
(l.a) If the set 96 is nonempty, the minimum rank of any matrix, A, in 96 is p.
Next, define 96p to be a subset of 96 comprised of all A such that rank(A) =p.
Then
(l.b) If the matrix YTX is symmetric, then one member of 96p is given by
Ap = YHYt (6.2.2)
with H = (YTX)_1
and
(1 .c) The matrix defined by Eq. (6.2.2) is the unique member of 96p.
Proof:
To prove Proposition (6.l.a), note that Eq. (6.2.1) is exactly satisfied if and only if
range(Y) is included in range(A), which is also the range(AT) by symmetry. This implies that
rank(Y) = p < rank(A).
Ill
To investigate Proposition (6.l.b), assume that the expanded singular value
decomposition of one member, Ap,j, of 96p to be of the form
ApJ = UjSjlJjT
(6.2.3)
IP = k, — »uJn
1.-1 -2’ ’ —pj
2J = diagJcP, oJ2, ...
■°,p)
where

70
where the superscript j indicates the jth family member of 9GP, the uj are the left and right
singular vectors and the o| are the non-zero singular values of ApJ. In the expanded singular
value decomposition, the (p+1) to n singular vectors are not shown in the factorization
because of their corresponding zero singular values. Note that the left and right singular
vectors are the same because ApJ is restricted to be symmetric. For Eq. (6.2.1) to be satisfied,
the range of Y, Ap,J, ApjT and UJ must be equal. Thus, any column of Y can be written as
a linear combination of the uj’s. The matrices Y and UJ are then related by a unique pxp
invertible matrix Q¡,
Y = UJ QJ (6.2.4)
Substituting Eq. (6.2.4) into Eq. (6.2.3) gives
Apo = Y(QJ"1ZjQj'T)YT = YHJYt (6.2.5)
Thus, each family member is uniquely defined by the factorization of Eq. (6.2.3). From Eq.
(6.2.5), it is evident that H-i is of full rank because its inverse exists (H> ' = QJTP QJ).
Inspection of Eq. (6.2.5) reveals that the only unknown term in the factorization is Hk
By using the factorization of Ap,J as defined by Eq. (6.2.5), Eq. (6.2.5) can be rewritten as
Y = ApJ'X = (YHJYT)X = Y(HJYTX) (6.2.6)
Equation (6.2.6) is satisfied if and only if HJYTX = Ipxp, where Ipxp is the pxp identity
matrix. This is true because Y and X are of full column rank. Thus, HJ is uniquely calculated
to be
Hj = (YTX)_1 (6-2.7)
III
Proposition (6.1 .c) follows immediately by inspecting the right hand side of Eq. (6.2.7).
Inspection reveals that H-i is the same for all members of 9GP. This fact, in conjunction with

71
Eq. (6.2.5) leads to the conclusion that Ap j is the unique member of the set %p. This member
is given by Eq. (6.2.2).
Ill
At this point, the MRPT as defined in Proposition 6.1 assumes that the matrices X and Y
are of full rank. In practical uses, as will be seen later, matrix X is usually of full rank. The
rank requirement on matrix Y can be of some concern since it is directly related to the rank of
matrix A. The next proposition addresses the case in which matrix Y is rank deficient.
PROPOSITION 6.2 Suppose that X, Y E Rnxm are given and rank(X)=m and rank(Y)=p,
where p < m < n. Further, suppose that the matrix YTX is symmetric.
Define CU to be the set of matrices A in Rnxn that satisfies the problem,
APXP = Yp with (Ap)T = Ap (6.2.8)
where the superscript p indicates a rank p matrix. In Eq. (6.2.8), Xp, Yp E Rnxp are
corresponding full rank submatrices of X and Y.
Then
The set CU contains a single member, Ap, that can be calculated from Eq. (6.2.2) using
any corresponding Xp and YP.
Proof:
The jth member of the set CU is given by
AP-i = YpJHjYpJT (6.2.9)
with HJ = (YpJTXpj) 1
where the additional superscript ( )-J indicates the jth member of CU. Note that EP is
symmetric since YTX is symmetric. The range of any YP is equal to the range of Y, thus
the YP’1 and YPJ are related by

72
Yp,i = Ypj Q'J (6.2.10)
where Q'j G Rpxp and rank(Q'J) = p. By utilizing Eq. (6.2.10), Eq. (6.2.9) can be written
for the ith member of 01 as
AP.i = Ypj(Qi’jHiQiJT)YpjT (6.2.11)
with H' = (QiJTYpjTXp’i) 1
Again, H1 is symmetric because YTX is symmetric. In comparing Eqs. (6.2.9) and (6.2.11),
it is seen that Ap,J = Ap,‘ if
or equivalently,
Hj = Q'j H' QijT
(6.2.12)
HJ”1 = Qi'j-THi_TQi'j-1 (6.2.13)
where Eq. (6.2.13) makes use of the symmetry of H1. By using the definitions of Hl and HJ,
Eq. (6.2.13) can be written as
Yp’iTXp’j = Q‘j“TXp’iTYp’J (6.2.14)
Pre-multiplying Eq. (6.2.14) by Q’j7 and utilizing the relation in Eq. (6.2.10) gives the
condition such that ApJ = Ap’’, namely
Yp’iTXpJ = Xp’iTYp'j (6.2.15)
This condition is clearly satisfied since YTX is symmetric.
The actual uses of Proposition 6.1 and Proposition 6.2 will be clearly seen in the next
sections. The practical implication of Proposition 6.2 is discussed in detail in Section 6.6.
6.3 Damage Extent: Undamped Structures
In some cases, the damping of the system under consideration is assumed to be
negligible. For this type of system, MRPT based algorithms will be developed assuming that

73
the structural damage affects (i) only the mass properties, or (ii) only the stiffness properties,
or (iii) simultaneously the mass and stiffness properties.
6.3.1 Damage Extent: Mass Properties
In this case, it is assumed that the effect of damage on the stiffness properties of the
structure is negligible. With this assumption, Eq. (6.1.1) can be rewritten as
MVdA5 + KVd = AMdVdA2 = B (6.3.1)
Note that the eigenvectors are real and the eigenvalues are purely imaginary. Further, the
eigenvectors are linearly independent, which implies that the matrix product VdAd is of full
column rank if rigid body modes are not included. Assume, for the moment, that B is of full
rank (rank(B) = p). Then, Proposition 6.1 can be applied to determine the perturbation
matrix, AMd, as
AMd = B(BTVdAj) V (6-3.2)
by letting Y=B and X= VdAd . Note that the required inversion is that of a pxp matrix, where
“p” is the number of measured modes. As discussed in Proposition 6.1, this inversion is
feasible if matrix B is of full rank and the rigid body modes of the system are omitted in the
computations. When matrix B is rank deficient, Proposition 6.2 should be used to render
the computation possible.
The properties associated with AMd as computed in Eq. (6.3.2) are as follows:
PROPERTY 6.3.1 The perturbation matrix, AMd, defined in Eq. (6.3.2) will be symmetric if
the eigenvectors, Vd, are stiffness orthogonal, i.e., the eigenvectors are orthogonal with
respect to the original stiffness matrix, K.
Proof:
Proposition (6. l.c) in conjunction with Proposition (6.1 .b) implies that the existence of
the unique symmetric rank p AMd requires the symmetry of the matrix product BTVdAd.
The symmetric equivalence associated with this matrix product is

74
BTVdA2 ee AjvjB (6.3.3)
Substituting the expression for B from Eq. (6.3.1) into Eq. (6.3.3) gives
AjvjMV^ + VjKVdAd2 S A2V>fVdA2 + A^vJKV,, (6.3.4)
where the symmetry of M, K and Ad has been used in writing Eq. (6.3.4). From Eq. (6.3.4),
it is clear that the equivalence is true if
(vdKVd)Ad = Ad(vjKVd) (6.3.5)
Equation (6.3.5) will obviously be satisfied if the measured eigenvectors are stiffness
orthogonal. Baruch (1978) treated one approach to mass orthogonalize the measured
eigenvectors. A similar approach can be used to orthogonalize the measured eigenvectors
with respect to the stiffness matrix.
â– 
PROPERTY 6.3.2 The updated finite element model (FEM) defined by the original mass
and stiffness matrix along with the perturbation mass matrix computed using Eq. (6.3.2)
preserve the rigid body characteristics of the original FEM.
Proof:
This is apparent in that the original stiffness matrix is unchanged and that the rigid body
modes are defined as modes whose eigenvectors lie in the null space of the stiffness matrix.
6.3.2 Damage Extent: Stiffness Properties
Here, it is assumed that the effect of damage on the mass properties of the structure is
negligible. With this assumption, Eq. (6.1.1) can be rewritten as
MVdA2 + KVd = AKdVd = B (6.3.6)
For this problem, the eigenvectors are real and the eigenvalues are purely imaginary. The
eigenvectors are also linearly independent, which implies that matrix Vj is of full rank. If

75
matrix B is assumed to be of full rank (rank(B)=p), Proposition 6.1 can be used to determine
the perturbation to the original stiffness matrix,
AKd = B(BTVd)_lBT (6-3-7)
This expression for AK^ is determined by setting Y=B and X=Vd in Eq. (6.1.3).
The properties associated with AK^ as computed by Eq. (6.3.7) are as follow.
PROPERTY 6.3.3 The matrix AK¿ will be symmetric if the eigenvectors are mass
orthogonal, i.e., the eigenvectors are orthogonal with respect to the original mass matrix.
The proof of Property 6.3.3 follows very much the same pattern as the one presented for AMj
(Property 6.3.1).
PROPERTY 6.3.4 The updated FEM defined by the original mass and stiffness matrices and
the perturbation stiffness matrix, AK measured eigenvectors and the rigid body modes are mass orthogonal.
Proof:
The original rigid body modes of an undamped system are defined by the eigenvalue
problem,
Kvr = XrMvr = 0 (6.3.8)
where the subscript r denotes the rigid body mode(s) and \ is equal to zero. Thus, the rigid
body modes lie in the null space of the original stiffness matrix. The rigid body modes of
the system will be preserved in the updated model if the original rigid body modes lie in the
null space of the updated stiffness matrix,
e = (K — AKd)vr (6.3.9a)
= - AKdvr (6.3.9b)
where vector e is zero if the the rigid body modes are preserved. Equation (6.3.8) has been
used to arrive at the expression shown in Eq. (6.3.9b). Substituting Eq. (6.3.7) into (6.3.9b)
gives

76
e = - B(BTVd) ‘BTvr (6.3.10)
By utilizing the symmetry of the original mass and stiffness matrices, along with Eq. (6.3.6),
Eq. (6.3.10) can be expanded as
e
B(BTVd)
VjKv, + AdVjMvr
(6.3.11)
The first term in the parenthesis is zero because the matrix-vector product Kvr is zero by
definition. The second term will be zero if the rigid body modes and the measured mode
shapes are mass orthogonal.
6.3.3 Damage Extent: Mass and Stiffness Properties
In this case, it is assumed that the structural damage affects simultaneously the mass and
stiffness properties of the structure. With this assumption, Eq. (6.1.1) can be rewritten as
MVdA* + KVd = AMdVdAd + AKdVd = B (6.3.12)
6.3.3.1 Application of The MRPT
Assume that Eq. (6.1.12) can be decoupled as follows,
AMdVd = Bm (6.3.13a)
AKdVd = Bk (6.3.13b)
Then, the MRPT, as formulated in Proposition 6.1, can be applied to determine the
perturbation matrices AM¿ and AK^, as
AMd = Bm (BjvJ-1 Bj, (6.3.14a)
AKd = Bk (B^Vk) B[ (6.3.14b)
Note that the matrices BmVd and Bk Vd are invertible if Bm and B^ are of full rank. When
these rank requirements are not met, Proposition 6.2 can be used to make the computations
possible.

77
6.3.3.2 Decomposition of Matrix B
The decomposition problem as illustrated in the previous section is equivalent to the
problem of solving for the matrices Bm and B^. So far, the only constraint that these
unknown matrices must satisfy is given by the expression,
B = Bm + Bk (6.3.15)
which results from Eqs. (6.3.12) and (6.3.13). Naturally, there is an infinite set of solutions
(Bm, Bk) that satisfy Eq. (6.3.15). To arrive at a unique solution, additional physically
meaningful constraints can be enforced. The decomposition proposed herein exploits the
cross-orthogonality relations that arise from the symmetric nature of the property matrices
and the undamped assumption. By measuring mass normalized “damaged” eigenvectors
(which is possible if a driving point measurement is made), the cross-orthogonality relations
associated with the damaged structure can be written as
Vj(M - AMd)Vd = Ipxp (6.3.16a)
Vj(K - AKd)vd = diag(wd|2 o)d>2) = Qd (6.3.16b)
in which cod is the natural frequency of the ith mode of the “damaged” structure. Matrix
IpXp is the pxp identity matrix. A rearrangement of Eq. (6.3.16) yields
Vj AMd Vd = Vj M Vd - Ipxp = Vj Bm (6.3.17a)
Vj AKd Vd = Vj K Vd - Qd = Vj Bt (6.3.17b)
Clearly, the matrices Bm and B^ can be computed from Eqs. (6.3.17). In the rare situation
that the number of measured modes is equal to the number of DOFs in the FEM (p = n), these
can be computed by simply inverting matrix Vd. Unfortunately, as discussed earlier, the
number of measured modes is usually much less than the number of FEM DOFs (p « n).
In this case, the solution that naturally comes to mind is to use the pseudo-inverse of matrix
Vj. The inconvenience of this approach is that the sparsity pattern of matrix B will not be

78
reflected in the computed matrices Bm and B^. Remember that the sparsity pattern of B, as
discussed in Chapter 5, indicates the location of the damage affecting the structure. A more
physically intuitive approach is to constrain Bm and B^ to exhibit the same sparsity pattern
as matrix B. This is done by casting B in an equation similar to the expressions of Eqs.
(6.3.17). The problem in question is then to find an nxp matrix P that satisfies
P(VJB) = B
(6.3.18)
Matrix P can be computed as
p = b(vJb)-'
(6.3.19)
The inverse involved in this computation is that of a pxp matrix which is invertible if matrix
B is of full rank. Now that P is computed, Bm and B^ can be computed using Eq. (6.3.19)
as
Bm = P (vj M Vd - Ipxp)
(6.3.20a)
Bt = P(vjKVd - Qd)
(6.3.20b)
It is clear from Eq. (6.3.19) that P will have the same sparsity pattern as matrix B. Hence
Bm and B^ will also reflect the important sparsity pattern of B. The computed matrices Bm
and Bk can also be used to determine the effect of the damage, respectively, on the mass, and
stiffness properties. As in Chapter 5, cumulative vectors associated to Bm and B^can also be
defined when more than one measured mode is available.
H = I V ~mil
P II vdi II
^ i P.tJIVd,
(6.3.21a)
(6.3.21b)
where dm and dk are, respectively, the ith column of matrix Bm and B^.

79
PROPERTY 6.3.5 The perturbation matrices (AMj, AKd) computed from the MRPT using
the Bm and resulting from the decomposition discussed above will be symmetric.
Proof:
The perturbation matrices AMj and AKd will be symmetric because they satisfy the
relationships in Eqs. (6.3.17) and the right hand sides of these equations are symmetric.
6.4 Damage Extent: Proportionally Damped Structures
Since many structures have non-negligible damping, it is of practical interest to extend
the MRPT to address damped structures. In this analysis, the structure under consideration is
assumed to exhibit proportional damping.
6.4.1 Damage Extent: Stiffness and Damping Properties
It is assumed that the effect of the structural damage on the mass properties is negligible.
In this context, Eq. (6.1.1) is rewritten as
MVdA2 + DVdAd + KVd = ADdVdAd + AKdVd ee B (6.4.1)
The complex conjugate of Eq. (6.4.1) is
ADdVdAd + AKdVd = B (6.4.2)
where the overbar indicates the complex conjugate operator, and the fact that ADj, AKd and
Vj are real has been used in writing Eq. (6.4.2). Subtracting Eq. (6.4.2) from Eq. (6.4.1)
gives
AD„Vd(Ad - Ad) = (B - B) (6.4.3)
If (B — B) is assumed to be of full rank, Proposition 6.1 can be applied to determine the
perturbation matrix, ADd, as
ADd = (B - B)Hd(B - B)T
with Hd
(B - B)TVd(Ad - Ad)
(6.4.4)

80
Note that ADd as defined by Eq. (6.4.4) is real. Post-multiplying Eq. (6.4.1) by Ad and
Eq. (6.4.2) by Ad and subtracting the two equations leads to
AK„Vd(Ad - Ad) = (BAd - BAd) (6.4.5)
where the fact that Ad and Ad are diagonal matrices has been used in writing Eq. (6.4.5).
If (BAd - BAd) is assumed to be of full rank, Proposition 6.1 can also be applied to
determine the perturbation matrix, AK^j, as
AKd = (BAd - BAd)Hk(BAd - BAd)T
with Hk =
— \T
-1
(BAd - BAd) Vd(Ad - Ad)
(6.4.6)
Note that AKd as defined by Eq. (33) is also real.
PROPERTY 6.4.1 The perturbation matrices ADd and AK symmetric if the measured eigenvectors, Vd, are mass orthogonal; i.e., the eigenvectors are
orthogonal with respect to the original unperturbed mass matrix.
Proof:
Matrix ADd is symmetric if Hd is symmetric or, equivalently, if H J1 is symmetric.
Hence, to get a symmetric ADd, the following equivalence must be satisfied.
(B - B)TVd(Ad - Ad) = (Ad - Ad)vJ(B - B) (6.4.7)
Substituting the expressions for B and B, from Eqs. (6.4.1) and (6.4.2), respectively, into
Eq. (6.4.7) yields
(AjVjM + AdVjD - AÃœvjM - AdVjD) Vd(Ad - Ad)
= (Ad - Ad)Vj (MVdAd + DVdAd - MVdAd + DVdAd)
(6.4.8)

81
Note that in Eq. (6.4.8) the terms involving matrix AKd canceled out. A further expansion
and simplification of Eq. (6.4.8) yields
(a3 - Ad) VjMV„ (Ad - Ad) = (Ad - Ad) VjMVd (aJ - Ad) (6.4.9)
which is clearly satisfied if the measured “damaged” eigenvectors, Vd, are mass orthogonal.
Ill
Likewise, the perturbation matrix AKd as computed in Eq. (6.4.6) is symmetric if is
symmetric or, equivalently, if Hk~1 is symmetric. This symmetry requirement yields the
following equivalence.
(BAd - BAd)TVd(Ad - Ad) s (Ad - Ad)Vj(BAd - BAd) (6.4.10)
Substitution of the expressions for B and B into Eq. (6.4.10) yields
(ájAJvJM + AdVdK - AdAdVdM - A.vJk) Vd(Ad - Ad)
(6.4.11)
S (Ad - Ad)Vd (MVdAdAd + KVdAd - MV^ - KVdAd)
in which the terms involving matrix ADj cancel. Manipulating and simplifying Eq. (6.4.11)
yields
(a„A3 - AdXd)vjMVd(Ad - Ad) S (Ad - Ad)vjMVd(AdA3 - AdÁÍ) (6.4.12)
This equivalence is obviously satisfied if the eigenvectors are mass orthogonal.
Ill
â– 
PROPERTY 6.4.2 The updated FEM, defined by the original FEM and the perturbation
matrices ADd and AKd computed from Eqs. (6.4.4) and (6.4.6), preserves the original rigid
body modes if the measured eigenvectors and the rigid body modes are mass orthogonal.

82
Proof:
As discussed earlier, a rigid body mode is defined as a mode whose eigenvalue is equal to
zero and whose eigenvector lies in the null space of the FEM stiffness matrix. Hence, the
rigid body modes of the original system are preserved in the updated FEM if they lie in the
null space of the perturbed stiffness matrix. Consider the relationship
e = (K - AKd)vr (6.4.13)
where vr is a rigid body mode eigenvector. Clearly, the rigid body mode associated to
eigenvector vr is preserved if e = 0. By definition, vr is a rigid body eigenvector of the
original system, hence Eq. (6.4.13) can be simplified as
e = - AKdvr (6.4.14)
Substituting the expression for AK¿ as defined in Eq. (6.4.6), into Eq. (6.4.14) gives
(6.4.15)
e = - (BAd - BAd)Hk(BAd - BAd)Tvr
Substitution of the expressions for B and B into this equation yields
S = - (BAd - BAd)H,
AdAjVjM + AdVjK - AdAdVdM - AdVdK
vr (6.4.16)
By using the fact that vr is a rigid body eigenvector of the original system (i.e. Kvr = 0), Eq.
(6.3.16) can be simplified as
e = - (BAd - BAd)Ht( AdAd - AdAS )vjMy, (6.4.17)
It is clear from Eq. (6.4.17) that e = 0 if the rigid body mode vr and the measured
eigenvectors Vd are mass orthogonal (i.e. VdMvr = 0 ).
m
6.4.2 Damage Extent: Mass and Damping Properties
In this case it is assumed that the effect of the structural damage on the stiffness
properties is negligible. In this context, Eq. (6.1.1) is rewritten as

83
MVdA2 + DVdAd + KVd = AMdVdA2 + ADdVdAd = B (6.4.18)
By using an approach similar to one used in the preceding section, Eq. (6.4.18) and its
complex conjugate can be manipulated to yield the following decomposition
AMdVd(A^Ad - AdAd) = (BAd - BAd) (6.4.19)
ADdVd(AdAÍ - AdAdj = (riA,i - BAdj (6.4.20)
Again by applying the MRPT to the preceding equations, AMd and ADd are determined to
be
AMd = (BAd - BAd)Hm(BAd - BAd)
with Hm =
1 -l
(BAd - BAd)TVd^AdAd - AdAd)
(6.4.21)
ADd = (BAd - BA^H^BAj - BAdj
with Hd =
BAd - BAj>) Vd(AdAd - AdAdj
1 — l (6.4.22)
Clearly, the perturbation matrices AMd and ADd as defined by Eqs. (6.4.21) and (6.4.22) are
real.
PROPERTY 6.4,3 The perturbation matrices AMd a°d ADd, as computed above, will be
symmetric if the measured eigenvectors, Vd, are stiffness orthogonal; i.e., the eigenvectors
are orthogonal with respect to the original unperturbed stiffness matrix.
PROPERTY 6.4.4 The updated FEM, defined by the original FEM and the perturbation
matrices, AMd and ADd, preserves the original rigid body modes.
The proof of Property 6.4.4 is straightforward since the original stiffness matrix is
unchanged (see Property 6.3.2). The proof of Property 6.4.4 follows very much the same
pattern as the proof of Property 6.4.2 .

84
6.4.3 Damage Extent: Mass and Stiffness Properties
In this problem, it is assumed that the effect of the structural damage on the damping
properties is negligible. For this situation, the general eigenvalue problem defined in Eq.
(6.1.1) associated to this case can be simplified as
MVdA3 + DVdAd + KVd = AMdVdAd + AKdVd = B (6.4.23)
Algebraic manipulations of Eq. (6.4.23) and its complex conjugate yield the following
decomposition
AMdVd(A3 - Adj = (B - B)
AKdVd(AÍ - Aj) = (BAd - BAdj
(6.4.24)
(6.4.25)
The perturbation matrices AMtj and AK AMd = (B - B)Hm(B - B)T
with Hm =
(B - B)TVd(A^ - Ad)l
AKd = (BA, - BAj)H,.(BAd - BAdj
with Hk =
T -i-i
BAS - BAd| Vd(Ad - Aj)
(6.4.26)
(6.4.27)
Note that AMj and AKj as defined by Eqs. (6.4.26) and (6.4.27) are real.
PROPERTY 6.4.5 The perturbation matrices AMj and AK¿, as computed above, will be
symmetric if the measured eigenvectors, Vj, are damping orthogonal; i.e., the eigenvectors
are orthogonal with respect to the original unperturbed damping matrix.
PROPERTY 6.4.6 The updated FEM, defined by the original FEM and the perturbation
matrices, AM eigenvectors and the rigid body modes are damping orthogonal.

85
These proofs of the above two properties are not reported here. They follow very much the
same pattern as the proofs in Section 6.4.2 .
6.4.4 Damage Extent: Mass. Damping and Stiffness Properties
The eigenvalue problem of a proportionally damped system with all property matrices
simultaneously affected by damage can be rearranged into the form
MVdA5 + DVdAd + KVd = AMdVdA5 + ADdVdAd + AKdVd s B (6.4.28)
The theory developed in Section 6.3.3 can be expanded to address this particular problem.
The cross-orthogonality relationships associated with this type of structures are
Vj(M - AMd)Vd = Ipxp (6.4.29a)
Vj(D - ADd)Vd - diag(2i;d|Wd] = 2d (6.4.29b)
VJ(K - AKd)Vd = djag(ojd|2 wd(2) = Qd (6.4.29c)
Notice that the cross-orthogonality relationships in Eqs. (6.4.29a) and (6.4.29c) are exactly
the same as the ones associated with undamped systems reported in Eqs. (6.3.16a) and
(6.3.16b). As before, these cross-orthogonality conditions can also be rearranged as
vj AMd Vd = Vj M Vd -
Ipxp
- VjBm
(6.4.30a)
VjADdVd = VjDVd -
- VjB,
(6.4.30b)
Vj AKd Vd = Vj K Vd -
Qd
s VIBk
(6.4.30c)
Following the exact same argument discussed for undamped systems in Section 6.3.3, an nxp
matrix P that satisfies the relation,
P(vjB) = B (6.4.31)
is sought, where B is computed using Eq. (6.4.18) and (vdB) is a pxp matrix. Although B
is a complex matrix, the nxp matrix P is real, since Vd is real. Hence, for computational
efficiency, matrix P can computed from

86
P = Br(vjBr) 1 (6.4.32)
where Br is the real part of B. In Eq. (6.4.32), it is assumed that matrix (v^B j is invertible.
With P computed, the next step is to determine the decomposed damage vectors that indicate
the effects of the damage on the mass, damping and stiffness matrices,
Bm
= AMVd
= P (Vj M Vd
Ipxp)
(6.4.33a)
Bd
3 ADVd
= P (Vj D Vd
- 2d)
(6.4.33b)
Bk
= AKVd
= P (Vj K Vd
- Qd)
(6.4.33c)
The minimum rank perturbation theory (MRPT), as formulated in Proposition 6.1, can again
be applied to determine the perturbation matrices, AM AMd = Bm (B^Vj) “1 Bl (6.4.34a)
ADd = Bd (BdVd) Bj (6.4.34b)
AKd = Bk (B^r1 Bj (6.4.34c)
Note that the matrices BmVd , BdrVd and BkVd are pxp matrices that are invertible if Bm,
Bj and Bk are of full rank. As in all other cases already studied, Proposition 6.2 can be used
to deal with the situation when any one of these matrices are rank deficient. The cumulative
damage location vector associated to Bm and Bk, defined in Eqs. 6.3.21, are also applicable
to this problem. An additional cumulative damage vector associated to the perturbations in
the damping properties can be similarly defined as
dd
(6.4.35)
where dd is the ith column of matrix Bd-

87
PROPERTY 6.4.5 The perturbation matrices (AMd, ADd, AKd) as computed above will be
symmetric.
Proof:
This is true since these perturbation matrices are constrained to satisfy the relationships in
Eqs. (6.4.30). Notice that the right hand sides of Eqs. (6.4.30) are symmetric.
6.5 Damage Extent: Nonproportionallv Damped Structures
As reported earlier in the beginning of the chapter, the eigenvalue problem associated
with the dynamic differential equation of motion of a damaged nonproportionally damped
system can be written as
MVdA;¡ + DVdAd + KVd = AMdVdAd + ADdVdAd + AKdVd s B (6.5.1)
where the matrices M, D, K, AMd, ADd, AK^, Vd, and have the same definitions as in
the previous sections. A number of cross-orthogonality relationships associated to general
nonproportionally damped systems have been reported in Chapter 2. The
cross-orthogonality conditions that are relevant to the following formulations are
'VcAd'
Vd _
‘VdAd'
_ vd _
"Yd Ad"
.Vd.
In the above equations, the notation [ J denotes the complex conjugate transpose operator.
Expansion of the above three equations yield, respectively, the following relations,
1
o
<
i
.S.
rvdAd-
"o'
i
<
1
l
o
[ vd
0
(D - ADd) (K _ AKJ
â– VdAd'
0"
(K - AKd) 0 J
vd
0
0 (M - AMd)"
rvdAd-
'o'
(M - AMd) (D _ ADd)
i
<
o.
0
(6.5.2)
(6.5.3)
(6.5.4)
*
AdVd(M - AMd)VdAd - Vd(K - AKd)Vd = [0]
(6.5.5)

88
AdVd(D - ADd)VdAd + AdVd(K - AKd)Vd + V¿(K - AKd)VdAd = [0] (6.5.6)
Vd(M - AM)VdAd + AdVd(M - AMd)Vd + V^(D - AD)Vd = [0] (6.5.7)
These equations can be further manipulated to become, respectively,
W, = AdVd M VdAd - Vd K Vd
(6.5.8)
= A^Vd AMd VdAd - VS AKd Vd
W2 = AdVd D VdAd + AdVd K Vd + Vd K VdAd
(6.5.9)
= AdVd ADd VdAd + A^Vd AKd Vd + VS AKd VdAd
W3 = V^ M VdAd + AdVd M Vd + Vd D Vd
(6.5.10)
= Vd AMd VdAd + A>d AMd Vd + Vd ADd Vd
Notice that W i, W2 and W3 can be computed using the equations in terms of known matrices
M, D, K, Vd, and Ad-
The above cross-orthogonality conditions are derived from two different state space
representations of the equation of motion shown in Eq. (6.5.1) (see Chapter 2). From each
representation, two cross-orthogonality condition were derived. The two sets provide
exactly the same information about Eq. (6.5.1) but in a different format. Hence, only two of
the three cross-orthogonality relations are linearly independent. In fact, Eq. (6.5.8) can be
generated by subtracting Eq. (6.5.9) from (6.5.10). Any set of two linearly independent
cross-orthogonality relationships can be viewed as a decoupling of the equation of motion
(Eq. (6.5.1)). Hence, for the present problem, there are only two distinct (linearly
independent) matrix equations for the three unknown perturbation matrices (AMd, ADd,
AKd). Naturally, if the problem consists of solving for all three unknown matrices, there will
exist an infinite number of solutions. Additional constraints are hence needed to get a better
defined problem. In this formulation, it is assumed that the damage only affects two of the
three property matrices. Mathematically, this can be translated as a constraint on one of three
perturbation matrices (either AMd or, ADd or, AKd)to be equal to zero.

89
6.5.1 Damage Extent: Damping and Stiffness Properties
In this case, it is assumed that the effects of the damage on the mass properties are
negligible (AMj = 0). The corresponding eigenvalue equation and the cross-orthogonality
conditions relevant to this problem are
MVdAj + DVdAd + KVd = ADdVdAd + AKdVd = B
(6.5.11)
<
CL *
>
*
O.
<
Cl
II
1
jS
(6.5.12)
VS ADd Vd = W3
(6.5.13)
The approach used in this formulation follows very much the same pattern as the one
developed in Section 3.3.3. The basic ideáis to generate fromEqs. (6.5.11-13) problems that
can take the advantage of the already existing MRPT concept. Based on the same argument
as in Section 3.3.3, a matrix P £ Cnxp is sought such that
P(VdB) = B
(6.5.14)
It is Assumed that (vJb) is invertible; then matrix P is given by
P = B (YdB)
(6.5.15)
Matrix P can then be used in Eqs. (6.5.11) and (6.5.12) to yield
AKd Vd = — PW,
(6.5.16a)
ADd Vd = PW3
(6.5.16b)
In order to force the perturbation matrices ADj and AK^ to be real, the complex conjugates
of Eqs. (6.5.16a) and (6.5.16b) are introduced,
AKd Vd = - PW,
(6.5.17a)
ADd Vd = PW3
(6.5.17b)
A combination of Eqs. (6.5.16a, b) and (6.5.17a, b) implies

90
AK
d
AD
(Vd)R I (vd)j = - f(PW,)R I (PW,),
(vd)R ' (Vd),l = i(PW3) I (PW3),
(6.5.18a)
(6.5.18b)
where the notation (X)R and (X), indicate, respectively, the real and imaginary part of matrix
X. The MRPT can now be applied to determine the perturbation matrices ADd and AK ADd = Fd (FjUd) Fj (6.5.19a)
AKd = Fk(FjUd)"' FT (6.5.19b)
where
Ud =
Fk = -
Fd =
WR ' (Vd),
(PW,)R I (PW,),
(pw3)r I (PW3)
Remember that Wj and W3 can be computed in terms of known matrices as shown in Eqs.
(6.5.8) and (6.5.10). The computations in Eqs. (6.5.19a) and (6.5.19b) require that (FdUd)
and (FdUd) be invertible. As discussed in the earlier sections, Proposition 6.2 can be used
when these inverse requirements are not meet.
6.5.2 Damage Extent: Mass and Damping Properties
Here, it is assumed that the damage effects on the stiffness properties are negligible
(AKd = 0). By considering this assumption, the eigenvalue equation associated with this
problem is found to be
MVdA2 + DVdAd + KVd = AMdVdA2 + ADdVdAd s B (6.5.20)
The cross-orthogonality conditions relevant to the present development are obtained from
Eqs. (6.5.8) and (6.5.9) by setting AK^ to zero.
AdVd AMd VdAd = Wj
(6.5.21)

91
AdVj ADd VdAd = W2 (6.5.22)
The solution procedure to the problem of determining AMd and ADd is exactly the same as
the one formulated in the preceding section. First, equations (6.5.21) and (6.5.22) are cast
in the following form.
AMd VdAd = PW, (6.5.23a)
ADd VdAd = PW2 (6.5.23b)
where P = B (AdVdB)
The above two equations involve complex matrices. To guarantee that the perturbation
matrices AMd and ADd are real, Eqs. (6.5.23a) and (6.5.23b) are used in conjunction with
their corresponding complex conjugates. The combinations of the two sets of complex
conjugate pairs yield
AM,
AD,
(VdAd)R I (VdAd),] = [(PW,)R I (PW,),
(VdAd)R I (VdAd),| = Í(PW2) I (PW2),
(6.5.24a)
(6.5.24b)
Finally, the perturbation matrices AMd and ADd are, respectively, determined from Eqs.
(6.5.24a) and (6.5.24b) by using the MRPT.
AMd = Fm (FmUd) ' Fl (6.5.25a)
ADd = Fd (FjUd)"' Fj (6.5.25b)
U„ =
Fm =
Fd =
(VdAd)R I (VdAd),
(PW,)R I (PW,),
(PW2)r I (PW2),]
where

92
Again, the computations in Eq. (6.5.25a) and (6.5.25b) are feasible if and only if (FmUd) and
(FdUd) are both invertible.
6.5.3 Damage Extent: Mass and Stiffness Properties
In some situations, structural damage is such that it only affects the mass and stiffness
properties of the system. The following expression are the results of setting the damping
perturbation to zero (ADj = 0) in Eqs. (6.5.1), (6.5.9) and (6.5.10).
MVdA2 + DVdAd + KVd = AMdVdA2 + AKdVd = B (6.5.26)
AdVd AKd Vd + Vd AKd VdAd = W2 (6.5.27)
AdVd AMd Vd + Vd AMd VdAd = W3 (6.5.28)
Equation (6.5.26) is the rearranged eigenvalue problem associated with the present case.
The expressions in Eqs. (6.5.12) and (6.5.13) are the “rearranged” cross-orthogonality
conditions that are relevant to the forthcoming formulation. First, Eqs. (6.5.27) and (6.5.28)
are conveniently simplified to the standard format encountered in the previous two sections.
Since Ad and Ad are diagonal matrices, Eqs. (6.5.27) can be written as
Vd AKd Vd = Q2 (6.5.29)
where q2.. = w2.. /
The variables q2. and w2 are the components in the ith row and jth column of matrix Qk and
W2, respectively; A.d is the ith diagonal component of the eigenvalue matrix Ad. Similarly,
Eq. (6.5.28) can also be written as
Vd AMd Vd = Q3 (6.5.30)
where q3.. = w3.. / (^d, + ^djj)
Equations (6.5.29) and (6.5.30) can then be transformed as

93
AKd Vd = PQ2 (6.5.31a)
AMd Vd = PQ3 (6.5.31b)
where P = B (vJ¡b) '
Notice that the above computations require that matrix (VdB) be invertible. The sought
perturbation matrices, AMd and AKd,are ensured to be real by supplementing Eqs. (6.5.3 la)
and (6.5.31b) with their complex conjugates. The complex conjugate pairs are then
combined together in the following fashion
AKd
(Vd)R ' (v„V
= [(PQA
(PQj,’
AMd
'(Vd)R ' (vd),
= [(PQ3)
(pQ3),]
(6.5.32a)
(6.5.32b)
At this point, the MRPT is applied to Eqs. (6.5.32a) and (6.5.32b) and the perturbation
matrices, AMd and AKd, are determined
AKd = Fk (FjUd)"1 FÍ (6.5.33a)
AMd = Fm (Fj;ud)_l Fj (6.5.33b)
where
ud = [(vd)R I (Vd),
Fk = [(PQi)R 1 (PQ2),]
Fm = [(PQ3)r i (PQ3)J
Again, it is assumed that Proposition 6.2 has been applied, if necessary, to ensure that the
inverses of the matrices (FmUd) and (F^Ud) exist.
PROPERTY 6.5.1 In the above three damage configurations considered for
nonproportionally damped structures, the computed perturbation matrices (AMd, ADd or
AKd ) from the MRPT will be symmetric.

94
Proof:
This is true since the developed formulations require the perturbation matrices along
with the “healthy” FEM to satisfy the cross orthogonality conditions in Eqs. (6.5.2-4). These
cross-orthogonality conditions are satisfied if and only if the corresponding state matrices
are symmetric; that is
o
s
<1
1
‘(M - AMd) 0
1
o
l
''R
l
>
w
Q-
1
<
1
1
o
1
(6.5.34)
(D “ ADd) (K - AKd)
(K - AKd) 0
(D - ADd) (K _ AKd)
(K - AKd) 0
0 (M - AMd)‘
(M “ AMd) (D _ ADd)
0
(M - AMd)
(M - AMd)'
(D - ADd)
(6.5.36)
Since the matrices M, D, K are known to be symmetric, the above relationships are satisfied
if and only if the perturbation matrices AMd, ADd and AKd are symmetric.
ü
6.6 Practical Issues
In all cases discussed in the previous sections of this Chapter, the damage extent problem
boiled down to solving problems of the form
AX = Y (6.6.1)
in which A is an unknown nxn matrix representing one of the perturbation matrices (AMd,
ADd and AKd) and X and Y are known nxp complex matrices. Matrix X is a function of the
damaged structure eigenvector (Vd) and eigenvalue (Ad) matrices. Recall that p represents
the number of modes and n is the total number of DOFs in the structure’s FEM. As discussed
in Chapter 5, the columns of matrix Y provide useful information in determining the location

95
of the structural damage. For each problem, a cumulative damage vector associated to the
columns of matrix Y can be defined as
, p ly. I
, 1 V -di
á-pZ
(6.6.2)
i = i
-di
Recall that the DOFs of the FEM affected by the damage are associated with the components
of vector d which are substantially larger than the other components of d.
Considering the present generic problem, the extent problem consists of determining a
symmetric real matrix A that satisfies Eq. (6.6.1). A solution to this problem based on the
Minimum Rank Perturbation Theory (MRPT) was developed in Proposition 6.1. The
solution technique presented in Proposition 6.1 requires the pxp matrix (YTX) to be
invertible. Proposition 6.2 was then presented to deal with the case when matrix (YTX) is not
invertible. In the forthcoming discussions, two strategies to improve the extent calculation
are presented.
6.6.1 The Concept of “Best” Modes
Note that for the sake of generality, the forthcoming discussion is based on the generic
problem shown in Eq. (6.6.1); however it should be viewed as representative of all extent
problems discussed in the earlier sections of this chapter.
Although Proposition 6.2 deals with noise free measurements, its practical usefulness is
for the case when the eigenvalue / eigenvector measurements are corrupted by noise (either
through actual measurement error or eigenvector expansion induced errors). In general,
matrix Y will typically be of full column rank due to the presence of noise (rank( Y)=p), even
though the actual damage has only induced a rank r ( r < p ) change to the FEM property
matrix (M, D or K) associated to A (AM^, ADj or AK^j). Application of the extent algorithm
at this point will result in a rank r perturbation to the property matrix A. A better estimate of
the true extent of the damage can be obtained by estimating the “true” rank of Y (and thus A),
where the true rank of Y is defined by the limiting case when there are no

96
eigenvalue/eigenvector measurement errors. There are two possible techniques for
estimating the “true” rank of Y. The first technique, which amounts to a brute-force
numerical computation, is to calculate the Singular Value Decomposition (SVD) of Y.
Various rank criteria measures, as formulated by Juang and Pappa (1985) in their work on
system realization theory, in conjunction with engineering judgement, can be used to arrive
at an estimate of the true rank of Y and, hence, A. However, this would greatly increase the
computational burden of the extent algorithm, and may not even be practical if the number of
DOFs of the original FEM is large. A computationally inexpensive technique for estimating
the true rank of Y is to make use of the damage location results. The cumulative damage
vector associated to the perturbation matrix A defined in Eq. (6.6.2) indicates which
structural DOFs have been directly affected by the damage. Using connectivity information
from the original FEM in conjunction with knowledge of the damaged DOFs allows for the
determination of which elements of the FEM have been damaged. The “true” rank of Y can
be estimated by adding up the ranks of the element matrices associated with each damaged
element.
Once the true rank of Y has been estimated, corresponding columns of X and Y are
eliminated such that the actual rank of B is set to the estimated true rank of Y. In Proposition
6.2, it was shown that with noise free measurements it does not matter which r modes from
the set of p measured modes are used in the extent calculation. However, when the
measurements are corrupted by noise, it is best to use those modes which best reflect the
damage state as indicated by the cumulative damage vector Eq. (6.6.2). As will be illustrated
in the example problems presented in Chapter 7, the concept discussed here improves the
damage extent estimation. This is especially true when the rank of the perturbation matrix (r)
due to the damage is less than the number of measured modes (p). The reason is twofold.
First, the rank of the computed perturbation matrix is consistent with the rank of the actual
perturbation matrix. Second, the measurement errors in the experimental modes that do not
reflect the actual state of the damage are not included in the computation.

97
6.6.2 Application of the Eigenvector Filtering Algorithm
Another technique that can be used to improved the damage extent estimation is the
eigenvector fdtering algorithm developed in Chapter 5. The modes that should be filtered
and used in the extent computation are the ones that reflect the overall structure cumulative
damage vector defined in Eqs. (5.4.1) or (5.4.2) of Chapter 5. The characteristics of the
eigenvector filtering algorithm are illustrated in Chapter 7. It should be pointed out that it is
not a requirement of the extent algorithm to use the eigenvector filtering algorithm, but
certainly it does allow for more engineering judgement to enter into the extent calculation.
The choice of whether or not to use the filtering algorithm is dependent on (i) the errors in the
eigenvectors as seen from the damage vectors, (ii) the requirement on accuracy of the extent
calculation and (iii) the size of the FEM, which defines the additional computational burden.
6.7 Summary
A computationally attractive algorithm was developed to provide an insight to the extent
of structural damage. The developed theory, called the Minimum Rank Perturbation Theory
(MRPT), makes use of an existing finite element model of the “healthy” structure and a
subset of experimentally measured modal properties of the “damaged” structure. The MRPT
formulation is consistent with the effects of many classes of structural damage on a finite
element model. The MRPT was applied to assess the damage in undamped, proportionally
damped and nonproportionally damped models. For each type of model, several different
damage scenarios were considered. The performance of the MRPT will be investigated in
Chapter 7 along with the subspace rotation damage location algorithm using both simulated
and actual experimental data. As will be shown in Chapter 7, the MRPT can also be used to
refine finite element models.

CHAPTER 7
VALIDATION AND ASSESSMENT OF THE SUBSPACE ROTATION ALGORITHM
AND THE MINIMUM RANK PERTURBATION THEORY
7.1 Introduction
In this chapter, the characteristics of the subspace rotation algorithm (Chapter 5) and the
minimum rank perturbation theory (Chapter 6) are illustrated by using both simulated and
actual experimental data. The main objective is to show that the algorithms are suited to
assess damage in structures. Additionally, it is shown that the minimum rank perturbation
theory formulated in Chapter 6 is also suited to update finite element models. The computer
simulated example problems provide a controlled setting in which key points made
throughout the formulation of both algorithms are emphasized. Actual experimental
examples are used to demonstrate the practicality of the algorithms in handling “real-life”
systems.
7.2 Kabe’s Problem
7.2.1 Problem Description
Kabe’s eight degree of freedom spring-mass problem is shown in Figure 7.1, which
includes the stiffness and mass values for the exact model. The main objective of this
problem is to illustrate the damage location algorithm formulated in Chapter 5. The
Subspace Rotation algorithm direct method is compared to the Angle Perturbation Method.
Key points discussed in the formulation will be highlighted. Application of the damage
98

99
extent algorithm of Chapter 6 is also addressed. Both algorithms will be evaluated using
noisy modal measurements in an attempt to reproduce a practical testing situation.
mi = 0.001 mg = 0.002 mj = 1.0 j = 2,..., 7
ki = 1000 k2=10 k3 = 900 k4=100 k5 = 1.5 k6 = 2.0
Figure 7.1 Kabe’s Problem.
A variation of Kabe’s original problem is used here. Rather than the standard model
commonly used, which has incorrect values for all of the connecting springs, only a single
spring constant is changed. This is reflective of the fact that damage may occur as a large
local change in the stiffness of a structural member. In this problem, the initial model is
incorrect for the spring between masses seven (7) and eight (8). Damage will be modeled
as a change in spring constant from 1.5 to 0.1. Changing the spring in this fashion causes
the damage to occur at degrees of freedom which have a small || Zd || in comparison to other
DOFs. Thus, it would be expected that damage would best be located using the Angle
Perturbation method as defined in Eq. (5.3.1). In this problem, it is assumed that only the
first two modes of vibration are measured. Note that the damage does not affect the mass
properties of the structure.

100
7.2.2 Damage Location
Figures 7.2 and 7.3 show, respectively, the damage location results of using the
Subspace Rotation direct (d¡) method and the Angle Perturbation Method (a,). Figure 7.4
provides the damage vector as defined by Lin’s algorithm (1990). These results were
generated using only the fundamental mode of vibration eigendata. In these figures, the
x-coordinate represents the elements of the damage vector d¡ (or a¡), where the ith element
of d¡ (a¡) corresponds to the ith degree of freedom.
The upper left plot represents the damage vector if the exact perturbation matrices are
used. The upper right plot corresponds to the case where the exact eigenvector information
is provided to the algorithm. The lower left and lower right plots correspond to the case
where the exact eigenvector has been corrupted with five and ten percent random noise,
respectively. In practice, the noise in eigenvector information could be due to both
measurement and/or expansion induced errors.
As shown in Figures (7.2) and (7.3), the location algorithm is able to exactly locate the
correct damage when provided with only a single correct eigenvalue/eigenvector pair as
guaranteed by Eq. (5.2.4). From the lower plots, it is seen that the algorithm experiences
a degradation in damage detection capability as the error in the eigenvector is increased. In
fact, when presented with 10% eigenvector noise, the direct subspace rotation method
incorrectly identifies the damage. However, when using the angle perturbation method the
location algorithm is able to discern damage with a 10% noise level. As shown in Figure
7.4, Lin’s algorithm is unable to discern damage in this particular test problem.
Figures 7.5 and 7.6 show the results of using the angle perturbation method and Lin’s
algorithm when provided with complete first and second mode eigenvalue/eigenvector
information. Again, the angle perturbation method is able to cleanly detect damage even
when presented with eigenvector information corrupted with ten percent noise. Again, Lin’s
algorithm is unable to discern damage.

101
Exact damage
1 h
0.5
O' ‘ *
0 2 4 6 8
DOF
1st mode
1
0.5
01 1 1 l-J
0 2 4 6 8
DOF
1 st mode + 5% error
1st mode + 10% error
1 '
0.5
01 —'-1 *— *—
0 2 4 6 8
DOF
Figure 7.2 Kabe’s Problem: Damage Location Results using the Subspace
Rotation Direct Method with the Eigendata of the 1st Mode.
Exact damage
20
a
10
0 1 1 1 ■xr~T^—1
0 2 4 6 8
DOF
Figure 7.3 Kabe’s Problem: Damage Location Results using the Angle
Perturbation Method with the Eigendata of the 1st Mode.

102
Exact Damage
1 -
0.5
O' ‘ * ‘-J
0 2 4 6 8
DOF
1st mode + 5% error
1 st mode
Figure 7.4 Kabe’s Problem: Damage Location Results using Lin’s
Algorithm with the Eigendata of the 1 st Mode.
Exact damage
a
Figure 7.5 Kabe’s Problem: Damage Location Results using the Angle Perturbation
Method with the Eigendata of the 1st and 2nd Modes.

103
0.5 -
Exact Damage
modes 1 & 2
2 4 6 8
DOF
modes 1 & 2 + 10% error
Figure 7.6 Kabe’s Problem: Damage Location Results using Lin’s
Algorithm with the Eigendata of the 1st and 2nd Modes.
7.2.3 Damage Extent
In the previous two-mode noisy test case, inspection of the cumulative damage vector
clearly indicates that the 7th and 8th degrees of freedom have experienced damage.
Inspecting the structural connectivity, it can be deduced that the spring between mass 7 and
8 has been damaged. The rank of the single spring “element” matrix is one. Thus, the rank
of the “true” perturbation to the stiffness matrix due to damage is one. From Proposition 6.1
and 6.2, it is clear that only experimental data from one mode of vibration should be used
to compute the extent of damage. In the noisy cases, as discussed in Section 6.6.1, the mode
that should be used in the extent calculation is the one associated with a d¡ that most cleanly
demonstrates the damage location shown in Figure 7.5. A simple inspection of the individual
dj of the two measured modes indicates that mode 2 provides the best insight into the state
of damage. Figure 7.7 presents element-by-element stiffness perturbation matrix results
from the application of the extent algorithm, formulated in Section 6.3.3, using mode 2

104
eigendata. The x-coordinates on each plot are the indices of a column vector constructed by
storing the upper triangular portion of the perturbation stiffness matrix (AK^j) in a column
vector. The y-coordinate on all plots consists of the difference between the updated stiffness
matrix and the original stiffness matrix (i.e. AK that the exact damage would be computed using any single set of eigenvalue/eigenvector
measurement. The extent algorithm experiences a modest degradation as the measurements
are corrupted by noise.
Exact damage 2nd mode
1 1 1
1
0
-1
-
1 1 1
I I l I I I
0 10 20 30 0 10 20 30
Indices
Indices
2nd mode + 5% error
Indices
Figure 7.7 Kabe’s Problem: Damage Extent Results using the MRPT
with the Eigendata of Mode 2.
To contrast the proposed minimum rank perturbation approach to the optimal matrix
update approaches, consider the application to the current problem of the commonly used
method formulated by Baruch and Bar Itzhack (1978). Although this algorithm was
developed from a model refinement viewpoint, it has been investigated for possible uses in
damage detection (Smith, 1992). It is evident from Figure 7.8 that this algorithm is unable
to discern the damage for this particular example when provided with (i) the second mode’s
exact eigendata, or (ii) the first six modes’ exact eigendata. In fact, it can be seen in the upper

105
Exact damage
1 -
0
-1 -
I l 1—
0 10 20 30
Indices
modes 1 to 6
n
â–¡
-
2nd mode
-
ir -y
IT
1 1 1
0 10 20 30
Indices
modes 1 to 7
1 1
1 -
0
-1 -
0 10 20 30 0 10 20 30
Indices
Indices
Figure 7.8 Kabe’s Problem: Damage Extent Results using Baruch’s Method.
right plot of Figure 7.8 that, consistent with the minimum Frobenius norm change
formulation, the approach tends to “smear” the changes throughout the entire stiffness
matrix instead of localizing them at the damage location. In this problem, this method is only
able to cleanly ascertain the damage when provided with the exact eigendata of the first
seven modes.
7.3 Damage Detection: Fifty-Bay Two-Dimensional Truss: Undamped FEM
7.3.1 Problem Description
The 50-bay, 2-dimensional truss used in this example is shown in Figure 7.9. The
geometric and material properties of the truss are also given in the figure. Each truss member
was modeled as a rod element. The finite element model of the structure has 201 degrees
of freedom and is undamped. In this example, damage is simulated by reducing the Young’s
modulus of two members. One of the damaged members is the upper longeron of the third

106
bay. In this member, the Young’s modulus is decreased from E=2.0xl0n Pa (29xl06 psi)
to 6.895xl06 Pa (lxlO3 psi). The other member is the lower longeron of bay forty. This
member is subjected to a complete loss of stiffness (Young’s modulus equal to zero). It is
assumed that only the first ten modes of vibration are measured.
E = 2.0xl0u N/m2 (29xl06 psi) p = 7.833xl03 kg/m3 (0.283 lb/in3)
A = 6.452x lO-4 m2 (1.0 in2) L = 0.1270 m (5.0 in)
Figure 7.9 Fifty-Bay Two-Dimensional Truss.
7.3.2 Damage Location
The first task in this analysis is to use the damage location algorithm to determine the
location of the damage. Since the zJd row norms (defined in Section 5.3) associated with this
particular FEM are of the same order of magnitude, either the subspace rotation direct
method or the angle perturbation method can be used to locate the damage. The results from
the application of the subspace rotation direct method are summarized in Figure 7.10. The
upper left plot of Figure 7.10 represents the exact damage location. The upper right plot
corresponds to the case where the exact (noise free) “damaged” eigenvector information is
provided to the algorithm. The lower left and right plots correspond to the case where the
exact “damaged” eigenvectors have been corrupted with 2.5% and 5% random noise,
respectively. As shown in Figure 7.10, the location algorithm is able to exactly locate the
damage when presented with noise free information. It is also seen that the algorithm

107
experiences a degradation in detecting damage as the error in the eigenvectors is increased;
however the damage is still locatable.
Exact Damage
Modes 1 to 10
Cd
.Sh
*3
c
W)
a
1
03
Q
100 200
DOF
Modes 1 to 10 + 2.5% error
100 200
DOF
Modes 1 to 10 + 5% error
200
200
Figure 7.10 Fifty-Bay Truss: Damage Location Results using the Subspace
Rotation Algorithm with the Eigendata of the First Ten Modes.
7.3.3 Damage Extent
With knowledge of the location of damage from the previous analysis, the rank of the
“true” perturbation to the stiffness matrix, AIQ can be found by adding the rank of the
element stiffness matrix of the damaged members. Hence, the rank of the perturbation to
the stiffness matrix due to damage is two because two members having rank one element
matrices are damaged. From Proposition 6.1 and 6.2, it is clear that only experimental
eigendata from two modes of vibration should be used to compute the extent of the damage.
In the noisy cases, the two modes that should be used are the ones with d¡’s that most cleanly
demonstrate the damage shown in Figure 7.10. These modes can be determined by

108
inspecting the individual damage vectors d¡ associated with the measured modes. In
reviewing the individual d¡, it was determined that modes 8 and 9 provide the best insight
into the state of the damage. The results of applying the extent algorithm to determine the
perturbations to the stiffness matrix due to the damage, AK^, are shown in Figure 7.11. The
mesh plots are 3-dimensional representations of the perturbation matrices. The rows and
columns of the mesh plots correspond to the rows and columns of the perturbation matrices.
The “height” of each peak represents the magnitude of the perturbation made to each matrix
element. In Figure 7.11, the upper left mesh plot represents the exact damage. The upper
right plot corresponds to the case where exact eigendata are used in computing the extent of
damage. Note that with only two noise free eigenvalues / eigenvectors (modes 8 and 9), the
algorithm is able to reproduce the exact damage. As stated and proved in Proposition 6.2,
the exact damage can be computed using any two noise free modes that have a corresponding
B (defined in Section 6.3.2) of rank two. The left and right mesh plots of the second row
of Figure 7.11 correspond to the cases where the eigenvectors have been corrupted with 2.5%
and 5% random noise, respectively. The algorithm again demonstrates good performance
when faced with noisy eigendata (middle plots). The results of applying the eigenvector
filtering algorithm described in Section 6.6.2 to predict the extent using modes 8 and 9 are
shown in the left and right lower plots of Figure 7.11. It is obvious that the filtering process
greatly enhances the accuracy of the extent algorithm. Figure 7.12 displays the MRPT’s
assessment of the damage extent, for the noisy eigenvector cases, when the eigendata of the
first ten modes of vibration are used. A comparison of the results in Figure 7.12 and those
in the second row of Figure 7.11 shows that the above procedure which resulted in the use
of only modes 8 and 9 to assess the extent provided cleaner results. In the mesh plots of
Figure 7.12, additional numerical values appear at the “undamaged” DOFs which can be
attributed to the noise added by the addition of the extra modes. The resultant perturbation
matrices when using all ten modes are of rank 10 which is not consistent with the “actual”
damage perturbation which is of rank 2. Furthermore, this “ten modes procedure” is less

109
Modes 8 & 9 + 2.5% error
Modes 8 & 9 + 2.5% error Modes 8 & 9 + 5% error
Filtered Filtered
Figure 7.11 Fifty-Bay Truss: Damage Extent Results using the MRPT
with the Eigendata of Modes 8 and 9.
efficient since it requires the inversion of a 10-by-10 matrix, contrasted to the 2-by-2 inverse
used in the “two modes procedure.” The percentage errors of the predicted damage extent
with respect to the exact stiffness damage for all studied cases, with and without filtering,

110
are listed in Table 7.1. It can be seen from the table, the results obtained when the eigenvector
filtering process is used are much more accurate. The results from the “two modes
procedure” are, in all cases, but one, better than the “ten modes procedure”.
Modes 1 to 10 + 2.5% error Modes 1 to 10 + 5% error
Figure 7.12 Fifty-Bay Truss: Damage Extent Results using the MRPT
With the Eigendata of the First Ten Modes.
Modes 8 & 9 Modes 1 to 10
Figure 7.13 Fifty-Bay Truss: Damage Extent Results using Baruch’s Algorithm.
For the sake of comparison, results from the application of Baruch and Bar Itzhack’s
algorithm (1978) are provided in Figure 7.13. These results were obtained using (i) noise
free eigendata from modes 8 and 9, and (ii) noise free eigendata from the first ten modes.
In both cases, Baruch’s algorithm fails to ascertain the extent of the damage. In the first case,
the approach completely “smeared” the changes throughout the entire stiffness matrix. In
the second case, the changes are more localized around the damaged areas.

Table 7.1 Fifty-Bay Truss: Summary of Damage Extent Results using the MRPT.
Percentage Error with Respect to the Exact Damage
Without Filtering - Modes 1 to 10
Without Filtering - Modes 8 & 9
With Filtering - Modes 8 & 9
Eigenvectors
Error
Upper Longeron
of Bay Three
Lower Longeron
of Bay Forty
Upper Longeron
of Bay Three
Lower Longeron
of Bay Forty
Upper Longeron
of Bay Three
Lower Longeron
of Bay Forty
0.0%
Not Applicable
Not Applicable
0.00%
0.00%
Not Applicable
Not Applicable
2.5%
7.90%
13.03%
3.49%
7.89%
0.00%
0.07%
5.0%
14.70%
20.30%
13.31%
24.89%
0.00%
0.00%

112
7.4 Experimental Study: The NASA 8-bav Truss
7.4.1 Problem Description
The eight bay hybrid-scaled truss used in this investigation is part of a series of structures
designed for research in dynamic scale model ground testing of large structures at the NASA
Langley Research Center. Among other studies, a complete analytical and experimental
analysis of this truss was performed to generate a realistic testbed for structural damage
location/extent algorithms (Kashangaki et al., 1992; Kashangaki, 1992). The truss
configuration used in this analysis was cantilevered and instrumented with ninety-six
accelerometers to measure all three translational degrees of freedom at each of the thirty-two
unconstrained nodes. A schematic of this truss is shown in Figure 7.14. Figure 7.15 shows a
close-up of the truss lacing pattern along with strut definitions that will be used in the
remainder of this study.
Figure 7.14 The NASA Eight-Bay Hybrid-Scaled Truss: Damage Cases.

113
Figure 7.15 The NASA 8-Bay Truss: Lacing Pattern.
A ninety-six degree of freedom undamped finite element model (FEM) of the original
“healthy” truss was generated using MSC/NASTRAN. In this FEM, each truss strut was
modeled as a rod element. Concentrated masses were added at each node to account for the
joint and instrumentation mass properties (Kashangaki, 1992). The mass and strut properties
of the truss are summarized in Tables 7.2 and 7.3.
Table 7.2 Mass Properties of the Eight Bay Truss.
Weight
(Pounds)
Total Number in
Eight Bay Truss
Total Weight
(Pounds)
Node Ball
0.0128
36
2.6935
Longeron Strut
0.0396
68
2.4524
Diagonal Strut
0.0598
41
0.4608
Joint Assembly
0.0170
218
3.7001
Triax + Block
0.0243
32
0.7760
Total Weight of the Truss
10.0776

114
Table 7.3 Strut Properties of the Eight Bay Truss.
Longeron
Diagonal
Strut Length
19.685 in
27.839 in
Strut Stiffness (EA/L)eff
13,040 lb/in
9,013 Ib/in
Experimental modal analysis of the truss was performed for the “no damage” and
sixteen damage cases. In the testing, the excitation source was provided by two shakers.
Modal parameters were identified from the measured frequency response functions using the
Polyreference complex exponential technique. For each case, five modes of vibration were
identified. Each measured mode consists of a natural frequency and its corresponding mode
shape with measurements at all ninety-six FEM degrees of freedom. For the “no damage”
case, the first, second, fourth and fifth mode are bending modes; and the third mode is the
first torsional mode. In the damage cases, the same five modes were measured; however
their order varies, since in some cases, the damage could cause mode switching. It should be
noted that during the testing process, two accelerometers at node fifteen failed. These
defective accelerometers would affect the component of the measured modes associated
with DOFs 44 and 45.
Data for fifteen of the sixteen damage cases tested were received from NASA. The
fifteen damage cases that were received are identified in the schematic of the truss shown in
Figure 7.14. For each damage case, the type of element and the FEM degrees of freedom
affected by damage are shown in Table 7.4.
In damage cases A to N, structural damage consisted of the full removal of one strut
from the truss. Case O damage consists of the full removal of two struts. In Case P, one of the
struts was buckled to illustrate a partial damage scenario. Note that the forthcoming analysis
of the fifteen damage cases was performed with no a priori knowledge of the actual damage
locations.

115
Table 7.4 NASA 8-Bay Truss Damage Case Definitions.
Damage Case
Element Type
Damaged DOFs
A
Longeron
86
C
Longeron
74, 86
D
Upper Diagonal
73, 74, 88, 89
E
X-Batten
73,76
F
Face Diagonal
76, 78, 79,81
G
Longeron
65,77
H
Longeron
62, 74
I
Left Diagonal
68, 69, 74, 75
J
Longeron
59,71
K
Right Diagonal
41,42, 59, 60
L
Longeron
44, 56
M
Longeron
29,41
N
Longeron
2, 14
0*
Longeron & Diagonal
62, 68, 69, 74, 75
Z-Batten
75,81
* Two struts removed
** Buckled strut
7.4.2 Refinement of the Original FEM
The original FEM provided by NASA is a first generation analytical model that was not
modified to match the experimentally measured modal parameters. It was found that the
original FEM does not accurately predict the dynamic behavior of the actual structure. As
an illustration to the inaccuracy of the original FEM, Table 7.5 shows a comparison between
the analytically computed and experimentally measured natural frequencies. Another
commonly used criterion to judge the accuracy of the original analytical FEM is to compare

116
its frequency response functions to those measured experimentally. One such comparison is
shown in Figure 7.16. It is clear from this typical frequency response comparison that the
original FEM is lacking in accuracy.
Table 7.5 Comparison of Analytical and Experimental Frequencies
Natural Frequencies (Hz)
MODE#
Analytical
Experimental
1
13.9245
13.8757
2
14.4407
14.4783
3
46.7445
48.4122
4
66.0067
64.0332
5
71.1420
67.4631
Hence, the first step in this study was to refine the original FEM. For the refinement
process, it was assumed that the original mass matrix is an accurate representation of the
truss’ mass properties. The inaccuracy of the original FEM was believed to be solely due to
modeling errors in the stiffness properties. The minimum rank perturbation theory discussed
in Chapter 6 (Section 6.3.2) was used to correct the original stiffness matrix. In order to get a
symmetric updated stiffness matrix, the measured mode shapes (eigenvectors) were mass
orthogonalized using the Optimum Weighted Orthogonalization technique (Baruch 1978).
A mesh plot of the changes made to the stiffness matrix (AK) from the refinement
process is shown in Figure 7.17. As can be seen in the figure, there are three areas of major
change made to the original stiffness matrix. One area of change corresponds to the
cantilever end of the truss. This change was expected since a perfect cantilever condition was
assumed in generating the original FEM. Note that it is common knowledge that perfect
cantilever conditions cannot be produce in practice. Other changes can be seen in the middle
of the mesh plot. These changes occur at and around DOFs 44 and 45 which are the location
of the bad sensors. Hence, these changes were attributed to the effect of the bad sensors. Yet,

117
another area of changes can be seen near the free end. These areas coincide with the locations
of the two shakers. The reason for the perturbation at these shaker locations is that their
localized stiffness effects were not modelled in the original FEM. The improvement made to
the original analytical FEM from the MRPT refinement process can also be seen in Figure
7.16. It is apparent from comparing the frequency response functions corresponding to
experimental measurement, original FEM and the refined FEM that the MRPT provided a
great deal of improvement to the original FEM. The amplitude mismatch between the
experimental data and refined FEM prediction can be attributed to the fact that the actual
effects of the damping were not considered in the analytical models. Note that the actual
damping of the truss is small and negligible but it is nonzero.
Figure 7.16 NASA 8-bay Truss: Typical Frequency Response Comparison.

118
ROOT
Figure 7.17 NASA 8-bay Truss: Perturbation to the Original Stiffness
Matrix that Resulted from the Refinement Process.
7.4.3 Damage Location
For each damage case, the modal parameters of the five measured modes were used to
compute a cumulative damage location vector (CDLV) to determine the location of the
structural damage. Since the values of || zJd ||, as defined in Section 5.3, are of the same orders
of magnitude for all DOFs and all five measured modes, the cumulative damage location
vector (CDLV) could be computed using either Eq. (5.4.1) or Eq. (5.4.2). The angle
perturbation CDLV (Eq. (5.4.2)) associated to damage case F is shown in Figure (7.18). The
upper left corners of Figures (7.19) to (7.32) display plots of the cumulative damage location
vectors as calculated by Eq. (5.4.2) for all cases where the damage was successfully located.
In these plots, the CDLV were unit normalized and their elements were plotted versus the
FEM DOFs. The performance of the location algorithm for each damage case is summarized
in the second column of Table 7.7. A summary of the procedures used in the interpretation of
the CDLV of each damage case is as follows:

119
(1) Initially, a comparative study of the CDLV for all cases was performed to deduce the
effects of the bad accelerometers. A fairly large numerical component at DOF 44 was
detected in most of the fifteen CDLV. This component was believed to be due to the
“bad” sensor located at DOF 44. Hence, for all cases, it was decided to ignore the
component of the CDLV at DOF 44.
(2) For damage cases A, C, D, G, H, I, J, K, M, N and O, the location of the damage was
determined by simply considering the DOFs associated with the substantially larger
numerical components of the CDLV’s as being the damaged DOFs. The smaller
numerical components of the CDLV’s at the other DOFs can be attributed to
measurement errors. Notice that for cases D, I, J, M and N, the component of the CDLV
at DOF 44 is fairly small and hence negligible. It should be also noted that the damage in
Case A affects only one degree of freedom (86) since the strut involved is a longeron
connected to the cantilevered end.
(3) For damage case E, the DOFs associated with the CDLV components of greater order of
magnitude are 42,73,75,76 and 81. By utilizing the connectivity of the original FEM,
the combinations of these DOFs that are physically meaningful, i.e. bound a strut, are
(73,76) and (75, 81). Based on these results, it was decided that two struts connecting
DOF (73,76) and (75,81) were damaged. However, in actuality the only damaged strut
is the one connecting DOFs (73, 76).
(4) For damage case L, by ignoring DOF 44 (the “bad” sensor), the only DOF that is clearly
damaged is 56. No strut with DOF 56 is connected to the wall. The damage of a strut not
connected to the cantilevered end must affect at least two DOFs. Since DOF 56 is in the
y-direction, it was deduced that the most probable damaged strut is a longeron
connected at one end to DOF 56. The two candidate struts are the ones bounded by
DOFs (44, 56) and (56, 68). However, the component of the CDLV associated with

120
DOF 68 is small and is in no way affected by damage. This prompted the deduction that
the strut connecting DOFs 44 and 56 is the damaged member in Case L.
(5) For damage case P, the substantially larger components of the CDLV occur at DOFs 42,
75 and 81 (DOF 44 ignored). Again, with the use of the connectivity of the original
FEM, it is deduced that the only combination of these three DOFs that are connected by a
strut is (75, 81). These DOFs are exactly the damaged DOFs of Case P.
(6) Damage Case F is the only case where the location algorithm failed to locate the damage.
The cumulative damage vector for this case is shown in Figure 7.18. The reason the
damage was not located in this case is that it involves a face diagonal strut. Face
diagonal struts, in general, affect only the axial modes and have slight or no effects on
the bending and torsional modes. As reported earlier, the only available (measured)
modes are bending and torsional modes. It was determined analytically that the first
axial mode of the truss occurs at the sixth mode.
Kashangaki et al. (1992) performed a pure analytical study to compile a list of the
elements of the 8-bay truss that have a substantial contribution to the total strain energy for
each of the five measured modes. The present study confirms that the damage in struts that
have a substantial contribution to the total modal strain energy for the measured modes is
detectable. However, our study also shows that damage of non-highly strained struts for the
measured modes can also be located as illustrated in cases G and N. Note that the damage in
cases E and F involves struts with low strain energy for the measured modes.
7.4.4 Damage Extent
Because the truss under investigation is light weight and very flexible, the effect of
structural damage is considered (i) negligible on the overall inertial properties, and (ii) a
substantial stiffness loss. Using the extent algorithm discussed in Section 6.3.2, the
perturbation to the stiffness matrix (AKj) due to the damage is calculated for each damage
case where the damage was successfully located. To insure the symmetry of AK^, the

121
eigenvectors used in these calculations were first mass orthogonalized using the Optimum
Weighted Orthogonalization technique (Baruch and Bar Itzhack, 1978).
7.4.4.1 The Brute Force Method
The upper right comers of Figures 7.19 to 7.32 display mesh plots of the calculated
A IQ’s using the modal parameters of all five measured modes. From these mesh plots, it is
clear that on the average the extent algorithm failed to concentrated the major changes at the
damaged DOFs. This unsatisfactory performance of the extent algorithm can be attributed to
the measurement error of the defective accelerometers. In most cases, the major changes are
concentrated at DOF 44, which corresponds to the location of a defective accelerometer. In
addition, for any given damage case, as discussed in Section 6.6.1, only a subset of the
measured modes are highly affected by damage. Using modes that are not highly affected by
damage only introduces their associated measurement noise in the extent calculation.
7.4.4.2 The Damage Consistent Method
As discussed in Section 6.6.1 and illustrated in the previous example, for each damage
case, the performance of the extent algorithm can be improved by only using as many modes
as the rank of the “actual” AIQ. The rank of the “actual” AK rank of all element stiffness matrices which connect damaged DOFs. Since the elemental
stiffness matrix of the truss under investigation is rank one, the rank of the “actual” AK each damage case, is equal to the number of damaged struts. Hence, only modal parameters
from one mode are needed to compute the damage extent of all cases featuring one damaged
strut. For Case O, since two struts are damaged, data from two modes are needed. For each
case, the modes that should be used are the ones that most cleanly demonstrate the state of
damage of that case. These modes can be simply determined by inspecting the individual
damage location vector d¡. The modes that provide the best insight into the state of the
damage for each damage case are reported in the third column of Table 7.7. The unit
normalized damage location vectors associated with the “best” modes for all cases are

122
plotted versus the FEM DOFs in the middle left of Figures 7.19 to 7.32. The mesh plots of the
AKd’s computed using only the “best” mode(s) for all damage cases are shown in the middle
right comers of Figures 7.19 to 7.32. It is clear that the performance of the extent algorithm
for all cases has been greatly improved in comparison to its performance when all five
measured modes were used. The estimated effective stiffness loss (lb/in) for all damage
cases, computed from this process, are reported in the fourth column of Table 7.7. For each
damage case, the effective stiffness loss was calculated by averaging the stiffness changes
(AK and Z-battens have an effective stiffness of 13,040 lb/in, while the effective stiffness of the
diagonal struts is 9,013 lb/in (Kashangaki, 1992). The removal of any strut results in the
complete loss of the stiffness of that strut. The fifth column of Table 7.7 reports the
percentage error of the estimated stiffness losses with respect to the original FEM. The
percentage error for Case P (buckled strut) was not computed since the damage extent is
unknown.
7.4.4.3 Application of the Eigenvector Filtering Algorithm
The damage extent assessment can be further improved by using the eigenvector
filtering algorithm discussed in Section 5.4.2. For consistency, the mode(s) used in the
filtering process are the “best” mode(s) as determined above and reported in the third column
of Table 7.7. For each case, the “undamaged” components of the damage vectors d¡ are first
set to zero. The components of d¡ associated to the “damaged” DOFs are then constrained to
be consistent with the actual effect of the damage as dictated by the element stiffness matrix
of the damage struts. The damage in X-battens, longerons or Z-battens affects two DOFs
(one DOF at each of the two nodes of the strut). In order for the damage to be finite element
consistent, the component associated with these two DOFs should be equal in magnitude and
of opposite sign. Damage in upper, face, left or right diagonal struts affects four DOFs (two
at each node of the strut). These four DOFs can be classified into two sets of two DOFs in the
same direction (x, y or z). The DOFs of a given set are, in general, equal in magnitude and of

123
opposite signs. In the 8-bay truss, all upper, face, left or right diagonals are either at a 45° or
-45° angle with respect to their in-plane global coordinate axis. This constrains the DOFs of
the two sets to have the same magnitude in order to be consistent with the FEM. The DOFs
associated with a given node have the same sign when the angle is 45°; they are of opposite
sign when the angle is -45°. For the cases where a single strut is damaged the magnitude of a
DOF could be set arbitrarily. The reason is that eigenvectors are unique in a relative sense;
i.e., if Uj is an eigenvector associated to a given mode then au¡ is also an eigenvector
associated to that same mode, a being any scalar. Thus, the components of the damage vector
are also a function of the scalar a. These components can be set to any values by varying a.
For the single member damage cases under study, this process is summarized in Table 7.6.
Table 7.6 Summary of the Filtering Process for Single Member Damage Cases.
Node 1
Node 2
DOF
X
Y
z
X
Y
z
X-Batten
P
0
0
-P
0
0
Longeron
0
P
0
0
-p
0
Z-Batten
0
0
p
0
0
-p
Case D (-45°)
p
-p
0
-p
p
0
Case I (45°)
p
0
p
-p
0
-p
Case K (-45°)
p
0
-p
-p
0
p
where (3 is any scalar
Note that cases A, C, E, G, H, J, L, M, N and P involve the damage of either an X-batten,
longeron or Z-batten. As discussed earlier, all DOFs of the fdtered damage vector df,
associated to a given damage member have the same magnitude, (3. Experience gained by
using the eigenvector filtering algorithm indicates that in the multiple member damage
scenario (case O), the ratio between the (3’s from different damaged struts contains important

124
information about the relative damage extent among the damaged struts. Hence, in case O, a
given P associated with a given damaged strut cannot be chosen arbitrary. This p should be
estimated by averaging the components of the unfiltered damage vector df corresponding to
the damaged strut in question. The absolute value of the unit normalized filtered damage
location vectors associated with all cases are plotted versus the FEM DOFs in the bottom left
corners of Figures 7.19 to 7.32. Now that the “filtered” damage vectors are generated, the
next step is to compute the corresponding filtered damage eigenvector using Eq. 5.4.3. The
process of computing the perturbation due to the damage using the filtered eigenvectors is as
in the previous section. For all damage cases, the mesh plots of the AK^’s computed using
the “filtered” mode(s) are displayed in the bottom right comers of Figures 7.19 to 7.32. The
effective stiffness loss computed using the filtering process, for each damage case, is
reported in the sixth column of Table 7.7. The seventh column of Table 7.7 reports the
percentage error of the estimated stiffness losses with respect to the original FEM. A
comparison of these results to the one acquired from the previous section shows that the
eigenvector filtering algorithm greatly enhances the extent estimate.
Damage Location: Modes 1-5
Figure 7.18 NASA 8-bay Truss: Cumulative Damage Vector
Associated to Case F.

Damage Vector Damage Vector Damage Vector
125
Damage Location - Modes 1-5
Damage Location - Mode 1
Damage Location - Mode 1 Filtered
DOF
Damage Extent - Modes 1-5
Damage Extent - Mode 1
Damage Extent - Mode 1 Filtered
Figure 7.19 NASA 8-bay Truss: Damage Assessment of Case A.

Damage Vector Damage Vector Damage Vector
126
Damage Location - Modes 1-5
0.5 -
Damage Location - Mode 1
DOF
Damage Extent - Modes 1-5
Damage Extent - Mode 1
Damage Extent - Mode 1 Filtered
Figure 7.20 NASA 8-bay Truss: Damage Assessment of Case C.

Damage Vector Damage Vector Damage Vector
127
Damage Location - Modes 1-5
Damage Location - Mode 3
Damage Location - Mode 3 Filtered
1 - m m -
0.5
O' *—
0 50
DOF
Damage Extent - Modes 1-5
Damage Extent - Mode 3
Damage Extent - Mode 3 Filtered
Figure 7.21 NASA 8-bay Truss: Damage Assessment of Case D.

Damage Vector Damage Vector Damage Vector
128
Damage Location - Modes 1-5
Damage Location - Mode 4
Damage Location - Mode 4 Filtered
Damage Extent - Modes 1-5
Damage Extent - Mode 4
Damage Extent - Mode 4 Filtered
Figure 7.22 NASA 8-bay Truss: Damage Assessment of Case E.

Damage Vector Damage Vector Damage Vector
129
Damage Location - Modes 1-5
Damage Location - Mode 1
Damage Location - Mode 1 Filtered
Damage Extent - Modes 1-5
Damage Extent - Mode 1
Damage Extent - Mode 1 Filtered
Figure 7.23 NASA 8-bay Truss: Damage Assessment of Case G.

Damage Vector Damage Vector Damage Vector
130
] Damage Location - Mode 4 Filtered
Damage Extent - Modes 1-5
Damage Extent - Mode 4
Damage Extent - Mode 4 Filtered
Figure 7.24 NASA 8-bay Truss: Damage Assessment of Case H.

Damage Vector Damage Vector Damage Vector
131
Damage Location - Modes 1-5
1
ON
NO
„74
68
75
44
liflfllm íllli j 1
1 wtíÜUiiíwv
DOF
Damage Location - Mode 4
Damage Location - Mode 4 Filtered
DOF
Damage Extent - Modes 1-5
Damage Extent - Mode 4
Damage Extent - Mode 4 Filtered
Figure 7.25 NASA 8-bay Truss: Damage Assessment of Case I.

Damage Vector Damage Vector Damage Vector
132
Damage Location - Modes 1-5
1
59
71
llWnUI Lwi nil UlfllfMH
jl
illtlkiU-
0 50
DOF
Damage Location - Mode 4
Damage Location - Mode 4 Filtered
Damage Extent - Modes 1-5
Damage Extent - Mode 4
Damage Extent - Mode 4 Filtered
Figure 7.26 NASA 8-bay Truss: Damage Assessment of Case J.

Damage Vector Damage Vector Damage Vector
133
Damage Location - Modes 1-5
-
42
60
-
41
-
59
n44
-
himíhILé! * i il
mi
fll
lilttL
0 50
DOF
Damage Location - Mode 3 Filtered
50
DOF
Damage Extent - Modes 1-5
Damage Extent - Mode 3
Damage Extent - Mode 3 Filtered
Figure 7.27 NASA 8-bay Truss: Damage Assessment of Case K.

Damage Vector Damage Vector Damage Vector
134
Damage Location - Mode 4
Damage Location - Mode 4 Filtered
DOF
Damage Extent - Modes 1-5
Damage Extent - Mode 4
Damage Extent - Mode 4 Filtered
Figure 7.28 NASA 8-bay Truss: Damage Assessment of Case L.

Damage Vector Damage Vector Damage Vector
135
Damage Location - Modes 1-5
41'
29
44
rfblLfa
l tnTTJlrvllTf>Tffln4WfhvfTjTnlUDn_.
0 50
DOF
Damage Location - Mode 4 Filtered
Damage Extent - Modes 1-5
Damage Extent - Mode 4
Damage Extent - Mode 4 Filtered
Figure 7.29 NASA 8-bay Truss: Damage Assessment of Case M.

Damage Vector Damage Vector Damage Vector
136
DOF
Damage Extent - Modes 1-5
Damage Extent - Mode 5
Damage Extent - Mode 5 Filtered
Figure 7.30 NASA 8-bay Truss: Damage Assessment of Case N.

137
Damage Extent - Modes 1-5
Damage Extent - Modes 3 & 5
Damage Location - Modes 3 & 5 Filtered Damage Extent - Modes 3 & 5 Filtered
DOF
Figure 7.31 NASA 8-bay Truss: Damage Assessment of Case O.

Damage Vector Damage Vector Damage Vector
138
Damage Location - Mode 4 Filtered
DOF
Damage Extent - Modes 1-5
Damage Extent - Mode 4
Damage Extent - Mode 4 Filtered
Figure 7.32 NASA 8-bay Truss: Damage Assessment of Case P.

139
Table 7.7 NASA 8-Bay Truss: Summary of the Damage Assessment Results.
Without Filtering
With Filtering
Damage
Case
Location
Performance
Best
mode
Extent
(lb/in)
% error
Extent
(lb/in)
% error
A
++
1
11,896
8.77
12,865
1.34
C
++
1
11,901
8.73
12,866
1.34
D
++
3
9,789
8.61
8,806
2.30
E
o
4
8,094
37.90
13,454
3.17
F
-
Location not detected
G
++
1
10,859
16.70
12,971
0.53
H
++
4
12,601
3.37
13,154
0.87
I
++
4
7,568
16.00
10,220
13.39
J
++
4
12,015
7.86
13,176
1.04
K
++
3
6,955
22.80
8,851
1.79
L
+
4
8,421
35.40
12,859
1.39
M
++
4
12,009
7.91
12,882
1.21
N
++
5
10,037
23.00
13,169
0.99
O
++
3,5
(L) 12,347
(D) 8,451
5.31
6.78
(L) 12,807
(D) 8,878
1.79
1.50
P
+
4
8,815
not def.
13,461
not def.
++ Damage clearly located
+ Damage located with further analysis
- Damage not located
o location narrowed to within two members
(L) Longeron (D) diagonal

140
7.5 Experimental Study: Mass Loaded Cantilevered Beam
7.5.1 Problem Description
This experiment was designed to illustrate the scenario of damage in mass properties.
The structure used in this investigation is a cantilevered beam loaded with a non-structural
mass. A schematic of this beam is shown in Figure 7.33. The dimensions and properties of
the structure are summarized in Table 7.8.
Figure 7.33 The Mass Loaded Cantilevered Beam.
Table 7.8 Mass Loaded Cantilevered Beam Properties.
Beam Length - .86 m
Beam Mass/Length - 1.246 kg/m
Beam Moment of Inertia - 1.458xl0~9 m4
Beam Young’s Modulus - 69 GPa
Discrete Mass - 0.7938 kg
Discrete Mass Moment of Inertia - l.lxlO-3 kg-m2
An undamped FEM of the mass loaded beam was constructed using beam elements in
conjunction with the properties of the non-structural mass. The beam element has two
degrees of freedom (DOF) at each node: bending and rotation. A sixteen DOF undamped

141
FEM was generated using the eight equal length element discretization shown in Figure 7.33.
The effect of the non structural mass was considered non-stiffening and concentrated at node
3. This mass was modelled by adding its mass and moment of inertia to node 3 bending and
rotation DOFs, respectively. The mass loaded beam, as described above, was considered the
“healthy” configuration. Structural damage consisted of the removal of the non-structural
mass from the beam.
Experimental modal analysis of the beam was performed on the “healthy and
“damaged” beam configurations. Modal parameters were identified using frequency
domain techniques and the Rational Fraction Least Square single degree of freedom curve
fitting algorithm. The excitation source used was an impact hammer and the driving point
measurement was an accelerometer mounted at the free end of the beam. For each
configuration (healthy and damaged), four modes of vibration were measured. Each mode
consisted of a natural frequency and its corresponding mode shape with measurements at
only the eight FEM bending degrees of freedom.
7.5.2 Analytical and Experimental Models Dimension Correlation
The number of measured eigenvector components (8) is less than the number of DOFs in
the FEM (16). In fact, only the bending DOFs of the beam were measured experimentally.
As discussed in Chapter 2, two approaches are available to correlate these dimensions: (i)
expansion of the measured eigenvectors or (ii) reduction of the FEM. It was found that a
FEM reduction is better suited for this application. Thus, the FEM was reduced using the
improved reduction system (IRS) method (O’Callahan, 1989).
7.5.3 Refinement of the Original FEM
Table 7.9 shows a comparison between the analytically computed and experimentally
measured natural frequencies. From this comparison, it is clear that the original reduced
FEM does not accurately predict the dynamic behavior of the “healthy” beam. The first step
in this study was to refine the original FEM. For the refinement process, it was assumed that

142
the original mass matrix is an accurate representation of the structure’s mass properties. The
inaccuracy of the original FEM was believed to be solely due to modeling errors in the
stiffness properties. The algorithm discussed in Section 6.3.2 was used to correct the original
reduced stiffness matrix. In order to get a symmetric updated stiffness matrix, the measured
mode shapes (eigenvectors) were mass orthogonalized using the Optimum Weighted
Orthogonalization technique (Baruch and Bar Itzhack, 1978).
Table 7.9 Analytical and Experimental Frequencies
of the “Healthy” Structure.
Mode #
Analytical
Frequency (Hz)
Experimental
Frequency (Hz)
1
4.5
4.6
2
40.3
41.6
3
96.8
99.8
4
195.9
205.9
7.5.4 Damage Location
The next step of this analysis is to determine the location of the structural damage. Using
the refined FEM, and the modal parameters of the four modes of vibration measured from the
“damaged” beam, a cumulative damage location vector was calculated (Chapter 5). Since
the values of ||zJd ||, as defined in Section 5.3, are of different orders of magnitude the
cumulative damage location vector should be computed using Eq. (5.4.2). The upper left
comer of Figure 7.34 displays the plot of the unit normalized cumulative damage location
vector as calculated by Eq. (5.4.2). From this plot, it is clear that DOF 3 has been affected by
damage. This is exactly the bending DOF where the non-structural mass was mounted. The

143
small numerical elements at all other DOFs can be attributed to experimental measurement
noise.
DOF
tot)
s
DOF
Damage Extent - Mode 4
Figure 7.34 Mass Loaded Cantilevered Beam: Damage Assessment
7.5.5 Damage Extent
The final step of this analysis is to determine the extent of structural damage. With
knowledge of the damage location, the rank of the “true” mass perturbation matrix, AMd, is
one since there is only one DOF affected by the damage and that DOF is not connected to the
cantilevered end. Hence, in order to compute a rank one AMd only one mode of vibration is
needed. As discussed earlier, the mode that should be used is the one that most cleanly
demonstrates the damage detected by the cumulative damage location vector. The damage

144
vector Ú4 was determined to provide the best insight into the state of damage. The damage
vector, ¿4 is shown in the upper right corner of Figure 7.34. The calculated AMj using mode
4 data is shown in the lower plot of Figure 7.34. It is clear that the extent calculation has
concentrated the major changes at the DOF affected by the damage. From the extent
calculation, the mass loss was estimated to be 0.7947 kilograms, which is within 1.1 % of the
actual mass loss.
7.6 Fifty-Bav Two-Dimensional Truss: Proportionally Damped FEM
7.6.1 Problem Description
The structure used in this investigation is the same as the one used in Section 7.3. For
convenience, it is shown again in Figure 7.35. This example is used to illustrate the
characteristics of the MRPT when dealing with proportionally damped systems in which two
of the three property matrices are affected by structural damage. The analytical model of the
truss is a 201 DOF FEM which was generated using rod elements. The damping of the
“healthy” model is proportional and equal to 1 x 10 1 times the “healthy” mass matrix plus
5xl0-7 times the “healthy” stiffness matrix. As in the problem of Section 7.3, damage is
simulated by reducing the Young’s modulus of two members: the upper longeron of bay three
and the lower longeron of bay forty. The modulus of elasticity of the upper longeron of the
third bay is reduced from E=29xl06 psi to lxlO3 psi. The bay forty’s lower longeron is
subjected to a complete loss of stiffness (Young’s modulus equal to zero). The damping of
the “damaged” model is also proportional and equal to lxlCH times the “healthy” mass
matrix plus 5xl0~7 times the “damaged” stiffness matrix. Thus, the simulated damage is
only affecting the stiffness and damping properties of the structure. Note that the damage in
the damping properties is proportional to the damage in the stiffness properties. For our
damage assessment analysis, it is assumed that only the first ten “damaged” modes are
available. This problem is investigated for three different scenarios. Each scenario
corresponds to a different level of random noise added to the “damaged” eigenvectors. In

145
practice the noise in eigenvector information could be due to both measurement and/or
expansion errors.
E = 2-OxlO11 (29x106 psi) p = 7.833xl03 kg/m3 (0.283 lb/in3)
A = 6.452x 1(H m2 (1.0 in2) L = 0.1270 m (5.0 in)
Figure 7.35 50-Bay 2-Dimensional Truss.
7.6.2 Damage Location
Cumulative damage location vectors, as defined in Chapter 5, are first computed for all
three scenarios using the modal properties of the ten “damaged” modes. In this problem, all
rows of matrix Zd (defined in Chapter 5) are of the same order of magnitude; hence either
Eq. (5.4.1) or Eq. (5.4.2) can be used to compute the cumulative damage vectors. The upper
left plot of Figure 7.36 represents the exact damage computed from the exact damage
perturbation matrices (AIQ and ADj). The upper right plot corresponds to the case where the
exact “damaged” eigenvector information is provided to the subspace rotation damage
location algorithm. The lower left and right plots correspond to the cases where the exact
“damaged” eigenvectors have been corrupted with 2.5% and 5% random noise, respectively.
As shown in Figure 7.36, the location algorithm is able to exactly locate the damage when
presented with noise free information. Although not as clean, the damage can still be clearly
located in the noisy eigenvectors cases.

146
Exact Damage
200
Modes 1 to 10
DOF
o
+-»
O
• rH
T3
C
(D
oa
ctf
cd
Q
Modes 1 to 10 + 2.5% error
1
0.5
0
0 100 200
T
UUBÉNlÉW^Éi
DOF
Modes 1 to 10 + 5% error
DOF
Figure 7.36 50-Bay 2-Dimensional Truss: Damage Location.
7.6.3 Damage Extent
The damage location assessed in the previous section can be used in conjunction with the
truss finite element connectivity to determine the damaged truss struts. The rank of the
“true” perturbation matrix, AK^ (or ADj), can be found by adding the rank of the element
stiffness (or damping) matrix of the damaged struts. Hence, the rank of the perturbation to
the stiffness matrix due to damage is two because two struts of rank one element matrices are
damaged. Note that the element stiffness matrix of a strut is one since it is modelled as a rod
element. Because the damping of the structure is proportional and the mass properties are
unaffected by the damage, it is deduced that the rank of the perturbation to the damping
matrix is the same as the rank of the perturbation to the stiffness matrix. From propositions

147
6.1 and 6.2, it is clear that only experimental data from two modes of vibration are needed to
compute the extent of the damage. In the noisy situations, the two modes that should be used
are the ones that most cleanly demonstrate the damage shown in Figure 7.36. These modes
can be determined by inspecting the individual damage vectors. An inspection of the
individual damage vectors associated with each “noisy” eigenvector suggests that modes 8
and 9 provide the best insight into the state of the damage. The results of applying the MRPT
( Section 6.4.1, Eq. (6.4.6)) to determine the perturbations to the stiffness matrix due to the
damage are shown in Figure 7.37. Clearly, from judging the upper right mesh plot of Figure
7.37, the MRPT is able to reproduce the exact damage with only two noise free modes. The
algorithm demonstrates good performance when faced with noisy eigendata (lower plots).
The percentage errors with respect to the exact stiffness damage for all studied cases are
listed in Table 7.10. The perturbations to the damping matrix, ADj, due to the damage are
estimated by the extent algorithm (Section 6.4.1, Eq. (6.4.4)) with exactly the same accuracy
as for AK shown in Figure 7.37. This would be expected since, as reported earlier, the damage in the
damping properties is proportional to the damage in the stiffness properties.
Table 7.10 50-Bay 2-Dimensional Truss: Summary of Percentage
Error With Respect to the Exact Damage.
Percentage Error with respect to
exact stiffness (or damping)
Eigenvectors
Error
Upper Longeron Of
Bay Three
Lower Longeron of
Bay Forty
0.0%
0.00
0.00
2.5%
6.60
7.42
5.0%
22.60
15.30

148
Exact Damage Modes 8 & 9
Figure 7.37 50-Bay 2-Dimensional Truss: Damage Extent.
7.7 Eight-Bay Two-Dimensional Mass-Loaded Cantilevered Truss
7.7.1 Problem Description
The structure under investigation is the eight-bay two-dimensional mass-loaded
cantilevered truss shown in Figure 7.38. This structure was designed (Rides, 1991) to
emulate typical properties of space structures: low frequency modes with large
non-structural mass. The geometric and material properties of the truss are given in Figure
7.38. The truss consists of 40 struts and 16 nodes. Fourteen of the nodes are loaded with
concentrated non-structural mass of magnitude 2.3 lbs-sec2/in and the remaining two nodes
have large lumped masses of magnitude 47.4 lbs-sec2/in. Each truss strut was modeled as a

149
rod element with negligible mass. The finite element model has 32 translational DOFs ( 2
DOFs per node). The truss, as described above, is considered the healthy (undamaged)
configuration in our study. Two problems based on this structure are presented to illustrate
the characteristics of the procedure of applying the MRPT to simultaneously determine the
damage extent in all property matrices of undamped (M, K) and proportioned damped (M, D,
K) structures.
Figure 7.38 The Eight-Bay Two-Dimensional Mass-Loaded
Cantilevered Truss.
7.7.2 Proportionally Damped Configuration: Damage of Small Order of Magnitude
In this first problem, it is assumed that the damping of the truss is proportional and equal
to lxlO-1 times the “healthy” mass matrix plus lxlO-6 times the “healthy” stiffness matrix.
In this example the damage is simulated by a 10% stiffness reduction of the darkened strut
and a 5% reduction of mass M). The damping of the “damaged” model is also assumed
proportional and equal to lxlO-1 times the “damaged” mass matrix plus lxlO-6 times the
“damaged” stiffness matrix. For the damage analysis, it is assumed that only the first four
modes of vibration are available. Each available mode consists of an eigenvalue and an
eigenvector with entries at all FEM DOFs. This present problem is similar to a problem

150
investigated by Ricles (1991) with the exception that in Ricles (1991) the truss was assumed
undamped.
7.7.2.1 Damage Location
The first step in the analysis is to use the damage location algorithm as discussed in
Chapter 5 to determine the location of damage. The left plot of Figure 7.39 represents the
unit normalized exact cumulative damage location vector (CDLV) (Eqs. (5.2.4b) and
(5.4.2)) computed from the exact perturbation matrices AMj, ADj and AK^. The unit
normalized CDLV computed from the subspace damage location algorithm (Eqs. (5.2.4b)
and (5.4.2)) is shown in the right plot of Figure 7.39. As previously proven, the location
algorithm is able to exactly locate the damage when presented with noise free data.
Exact Damage
Modes 1 to 4
Figure 7.39 Problem 7.7: Cumulative Damage Location Vector: First Four Modes.
1.12.2 Decomposition of Matrix B
The next step is to decompose matrix B as defined by Eq. (6.4.28) into Bm, B^, and B^.
As discussed in Section 6.4.4, the proposed decomposition requires that matrix B be of full
rank. A simple singular value decomposition of matrix B computed using all 4 available
modes shows that its rank is 3. Therefore, only three of the four available modes should be
used. A further investigation reveals that any three of the four modes result in a full rank B.

151
Hence, any combination of three modes can be used in the decomposition. The three plots in
the upper half of Figure 7.40 show the unit normalized exact cumulative vector (Eq. (6.4.35),
Section 6.4.4) computed using the exact perturbation matrices, AMj, ADj and AK,j, and the
first three modes. The unit normalized cumulative vector associated with the computed Bm,
Bd, and B^ using the proposed decomposition procedure, for the first three modes, are
displayed in the lower half of Figure 7.40. A comparison between upper and lower plots
shows that the decomposition procedure resulted in the exact Bm, Bj, and B^ for the
associated modes. This is also true if any other combination of three modes was used. Note
that the cumulative vectors in Figure 7.40 give information on the effect of the damage on the
mass, damping and stiffness properties.
Exact Br
a
o
•3
c
Ml
03
DOF
DOF
DOF
Figure 7.40
Problem 7.7: Cumulative Vectors Associated With the
Exact and Computed Bm, B¿, B^: First Three Modes.

152
7.7.2.3 Damage Extent
Now that Bm, Bj, and B^ have been computed, the final step is to compute the
perturbation to the property matrices due to the damage (AMd, ADd, AKd) using the
minimum rank perturbation theory (MRPT) as defined in Section 6.4.4. Using the
Computed AMd: Computed ADd:
Any Two Modes Any Three Modes
Computed AKd:
Any One Mode
Figure 7.41 Problem 7.7: Exact and Computed AMd, ADd, AKd
connectivity information of the original FEM along with the information of the damage
effect on the three property matrices (shown in Figure (7.39)), it can be deduced that the rank
of the “true” AMd is two and the rank of the true AKd is one. Because of the fact that the
damage in the damping is proportional to the damage in the mass and stiffness, it can also be
deduced that the rank of the “true” ADd is three. The rank information is ascertained by
identifying which structural elements connect the damaged DOFs, and then adding up the
rank of each damaged structural element. The rank estimation could also be obtained by
performing a singular value decomposition on Bm, Bd and B^. From the rank information

153
and propositions 6.1 and 6.2, it is clear that only two modes are needed to computed AMd
using the MRPT. Likewise, only three modes are needed to computed ADd and only one
mode is needed to compute AKd. Since we are dealing with noise free eigendata, any two,
three modes, or any one mode can be used in the computation AMd, ADd or AKd,
respectively. Mesh plots of the exact and computed AMd, ADd and AKd matrices are shown
in Figures 7.41. Again, a comparison between exact and computed AMd, ADd and AKd
shows that the MRPT is able to reproduce the exact damage effect.
In this problem, as shown in this investigation, the proposed damage detection
technique is able to assess the damage exactly when three noise free
eigenvalues/eigenvectors are used. Unfortunately, in practice, the measured eigendata are
always corrupted by noise. To simulate some kind of a practical situation, random noise was
added to the eigendata. When using noisy eigendata ( first four modes), the present
algorithm was unable to locate the damage. The reason is that the damage as simulated is of
small order of magnitude which results in only a small change in the eigendata of the healthy
model. In this case, the noise totally masks the damage as reflected in the “measured” noisy
eigendata.
7.7.3 Undamped Configuration: Damage of Large Order of Magnitude
In this problem, the truss is assumed undamped. The damage is simulated by a 60%
stiffness reduction of the darkened strut and a 40% reduction of mass Mj. For the analysis, it
is assumed that only the first five modes of vibration are available. Each available mode
consist of an eigenvalue and an eigenvector with entries at all FEM DOFs. This problem is
investigated for four different cases. Each case represents a different amount of random
noise added to the eigenvectors of the available modes: 0 % , 2.5 %, 5% and 10%.
7.7.3.1 Noise Free Eigendata
The procedures used in the investigation of this case are similar to the one used in the
previous problem. The computed/exact cumulative damage location vector and the

154
cumulative vector associated with Bm and Bk are displayed in the first row of plots of Figure
7.42. Likewise, the first row of plots of Figure 7.43 show the computed/exact AMj and AKj
when using noise free data. Again, as previously shown, exact data provides exact results.
7.7.3.2 Noisy Eigendata
The effect of introducing noise into the “measured” eigenvectors is shown in the lower
three rows of plots in Figures 7.42 and 7.43. In Figure 7.42, it is obvious that as noise is
increased, the cumulative damage vector B (first column) becomes corrupted, to the point
that when 10% noise is added it is difficult to ascertain information concerning the state of
damage. In performing the decomposition of B into Bm and Bk one should only use those
modes of vibrations whose damage vector d, reflects the same “nature of damage” as the
cumulative damage vector B. For this particular example, only modes 3,4 and 5 meet this
criterion for the cases of 2.5% and 5% random noise. For the 10% noise case, the cumulative
damage vector and the individual mode damage vectors did not clearly indicate damage. To
maintain a level of consistency, the same mode set (3, 4, 5) was used for the case of 10%
noise. The decomposition of B into Bm and Bk for the case of noisy measurements is shown
in the remaining rows and columns of Figure 7.42. As shown in Figure 7.42, the added noise
has two effects in the decomposition process. The first, which is rather obvious, is that noise
causes the appearance of small numerical component at all DOFs, although in the low level
noise cases (2.5% and 5%) it is still clear which DOFs have been actually damaged. The
second more subtle effect is that the actual decomposition process is no longer exact. This
manifests itself in that the computed Bm has substantial indicators of damage at DOFs in
which there is actually only stiffness damage. The same is true for Bk, in that damage is also
indicated at DOFs in which there is only mass damage. These indicators are shown darkened
in Figure 7.42.
Figure 7.43 shows the effect of noise on the computed perturbation matrices. Mode
selection was based on (i) rank calculation of Bm and Bk, and (ii) comparison of individual
columns of Bm and Bk to their associated cumulative vector. For the cases of 2.5% and 5%

Damage Indicator Damage Indicator Damage Indicator Damage Indicator
155
Noise Free Eigendata
Computed
Q LJJ Lru^Jl M in II n-r-yj-t
0 10 20 30
2.5% Noise Added to the Eigenvectors
5% Noise Added to the Eigenvectors
10% Noise Added to the Eigenvectors
Figure 7.42 Problem7.7: Cumulative Vectors Associated With B, Bm, Bd, B^
B Computed Using Modes 1-4
Bm, Bd, Bk Computed Using modes 3, 4 & 5

156
noise, the extent calculation stills provides a reasonable indication of damage, although the
magnitude of damage is in error. The percentage errors of the estimated mass and stiffness
damage with respect to the exact damage are listed in Table 7.11. Figure 7.43 shows the
extent calculation with no enhancements.
For the case in which 10% noise was added, the damage location and extent algorithm
failed. In actual practice, the analysis would have halted after the calculation of the
individual and cumulative damage vectors. At that point, inspection of the damage vectors
would give no clear indication as to the state of damage. The decomposition of B into Bm and
Bk as well as the calculation of the perturbation matrices was just carried out for
completeness of the results.
Table 7.11 Problem 7.7: Percentage Error of Damage Estimate with
respect to Exact Damage
Percentage Error With Respect to
Exact Damage
Noise Added
Mass
Stiffness
0.0%
0.00
0.00
2.5%
6.60
9.10
5.0%
99.10
24.60
7.8 Summary
A technique that approaches the structural damage assessment problem in a decoupled
fashion was demonstrated and evaluated using numerical and actual experimental test data.
In all presented examples, the structural damage was first located by using the subspace
rotation damage location algorithm formulated in Chapter 5. With location determined, the

157
Exact / Computed AMd; Any two Modes Exact / Computed AK^ Any One Mode
Computed AMd; Modes 3 & 5
Computed AK 2.5% Noise Added to the Eigenvectors
Computed AMd; Modes 3 & 5 Computed AK* Mode 3
Computed AMd: Modes 3 & 5 Computed AK^ Mode 3
10% Noise Added to the Eigenvectors
Figure 7.43 Problem7.7: Exact and Computed AMd, ADd, AK
158
minimum rank perturbation theory (MRPT) developed in Chapter 6 is then applied to assess
the extent of the damage. As illustrated, the decomposition of the damage assessment
problem has two distinct advantages. First, each subproblem is shown to be computationally
attractive. Second, the decoupling allows engineering judgement to enter into the extent
algorithm. By making use of the results of the damage location algorithm, the modes which
do not exhibit the damage may be eliminated before the application of the MRPT extent
algorithm. In general, the subspace rotation damage location algorithm and the MRPT
performed well in assessing damage in all studied examples. The eigenvector filtering
algorithm developed in Chapter 5 was shown to be useful in improving the accuracy of the
damage extent assessment. The MRPT was also demonstrated to be applicable in model
refinement problems.

CHAPTER 8
CONCLUSION AND SUGGESTIONS FOR FUTURE WORK
This study investigated the development of four algorithms relevant to the areas of finite
element model refinement and structural damage assessment. All four algorithms make use
of an original finite element model (FEM) and a subset of experimentally measured
eigenvalues and eigenvectors. The first algorithm, termed the inverse/hybrid approach, is a
model refinement algorithm with basis in the standard framework of the inverse problem. In
the formulation of the inverse/hybrid approach, it is proposed to approximate the
unmeasured experimental modes with the corresponding analytical modal information.
With a complete hybrid set of experimental and analytical modal parameters, the symmetry
of the inaccurate finite element property matrices is enforced by mass orthogonalizing the
eigenvectors. An orthogonalization strategy that assigns more credibility to the measured
eigenvectors was discussed. Through an example, it was shown that the performance of the
inverse/hybrid approach is similar to the algorithm formulated by Baruch and Bar Itzhack
(1978).
Next, a subspace rotation technique to improve the performance of an existing model
refinement algorithm, termed SEAMRA (Zimmerman and Widengren, 1989), was
presented. The mathematical formulation of SEAMRA is based on a control concept known
as the eigenstructure assignment. The proposed improvement results in both an
enhancement of the eigenvector assignability and a substantial decrease of the computational
burden. The enhanced SEAMRA was shown to be suited for both model refinement and
structural damage assessment.
An efficient structural damage location algorithm that bypasses the general framework
of the model refinement problem was then formulated. The algorithm, also termed the
159

160
subspace rotation, is similar to the Modal Force Error Criteria formulated by several
researchers. The effects of measurement error on the performance of the subspace rotation
were discussed. A new viewpoint that reduces the effects of measurement error for certain
classes of structure was developed. Additionally, the use of multi-mode measurements as a
technique to overcome the effects of these errors is also discussed. Furthermore, an
eigenvector filtering technique was formulated.
A minimum rank perturbation theory (MRPT) was then developed. A number of
computationally attractive damage extent algorithms based on the MRPT were formulated.
The formulations of the presented MRPT based algorithms are consistent with the effects of
most structural damage on FEMs. The formulated algorithms address several damage
scenarios for undamped, proportionally damped and nonproportionally damped FEMs. A
damage scenario is defined by the effect of the structural damage on the FEM. Two
techniques to improve the performances of the MRPT based extent algorithm were
discussed. The first technique suggests the exclusion in the damage extent calculations of the
measured “damaged” modes that do not reflect the state of the damage as determined by the
subspace rotation damage location algorithm. Alternatively, the eigenvector filtering
algorithm discussed earlier was shown to be also useful in improving the damage estimate.
Finally, the subspace rotation damage location algorithm and the MRPT based damage
extent algorithms were evaluated using both computer simulated and actual experimental
data. All issues raised in the formulation of these algorithms were demonstrated. In every
problem, the damage assessment process was approached in a decoupled fashion. The
location of the damage was first determined. The extent of the damage was then assessed by
making use of the results of the damage location. The decoupled approach improves both the
accuracy and the efficiency of the extent computations. In general, all evaluated algorithms
performed very well in assessing the damage in the examples studied. The algorithms show
great promises in handling “real life” structures. This was demonstrate in the investigation of
the experimental data and, in particular, the NASA 8-bay truss examples.

161
To fully demonstrate the practicality of the MRPT based damage extent algorithms and
the subspace rotation damage location algorithm in handling “real life” structures, the issue
of incomplete eigenvector measurement must be intensively investigated. This problem was
pointed out in Chapter 2. Two alternative approaches, model reduction and eigenvector
expansion, were discussed as possible solutions to the incomplete measurement problem.
These approaches have only been investigated in the context of the model refinement
problem. An investigation of their performance vis-a-vis the damage assessment problem is
of order. Additionally, formulations of damage assessment algorithms that do not require the
use of a finite element model should be considered. This could be useful when dealing with
“old” structures that have no analytical model.

REFERENCES
Adelman, H. M. and Haftka, R. T. (1986), “Sensitivity Analysis of Discrete Structural
Systems,” AMA Journal, Vol. 24, No. 5, pp. 823-832.
Andry, A. N., Shapiro, E. Y., and Chung, J. C. (1983), “Eigenstructure Assignment For
Linear Systems,” IEEE Transactions on Aerospace and Electronic Systems, Vol.
AES-19, No. 5, pp. 711-729.
Baruch, M., (1979), “Optimum Weighted Orthogonalization of Measured Modes,” AIAA
Journal, Vol. 17, No. 1, pp. 120-121.
Baruch, M., and Bar Itzhack, I. Y. (1978), “Optimum Weighted Orthogonalization of
Measured Modes,” AIAA Journal, Vol. 16, No. 4, pp. 346-351.
Berman, A., and Nagy, E. J. (1983), “Improvements of a Large Analytical Model Using Test
Data,” AIAA Journal, Vol. 21, No. 8, pp. 1168-1173.
Brock, J. E. (1968), “Optimal Matrices Describing Linear Systems,” AMA Journal, Vol. 6,
No. 7, pp. 1292-1296.
Chen, J. C. and Garba, J. A., (1980), “Analytical Model Improvement Using Modal Test
Results,” AIAA Journal, Vol. 18, No. 6, pp. 684-690.
Chen, J. C. and Garba, J. A., (1988), “On-Orbit Damage Assessment for Large Space
Structures,” AIAA Journal, Vol. 26, No. 12, pp. 1119-1126.
Collins, J. D., Hart, G. C., Hasselman, T. K. and Kennedy, B. (1974), “Statistical
Identification of Structures,” AIAA Journal, Vol. 12, No. 2, pp. 185-190.
Creamer, N. G., and Hendricks, S. L. (1987), “Structural Parameter Identification Using
Modal Response Data,” Proceedings of the 6th VPI&SU/AIAA Symposium on
Dynamics and Controls for Large Structures, Blacksburg, VA, pp. 27-38.
Ewins, D. J. (1986), Modal Testing: Theory and Practice, Bruel & Kjaer, Letchworth,
Hertfordshire, England.
Flanigan, C. C. (1991), “Correction of Finite Element Models Using Mode Shape Design
Sensitivity,” Proceedings of the 9th International Modal Analysis Conference, Firenza,
Italy, pp. 151-159.
162

163
Freed, A. M., and Flanigan, C. C. (1991), “A Comparison of Test-Analysis Model Reduction
Methods,” Sound and Vibration, March, pp. 30-35.
Fuh, J., Chen S. and Berman A. (1984), “System Identification of Analytical Models of
Damped Structures,” Proceedings of the 25th AIAA Structures, Structural Dynamics
and Materials Conference, Palm Springs, CA, pp. 112-122.
Golub, G. H. and Van Loan, C. F. (1989), Matrix Computations, The Johns Hopkins
University Press, Baltimore, MD.
Guyan, R. J. (1965), “Reduction of Stiffness and Mass Matrices,” AIAA Journal, Vol. 3, No.
2, p. 380.
Gysin, H. (1990), “Comparison of Expansion Methods for FE Modeling Error
Localization,” Proceedings of the 8th International Modal Analysis Conference,
Kissimmee, FL, pp. 195-204.
Hajela, P. and Soeiro, F. (1990), “Recent Developments in Damage Detection Based on
System Identification Methods,” Structural Optimization, Vol. 2, pp. 1-10.
Hanagud, S., Meyyappa, M. Cheng, Y. R, and Graig, J. I. (1984), “Identification of
Structural Dynamic Systems with Nonproportional Damping,” Proceedings of the 25th
AIAA Structures, Structural Dynamics and Materials Conference, Palm Springs, CA,
pp. 283-291.
Heylen, W., and Saas, P. (1987), “Correlation of Analysis and Test in Modeling of Structures,
Assessment and Review,” Proceedings of the 5th IMAC, London, England, pp.
1177-1182.
Hughes, T. J. R. (1987), The Finite Element Method, Prentice Hall, Englewood Cliffs, NJ.
Ibrahim, S. R. and Saafan, A. A., (1987), “Correlation of Analysis and Test in Modeling of
Structures, Assessment and Review,” Proceedings of the 5th IMAC, London, England,
pp. 1651-1660.
Inman, D. J., (1989), Vibration With Control, Measurement, and Stability, Prentice Hall,
Englewood Cliffs, NJ.
Inman, D. J. and Minas, C. (1990), “Matching Analytical Models with Experimental Modal
Data in Mechanical Systems,” Control and Dynamics Systems, Vol. 37, pp. 327-363.
Juang, J.-N., and Pappa, R. S. (1985), “An Eigensystem Realization Algorithm for Modal
Parameter Identification and Model Reduction,” Journal of Guidance, Control and
Dynamics, Vol. 8, No. 5, pp. 620-627.
Kabe, A. M. (1985), “Stiffness Matrix Adjustment Using Mode Data,” AIAA Journal, Vol.
23, No. 9, pp. 1431-1436.

164
Kammer, D. C. (1987). “Test-Analysis Model Development Using an Exact Modal
Reduction,” International Journal of Analytical and Experimental Modal Analysis, Vol.
2, No. 4, pp. 175-179.
Kammer, D. C. (1988), “Optimum Approximation for Residual Stiffness in Linear System
Identification,” AIAA Journal, Vol. 26, No. 1, pp. 104-112.
Kaouk, M. and Zimmerman D. C. (1993a), “Evaluation of the Minimum Rank Update in
Damage Detection: An Experimental Study,” Proceedings of the 11th IMAC,
Kissimmee, FL, pp. 1061-1068.
Kaouk, M. and Zimmerman D. C. (1993b), “Structural Damage Assessment Using a
Generalized Minimum Rank Perturbation Theory,” Proceedings of the 34th AIAA
Structures, Structural Dynamics and Materials Conference, La Jolla, CA, pp.
1529-1539.
Kashangaki, T. A.-L. (1992), “Ground Vibration Tests of a High Fidelity Truss For
Verification of on Orbit Damage Location Techniques,” NASA Technical
Memorandum 107626.
Kashangaki, T. A.-L., Smith, S. W., and Lim, T. W. (1992), “Underlying Modal Data Issues
for Detecting Damage in Truss Structures,” Proceedings of the 33rd AIAA Structures,
Structural Dynamics and Materials Conference, Dallas, TX, pp. 1437-1446.
Kidder, R. L. (1973), “Reduction of Structural Frequency Equations,” AIAA Journal, Vol.
11, No. 6, pp. 119-126.
Lim, T. W. (1990), “Submatrix Approach to Stiffness Matrix Correction Using Modal Test
Data,” AIAA Journal, Vol. 28, No. 6, pp. 1123-1130.
Lin C. S. (1990), “Location of Modeling Errors Using Modal Test Data,” AMA Journal, Vol.
28, No. 9, pp. 1650-1654.
Martensson, K. (1971), “On the Matrix Riccati Equation,” Information Sciences, Vol. 3, pp.
17—49.
Martinez, D., Red-Horse, J. and Allen, J. (1991), “System Identification Methods for
Dynamic Structural Models of Electronic Packages,” Proceedings of the 32nd AIAA
Structures, Structural Dynamics and Materials Conference, Baltimore, MD, pp.
2336-2346.
McGowan, P. E. (1991), Dynamic Test/Analysis Correlation Using Reduced Analytical
Models, M. S. Thesis, Engineering Mechanics, Old Dominion University, Norfolk, VA.
Meirovitch, L. (1980), Computational Methods in Structural Dynamics, Sijhoff &
NoorDhoff, Alphen aan den Rijn, The Netherlands.

165
Meirovitch, L. (1986), Elements of Vibration Analysis, McGraw-Hill Book Company, New
York.
Miller, C. A. (1980), “Dynamic Reduction of Structural Model,” Journal of the Structural
Division ASCE, Vol. 106, pp. 2097-2108.
O’Callahan, J. C. (1989), “A Procedure for an Improved Reduced System (IRS) Model,”
Proceedings of the Seventh International Modal Analysis Conference, Las Vegas, NV,
pp. 17-21.
O’Callahan, J. C., Avitabile P. A. and Riemer, R. (1989), “System Equivalent Reduction
Expansion Process (SEREP),” Proceedings of the Seventh International Modal
Analysis Conference, Las Vegas, NV, pp. 29-37.
Ojalvo, I. U. and Pilón, D. (1988), “Diagnostics for Geometrically Locating Structural Math
Model Errors From Modal Test Data,”
Paz, M. (1984), “Dynamic Condensation,” AIAA Journal, Vol. 22, No. 5, pp. 724-727.
Potter, J. E. (1966), “Matrix Quadratic Solutions,” SIAM Journal of Applied Mathematics,
Vol. 14, No. 3, pp. 496-501.
Rides, J. M., (1991) “Nondestructive Structural Damage Detection in Flexible Space
Structures Using Vibration Characterization,” Report NDT-44-001-800 submitted to
NASA/JSC.
Rides, J. M. and Kosmatka, J. B. (1992), “Damage Detection in Elastic Structures Using
Vibratory Residual Forces and Weighted Sensitivity,” AIAA Journal, Vol. 30, No. 9, pp.
2310-2316.
Rodden, W. P. (1967), “A Method for Deriving Structural Influence Coefficients from
Ground Vibration Tests,” AIAA Journal, Vol. 5, No. 5, pp. 991-1000.
Smith, S. W. (1992), “Iterative Use of Direct Matrix Updates: Connectivity and
Convergence,” Proceedings of the 33rd AIAA Structures, Structural Dynamics and
Materials Conference, Dallas TX, pp. 1797-1806.
Smith, S. W., Baker, J. R., Kaouk, M. and Zimmerman, D. C. (1993),“Mode shape
Expansion for Visualization and Model Correction,” Proceedings of the 9th VPI&SU
Symposium on Dynamics and Control of Large Space Structures, Blacksburg, VA
(proceedings to be published).
Smith, S. W. and Beattie, C. A. (1990), “Simultaneous Expansion and Orthogonalization of
Measured Modes for Structure Identification,” Proceedings of the AIAA Dynamic
Specialist Conference, Long Beach, CA, pp. 261-270.

166
Smith, S. W. and Beattie, C. A. (1991), “Secant-Method Adjustment for Structural Models,”
AIAA Journal, Vol. 29, No. 1, pp. 119-126.
Smith, S. W. and Hendricks, S. L. (1987), “Evaluation of Two Identification Methods for
Damage Detection in Large Space Trusses,” Proceedings of the 6th VPI&SU/AIAA
Symposium on Dynamics, and Controls for Large Space Structures, Virginia
Polytechnic Institute and State University, Blacksburg, VA, pp. 127-142.
Soeiro, F. (1990), Structural Damage Assessment Using Identification Techniques, Ph.D.
Dissertation, Department of Aerospace Engineering, Mechanics, and Engineering
Science, University of Florida, Gainesville, FL.
Srinathkumar, S. (1978), “Eigenvalues/Eigenvectors Assignment Using Output Feedback,”
IEEE Transactions on Automatic Control, Vol. AC-23, 1, pp. 79-81.
Widengren, M. (1989), An Analytical Method for the Symmetric Correction of Mathematical
Models of Vibrating Systems Using Eigenstructure Assignment, M.S. Thesis,
Department of Mechanics, Royal Institute of Technology, Stockholm, Sweden.
White, C. W. and Maytum, B. D. (1976), “Eigensolution Sensitivity to Parametric Model
Perturbation,” Shock and Vibration Bulletin, Vol. 46, Pat 5, pp. 123-133.
Zimmerman, D. C., and Kaouk, M. (1992a), “Eigenstructure Assignment Approach for
Structural Damage Detection,” AIAA Journal, Vol. 21, No. 8, pp. 1848-1855.
Zimmerman, D. C. and Kaouk, M. (1992b), “Structural Damage Detection Using a Subspace
Rotation Algorithm,” Proceedings of the 33rd AIAA Structures, Structural Dynamics
and Materials Conference, Dallas, TX, pp. 2341-2350.
Zimmerman, D. C. and Smith, S. W., (1992), “Model Refinement and Damage Location for
Intelligent Structures,” chapter in Intelligent Structural Systems (H.S. Tzou, editor),
Kluwer Academic Publishers, Amsterdam, The Netherlands.
Zimmerman, D. C., and Widengren, W. (1989), “Equivalence Relations for Model
Correction of Nonproportionally Damped Linear Systems,” Proceedings of the Seventh
VPI&SU Symposium on the Dynamics and Control of Large Structures, Blacksburg,
VA, pp. 523-538.
Zimmerman, D.C., and Widengren, W. (1990), “Model Correction Using a Symmetric
Eigenstructure Assignment Technique,” AIAA Journal, Vol. 28, No. 9, pp. 1670-1676.

BIOGRAPHICAL SKETCH
Mohamed Kaouk received a Bachelor of Science in aerospace engineering from the
Department of Aerospace Engineering, Mechanics, and Engineering Science at the
University of Florida in December of 1988. From the same department, he then received a
Master of Science in aerospace engineering in August of 1991.
167

I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a
dissertation for the degree of Doctor of Philosophy.
David C. Smmerman, Chair
Associate Professor of Aerospace Engineering,
Mechanics, and Engineering Science
I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a
dissertation for the degree of Doctor of Philosophy.
Lawrence E. Malvern
Professor Emeritus of Aerospace Engineering,
Mechanics, and Engineering Science
I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a
dissertation for the degree of Doctor of Philosophy.
Bhavani Sankar
Associate Professor of Aerospace Engineering,
Mechanics, and Engineering Science
I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a
dissertation for the degree of Doctor of Philosophy.
-S
Norman Fitz-Coy
Assistant Professor of Aerospace Engineering,
Mechanics, and Engineering Science

I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a
dissertation for the degree of Doctor of Philosophy.
Associate Professor of Mathematics
This dissertation was submitted to the Graduate Faculty of the College of
Engineering and to the Graduate School and was accepted as partial fulfillment of the
requirements for the degree of Doctor of Philosophy.
August 1993
A
Winfred M. Phillips
Dean, College of Engineering
Dean, Graduate School

UNIVERSITY OF FLORIDA
3 1262 08554 0457




PAGE 1

6758&785$/ '$0$*( $66(660(17 $1' ),1,7( (/(0(17 02'(/ 5(),1(0(17 86,1* 0($685(' 02'$/ '$7$ %\ 02+$0(' .$28. $ ',66(57$7,21 35(6(17(' 72 7+( *5$'8$7( 6&+22/ 2) 7+( 81,9(56,7< 2) )/25,'$ ,1 3$57,$/ )8/),//0(17 2) 7+( 5(48,5(0(176 )25 7+( '(*5(( 2) '2&725 2) 3+,/2623+< 81,9(56,7< 2) )/25,'$

PAGE 2

$ 0HV 3DUHQWV 7RXW OH PULWH GH FH WUDYDLO VfLO HQ HVW YRXV UHYLHQW

PAGE 3

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
PAGE 4

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

PAGE 5

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fV 3UREOHP /RFDO WR *OREDO 0RGH &KDQJH &RQVLVWHQW 0RGHV 0RGHO 5HILQHPHQW RI D &DQWLOHYHU %HDP ([SHULPHQWDO 6WXG\ 0RGDO 7HVW 'HVFULSWLRQ )LQLWH (OHPHQW 0RGHO 'HVFULSWLRQ $SSOLFDWLRQ RI 6($05$ 'LVFXVVLRQ RI WKH 6($05$fV )RUPXODWLRQ 6XPPDU\ Y

PAGE 6

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f%HVWf 0RGHV $SSOLFDWLRQ RI WKH (LJHQYHFWRU )LOWHULQJ $OJRULWKP 6XPPDU\ YL

PAGE 7

9$/,'$7,21 $1' $66(660(17 2) 7+( 68%63$&( 527$7,21 $/*25,7+0 $1' 7+( 0,1,080 5$1. 3(5785%$7,21 7+(25< ,QWURGXFWLRQ .DEHf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

PAGE 8

(LJKW%D\ 7ZR'LPHQVLRQDO 0DVV/RDGHG &DQWLOHYHUHG 7UXVV 3UREOHP 'HVFULSWLRQ 3URSRUWLRQDOO\ 'DPSHG &RQILJXUDWLRQ 'DPDJH RI 6PDOO 2UGHU RI 0DJQLWXGH 'DPDJH /RFDWLRQ 'HFRPSRVLWLRQ RI 0DWUL[ % 'DPDJH ([WHQW 8QGDPSHG &RQILJXUDWLRQ 'DPDJH RI /DUJH 2UGHU RI 0DJQLWXGH 1RLVH )UHH (LJHQGDWD 1RLV\ (LJHQGDWD 6XPPDU\ &21&/86,21 $1' 68**(67,216 )25 )8785( :25. 5()(5(1&(6 %,2*5$3+,&$/ 6.(7&+ YLLL

PAGE 9

/,67 2) 7$%/(6 7DEOH 3DJH .DEHf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f+HDOWK\f 6WUXFWXUH %D\ 'LPHQVLRQDO 7UXVV 6XPPDU\ RI WKH 3HUFHQWDJH (UURU ZLWK 5HVSHFW WR WKH ([DFW 'DPDJH 3UREOHP 3HUFHQWDJH (UURU RI 'DPDJH (VWLPDWH ZLWK 5HVSHFW WR ([DFW 'DPDJH ,;

PAGE 10

/,67 2) ),*85(6 )LJXUH 3DJH 2YHUYLHZ RI )LQLWH (OHPHQW 0RGHO 5HILQHPHQW 2YHUYLHZ RI )(0 0RGHO 5HILQHPHQW 3URFHVV 8VHG IRU 'DPDJH $VVHVVPHQW &RPSRQHQWV RI D 9LEUDWLRQ 0HDVXUHPHQW 6\VWHP IRU 0RGDO $QDO\VLV $ VLPSOH ([SHULPHQWDO 0RGDO $QDO\VLV 6HWXS )ORZ &KDUW RI WKH ,WHUDWLYH /RDG 3UHVHUYDWLRQ 3DWK $OJRULWKP .DEHfV 3UREOHP $QDO\WLFDO 7HVW 6WUXFWXUH %HVW $FKLHYDEOH (LJHQYHFWRU 3URMHFWLRQ 5RWDWLRQ RI WKH $FKLHYDEOH 6XEVSDFH .DEHfV 3UREOHP 5HVXOWV IRU .DEHfV 3UREOHP XVLQJ WKH VW 0RGH )XOO (LJHQYHFWRU 5HVXOWV IRU .DEHfV 3UREOHP 0RGHV DQG (LJHQYHFWRUV &RPSRQHQWV 5HVXOWV IRU .DEHfV 3UREOHP XVLQJ /RDG 3DWK 3UHVHUYDWLRQ 0RGHV DQG (LJHQYHFWRUV &RPSRQHQWV 5HVXOWV IRU .DEHfV 3UREOHP XVLQJ WKH VW 0RGH )XOO (LJHQYHFWRU 5HVXOWV IRU .DEHfV 3UREOHP 0RGHV DQG (LJHQYHFWRUV &RPSRQHQWV ([SHULPHQWDO &DQWLOHYHU %HDP ([SHULPHQWDO DQG $QDO\WLFDO )UHTXHQF\ 5HVSRQVH )XQFWLRQ RI WKH &DQWLOHYHU %HDP [

PAGE 11

.DEHfV 3UREOHP .DEHfV 3UREOHP 'DPDJH /RFDWLRQ 5HVXOWV XVLQJ WKH 6XEVSDFH 5RWDWLRQ 'LUHFW 0HWKRG ZLWK WKH (LJHQGDWD RI WKH VW 0RGH .DEHfV 3UREOHP 'DPDJH /RFDWLRQ 5HVXOWV XVLQJ WKH $QJOH 3HUWXUEDWLRQ 0HWKRG ZLWK WKH (LJHQGDWD RI WKH VW 0RGH .DEHfV 3UREOHP 'DPDJH /RFDWLRQ 5HVXOWV XVLQJ /LQfV $OJRULWKP ZLWK WKH (LJHQGDWD RI WKH VW 0RGH .DEHfV 3UREOHP 'DPDJH /RFDWLRQ 5HVXOWV XVLQJ WKH $QJOH 3HUWXUEDWLRQ 0HWKRG ZLWK WKH (LJHQGDWD RI WKH VW DQG QG 0RGHV .DEHfV 3UREOHP 'DPDJH /RFDWLRQ 5HVXOWV XVLQJ /LQfV $OJRULWKP ZLWK WKH (LJHQGDWD RI WKH VW DQG QG 0RGHV .DEHfV 3UREOHP 'DPDJH ([WHQW 5HVXOWV XVLQJ WKH 0537 ZLWK WKH (LJHQGDWD RI 0RGH .DEHfV 3UREOHP 'DPDJH ([WHQW 5HVXOWV XVLQJ %DUXFKfV 0HWKRG )LIW\%D\ 7ZR'LPHQVLRQDO 7UXVV )LIW\%D\ 7UXVV 'DPDJH /RFDWLRQ 5HVXOWV XVLQJ WKH 6XEVSDFH 5RWDWLRQ $OJRULWKP ZLWK WKH (LJHQGDWD RI WKH )LUVW 7HQ 0RGHV )LIW\%D\ 7UXVV 'DPDJH ([WHQW 5HVXOWV XVLQJ WKH 0537 ZLWK WKH (LJHQGDWD RI 0RGHV DQG )LIW\%D\ 7UXVV 'DPDJH ([WHQW 5HVXOWV XVLQJ WKH 0537 ZLWK WKH (LJHQGDWD RI WKH )LUVW 7HQ 0RGHV )LIW\%D\ 7UXVV 'DPDJH ([WHQW 5HVXOWV XVLQJ %DUXFKfV $OJRULWKP 7KH 1$6$ (LJKW%D\ +\EULG6FDOHG 7UXVV 'DPDJH &DVHV 7KH 1$6$ %D\ 7UXVV /DFLQJ 3DWWHUQ 1$6$ %D\ 7UXVV 7\SLFDO )UHTXHQF\ 5HVSRQVH &RPSDULVRQ 1$6$ %D\ 7UXVV 3HUWXUEDWLRQ WR WKH 2ULJLQDO 6WLIIQHVV 0DWUL[ WKDW 5HVXOWHG )URP WKH 5HILQHPHQW 3URFHVV 1$6$ %D\ 7UXVV &XPXODWLYH 'DPDJH 9HFWRU $VVRFLDWHG ZLWK &DVH ) 1$6$ %D\ 7UXVV 'DPDJH $VVHVVPHQW RI &DVH $ 1$6$ %D\ 7UXVV 'DPDJH $VVHVVPHQW RI &DVH & [L

PAGE 12

1$6$ %D\ 7UXVV 'DPDJH $VVHVVPHQW RI &DVH 1$6$ %D\ 7UXVV 'DPDJH $VVHVVPHQW RI &DVH ( 1$6$ %D\ 7UXVV 'DPDJH $VVHVVPHQW RI &DVH 1$6$ %D\ 7UXVV 'DPDJH $VVHVVPHQW RI &DVH + 1$6$ %D\ 7UXVV 'DPDJH $VVHVVPHQW RI &DVH 1$6$ %D\ 7UXVV 'DPDJH $VVHVVPHQW RI &DVH 1$6$ %D\ 7UXVV 'DPDJH $VVHVVPHQW RI &DVH 1$6$ %D\ 7UXVV 'DPDJH $VVHVVPHQW RI &DVH / 1$6$ %D\ 7UXVV 'DPDJH $VVHVVPHQW RI &DVH 0 1$6$ %D\ 7UXVV 'DPDJH $VVHVVPHQW RI &DVH 1 1$6$ %D\ 7UXVV 'DPDJH $VVHVVPHQW RI &DVH 2 1$6$ %D\ 7UXVV 'DPDJH $VVHVVPHQW RI &DVH 3 7KH 0DVV /RDGHG &DQWLOHYHUHG %HDP 0DVV /RDGHG &DQWLOHYHUHG %HDP 'DPDJH $VVHVVPHQW %D\ 'LPHQVLRQDO 7UXVV %D\ 'LPHQVLRQDO 7UXVV 'DPDJH /RFDWLRQ %D\ 'LPHQVLRQDO 7UXVV 'DPDJH ([WHQW 7KH (LJKW%D\ 7ZR'LPHQVLRQDO 0DVV/RDGHG &DQWLOHYHUHG 7UXVV 3UREOHP &XPXODWLYH 'DPDJH /RFDWLRQ 9HFWRU )LUVW )RXU 0RGHV 3UREOHP &XPXODWLYH 9HFWRUV $VVRFLDWHG ZLWK WKH ([DFW DQG &RPSXWHG %P %G %A )LUVW 7KUHH 0RGHV 3UREOHP ([DFW DQG &RPSXWHG $0M $'M $. 3UREOHP &XPXODWLYH 9HFWRU $VVRFLDWHG ZLWK % %P %M %A % &RPSXWHG XVLQJ 0RGHV %P %M %N &RPSXWHG XVLQJ 0RGHV t 3UREOHP ([DFW DQG &RPSXWHG $0G $'G $.G [LL

PAGE 13

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f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

PAGE 14

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f LV GHYHORSHG 7KH IRUPXODWLRQ RI WKH 0537 LV FRQVLVWHQW ZLWK WKH HIIHFW RI PRVW VWUXFWXUDO GDPDJH RQ )(0V 7KH FKDUDFWHULVWLFV RI WKH VXEVSDFH URWDWLRQ DOJRULWKP DQG WKH PLQLPXP UDQN SHUWXUEDWLRQ WKHRU\ DUH LOOXVWUDWHG XVLQJ VLPXODWHG DQG H[SHULPHQWDO WHVWEHGV 7KH GHFRPSRVLWLRQ RI WKH GDPDJH DVVHVVPHQW SUREOHP LQWR ORFDWLRQ DQG H[WHQW VXESUREOHPV LV VKRZQ WR EH DGYDQWDJHRXV IURP ERWK IRU FRPSXWDWLRQDO HIILFLHQF\ DQG IRU HQJLQHHULQJ LQVLJKW [LY 7

PAGE 15

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f FDQ EH LPSURYHG WR VRPH H[WHQW E\ LQFUHDVLQJ WKH QXPEHU RI GHJUHHV RI IUHHGRP '2)Vf LQFOXGHG LQ WKH PRGHO +RZHYHU DFFXUDF\ RI WKH )(0 PD\ VWLOO EH ODFNLQJ GXH WR XQFHUWDLQWLHV LQ PDWHULDO SURSHUWLHV JHRPHWULF OD\RXW DQG IDEULFDWLRQ LQGXFHG HUURUV 7KXV LW LV HVVHQWLDO WR fYDOLGDWHf WKHVH )(0V SULRU WR WKHLU DFFHSWDQFH DV D EDVLV IRU DQDO\VLV 2QH ZD\ WR YDOLGDWH WKHP LV WR FRPSDUH WKHLU PRGDO SURSHUWLHV HLJHQYDOXHV HLJHQYHFWRUVf ZLWK WKH PHDVXUHG PRGDO SDUDPHWHUV RI WKH DFWXDO VWUXFWXUH DV REWDLQHG IURP H[SHULPHQWDO PRGDO DQDO\VLV (0$f $ )(0 LV DFFHSWDEOH ZKHQ WKHVH WZR VHWV RI PRGDO SURSHUWLHV DUH LQ DJUHHPHQW 8QIRUWXQDWHO\ WKLV DJUHHPHQW UDUHO\ RFFXUV $V D UHVXOW WKH

PAGE 16

)(0 PXVW EH DGMXVWHG WR SURGXFH D FRUUHODWLRQ RI DQDO\WLFDO DQG H[SHULPHQWDO PRGDO SURSHUWLHV $Q RYHUYLHZ RI WKLV SURFHGXUH LV VFKHPDWLFDOO\ LOOXVWUDWHG LQ )LJXUH =LPPHUPDQ DQG 6PLWK f ([SHULPHQWDO )LJXUH 2YHUYLHZ RI )LQLWH (OHPHQW 0RGHO 5HILQHPHQW =LPPHUPDQ DQG 6PLWK f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

PAGE 17

FRPPRQO\ UHIHUUHG WR LQ WKH OLWHUDWXUH DV )(0 UHILQHPHQW )(0 DGMXVWPHQW )(0 FRUUHFWLRQ )(0 FRUUHODWLRQ DQG )(0 LGHQWLILFDWLRQ /LWHUDWXUH 6XUYH\ &RPSUHKHQVLYH OLWHUDWXUH VXUYH\V FRYHULQJ D ODUJH SRUWLRQ RI WKH ZRUN WKDW DGGUHVVHV WKH PRGHO UHILQHPHQW SUREOHP FDQ EH IRXQG LQ WKH ERRN FKDSWHU E\ =LPPHUPDQ DQG 6PLWK f DQG LQ WKH SDSHUV E\ ,EUDKLP DQG 6DDIDQ f DQG +H\OHQ DQG 6DDV f )RU FRPSOHWHQHVV DQG WR SURSHUO\ XQGHUOLQH WKH REMHFWLYH RI WKH FXUUHQW VWXG\ LW LV DSSURSULDWH WR LQFOXGH D EULHI VXUYH\ RI WKH GHYHORSPHQW PDGH LQ WKLV DUHD 7KH FRQFHSW RI XVLQJ H[SHULPHQWDO PRGDO GDWD LQ DQDO\WLFDO VWXGLHV ZDV LQLWLDWHG E\ DQ HDUO\ ZRUN SUHVHQWHG E\ 5RGGHQ f ,Q KLV ZRUN 5RGGHQ H[SORUHG WKH SRVVLELOLWLHV RI JHQHUDWLQJ PDVV DQG VWLIIQHVV SURSHUW\ PDWULFHV E\ XVLQJ H[SHULPHQWDOO\ PHDVXUHG PRGDO GDWD 7KH H[SHULPHQWDOO\ JHQHUDWHG PDVV DQG VWLIIQHVV PDWULFHV ZHUH QRQV\PPHWULF %URFN f LPSURYHG WKH ZRUN RI 5RGGHQ E\ SURSRVLQJ D VWUDWHJ\ WR LQVXUH V\PPHWU\ RI SURSHUW\ PDWULFHV 7KH HVVHQFH RI WKH PRGHO UHILQHPHQW FRQFHSW DV DGRSWHG E\ PRVW UHVHDUFKHUV LV WR PRGLI\ WKH ILQLWH HOHPHQW PRGHO )(0f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

PAGE 18

7KH EDVLF SKLORVRSK\ RI WKH RSWLPDOPDWUL[ XSGDWH LV WR PLQLPL]H WKH FRUUHFWLRQ WR WKH )(0 SURSHUW\ PDWULFHV WR DFFRPSOLVK WKH DQDO\WLFDOWHVW PRGDO FRUUHODWLRQ 7KH SLRQHHULQJ ZRUN LQ WKLV DUHD FDQ EH FUHGLWHG WR %DUXFK DQG %DU ,W]KDFN f ,Q WKHLU IRUPXODWLRQ E\ DVVXPLQJ WKDW WKH PDVV PDWUL[ LV FRUUHFW WKH UHILQHPHQW RI WKH VWLIIQHVV PDWUL[ RI DQ XQGDPSHG )(0 LV FDVW DV D FRQVWUDLQHG PLQLPL]DWLRQ SUREOHP 7KH REMHFWLYH RI WKHLU IRUPXODWLRQ LV WR GHWHUPLQH WKH PLQLPDO )UREHQLXVQRUP V\PPHWULF VWLIIQHVV DGMXVWPHQW WKDW VDWLVILHV WKH HLJHQYDOXH SUREOHP LQ WHUPV RI WKH H[SHULPHQWDOO\ PHDVXUHG HLJHQYDOXHV DQG HLJHQYHFWRUV $ FRPSXWDWLRQDOO\ HIILFLHQW FORVHG IRUP VROXWLRQ ZDV GHYHORSHG IRU WKH XSGDWHG VWLIIQHVV PDWUL[ %HUPDQ DQG 1DJ\ f H[WHQGHG WKH %DUXFK DQG %DU ,W]KDFN DSSURDFK WR WKH UHILQHPHQW RI ERWK WKH PDVV DQG VWLIIQHVV PDWULFHV 7KLV VDPH PHWKRGRORJ\ ZDV IXUWKHU H[WHQGHG E\ )XK HW DO f WR XSGDWH WKH PDVV GDPSLQJ DQG VWLIIQHVV PDWULFHV RI GDPSHG )(0V )XK DQG KLV FROOHDJXHV XVHG FURVVRUWKRJRQDOLW\ UHODWLRQVKLSV WR FRUUHFW WKH PDVV DQG GDPSLQJ PDWULFHV DQG D FRQVWUDLQHG PLQLPL]DWLRQ SUREOHP VLPLODU WR WKH RQH SURSRVHG E\ %DUXFK DQG %DU ,W]KDFN WR GHWHUPLQH WKH UHILQHG VWLIIQHVV PDWUL[ 7KH SUREOHP RI DGMXVWLQJ WKH PDVV GDPSLQJ DQG VWLIIQHVV PDWULFHV ZDV DOVR DWWHPSWHG E\ +DQDJXG HW DO f ,Q WKHLU DSSURDFK DOO WKUHH SURSHUW\ PDWULFHV RI D QRQSURSRUWLRQDOO\ GDPSHG )(0 DUH LQFRUSRUDWHG LQ WKH FRQVWUDLQHG PLQLPL]DWLRQ SUREOHP 7KH SUHYLRXVO\ VWDWHG DOJRULWKPV GR QRW SUHVHUYH WKH ORDG SDWK VSDUVLW\ SDWWHUQf RI WKH RULJLQDO DQDO\WLFDO SURSHUW\ PDWULFHV .DEH f SURSRVHG D UHIRUPXODWLRQ RI WKH %DUXFK DQG %DU ,W]KDFN DOJRULWKP WKDW FRQVWUDLQV WKH XSGDWHG VWLIIQHVV PDWUL[ WR SUHVHUYH WKH ORDG SDWK RI WKH RULJLQDO VWLIIQHVV PDWUL[ ,Q DGGLWLRQ LQ KLV UHIRUPXODWLRQ KH DGRSWHG D SHUFHQW ZHLJKWLQJ RQ WKH VWLIIQHVV FKDQJHV LQVWHDG RI PDVV PDWUL[ ZHLJKWLQJ DV XVHG LQ WKH %DUXFK DQG %DU ,W]KDFN DOJRULWKP .DPPHU f SURSRVHG DQ DOWHUQDWLYH VROXWLRQ DSSURDFK WR WKH SUREOHP GHILQHG E\ .DEH WKDW XVHV SURMHFWLRQ PDWUL[ WKHRU\ DQG WKH 0RRUH3HQURVH JHQHUDOL]HG LQYHUVH 7KH DOJRULWKPV IRUPXODWHG E\ .DEH DQG .DPPHU LQYROYH DQ H[FHVVLYH DPRXQW RI FRPSXWDWLRQDO HIIRUW $QRWKHU DOWHUQDWLYH DQG PRUH HIILFLHQW IRUPXODWLRQ RI WKH

PAGE 19

.DEH SUREOHP WKDW XWLOL]HG D JHQHUDOL]DWLRQ RI 0DUZLOO7RLQW XSGDWHV ZDV GHYHORSHG E\ 6PLWK DQG %HDWWLH f 7KH RWKHU VXEFODVV RI SURSHUW\ PDWUL[ XSGDWHV LV EDVHG RQ WKH JHQHUDO IUDPHZRUN RI FRQWUROEDVHG HLJHQVWUXFWXUH DVVLJQPHQW DOJRULWKPV 7KH HVVHQFH RI WKLV DSSURDFK LV WR GHWHUPLQH SVHXGRFRQWUROOHUV WKDW ZRXOG DVVLJQ WKH H[SHULPHQWDOO\ PHDVXUHG PRGDO SDUDPHWHUV WR WKH RULJLQDO DQDO\WLFDO )(0 7KH SVHXGRFRQWUROOHUV DUH WKHQ WUDQVODWHG LQWR PDWUL[ DGMXVWPHQW WR WKH RULJLQDO SURSHUW\ PDWULFHV 7KLV DSSURDFK ZDV LQLWLDOO\ IRUPXODWHG E\ ,QPDQ DQG 0LQDV f WR DGMXVW WKH GDPSLQJ DQG VWLIIQHVV PDWULFHV RI WKH RULJLQDO )(0 ,Q WKHLU IRUPXODWLRQ WKH V\PPHWU\ RI WKH DGMXVWHG SURSHUW\ PDWULFHV ZDV HQIRUFHG WKURXJK DQ LWHUDWLYH SURFHVV WKDW LQYROYHV D QXPHULFDO QRQOLQHDU RSWLPL]DWLRQ SURFHVV =LPPHUPDQ DQG :LGHQJUHQ f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f &KHQ DQG *DUED f $GHOPDQ DQG +DIWND f &UHDPHU DQG +HQGULFNV f )ODQLJDQ f 0DUWLQH] HW DO f WR QDPH RQO\ D IHZ ,Q WKHVH VWXGLHV GLIIHUHQW VHQVLWLYLW\ IRUPXODWLRQV DQG LWHUDWLYH RSWLPL]DWLRQ VWUDWHJLHV DUH SURSRVHG $ GLUHFW DSSURDFK WKDW

PAGE 20

E\SDVVHV WKH XVH RI FORVHGIRUP VHQVLWLYLW\ GHULYDWLYHV ZDV SUHVHQWHG LQ WKH SDSHUV RI +DMHOD DQG 6RHLUR f DQG 6RHLUR f :KLWH DQG 0D\ WXUQ f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f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

PAGE 21

.RVPDWND =LPPHUPDQ DQG .DRXN DE .DRXN DQG =LPPHUPDQ DEf 7KHVH WHFKQLTXHV XWLOL]H WKH YLEUDWLRQ VLJQDWXUH RI WKH SUH DQG SRVW GDPDJHG VWUXFWXUH LQ FRQMXQFWLRQ ZLWK DQ DQDO\WLFDO PRGHO RI WKH RULJLQDO VWUXFWXUH WR DVVHVV ERWK WKH ORFDWLRQ DQG H[WHQW RI WKH VWUXFWXUDO GDPDJH 7KH SUHGDPDJH PRGDO SDUDPHWHUV DUH XVHG WR FRUUHFW UHILQHf WKH RULJLQDO ILQLWH HOHPHQW PRGHO )(0f WR GHWHUPLQH DQ fDFFXUDWHf UHIHUHQFH EDVHOLQH 2QFH GDPDJHG WKH SRVWGDPDJH PRGDO SURSHUWLHV RI WKH VWUXFWXUH DUH XVHG WR IXUWKHU fUHILQHf WKH UHILQHG DQDO\WLFDO PRGHO 7KLV UHVXOWV LQ SHUWXUEDWLRQV WR WKH UHILQHG DQDO\WLFDO PRGHO $QDO\VLV RI WKH SHUWXUEDWLRQV FRXOG LQGLFDWH WKH GDPDJH ORFDWLRQ DQG H[WHQW $Q RYHUYLHZ RI WKH DSSOLFDWLRQ RI PRGHO UHILQHPHQW DOJRULWKPV LQ DVVHVVLQJ VWUXFWXUDO GDPDJH LV VXPPDUL]HG LQ )LJXUH )LJXUH 2YHUYLHZ RI )LQLWH (OHPHQW 0RGHO 5HILQHPHQW 3URFHVV 8VHG IRU 'DPDJH $VVHVVPHQW

PAGE 22

1RWDEOH H[FHSWLRQV WR WKH GLUHFW XVH RI )(0 UHILQHPHQW DOJRULWKPV WR WKH GDPDJH GHWHFWLRQ SUREOHP DUH WKH ZRUN RI /LQ f 2MDOYR DQG 3LOQ f DQG *\VLQ f ,Q WKH ZRUN RI /LQ D IOH[LELOLW\ PDWUL[ LV GHWHUPLQHG XVLQJ H[SHULPHQWDO GDWD 7KLV PDWUL[ LV WKHQ PXOWLSOLHG E\ WKH RULJLQDO VWLIIQHVV PDWUL[ ZLWK WKRVH URZV DQGRU FROXPQV WKDW GLIIHU VLJQLILFDQWO\ IURP D URZ DQGRU FROXPQ RI WKH LGHQWLW\ PDWUL[ LQGLFDWLQJ ZKLFK GHJUHHV RI IUHHGRP KDYH EHHQ PRVW DIIHFWHG E\ WKH GDPDJH ,W LV WKHQ DVVXPHG WKDW GDPDJH KDV RFFXUUHG LQ VWUXFWXUDO HOHPHQWV FRQQHFWLQJ WKRVH GHJUHHV RI IUHHGRP $Q RYHUYLHZ RI WKH ZRUN E\ 2MDOYR DQG 3LOQ f DQG *\VLQ f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fUHDO OLIHf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

PAGE 23

PHDVXUHPHQWV LV WKHQ LQYHVWLJDWHG DQG DOUHDG\ H[LVWLQJ WHFKQLTXHV WR GHDO ZLWK WKLV SUREOHP DUH SUHVHQWHG )XUWKHUPRUH HLJHQYHFWRU RUWKRJRQDOL]DWLRQ WHFKQLTXHV XVHIXO WR D ODUJH QXPEHU RI PRGHO UHILQHPHQW DOJRULWKPV DUH UHSRUWHG )LQDOO\ DQ LWHUDWLYH DSSURDFK WR WKH SUREOHP RI ORDG SDWK SUHVHUYDWLRQ HQFRXQWHUHG LQ D ODUJH QXPEHU RI SURSHUW\ PDWUL[ XSGDWH DOJRULWKPV LV GLVFXVVHG ,Q FKDSWHU WKH DXWKRU SURSRVHV DQG IRUPXODWHV D QHZ PRGHO UHILQHPHQW DOJRULWKP 7KH QHZO\ GHYHORSHG DOJRULWKP QDPHG WKH LQYHUVHK\EULG PHWKRG LV QDWXUDOO\ EDVHG RQ WKH LQYHUVH SUREOHP 7KH PRGHO W\SH XQGHU FRQVLGHUDWLRQ LV SURSRUWLRQDOO\ GDPSHG DQG WKH LQDFFXUDFLHV RI WKH DQDO\WLFDO PRGHO LV DVVXPHG WR EH GXH WR HUURUV LQ WKH PRGHOOLQJ RI WKH GDPSLQJ DQG VWLIIQHVV SURSHUWLHV $ FRPSOHWH K\EULG VHW RI PRGDO GDWD LV JHQHUDWHG E\ DSSUR[LPDWLQJ WKH XQPHDVXUHG PRGDO LQIRUPDWLRQ E\ WKH FRUUHVSRQGLQJ DQDO\WLFDO PRGHV 6\PPHWU\ RI WKH FRUUHFWHG VWLIIQHVV DQG GDPSLQJ PDWULFHV LV HQIRUFHG E\ PDVV RUWKRJRQDOL]DWLRQ WKH FRPSOHWH K\EULG HLJHQYHFWRU PDWUL[ $Q RUWKRJRQDOL]DWLRQ SURFHGXUH WKDW DVVLJQV PRUH FRQILGHQFH RQ WKH PHDVXUHG HLJHQYHFWRUV LV SURSRVHG $ FRPSDUDWLYH VWXG\ RI WKH LQYHUVHK\EULG PHWKRG DQG WKH DOJRULWKP SURSRVHG E\ %DUXFK DQG %DU ,W]KDFN f VKRZV WKDW ERWK DOJRULWKPV JLYH VLPLODU UHVXOWV +RZHYHU LW LV VKRZQ WKDW WKH LQYHUVHK\EULG DSSURDFK LV OHVV FRPSXWDWLRQDOO\ HIILFLHQW &KDSWHU SURSRVHV LPSURYHPHQWV WR WKH V\PPHWULF HLJHQVWUXFWXUH DVVLJQPHQW PRGHO UHILQHPHQW DOJRULWKP 6($05$f IRUPXODWHG E\ =LPPHUPDQ DQG :LGHQJUHQ f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

PAGE 24

,Q &KDSWHU DQ HIILFLHQW GDPDJH ORFDWLRQ DOJRULWKP WKDW XWLOL]HV PRGDO GDWD LQIRUPDWLRQ EXW E\SDVVHV WKH JHQHUDO IUDPHZRUN RI WKH PRGHO UHILQHPHQW SUREOHP LV SUHVHQWHG 7KLV ORFDWLRQ DOJRULWKP LV DQ RXWJURZWK RI WKH VXEVSDFH URWDWLRQ PHWKRG XVHG WR HQKDQFH HLJHQYHFWRU DVVLJQDELOLW\ RI WKH 6($05$ 7KH SURSRVHG ORFDWLRQ DOJRULWKP LV VLPLODU WR WKH PRGDO IRUFH HUURU FULWHULD SUHVHQWHG E\ VHYHUDO UHVHDUFKHUV 2MDOYR DQG 3LOQ *\VLQ f )XUWKHU LQWHUSUHWDWLRQ RI WKH DOJRULWKP RSHUDWLRQ LV JLYHQ $GGLWLRQDOO\ WKH DXWKRU SURSRVHV DQG IRUPXODWHV D QHZ YLHZSRLQW WKDW UHGXFHV WKH HIIHFW RI PHDVXUHPHQW QRLVH IRU FHUWDLQ W\SHV RI VWUXFWXUHV %DVHG RQ WKLV IRUPXODWLRQ DQ HLJHQYHFWRU ILOWHULQJ DOJRULWKP LV DOVR GHYHORSHG &KDSWHU SUHVHQWV WKH IRUPXODWLRQ RI FRPSXWDWLRQDOO\ DWWUDFWLYH GDPDJH H[WHQW DOJRULWKPV WKDW DUH EDVHG RQ D PLQLPXP UDQN SHUWXUEDWLRQ WKHRU\ 0537f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

PAGE 25

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

PAGE 26

7KLV FRQFHSW LV NQRZQ DV VSDWLDO GLVFUHWL]DWLRQ ZKLFK HOLPLQDWHV WKH FRQWLQXRXV VSDWLDO GHSHQGHQFH RI WKH GLVWULEXWHG SURSHUWLHV 7KH GLVFUHWH PRGHO LV RI ILQLWH RUGHU DQG LV GHVFULEHG E\ D ILQLWH QXPEHU RI YDULDEOHV NQRZQ DV GHJUHHV RI IUHHGRP '2)Vf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f 0RUH HODERUDWH GLVFXVVLRQV RI WKH FRQFHSWV RI FRQWLQXRXV DQG GLVFUHWH PRGHOV DV ZHOO DV ILQLWH HOHPHQW PHWKRGV DUH FRYHUHG LQ GHWDLO LQ WKH ERRNV RI 0HLURYLWFK f ,QPDQ f DQG +XJKHV f &RPPRQO\ WKH IUHH YLEUDWLRQ PRWLRQ RI D VWUXFWXUH LQ WHUPV RI DQ QWK RUGHU GLVFUHWH PRGHO LV UHSUHVHQWHG E\ WKH IROORZLQJ VHW RI VLPXOWDQHRXV RUGLQDU\ GLIIHUHQWLDO HTXDWLRQV 0ZWf 'ZWf .ZWf f ZKHUH 0 DQG DUH WHUPHG UHVSHFWLYHO\ WKH PDVV GDPSLQJ DQG VWLIIQHVV PDWULFHV 7KH\ DUH PRGHOV RI WKH PDVV GDPSLQJ DQG VWLIIQHVV SURSHUWLHV RI WKH VWUXFWXUH 6LQFH WKH V\VWHP PRGHO LV RUGHU Q WKHVH PDWULFHV DUH RI GLPHQVLRQ Q[Q DQG DUH JHQHUDOO\ UHDO 7KH YDULDEOH ZWf UHSUHVHQWV WKH Q GLVSODFHPHQWV RI WKH Q'2) PRGHO RI WKH VWUXFWXUH 7KH RYHUGRWV UHSUHVHQW GLIIHUHQWLDWLRQ ZLWK UHVSHFW WR WLPH 7KH PDVV PDWUL[ 0 LV DOZD\V V\PPHWULF SRVLWLYH GHILQLWH 7KH VWLIIQHVV .f DQG GDPSLQJ 'f PDWULFHV RI QRQJ\URVFRSLF DQG QRQFLUFXODWRU\ V\VWHPV DUH V\PPHWULF ,Q JHQHUDO WKH PRGHOOLQJ RI WKH PDVV DQG VWLIIQHVV SURSHUWLHV RI WKH VWUXFWXUH LV VLPSOHU DQG PRUH DFFXUDWH WKDQ WKH PRGHOOLQJ RI WKH GDPSLQJ SURSHUWLHV ,Q WKH IRUWKFRPLQJ GLVFXVVLRQ LW LV DVVXPHG WKDW WKH V\VWHP XQGHU FRQVLGHUDWLRQ LV QRQJ\URVFRSLF DQG QRQFLUFXODWRU\ 7KH VWDQGDUG VROXWLRQ WR (T f LV RI WKH IRUP ZWf YH;W f

PAGE 27

ZKHUH Y LV D FRQVWDQW Q[O YHFWRU DQG ; LV D FRQVWDQW VFDODU 6XEVWLWXWLQJ (T f LQWR (T f DQG GLYLGLQJ WKH UHVXOWDQW HTXDWLRQ E\ HA \LHOGV WKH FRQGLWLRQ 0\ ; 'Y ; .Y f 7KHUH DUH Q VHWV RI QRQWULYLDO FRPSOH[ FRQMXJDWH VROXWLRQV ;A Ycf WR (T f 1RWH WKDW VLQFH WKH SURSHUW\ PDWULFHV 0 .f DUH UHDO LI ;A Ycf LV D VROXWLRQ VHW WR (T f WKH FRPSOH[ FRQMXJDWH RI WKDW VHW LV DOVR D VROXWLRQ 7KH SUREOHP RI VROYLQJ IRU WKHVH VROXWLRQV LV FRPPRQO\ NQRZQ DV WKH HLJHQYDOXH SUREOHP DQG LV VRPHWLPHV UHIHUUHG WR DV WKH FKDUDFWHULVWLF YDOXH SUREOHP 7KH VFDODU ;c DQG WKH YHFWRU Yc DUH NQRZQ UHVSHFWLYHO\ DV WKH HLJHQYDOXH DQG HLJHQYHFWRU RI WKH LWK PRGH RI YLEUDWLRQ RI WKH VWUXFWXUH )RU D JHQHUDO GDPSHG V\VWHP HLJHQYDOXHV DQG HLJHQYHFWRUV DUH ERWK FRPSOH[ 1RWH WKDW (T f FDQ EH HDVLO\ UHDUUDQJHG LQ WKH PRUH JHQHUDO PDWKHPDWLFDO HLJHQYDOXH SUREOHP IRUPDW DV 0an. ,Q[Q 0fn' f ZKHUH ,Q[Q LV WKH Q[Q LGHQWLW\ PDWUL[ (TXDWLRQ f LV FDOOHG D VWDWH VSDFH UHSUHVHQWDWLRQ RI (T f 7KH HLJHQYDOXH DQG HLJHQYHFWRU FDQ EH UHODWHG WR VRPH SK\VLFDO FKDUDFWHULVWLF SURSHUWLHV RI VWUXFWXUHV )RU WKDW PDWWHU WKH LWK HLJHQYDOXH LV ZULWWHQ DV ? &M:c M fc ^f ZKHUH M f§ 1RWH WKDW LQ ZULWLQJ WKLV HTXDWLRQ LW LV DVVXPHG WKDW WKH V\VWHP LV XQGHUGDPSHG 7KH UHDO VFDODU fc DQG DUH WKH QDWXUDO IUHTXHQF\ DQG GDPSLQJ UDWLR RU GDPSLQJ IDFWRUf UHVSHFWLYHO\ RI WKH LWK PRGH RI WKH VWUXFWXUH 7KH HLJHQYHFWRU Yc LQGLFDWHV WKH fVKDSHf RI YLEUDWLRQ RI WKH LWK PRGH RI WKH VWUXFWXUH 7KH VHWV RI IUHTXHQFLHV GDPSLQJ UDWLRV DQG PRGH VKDSHV DUH VRPHWLPHV UHIHUUHG WR DV PRGDO SDUDPHWHUV 7KH V\PPHWULF QDWXUH RI WKH SURSHUW\ PDWULFHV 0 .f FRQVWUDLQV WKH HLJHQYHFWRUV WR VDWLVI\ VRPH FURVVRUWKRJRQDOLW\ UHODWLRQVKLSV )RU WKH SXUSRVH RI GLVFXVVLQJ WKHVH

PAGE 28

FURVVRUWKRJRQDOLW\ UHODWLRQVKLSV FRQVLGHU WKH IROORZLQJ DOWHUQDWLYH VWDWH VSDFH UHSUHVHQWDWLRQ RI WKH HLJHQYDOXH SUREOHP LQ (T f >0 RO 9$ 9$ >D RO >' 9$ 9$ UrL f§ 9 9 R 9 9 ZKHUH 9 > YS Y YQ @ $ GLDJI;M ; ;Qf ZKHUH WKH RYHUEDU GHQRWHV WKH PDWUL[ FRPSOH[ FRQMXJDWH RSHUDWRU %DVHG RQ WKLV SDUWLFXODU UHSUHVHQWDWLRQ DQG WKH IDFW WKDW WKH VWDWH PDWULFHV RI (T f DUH V\PPHWULF E\ SURSHU QRUPDOL]DWLRQ RI WKH HLJHQYHFWRUV WKH FURVVRUWKRJRQDOLW\ UHODWLRQVKLS DVVRFLDWHG ZLWK WKH V\VWHP DUH JLYHQ E\ Q 7B 0 9$ 9$ 9 9 9$ 9$ 9 9 Q[Q f 7 9$ 9$ 9$ 9$ $ 9 9 9 Y $ f (TXDWLRQV f DQG f UHVSHFWLYHO\ FOHDUO\ LPSO\ WKH IROORZLQJ UHODWLRQV r 9$n 0 Rn 9$ nRn 9 9 r n9$n .n n9$n nRn 9 9 f ZKHUH > @ GHQRWHV WKH FRPSOH[ FRQMXJDWH WUDQVSRVH RSHUDWRU > @ A f 1RWH WKDW FRQWUDU\ WR (TV f QR SDUWLFXODU QRUPDOL]DWLRQ RI WKH HLJHQYHFWRU LV QHHGHG LQ RUGHU IRU (TV f WR EH VDWLVILHG $QRWKHU VWDWH VSDFH UHSUHVHQWDWLRQ RI (T f LV JLYHQ E\

PAGE 29

2 0 0 9$ 9$ 9 9 $ 2 2 $ 2 9$ 9$ 9 9 f %DVHG RQ WKH VDPH DUJXPHQW GLVFXVVHG HDUOLHU WKH FURVVRUWKRJRQDOLW\ FRQGLWLRQV WKDW DULVH IURP WKLV UHSUHVHQWDWLRQ DUH 7 9$ 9$ 9 9 0 0 9$ 9$ 9 9 AQ[Q U 7 B 9$ 9$ 0 9$ 9$ $ 9 9 9 9 $ $JDLQ WKH IROORZLQJ WZR UHODWLRQVKLSV IROORZ r 9$ nR Pn n9$n nRn 9 0 9 r n9$r n 0 Rn 9$ nRn 9 9 f f f f 8QGDPSHG 0RGHOV ,Q WKH PRGHOOLQJ RI VWUXFWXUHV LW LV RIWHQ DVVXPHG WKDW WKH GDPSLQJ LV QHJOLJLEOH DQG KHQFH LV VHW WR ]HUR )RU WKHVH W\SH RI V\VWHP PRGHOV WKH HLJHQYDOXH SUREOHP LQ PDWUL[ IRUP LV JLYHQ E\ 09 $ .9 >@ f 7KH PDWULFHV 9 DQG $ DUH DV GHILQHG HDUOLHU )RU XQGDPSHG V\VWHPV WKH HLJHQYDOXHV DUH SXUH LPDJLQDU\ DQG WKH HLJHQYHFWRUV Yc DUH UHDO 1RWH WKDW WKH HLJHQYDOXHV DUH UHODWHG WR WKH V\VWHP QDWXUDO IUHTXHQF\ E\ ; FR f )XUWKHUPRUH E\ SURSHU QRUPDOL]DWLRQ RI WKH HLJHQYHFWRUV WKH FURVVRUWKRJRQDOLW\ UHODWLRQV DVVRFLDWHG ZLWK WKLV W\SH RI V\VWHPV DUH

PAGE 30

9W09 ,Q[Q f 97.9 GLDJZM XfA 2Qf f 3URSRUWLRQDOO\ 'DPSHG 0RGHOV :KHQ WKH GDPSLQJ RI WKH VWUXFWXUH LV DFFRXQWHG IRU LW LV VRPHWLPHV PRGHOOHG WR EH SURSRUWLRQDO D0 S. f ZKHUH D DQG 3 DUH UHDO VFDODUV 7KH HLJHQYHFWRUV RI D SURSRUWLRQDOO\ GDPSHG V\VWHP DUH WKH VDPH DV WKH HLJHQYHFWRUV DVVRFLDWHG ZLWK WKH FRUUHVSRQGLQJ XQGDPSHG V\VWHP 7KH FURVVRUWKRJRQDOLW\ UHODWLRQVKLSV DVVRFLDWHG ZLWK SURSRUWLRQDOO\ GDPSHG PRGHOV DUH 9W09 ,Q[Q f 9W'9 GLDJAM:M e& f ‘ AQRfQf f 9W.9 GLDJWRPM f :Qf f ZKHUH DOO YDULDEOHV KDYH WKH VDPH GHILQLWLRQV DV LQ WKH SUHYLRXV GLVFXVVLRQ $ PRUH GHWDLOHG GHYHORSPHQW RI WKH HLJHQYDOXH SUREOHP FDQ EH IRXQG LQ WKH ERRNV RI 0HLURYLWFK f DQG ,QPDQ f ([SHULPHQWDO 0RGDO $QDO\VLV 7KH YLEUDWLRQ FKDUDFWHULVWLFV RI VWUXFWXUHV FDQ DOVR EH PHDVXUHG H[SHULPHQWDOO\ 5HFDOO WKDW WKH VWUXFWXUDO YLEUDWLRQ FKDUDFWHULVWLFV DUH JLYHQ E\ QDWXUDO IUHTXHQFLHV GDPSLQJ UDWLRV DQG PRGH VKDSHV HLJHQYHFWRUVf 7KH SURFHVV RI PHDVXULQJ WKH PRGDO SDUDPHWHUV LV NQRZQ DV PRGDO WHVWLQJ RU H[SHULPHQWDO PRGDO DQDO\VLV $Q LQWURGXFWRU\ WUHDWPHQW RI WKH FRQFHSW RI H[SHULPHQWDO PRGDO DQDO\VLV FDQ EH IRXQG LQ WKH ERRN RI ,QPDQ f D PRUH ULJRURXV FRYHUDJH LV WUHDWHG LQ WKH ERRN RI (ZLQV f 7KH KDUGZDUH FRPSRQHQWV QHHGHG LQ D PRGDO DQDO\VLV H[SHULPHQW DUH LGHQWLILHG LQ )LJXUH ,QPDQ f $ VFKHPDWLF RI D VLPSOH PRGDO YLEUDWLRQ PHDVXUHPHQW WHVW VHWXS

PAGE 31

LV VKRZQ LQ )LJXUH %ULHI GHVFULSWLRQV RI VRPH RI WKH FRPSRQHQWV DQG WKHLU IXQFWLRQV DUH JLYHQ EHORZ 7 7UDQVGXFHU 6& 6LJQDO &RQGLWLRQHU )LJXUH &RPSRQHQWV RI D 9LEUDWLRQ 0HDVXUHPHQW 6\VWHP )RU 0RGDO $QDO\VLV ,QPDQ f 7KH H[FLWHU LV XVHG WR GHOLYHU WKH GULYLQJ IRUFH WKDW SXWV WKH VWUXFWXUH LQ PRWLRQ 7KH WZR PRVW FRPPRQO\ XVHG H[FLWHUV DUH WKH VKDNHU DQG WKH LPSXOVH KDPPHU 7UDQVGXFHUV DUH GHYLFHV WKDW PHDVXUH WKH GULYLQJ IRUFH DV ZHOO DV WKH UHVSRQVH RI WKH VWUXFWXUH 7KH\ VHQVH PHFKDQLFDO IRUFHV RU PRWLRQV DQG WKHQ FRQYHUW WKHP LQWR HOHFWULFDO VLJQDOV 0HFKDQLFDO IRUFHV DUH XVXDOO\ UHFRUGHG E\ D IRUFH WUDQVGXFHUV &RPPRQO\ WKH YLEUDWLRQ UHVSRQVH RI VWUXFWXUHV LV PHDVXUHG E\ DFFHOHURPHWHUV $FFHOHURPHWHUV DUH PRXQWHG GLUHFWO\ RQ WKH VWUXFWXUH DQG QDWXUDOO\ UHFRUG WKH ORFDO DFFHOHUDWLRQV 6LJQDO FRQGLWLRQHUV DUH XVHG WR PDWFK WKH VLJQDO UHFHLYHG IURP WKH WUDQVGXFHUV WR WKH VSHFLILFDWLRQ RI WKH DQDO\]HU 7KHVH GHYLFHV FDQ DOVR EH XVHG WR DPSOLI\ WKH VLJQDOV

PAGE 32

6LJQDO $QDO\]HU )LJXUH $ 6LPSOH ([SHULPHQWDO 0RGDO $QDO\VLV 6HWXS 7KH VLJQDO DQDO\]HU SURFHVVHV WKH HOHFWULFDO VLJQDO UHFHLYHG IURP WKH VLJQDO FRQGLWLRQHUV 7KH VWDQGDUG W\SH RI DQDO\]HU DOORZV WLPH GRPDLQ VLJQDOV WR EH YLHZHG LQ WKH IUHTXHQF\ GRPDLQ YLD D )DVW )RXULHU 7UDQVIRUP ))7f DOJRULWKP ,Q DQ ))7 WKH VLJQDOV DUH ILUVW ILOWHUHG GLJLWL]HG DQG WKHQ WUDQVIRUPHG LQWR GLVFUHWH IUHTXHQF\ VSHFWUD 7KH IUHTXHQF\ VSHFWUD FDQ WKHQ EH PDQLSXODWHG WR FRPSXWH WKH PRGDO SURSHUWLHV RI WKH VWUXFWXUH ,W LV LPSRUWDQW WR QRWH WKDW WKH H[SHULPHQWDOO\ LGHQWLILHG PRGDO SDUDPHWHUV DUH XVXDOO\ DIIHFWHG E\ XQSUHGLFWDEOH PHDVXUHPHQW QRLVH 7\SLFDOO\ QDWXUDO IUHTXHQFLHV DUH LGHQWLILHG WR ZLWKLQ WR b UHSHDWDELOLW\ GDPSLQJ UDWLRV WR ZLWKLQ WR b UHSHDWDELOLW\ DQG PRGH VKDSHV WR ZLWKLQ WR b UHSHDWDELOLW\ ,Q SUDFWLFH GXH WR WHVWLQJ OLPLWDWLRQV WKH VHW RI e

PAGE 33

VWUXFWXUDO PRGDO SDUDPHWHUV LGHQWLILHG H[SHULPHQWDOO\ LV LQFRPSOHWH ZLWK UHVSHFW WR WKH DQDO\WLFDO PRGHO ([SHULPHQWDO LQFRPSOHWHQHVV LV PDQLIHVWHG LQ WZR IRUPV Lf D OLPLWHG QXPEHU RI PHDVXUHG PRGHV RI YLEUDWLRQ LLf D OLPLWHG QXPEHU RI HLJHQYHFWRU '2) PHDVXUHPHQWV $QDO\WLFDO DQG ([SHULPHQWDO 0RGHO 'LPHQVLRQV &RUUHODWLRQ 2QH PDMRU SUHUHTXLVLWH FRPPRQ WR PRVW PRGHO UHILQHPHQW DOJRULWKPV LV WR PDWFK WKH QXPEHU RI GHJUHHV RI IUHHGRP '2)Vf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f 7KH VXEMHFW RI HLJHQYHFWRU H[SDQVLRQ LV GLVFXVVHG LQ IDLU GHWDLO LQ WKH SDSHUV E\ *\VLQ f DQG =LPPHUPDQ DQG .DRXN Df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f

PAGE 34

ZKHUH DV GHILQHG HDUOLHU 0 DQG DUH UHVSHFWLYHO\ WKH PDVV DQG VWLIIQHVV PDWUL[ 9 LV WKH HLJHQYHFWRU PDWUL[ DQG $ LV WKH GLDJRQDO HLJHQYDOXH PDWUL[ $VVXPH WKDW RQO\ D VXEVHW RI WKH HLJHQYHFWRU '2)V KDV EHHQ H[SHULPHQWDOO\ PHDVXUHG (TXDWLRQ f FDQ EH UHRUGHUHG VXFK WKDW WKH '2)V DVVRFLDWHG ZLWK WKH PHDVXUHG '2)V DUH LQ WKH XSSHU URZV RI WKH HTXDWLRQ 09$ .9 >@ 9P .PP .PX 0PP 0PX ZKHUH 9 9X .XP .XX 0 0XP 0XX 7KH PDWULFHV 0 DQG 9 DUH UHVSHFWLYHO\ WKH UHRUGHUHG PDVV VWLIIQHVV DQG HLJHQYHFWRU PDWULFHV ,Q WKH DERYH HTXDWLRQ WKH VXEVFULSWV fPf DQG fXf GHQRWH UHVSHFWLYHO\ WKH FRPSRQHQWV DVVRFLDWHG ZLWK WKH PHDVXUHG DQG XQPHDVXUHG '2)V $ WUDQVIRUPDWLRQ PDWUL[ 3 WKDW UHODWHV PDWULFHV 9X DQG 9P FDQ EH GHILQHG DV 9X 39P f $ VXEVWLWXWLRQ RI WKLV UHODWLRQVKLS LQ WKH UHRUGHUHG HLJHQYHFWRU PDWUL[ 9 UHVXOWV LQ 9 79P f 6XEVWLWXWLQJ (T f LQWR (T f DQG SUHPXOWLSO\LQJ E\ 77 \LHOGV WKH HLJHQYDOXH SUREOHP RI WKH UHGXFHG PRGHO 0U9P$ .U9P >@ f ZKHUH 0U 7W07 .U 7W.7 ZKHUH 0U DQG .U DUH WKH UHGXFHG PDVV DQG VWLIIQHVV PDWULFHV UHVSHFWLYHO\ ,Q WHUPV RI WKH SDUWLWLRQHG PDWULFHV WKH UHGXFHG PDWULFHV DUH GHILQHG DV

PAGE 35

0U 0PP 370XP 0PX3 SWPXXS f .U .PP 37.XP .PX3 37.XX3 f $W WKLV SRLQW WKH RQO\ FRQGLWLRQ SODFHG RQ 3 LV WKH UHODWLRQVKLS RI (T f 1DWXUDOO\ PDWUL[ 3 FDQ EH FRPSXWHG GLUHFWO\ IURP (T f LI WKH HLJHQYHFWRUV RI WKH V\VWHP DUH DYDLODEOH 7KLV DSSURDFK LV NQRZQ DV WKH H[DFW UHGXFWLRQ PHWKRG DQG KDV EHHQ GLVFXVVHG LQ WKH SDSHUV E\ .DPPHU f DQG 2f&DOODKDQ HW DO f 7KH H[DFW UHGXFWLRQ PHWKRG UHTXLUHV VROYLQJ IRU D ODUJH QXPEHU RI HLJHQYHFWRUV ZKLFK FDQ EH FRPSXWDWLRQDOO\ H[SHQVLYH 7KH WKUHH UHGXFWLRQ PHWKRGV WKDW DUH SUHVHQWHG LQ WKH IRUWKFRPLQJ GLVFXVVLRQV SURSRVH DOWHUQDWLYH WHFKQLTXHV WR FRPSXWH PDWUL[ 3 7KH ILUVW WZR GR QRW UHTXLUH WKH FRPSXWDWLRQ RI WKH V\VWHP HLJHQYDOXH SUREOHP 7KH ODVW RQH UHTXLUHV WKH NQRZOHGJH RI RQH HLJHQYDOXH ZKLFK LV FRPSXWDWLRQDOO\ DGPLVVLEOH 6WDWLF 5HGXFWLRQ 7KLV UHGXFWLRQ PHWKRG LV RIWHQ UHIHUUHG WR DV *X\DQ f UHGXFWLRQ ,Q WKH VWDWLF UHGXFWLRQ WKH PDVV SURSHUWLHV DVVRFLDWHG WR WKH XQPHDVXUHG '2)V DUH DVVXPHG QHJOLJLEOH :LWK WKDW DVVXPSWLRQ (T f FDQ EH ZULWWHQ DV 0PP $ .PP .XP .PX L ( L r & & 9X 7KH VHFRQG URZ RI WKLV PDWUL[ HTXDWLRQ FDQ WKHQ EH PDQLSXODWHG DV 9X .M.XP9P f f )URP FRPSDULQJ (T f WR (T f LW FDQ GHGXFHG WKDW WKH WUDQVIRUPDWLRQ PDWUL[ 3 FRPSXWHG XVLQJ WKH *X\DQ DSSURDFK LV JLYHQ E\ 3J .XXn.XP f 7KH UHGXFHG PDVV DQG VWLIIQHVV PDWULFHV FDQ WKHQ EH FRPSXWHG E\ VXEVWLWXWLQJ PDWUL[ 3J IRU PDWUL[ 3 LQ (TV f DQG f 1DWXUDOO\ WKH *X\DQ DVVXPSWLRQ (T ff

PAGE 36

VXJJHVWV WKDW LI WKH PDVV SURSHUWLHV RI WKH RPLWWHG '2)V DUH QRW VPDOO WKH DFFXUDF\ RI WKH *X\DQ UHGXFHG PRGHO FRXOG EH ODFNLQJ ,56 5HGXFWLRQ 7KH LPSURYHG UHGXFWLRQ PHWKRG ,56f ZDV IRUPXODWHG E\ 2f&DOODKDQ f ,W LV DQ LPSURYHPHQW RYHU WKH *X\DQ UHGXFWLRQ LQ WKDW LW DFFRXQWV IRU WKH PDVV SURSHUWLHV RI WKH XQPHDVXUHG '2)V ,Q WKH IRUPXODWLRQ RI WKH ,56 PHWKRG WKH *X\DQ UHGXFHG PRGHO LV FRUUHFWHG WR LQFOXGH WKH PDVV LQIOXHQFH RI WKH XQPHDVXUHG '2)V 7KLV IRUPXODWLRQ LV VRPHZKDW OHQJWK\ DQG WKH LQWHUHVWHG UHDGHU LV UHIHUUHG EDFN WR WKH SDSHU RI 2f&DOODKDQ f RU WKH WKHVLV RI 0F*RZDQ f 7KH WUDQVIRUPDWLRQ PDWUL[ 3 FRPSXWHG XVLQJ WKH ,56 UHGXFWLRQ LV 356 3J .M0XP 0XX3J@ 0n.UJ f 7KH UHGXFHG ,56 PRGHO LV WKHQ FRPSXWHG E\ VXEVWLWXWLQJ PDWUL[ 3LUV LQ (TV f '\QDPLF 5HGXFWLRQ 7KH G\QDPLF UHGXFWLRQ ZDV SURSRVHG DV DQRWKHU LPSURYHPHQW WR WKH *X\DQ UHGXFWLRQ .LGGHU 0LOOHU DQG 3D] f 7KLV UHGXFWLRQ XWLOL]HV WKH G\QDPLF HTXDWLRQ DVVRFLDWHG ZLWK D VLQJOH PRGH RI YLEUDWLRQ WR FRPSXWH WKH WUDQVIRUPDWLRQ PDWUL[ 3 ,Q WKLV WHFKQLTXH WKH WUDQVIRUPDWLRQ PDWUL[ 3 LV DUULYHG DW E\ FRQVLGHULQJ WKH UHRUGHUHG G\QDPLF HTXDWLRQ DVVRFLDWHG ZLWK WKH LWK PRGH AfM 0PUQ .PP 0PX .PX f§Pc n2n .XP AI0XX .XX r}B ZKHUH LV WKH LWK HLJHQYDOXH YP DQG YX DUH UHVSHFWLYHO\ WKH PHDVXUHG DQG XQPHDVXUHG HLJHQYHFWRU '2)V DVVRFLDWHG ZLWK WKH LWK PRGH RI YLEUDWLRQ %DVHG RQ WKLV SDUWLWLRQ WZR HTXDWLRQV FDQ EH JHQHUDWHG %\ XVLQJ WKH URZV WKDW FRUUHVSRQG WR WKH XQPHDVXUHG '2)V ORZHU URZVf WKH IROORZLQJ UHODWLRQVKLS LV REWDLQHG

PAGE 37

YXc L 088;" .f 0fP;" f§Pc f +HQFH WKH WUDQVIRUPDWLRQ PDWUL[ DVVRFLDWHG ZLWK WKH LWK PRGH LV GHILQHG E\ 3G 0QQ;" .ff 0OLP;" f 7KH UHGXFHG PDVV DQG VWLIIQHVV PDWULFHV DUH WKHQ FRPSXWHG E\ XVLQJ (TV f DQG f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f LV RQH RI WKH PRVW FRPPRQO\ XVHG HLJHQYHFWRU H[SDQVLRQ DOJRULWKPV $ VOLJKW PRGLILFDWLRQ RI WKH %HUPDQ DQG 1DJ\ IRUPXODWLRQ LV SUHVHQWHG KHUH WR DFFRPPRGDWH GDPSHG V\VWHPV )XK HW DO f ,Q WKH IRUPXODWLRQ RI WKH G\QDPLF H[SDQVLRQ LW LV DVVXPHG WKDW WKH PHDVXUHG PRGHV VDWLVI\ WKH HLJHQYDOXH SUREOHP LQYROYLQJ WKH SURSHUW\ PDWULFHV RI WKH RULJLQDO PRGHO 0 .f )RU WKH LWK PHDVXUHG PRGH WKLV DVVXPSWLRQ WDNHV WKH IRUP ;_0 ;H' .f \H f ZKHUH ;H DQG YH DUH UHVSHFWLYHO\ WKH LWK H[SHULPHQWDO HLJHQYDOXH DQG HLJHQYHFWRU 7KH PDWULFHV 0 DQG KDYH WKH VDPH GHILQLWLRQV DV LQ WKH HDUOLHU VHFWLRQV $VVXPH WKDW RQO\ D VXEVHW RI WKH '2)V RI HLJHQYHFWRU YH KDV EHHQ PHDVXUHG (TXDWLRQ f FDQ EH UHRUGHUHG DV LQ 6HFWLRQ VXFK WKDW WKH PHDVXUHG HLJHQYHFWRU '2)V UHVLGH LQ WKH XSSHU KDOI RI WKH HTXDWLRQ

PAGE 38

0PP 0PX ;H 'PP 'PX ;rS .PP .PX ! /HP 08P 0XX 'XP 'XX HL .XP .XX /HX r f f ZKHUH YHP DQG YHX DUH UHVSHFWLYHO\ WKH PHDVXUHG DQG XQPHDVXUHG '2)V RI YH 7KH VXEVFULSWV fPf DQG fXf GHQRWH PHDVXUHG DQG XQPHDVXUHG FRPSRQHQWV $ UHDUUDQJHPHQW RI (T f \LHOGV AHAPP E ;H'PUQ .PP AHL0PX E ;HM'PX E .PX UY L f§HPc f2n ;A0XP E AHM'XP E .8P ;H08X E ;H'XX E .XX Vn !L L )URP (T f WZR PDWUL[ HTXDWLRQV LQ IXQFWLRQ RI YHP DQG YHX FDQ EH JHQHUDWHG %\ XVLQJ WKH HTXDWLRQ DVVRFLDWHG ZLWK WKH VHFRQG URZ RI WKH SDUWLWLRQ WKH XQPHDVXUHG FRPSRQHQWV RI WKH LWK H[SHULPHQWDO HLJHQYHFWRU DUH FRPSXWHG WR EH /HX 08XAHc 7 'fX;Hc .XX 0XP;H 'XP;H f‘XP YHPL f 1RWH WKDW WKLV H[SDQVLRQ ZRUNV RQ D VLQJOH PRGH DW D WLPH $OVR QRWLFH WKDW LW LQYROYHV WKH RULJLQDO DQDO\WLFDO PRGHO 0 .f 7KLV LPSOLHV WKDW WKH DFFXUDF\ RI WKH H[SDQVLRQ GHSHQGV RQ WKH YDOLGLW\ RI WKH RULJLQDO DQDO\WLFDO PRGHO 2UWKRJRQDO 3URFUXVWHV ([SDQVLRQ $QRWKHU H[SDQVLRQ SURFHVV WKDW KDV VKRZQ JUHDW SURPLVH LV WKH RUWKRJRQDO 3URFUXVWHV 23f H[SDQVLRQ PHWKRG SUHVHQWHG LQ WKH SDSHUV RI 6PLWK DQG %HDWWLH f DQG =LPPHUPDQ DQG .DRXN Df 7KH WHFKQLTXH XVHV WKH JHQHUDO PDWKHPDWLFDO IUDPHZRUN RI WKH RUWKRJRQDO 3URFUXVWHV SUREOHP *ROXE DQG 9DQ /RDQ f /HW 9HP EH WKH H[SHULPHQWDOO\ PHDVXUHG HLJHQYHFWRU FRPSRQHQW PDWUL[ DQG 9DP EH WKH FRUUHVSRQGLQJ DQDO\WLFDO HLJHQYHFWRU FRPSRQHQW PDWUL[ 7KH HVVHQFH RI WKH 23 H[SDQVLRQ LV WR ILQG D XQLWDU\ WUDQVIRUPDWLRQ PDWUL[ 3RS WKDW FORVHO\ URWDWHV 9DP LQWR 9HP 7KLV LV DWWHPSWHG E\ VROYLQJ WKH IROORZLQJ SUREOHP

PAGE 39

0LQLPL]H __ 9HP 9DP3RS __) VXEMHFW WR 3MS3RS f 7KH VROXWLRQ WR WKLV JHQHUDO SUREOHP LV GLVFXVVHG LQ WKH ERRN RI *ROXE DQG 9DQ /RDQ DQG LV JLYHQ E\ 3RS <=W f ZKHUH < DQG = DUH UHVSHFWLYHO\ WKH OHIW DQG ULJKW VLQJXODU PDWULFHV RI PDWUL[ 2 GHILQHG E\ r 9MP9HP f /HW 9DX EH WKH HLJHQYHFWRU PDWUL[ DVVRFLDWHG ZLWK WKH XQPHDVXUHG '2)V ,Q WKH RUWKRJRQDO 3URFUXVWHV H[SDQVLRQ LW LV DVVXPHG WKDW WKH WUDQVIRUPDWLRQ PDWUL[ FRPSXWHG DERYH DOVR URWDWHV 9DX LQWR WKH XQPHDVXUHG fH[SHULPHQWDOf HLJHQYHFWRU FRPSRQHQW PDWUL[ 9HX DV
PAGE 40

UHTXLUHG D VHSDUDWHG RUWKRJRQDOLW\ DOJRULWKP FDQ EH XVHG 6HFWLRQ f ,Q WKH SDSHU E\ 6PLWK DW DO f LW LV VKRZQ WKDW IRU DFWXDO PRGHO UHILQHPHQW SUREOHPV ERWK YLHZSRLQWV JLYH HTXLYDOHQW HLJHQYHFWRU H[SDQVLRQ UHVXOWV +RZHYHU IRU GDPDJH GHWHFWLRQ SUREOHPV D SUHOLPLQDU\ VWXG\ LQGLFDWHV WKDW HLJHQYHFWRUV H[SDQGHG E\ XVLQJ WKH YLHZSRLQW GHILQHG LQ (T f JLYH EHWWHU DVVHVVPHQW RI WKH GDPDJH (LJHQYHFWRU 2UWKRJRQDOL]DWLRQ 0RVW PDWUL[ XSGDWH DOJRULWKPV UHTXLUH WKH PHDVXUHG HLJHQYHFWRUV WR VDWLVI\ D FURVVRUWKRJRQDOLW\ FRQGLWLRQ 7KLV LV HVSHFLDOO\ WUXH LQ PDWUL[ XSGDWH DOJRULWKPV LQ ZKLFK Lf WKH PRGHO RI WKH VWUXFWXUH LV DVVXPHG XQGDPSHG DQG WKH PRGHOOLQJ HUURUV DUH DVVXPHG WR EH LQ RQO\ RQH RI WKH WZR SURSHUW\ PDWULFHV 0 RU LV DVVXPHG FRUUHFWf RU LLf WKH V\VWHP LV PRGHOHG E\ D SURSRUWLRQDOO\ GDPSHG PRGHO ZLWK HUURUV LQ RQO\ WZR RI WKH WKUHH SURSHUW\ PDWULFHV 0 RU .f ,Q WKHVH VLWXDWLRQV LQ RUGHU WR LQVXUH V\PPHWU\ RI WKH XSGDWHG SURSHUW\ PDWULFHV LW LV UHTXLUHG WKDW WKH H[SDQGHG H[SHULPHQWDO HLJHQYHFWRUV EH RUWKRJRQDO ZLWK UHVSHFW WR WKH SURSHUW\ PDWUL[ DVVXPHG DFFXUDWH 7KLV VLWXDWLRQ LV HQFRXQWHUHG LQ WKH RSWLPDO XSGDWH DOJRULWKPV GHYHORSHG E\ %DUXFK DQG %DU ,W]KDFN f .DEH f .DPPHU f DQG 6PLWK DQG %HDWWLH f =LPPHUPDQ DQG .DRXN Ef .DRXN DQG =LPPHUPDQ Ef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f %DUXFK DQG %DU ,W]KDFN f DQG %DUXFK f ,Q WKH QH[W WZR VHFWLRQV WZR RUWKRJRQDOL]DWLRQ WHFKQLTXHV %DUXFK DQG %DU ,W]KDFN %DUXFK f DUH GLVFXVVHG %RWK WHFKQLTXHV DUH PDVV RUWKRJRQDOL]DWLRQ WHFKQLTXHV KRZHYHU ZLWK REYLRXV PRGLILFDWLRQV WKHVH WHFKQLTXHV FDQ EH

PAGE 41

DGRSWHG WR VROYH WKH RUWKRJRQDOL]DWLRQ SUREOHP RI WKH HLJHQYHFWRUV ZLWK UHVSHFW WR WKH VWLIIQHVV RU WKH GDPSLQJ PDWULFHV 2SWLPDO :HLJKWHG 2UWKRJRQDOL]DWLRQ 7KH HVVHQFH RI WKH VWDQGDUG PDVV RUWKRJRQDOL]DWLRQ WHFKQLTXH LV WR PRGLI\ WKH PHDVXUHG HLJHQYHFWRUV VXFK WKDW WKH PDVV FURVVRUWKRJRQDOLW\ FRQGLWLRQ LV VDWLVILHG %DUXFK DQG %DU ,W]KDFN f SURSRVHG DQ HOHJDQW VROXWLRQ WR WKDW SUREOHP $Q RYHUYLHZ RI WKHLU SUREOHP VWDWHPHQW DQG VROXWLRQ LV JLYHQ EHORZ $VVXPLQJ WKDW 9H LV D PDWUL[ RI H[SDQGHG H[SHULPHQWDO HLJHQYHFWRUV WKDW QHHG WR EH PDVV RUWKRJRQDOL]HG 7KH SUHVHQW IRUPXODWLRQ VHDUFKHV IRU WKH RSWLPDO PDVV ZHLJKWHG FKDQJH RI PDWUL[ 9H VXFK WKDW WKH PDVV FURVVRUWKRJRQDOLW\ FRQGLWLRQ LV VDWLVILHG 7KLV SUREOHP LV FDVW DV 0LQLPL]H __ 1 9HR 9Hf __S f VXEMHFW WR 9LR 0 9HR f ZKHUH 1 0 DQG 0 LV WKH PDVV PDWUL[ %\ PHDQV RI D /DJUDQJH PXOWLSOLHU (T f FDQ EH LQFRUSRUDWHG LQWR (T f WKHQ WKH DSSOLFDWLRQ RI WKH RSWLPDOLW\ FRQGLWLRQV \LHOGV WKH IROORZLQJ H[SUHVVLRQ IRU 9H 9HR 9H9c) 0 9Hfn f %HIRUH EHLQJ LQFRUSRUDWHG LQWR WKH RUWKRJRQDOL]DWLRQ SURFHVV WKH PHDVXUHG HLJHQYHFWRUV KDYH WR EH XQLW PDVV QRUPDOL]HG LH f ZKHUH YH LV WKH LWK H[SDQGHG H[SHULPHQWDO HLJHQYHFWRU LWK FROXPQ RI 9Hf 6HOHFWLYH 2SWLPDO 2UWKRJRQDOL]DWLRQ 6RPH VWUXFWXUHV H[KLELW ULJLG ERG\ PRGHV PRGHV ZLWK ]HUR HLJHQYDOXHVf ,W LV GHVLUDEOH WR SUHVHUYH WKHVH ULJLG ERG\ PRGHV LQ WKH UHILQHPHQW SURFHVV +RZHYHU VRPH PDWUL[ XSGDWH

PAGE 42

DOJRULWKPV UHTXLUH WKH ULJLG ERG\ PRGHV DQG WKH H[SHULPHQWDO HLJHQYHFWRUV WR EH PDVV RUWKRJRQDO WR LQVXUH V\PPHWU\ RI WKH XSGDWHG SURSHUW\ PDWULFHV VHH &KDSWHU f 1DWXUDOO\ WKH ULJLG ERG\ PRGHV ZLOO EH FRUUXSWHG LI WKH\ DUH LQFRUSRUDWHG DORQJ ZLWK WKH H[SDQGHG H[SHULPHQWDO HLJHQYHFWRUV LQ WKH DERYH RUWKRJRQDOL]DWLRQ SURFHVV 7KXV %DUXFK f SUHVHQWHG D PRGLILFDWLRQ RI WKH SURFHGXUH LQ 6HFWLRQ WR GHDO ZLWK VXFK D SUREOHP 7KH UHVXOWLQJ SUREOHP LV D VHOHFWLYH RUWKRJRQDOL]DWLRQ DQG LV IRUPXODWHG DV IROORZV PLQLPL]H __ 1 9HR 9Hf __) f 6XEMHFW WR 9LR 0 9HR f DQG 9c7 0 9U >@ f ,Q WKH DERYH HTXDWLRQV 9H DQG 9U DUH UHVSHFWLYHO\ WKH H[SDQGHG H[SHULPHQWDO HLJHQYHFWRU PDWUL[ DQG WKH ULJLG ERG\ PRGH PDWUL[ $JDLQ WKH /DJUDQJH PXOWLSOLHU LV XVHG DQG WKH RUWKRJRQDOL]HG H[SHULPHQWDO HLJHQYHFWRU PDWUL[ WKDW VDWLVILHV WKH FRQGLWLRQV LQ (TV f DQG f LV IRXQG WR EH 9HR 447 0 4f ZKHUH 4 9H 9U9"09H f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f DQ LWHUDWLYH DSSURDFK WR SUHVHUYH WKH ORDG SDWK RI WKH RULJLQDO SURSHUW\ PDWULFHV ZDV GHYHORSHG 7KH

PAGE 43

DSSURDFK ZDV SUHVHQWHG LQ WKH FRQWH[W RI WKH V\PPHWULF HLJHQVWUXFWXUH DVVLJQPHQW PRGHO UHILQHPHQW DOJRULWKP GLVFXVVHG LQ &KDSWHU f KRZHYHU LWV DSSOLFDWLRQ FDQ DOVR EH H[WHQGHG WR RWKHU PRGHO UHILQHPHQW DOJRULWKPV ,Q )LJXUH D IORZ FKDUW RI WKH LWHUDWLYH ORDG SUHVHUYDWLRQ DOJRULWKP LV SUHVHQWHG 7KH SURFHGXUH LV LOOXVWUDWHG IRU D JHQHUDO PRGHO UHILQHPHQW VFHQDULR LQ ZKLFK DOO WKUHH SURSHUW\ PDWULFHV 0 .f DUH EHLQJ XSGDWHG +RZHYHU LW FDQ EH HDVLO\ PRGLILHG WR DFFRPPRGDWH RWKHU UHILQHPHQW SUREOHPV )LJXUH )ORZ &KDUW RI WKH ,WHUDWLYH /RDG 3DWK 3UHVHUYDWLRQ $OJRULWKP ,Q WKH IORZ FKDUW WKH PDWULFHV 0D P 'D P DQG .D P DUH UHVSHFWLYHO\ WKH DGMXVWHG PDVNHG PDVV GDPSLQJ DQG VWLIIQHVV PDWULFHV GHILQHG E\

PAGE 44

0DP f§ 0D 2 0P 'DP 'D 2 'P f .DP .D 2 .P ZKHUH 0D 'D DQG .D DUH WKH DGMXVWHG PDVV GDPSLQJ DQG VWLIIQHVV PDWULFHV 7KH PDWULFHV 0P 'P DQG .P DUH WKH PDVNLQJ PDWULFHV DVVRFLDWHG ZLWK WKH RULJLQDO PDVV GDPSLQJ DQG VWLIIQHVV PDWUL[ %\ GHILQLWLRQ WKH PDVNLQJ PDWUL[ $P DVVRFLDWHG ZLWK PDWUL[ $ LV JLYHQ E\ $PLMf LI $LMf r f $PLMf LI $LMf ,Q (TV f WKH RSHUDWRU 2 LV WKH HOHPHQWE\HOHPHQW VFDODUf PDWUL[ PXOWLSOLFDWLRQ /HW % DQG & EH WZR Q[Q PDWULFHV WKHQ WKH HOHPHQWE\HOHPHQW PXOWLSOLFDWLRQ RI % DQG & LV JLYHQ E\ 6 %2& 6LMf %LMf r &LMf L M OQ f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

PAGE 45

&+$37(5 ,19(56( +<%5,' $3352$&+ )25 ),1,7( (/(0(17 02'(/ 5(),1(0(17 ,QWURGXFWLRQ 7KH LQYHUVH HLJHQYDOXH SUREOHP LV FRQFHUQHG ZLWK WKH FRQVWUXFWLRQ RI WKH SURSHUW\ PDWULFHV PDVV GDPSLQJ RU VWLIIQHVVf RI D G\QDPLF PRGHO XVLQJ H[SHULPHQWDOO\ PHDVXUHG PRGDO GDWD 7KHVH WHFKQLTXHV UHTXLUH FRPSOHWH PRGDO SURSHUWLHV 7KXV IRU DQ Q'2) PRGHO Q QDWXUDO IUHTXHQFLHV GDPSLQJ UDWLRV DQG PRGH VKDSHV HLJHQYHFWRUVf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f )(0 7KH IUHH YLEUDWLRQ PRWLRQ RI VXFK D G\QDPLF VWUXFWXUH FDQ EH DQDO\WLFDOO\ UHSUHVHQWHG E\ D GLIIHUHQWLDO HTXDWLRQ RI WKH IRUP 0ZWf 'ZWf .ZWf f ZKHUH WKH YDULDEOHV 0 DQG DUH Q[Q UHDO V\PPHWULF PDWUL[ PRGHOV RI WKH PDVV GDPSLQJ DQG VWLIIQHVV SURSHUWLHV RI WKH VWUXFWXUH 7KH Q[O WLPH YDU\LQJ YHFWRU ZWf UHSUHVHQWV WKH Q

PAGE 46

GLVSODFHPHQWV RI WKH Q'2) PRGHO RI WKH V\VWHP 7KH RYHUGRWV UHSUHVHQW GLIIHUHQWLDWLRQ ZLWK UHVSHFW WR WLPH 7KH HLJHQYDOXH SUREOHP DVVRFLDWHG ZLWK WKH GLIIHUHQWLDO HTXDWLRQ VKRZQ LQ (T f LV RI WKH IRUP 0Yc ;" '\M ;c .YM f ZKHUH ;M DQG Yc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f 97 9 GLDJAFR f f f }AQZQf A Ef 9W.9 GLDJRf fQf t Ff !7 LL YQ @ ZKHUH W2M DQG ec DUH WKH QDWXUDO IUHTXHQF\ DQG GDPSLQJ UDWLR UHVSHFWLYHO\ RI WKH LWK PRGH RI WKH VWUXFWXUH 7KH PDWUL[ ,Q[Q LV WKH Q[Q LGHQWLW\ PDWUL[ ,W LV LPSRUWDQW WR UHFRJQL]H WKDW (T Df UHSUHVHQWV QHFHVVDU\ DQG VXIILFLHQW FRQGLWLRQV IRU FRQVHUYLQJ V\PPHWU\ DQG GDPSLQJ SURSRUWLRQDOLW\ ZKHQ XSGDWLQJ WKH VWLIIQHVV DQG GDPSLQJ SURSHUWLHV RI SURSRUWLRQDOO\ GDPSHG V\VWHPV 6XSSRVH WKDW S S m Qf PRGHV RI DQ H[LVWLQJ VWUXFWXUH KDYH EHHQ H[SHULPHQWDOO\ LGHQWLILHG PRGH VKDSHV RU HLJHQYHFWRUV IUHTXHQFLHV DQG GDPSLQJ UDWLRVf $VVXPH WKDW WKH GLPHQVLRQ RI PHDVXUHG HLJHQYHFWRUV LV HTXDO WR WKH GLPHQVLRQ RI WKH )(0 LH DOO Q FRPSRQHQWV RI WKH PHDVXUHG HLJHQYHFWRUV DUH DYDLODEOH ,W LV ZLGHO\ DFFHSWHG WKDW LQ WKH

PAGE 47

DEVHQFH RI VSHFLILF H[SHULPHQWDO PHDVXUHPHQWV D JRRG DSSUR[LPDWLRQ WR WKH XQPHDVXUHG PRGHV LV WKHLU FRUUHVSRQGLQJ DQDO\WLFDO PRGDO LQIRUPDWLRQ :LWK WKDW LQ PLQG D FRPSOHWH K\EULG VHW RI PRGDO GDWD LV JHQHUDWHG E\ FRPELQLQJ H[SHULPHQWDO DQG DQDO\WLFDO LQIRUPDWLRQ 9HD >9H 94H 3 =H f ZKHUH 9 LV WKH HLJHQYHFWRU PDWUL[ e DQG ; DUH GLDJRQDO PDWULFHV RI IUHTXHQFLHV VTXDUHG DQG GDPSLQJ UDWLRV UHVSHFWLYHO\ 7KH VXEVFULSWV H DQG D GHQRWH UHVSHFWLYHO\ H[SHULPHQWDO DQG DQDO\WLFDO VHWV $W WKLV SRLQW WKH FRPSOHWH fK\EULGf VHW RI HLJHQYHFWRUV 9HD GRHV QRW VDWLVI\ WKH FURVVRUWKRJRQDOLW\ FRQGLWLRQV GHILQHG LQ (T Df WKXV WKH FRQGLWLRQV LQ (TV Ef DQG Ff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f ,I WKH SUREOHP LV VHW VXFK WKDW WKH H[SHULPHQWDO PRGHV 9H DUH FRUUHFWHG ILUVW LW LV FOHDU WKDW WKH DQDO\WLFDO PRGHV 9D ZLOO EH VXEMHFW WR ODUJHU FRUUHFWLRQ ZKHQ LQFRUSRUDWHG LQWR WKH RUWKRJRQDOL]DWLRQ SURFHVV VLQFH WKH\ ZLOO EH VXEMHFW WR PRUH FRQVWUDLQWV 7KH H[SHULPHQWDO PRGHV 9H DUH RUWKRJRQDOL]HG E\ XVLQJ WKH RUWKRJRQDOL]DWLRQ WHFKQLTXH IRUPXODWHG E\ %DUXFK DQG %DU ,W]KDFN f 7KH JHQHUDO IRUPXODWLRQ RI WKLV RUWKRJRQDOL]DWLRQ WHFKQLTXH LV GLVFXVVHG LQ &KDSWHU )RU WKLV SDUWLFXODU DSSOLFDWLRQ WKH SUREOHP FRQVLVWV RI ILQGLQJ WKH PDWUL[ 9HR WKDW VDWLVILHV WKH FURVVRUWKRJRQDOLW\ FRQGLWLRQ

PAGE 48

97 0 9 7 Y HR L9$ Y HR $S[S DQG WKDW PLQLPL]HV WKH ZHLJKWHG (XFOLGHDQ QRUP r} __ 1 9HR 9Hf __) ZKHUH 1 0 7KH VROXWLRQ WR WKLV SUREOHP DV UHSRUWHG LQ &KDSWHU LV 9HR 9HIY0 9Hff f f f 7KH QH[W VWHS LV WR LQYRNH WKH RUWKRJRQDOLW\ UHTXLUHPHQW RQ WKH DQDO\WLFDO HLJHQYHFWRU PDWUL[ 9D E\ VHDUFKLQJ IRU D PDWUL[ 9DR WKDW VDWLVILHV WKH IROORZLQJ WZR FRQGLWLRQV 9LR 0 9DR ,QBSf[QBSf f DQG 9/ 0 9HR >@ ZKLOH PLQLPL]LQJ WKH REMHFWLYH IXQFWLRQ U 1 9DR 9Df __S ZKHUH 1 0 f f $ VLPLODU SUREOHP ZDV DOVR WUHDWHG E\ %DUXFK f LQ D GLIIHUHQW FRQWH[W $ EULHI GLVFXVVLRQ RI WKH VROXWLRQ DSSURDFK FDQ EH IRXQG LQ &KDSWHU 7KH VHW RI HLJHQYHFWRUV 9DR WKDW VDWLVILHV WKLV SUREOHP LV JLYHQ E\ YDR TTW P Tf 4 9D 9HR9MR09D f &OHDUO\ WKH UHVXOWDQW PDWUL[ 9HDR > 9HR 9DR @ VDWLVILHV (T Df 7KH FRUUHFWHG VWLIIQHVV DQG GDPSLQJ PDWULFHV DUH WKHQ FRPSXWHG XVLQJ (TV Ef DQG Ff

PAGE 49

.D 09HDR 4 HD9MDR0 Df 'D 09HDR k Df ZKHUH HD QHD4HD 1RWH WKDW WKH PDWULFHV .D DQG 'D FRPSXWHG IURP (TV DEf ZLOO EH V\PPHWULF 7KH DERYH IRUPXODWLRQ VXJJHVWV WKDW WKH V\VWHP PRGHOOHG E\ WKH RULJLQDO PDVV PDWUL[ 0f DQG WKH VWLIIQHVV .Df DQG GDPSLQJ 'Df PDWULFHV FRPSXWHG IURP (TV DEf ZLOO KDYH HLJHQYHFWRUV 9HDR! IUHTXHQFLHV eAD DQG GDPSLQJ UDWLRV QHD 6RPH VWUXFWXUHV H[KLELW ULJLG ERG\ PRGHV RI YLEUDWLRQ &RPPRQO\ LW LV GHVLUDEOH WR SUHVHUYH WKHVH ULJLG ERG\ PRGHV LQ WKH XSGDWHG PRGHO 7KH DERYH IRUPXODWLRQ DOVR VXJJHVW WKDW WKH XSGDWHG PRGHO ZLOO SUHVHUYH WKH RULJLQDO ULJLG ERG\ PRGHV 7KH SURFHGXUH GHYHORSHG DERYH FDQ EH HDVLO\ FRQWUDFWHG WR DGGUHVV WKH FDVH ZKHQ WKH V\VWHP PRGHO GRHV QRW DFFRXQW IRU WKH HIIHFWV RI GDPSLQJ XQGDPSHG PRGHOf 7KH FRQWUDFWLRQ FDQ EH REWDLQHG E\ VHWWLQJ WR ]HURV PDWULFHV HD DQG QHD LQ (TV f 7KH FRPSXWDWLRQDO EXUGHQ OLPLWV WKH VL]H RI WKH )(0 ZKLFK FDQ EH XSGDWHG RUGHU RI f (VVHQWLDOO\ WKH OLPLWLQJ IDFWRU LV WKDW DOO PRGH VKDSHV RI WKH VWUXFWXUH WKDW DUH QRW DYDLODEOH IURP H[SHULPHQWDO PHDVXUHPHQWV PXVW EH FDOFXODWHG DQDO\WLFDOO\ E\ VROYLQJ WKH HLJHQYDOXH HLJHQYHFWRU SUREOHP 1XPHULFDO ,OOXVWUDWLRQ 7KH V\VWHP DGGUHVVHG LQ WKLV LQYHVWLJDWLRQ LV WKH FRPPRQO\ XVHG HLJKW GHJUHHV RI IUHHGRP PRGHO VKRZQ LQ )LJXUH 7KLV PRGHO ZDV GHYHORSHG E\ .DEH f WR JLYH D FRPPRQ WHVWEHG IRU WKH HYDOXDWLRQ RI WKH SHUIRUPDQFH RI PRGHO UHILQHPHQW DOJRULWKPV $Q RULJLQDO XQGDPSHG DQDO\WLFDO PRGHO RI WKH V\VWHP ZDV JHQHUDWHG E\ XVLQJ WKH PDVV DQG VWLIIQHVV SURSHUWLHV VKRZQ LQ )LJXUH 7KH HOHPHQWV RI WKH RULJLQDO DQDO\WLFDO VWLIIQHVV PDWUL[ DUH GLVSOD\HG LQ WKH VHFRQG FROXPQ RI 7DEOH ,Q WKLV SUREOHP LW LV DVVXPHG WKDW WKH RULJLQDO VWLIIQHVV PDWUL[ RI WKH PRGHO LV LQFRUUHFW 7KHVH LQDFFXUDFLHV ZHUH VLPXODWHG E\

PAGE 50

PL QLIW PM M NM N N N N A )LJXUH .DEHfV 3UREOHP $QDO\WLFDO 7HVW 6WUXFWXUH XVLQJ LQFRUUHFW VWLIIQHVV FRQVWDQWV IRU PRVW RI WKH VSULQJV 7KH HOHPHQWV RI WKH DFWXDO FRUUHFW VWLIIQHVV PDWUL[ DUH JLYHQ LQ WKH ODVW FROXPQ RI 7DEOH 1RWH WKDW WKH SUHVHQW PRGHO UHILQHPHQW SUREOHP LV YHU\ FKDOOHQJLQJ DQG EHFDXVH RI WKH ODUJH GLIIHUHQFH EHWZHHQ WKH VWLIIQHVV PDWULFHV RI WKH RULJLQDO DQG H[DFW PRGHO ,Q WKLV VWXG\ WZR FDVHV ZLOO EH FRQVLGHUHG ,Q WKH ILUVW FDVH LW LV DVVXPHG WKDW RQO\ WKH PRGDO SDUDPHWHUV HLJHQYDOXH DQG HLJHQYHFWRUf RI WKH ILUVW PRGH ZHUH PHDVXUHG ,Q WKH RWKHU WKH PRGDO SDUDPHWHUV RI ILUVW WKUHH PRGHV DUH DVVXPHG WR EH DYDLODEOH ,Q ERWK FDVHV WKH PHDVXUHG HLJHQYHFWRUV DUH VXSSRVHG WR EH IXOO DOO GHJUHHV RI IUHHGRP RI WKH HLJHQYHFWRUVf DUH PHDVXUHGf 7KH PDLQ REMHFWLYH RI WKLV LQYHVWLJDWLRQ LV WR FRPSDUH WKH SHUIRUPDQFH RI WKH LQYHUVHK\EULG PHWKRG WR WKH DOJRULWKP SURSRVHG E\ %DUXFK DQG %DU ,W]KDFN f 7KH %DUXFK DQG %DU ,W]KDFN PRGHO XSGDWH WHFKQLTXH LV RQH RI WKH PRVW FRPPRQO\ XVHG PRGHO UHILQHPHQW DOJRULWKP 2QH RI WKH PDLQ UHDVRQV IRU LWV FRPPRQ XVH LV EHFDXVH WKH DOJRULWKP LV FRPSXWDWLRQDOO\ HIILFLHQW 7KH XSGDWHG VWLIIQHVV PDWULFHV JHQHUDWHG E\ XVLQJ WKH %DUXFK DQG %DU ,W]KDFN DSSURDFK IRU WKH RQH PRGH DQG WKUHH PRGH FDVHV DUH VKRZQ LQ WKH WKLUG DQG

PAGE 51

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t f ZHUH FRQVLGHUHG

PAGE 52

7DEOH .DEHfV 3UREOHP (OHPHQWDO 6WLIIQHVV &RPSRQHQWV %DUXFK ,QYHUVH+\EULG (OHPHQW 2ULJLQDO 0RGH 0RGHV 0RGH 0RGHV ([DFW f f f f f f f f f f f f f f f f f A f f f f f f f f f f f f f f f f f f f

PAGE 53

&+$37(5 6<00(75,& (,*(16758&785( $66,*10(17 02'(/ 5(),1(0(17 $/*25,7+0 ,QWURGXFWLRQ (LJHQVWUXFWXUH DVVLJQPHQW LV D FRQWURO FRQFHSW XVHG WR DOWHU WKH WUDQVLHQW UHVSRQVH RI OLQHDU V\VWHPV 7KLV LV GRQH E\ IRUFLQJ WKH V\VWHP WR KDYH VRPH SUHGHWHUPLQHG HLJHQYDOXHV DQG HLJHQYHFWRUV $ GHWDLOHG RYHUYLHZ RI HLJHQVWUXFWXUH DVVLJQPHQW WKHRULHV FDQ EH IRXQG LQ WKH SDSHU E\ $QGU\ HW DO f ,QPDQ DQG 0LQDV f =LPPHUPDQ DQG :LGHQJUHQ f DQG :LGHQJUHQ f KDYH GHYHORSHG PRGHO UHILQHPHQW DOJRULWKPV EDVHG RQ WKH PDWKHPDWLFDO IUDPHZRUN RI HLJHQVWUXFWXUH DVVLJQPHQW 7KH EDVLF LGHD RI WKHVH PRGHO UHILQHPHQW WHFKQLTXHV LV WR GHVLJQ WKH SVHXGRFRQWUROOHU ZKLFK LV UHTXLUHG WR SURGXFH WKH PHDVXUHG PRGDO SURSHUWLHV QDWXUDO IUHTXHQFLHV GDPSLQJ UDWLRV DQG PRGH VKDSHVf ZLWK WKH RULJLQDO ILQLWH HOHPHQW PRGHO )(0f RI WKH VWUXFWXUH 7KH SVHXGRFRQWUROOHU LV WKHQ WUDQVODWHG LQWR PDWUL[ DGMXVWPHQWV DSSOLHG WR WKH LQLWLDO )(0 ,Q WKLV ZRUN WKH HLJHQVWUXFWXUH DVVLJQPHQW EDVHG PRGHO UHILQHPHQW DOJRULWKP SURSRVHG E\ =LPPHUPDQ DQG :LGHQJUHQ f LV H[WHQGHG WR EHWWHU DSSURDFK WKH GDPDJH DVVHVVPHQW SUREOHP $ VXEVSDFH URWDWLRQ DOJRULWKP LV GHYHORSHG WR HQKDQFH HLJHQYHFWRU DVVLJQDELOLW\ )LQDOO\ WKH HQKDQFHG DOJRULWKP LV WHVWHG DQG FRPSDUHG WR RWKHU WHFKQLTXHV RQ ERWK fVLPXODWHGf DQG DFWXDO H[SHULPHQWDO GDWD 3UREOHP )RUPXODWLRQ ,Q WKLV VHFWLRQ D UHYLHZ RI WKH =LPPHUPDQ DQG :LGHQJUHQ f UHILQHPHQW WHFKQLTXH ZKLFK LV WHUPHG WKH V\PPHWULF HLJHQVWUXFWXUH DVVLJQPHQW PRGHO UHILQHPHQW

PAGE 54

DOJRULWKP 6($05$f LV SUHVHQWHG 7KLV UHYLHZ LV HVVHQWLDO LQ RUGHU WR SURSHUO\ LQWURGXFH DQG GLVFXVV WKH H[WHQVLRQ DQG LPSURYHPHQW SURSRVHG LQ WKLV ZRUN 6WDQGDUG (LJHQVWUXFWXUH $VVLJQPHQW )RUPXODWLRQ &RQVLGHU WKH VWDQGDUG GLIIHUHQWLDO HTXDWLRQ RI PRWLRQ RI DQ Q GHJUHHV RI IUHHGRP GDPSHG QRQJ\URVFRSLF DQG QRQFLUFXODWRU\ VWUXFWXUH ZLWK FRQWURO IHHGEDFN 0ZWf 'ZWf .ZWf %JXWf f $JDLQ 0 DQG DUH QE\Q UHDO V\PPHWULF PDWUL[ PRGHOV RI WKH PDVV GDPSLQJ DQG VWLIIQHVV SURSHUWLHV RI WKH VWUXFWXUH $VVXPH WKDW WKHVH PDWULFHV ZHUH JHQHUDWHG XVLQJ WKH ILQLWH HOHPHQW PHWKRG 7KH Q[ WLPH YDU\LQJ YHFWRU ZWf UHSUHVHQWV WKH Q GLVSODFHPHQWV RI WKH Q'2) )(0 RI WKH V\VWHP 7KH RYHUGRWV UHSUHVHQW GLIIHUHQWLDWLRQ ZLWK UHVSHFW WR WLPH ,Q FRQWURO WHUPLQRORJ\ %R LV WKH Q[P P m Qf FRQWURO LQIOXHQFH PDWUL[ GHVFULELQJ WKH DFWXDWRU IRUFH GLVWULEXWLRQV DQG XWf LV WKH P[O YHFWRU RI RXWSXW IHHGEDFN FRQWURO IRUFHV GHILQHG E\ XWf )\Wf f ,Q (T f ) LV WKH P[U IHHGEDFN JDLQ PDWUL[ DQG \Wf LV WKH U[O RXWSXW RI VHQVRU PHDVXUHPHQWV GHILQHG E\ \Wf &T: &Z f LQ ZKLFK &R DQG &? DUH WKH U[Q RXWSXW LQIOXHQFH PDWULFHV FRUUHVSRQGLQJ WR SRVLWLRQ DQG YHORFLW\ UHVSHFWLYHO\ $ VXEVWLWXWLRQ RI (TV f DQG f LQWR (T f \LHOGV 0ZWf %)&_fZWf %)&fZWf f ,W LV FOHDU IURP (T f WKDW WKH IHHGEDFN FRQWUROOHU UHVXOWV LQ UHVLGXDO FKDQJHV %R)&R DQG %R)&M WR WKH VWLIIQHVV DQG GDPSLQJ PDWULFHV UHVSHFWLYHO\ 7KHVH FKDQJHV FDQ EH YLHZHG DV SHUWXUEDWLRQV WR WKH LQLWLDO ILQLWH HOHPHQW PRGHO )(0f VXFK WKDW WKH DGMXVWHG )(0 PDWFKHV FORVHO\ WKH H[SHULPHQWDOO\ PHDVXUHG PRGDO SURSHUWLHV 7KH DGMXVWHG )(0 FRQVLVWV RI WKH RULJLQDO PDVV PDWUL[ DQG WKH DGMXVWHG VWLIIQHVV DQG GDPSLQJ PDWULFHV JLYHQ E\

PAGE 55

.D f§ %)& 'D f§ %T)&M f $VVXPH WKDW PRGDO DQDO\VLV RI WKH VWUXFWXUH XQGHU FRQVLGHUDWLRQ KDV EHHQ SHUIRUPHG DQG WKDW S PRGHV S HLJHQYDOXHV ;Hc DQG S HLJHQYHFWRUV YHf KDYH EHHQ LGHQWLILHG $V GLVFXVVHG HDUOLHU LQ &KDSWHU LQ SUDFWLFH S LV W\SLFDOO\ PXFK OHVV WKDQ Q 7KH IHHGEDFN JDLQ PDWUL[ ) VXFK WKDW WKH DGMXVWHG )(0 HLJHQGDWD PDWFKHV WKH H[SHULPHQWDO PRGDO SDUDPHWHUV LV FRPSXWHG XVLQJ VWDQGDUG HLJHQVWUXFWXUH DVVLJQPHQW WKHRULHV $QGU\ HW DO f ) >= $89@ &R &@ f Z Z :$ :$ f ZKHUH $ ,Q[Q 0B. 0n' % 0B. 7 >% 3@ $ 7B$7 n$X ,P[P $O % 7B% f§ [ 9 7 : : :$ :$ $ GLDJ;H_;H ;H: MAgHDO!;HDff}f§HDS@ f§ F ] V U Z L f : :$ $ $ :$ 7 7KH RYHUEDU LQ WKH DERYH HTXDWLRQV LQGLFDWHV WKH FRPSOH[ FRQMXJDWH RSHUDWRU 7KH YHFWRUV YHD LQ PDWUL[ : DUH WKH H[SDQGHG fEHVW DFKLHYDEOHf HLJHQYHFWRUV DVVRFLDWHG ZLWK WKH H[SHULPHQWDOO\ PHDVXUHG HLJHQYHFWRUV YH $Q H[SODQDWLRQ RI WKH FRQFHSW RI fEHVW

PAGE 56

DFKLHYDEOHf HLJHQYHFWRUV LV GLVFXVVHG LQ 6HFWLRQ 7KH VXEPDWUL[ 3 RI PDWUL[ 7 LV DUELWUDU\ DV ORQJ DV 7 LV LQYHUWLEOH $W WKLV SRLQW WKH YDULDEOHV %R &R DQG &M DUH VWLOO DUELWUDU\ $ UDQGRP VHOHFWLRQ RI WKHVH YDULDEOHV ZLOO XVXDOO\ UHVXOW LQ QRQV\PPHWULF SHUWXUEDWLRQ PDWULFHV DQG FRQVHTXHQWO\ QRQV\PPHWULF DGMXVWHG VWLIIQHVV DQG GDPSLQJ PDWULFHV 7KLV FOHDUO\ FRQIOLFWV ZLWK WKH IXQGDPHQWDO V\PPHWU\ UHTXLUHPHQW RI PRVW VWUXFWXUHVf )(0 ,Q WKH IRUPXODWLRQ RI ,QPDQ DQG 0LQDV f WKH UHVXOWLQJ SHUWXUEDWLRQ PDWULFHV IURP WKH SVHXGRFRQWUROOHU DUH IRUFHG WR EH V\PPHWULF WKURXJK D QRQOLQHDU XQFRQVWUDLQHG RSWLPL]DWLRQ SUREOHP =LPPHUPDQ DQG :LGHQJUHQ f SURSRVHG D QRQLWHUDWLYH DQG FRPSXWDWLRQDOO\ PRUH HIILFLHQW DSSURDFK WR VDWLVI\ WKH V\PPHWU\ UHTXLUHPHQW 7KLV DSSURDFK LQ GLVFXVVHG LQ WKH IROORZLQJ VHFWLRQ 6\PPHWULF (LJHQVWUXFWXUH $VVLJQPHQW )RUPXODWLRQ 7KH SHUWXUEDWLRQ PDWULFHV DUH V\PPHWULF LI WKH IROORZLQJ FRQGLWLRQV DUH PHW %f)& &M)7%M Df %)& &`)7%M Ef f $W WKLV SRLQW WZR DGGLWLRQDO DVVXPSWLRQV DUH PDGH $V D SUHUHTXLVLWH WR WKH H[LVWHQFH RI WKH LQYHUVH RI VRPH PDWULFHV XVHG LQ WKH FRPSXWDWLRQV LW LV DVVXPHG WKDW WKH QXPEHU RI SVHXGR VHQVRUV DQG DFWXDWRUV LV HTXDO WR WZLFH WKH QXPEHU RI PHDVXUHG PRGHV P U Sf 7KH RWKHU DVVXPSWLRQ FRQVLVWV RI UHVWULFWLQJ WKH PDWULFHV &R DQG 4 E\ WKH FRQGLWLRQV Df Ef F JEf & *%M ZKHUH *R DQG *M DUH P[P LQYHUWLEOH PDWULFHV $ VXEVWLWXWLRQ RI (T f LQWR (T f VLPSOLILHV WKH V\PPHWU\ FRQGLWLRQV WR WKH IROORZLQJ UHODWLRQVKLSV Df Ef )* *M)7 f )* *`)W

PAGE 57

%\ XVLQJ WKH FRQGLWLRQV LQ (T f DORQJ ZLWK WKH H[SUHVVLRQ IRU WKH IHHGEDFN JDLQ PDWUL[ (T ff D QHFHVVDU\ EXW QRW VXIILFLHQW FRQGLWLRQ RQ *R DQG *M IRU V\PPHWULF SHUWXUEDWLRQ PDWULFHV LV H[SUHVVHG LQ WKH IRUP RI D JHQHUDOL]HG DOJHEUDLF 5LFFDWL HTXDWLRQ $; ;$ ;$; $ >@ f ZKHUH ; *Mn*R $ r B B r r ? r D D fD 0DW f§ [Df [ D r B L B L r $ W D D fD $ [ D fD 0DW WDOD $ D D nD nD ,P[P >:r% $:r% ; r D r :%R $: % D = $M9 7KH PDWULFHV $? : = DQG 9 DUH GHILQHG LQ (TV f 7KH VXSHUVFULSW far LQGLFDWHV WKH LQYHUVH RI WKH FRPSOH[ FRQMXJDWH WUDQVSRVH PDWUL[ (TXDWLRQ f FDQ EH VROYHG IRU ; E\ XVLQJ WKH WHFKQLTXHV GHVFULEHG LQ WKH SDSHUV RI 3RWWHU f RU 0DUWHQVVRQ f ,Q JHQHUDO WKHUH H[LVW PXOWLSOH VROXWLRQV ;fVf WR WKLV JHQHUDOL]HG DOJHEUDLF 5LFFDWL HTXDWLRQ :LWK DOO VROXWLRQV FRPSXWHG WKH QH[W VWHS LV WR GHFRPSRVH WKHVH VROXWLRQV LQWR *RfV DQG *MfV ,W LV VKRZQ LQ WKH SDSHU E\ =LPPHUPDQ DQG :LGHQJUHQ f WKDW IRU D JLYHQ VROXWLRQ ; DQ\ VHOHFWLRQ RI *R DQG *@ VDWLVI\LQJ ; n* UHVXOWV LQ WKH VDPH DGMXVWHG GDPSLQJ 'Df DQG VWLIIQHVV .Df PDWULFHV +HQFH HLWKHU *L RU *Rf FDQ EH FKRVHQ DUELWUDULO\ DV ORQJ DV LWV LQYHUVH H[LVWV 7KHQ *R RU *Mf LV FDOFXODWHG IURP WKH UHODWLRQVKLS ; *Mf n*T )RU HDFK VHW *R *Mf D IHHGEDFN JDLQ PDWUL[ ) LV FDOFXODWHG IURP (T f DQG WKH FRUUHVSRQGLQJ DGMXVWHG GDPSLQJ 'Df DQG VWLIIQHVV .Df PDWULFHV DUH FRPSXWHG XVLQJ (TV

PAGE 58

f $W WKLV SRLQW D UDWLRQDOH LV SURSRVHG WR FKRRVH WKH PRVW PHDQLQJIXO DGMXVWHG GDPSLQJ DQG VWLIIQHVV PDWULFHV $PRQJ DOO FRPSXWHG VHWV 'D .Df LW LV DSSDUHQW WKDW RQO\ WKH RQHV WKDW DUH UHDO DQG V\PPHWULF DUH DFFHSWDEOH :KHQ GHDOLQJ ZLWK D PRGHO UHILQHPHQW SUREOHP DPRQJ DOO DFFHSWDEOH VROXWLRQV WKH ILQDO VHOHFWLRQ FRXOG EH PDGH E\ FKRRVLQJ WKH VHW 'D .Df WKDW PLQLPL]HV WKH FRVW IXQFWLRQ T __ f§ .D __) __ ''D __) f &OHDUO\ WKLV SURFHVV VHOHFWV WKH VHW 'D .Df WKDW UHVXOWV LQ D PLQLPXP FKDQJH IURP WKH RULJLQDO VHW .f 7KH VFDOH IDFWRU T LQ (T f LV XVHG WR JLYH HTXDO ZHLJKW WR WKH FKDQJHV LQ DQG )RU WKH GDPDJH GHWHFWLRQ SUREOHP WKHUH LV QR XQLTXH UDWLRQDOH WR FKRRVH WKH fEHVWf VHW 'D .Df $ SK\VLFDOO\ LQWXLWLYH DSSURDFK LV WR XVH HQJLQHHULQJ MXGJHPHQW LQ VHOHFWLQJ WKH fEHVWf XSGDWHG PRGHO 7KXV DOO DFFHSWDEOH fDGMXVWHGf VHWV RI VROXWLRQV VKRXOG EH LQVSHFWHG WR GHWHUPLQH ZKLFK EHVW SURYLGHV LQIRUPDWLRQ FRQFHUQLQJ WKH VWDWH RI GDPDJH %HVW $FKLHYDEOH (LJHQYHFWRUV )URP VWDQGDUG HLJHQVWUXFWXUH DVVLJQPHQW WKHRU\ $QGU\ HW DO f LW LV VKRZQ WKDW WKH PHDVXUHG HLJHQYHFWRUV DUH QRW DOZD\V H[DFWO\ DVVLJQDEOH WR WKH DGMXVWHG ILQLWH HOHPHQW PRGHO ,Q IDFW LW FDQ EH VKRZQ WKDW WKH PHDVXUHG HLJHQYHFWRUV DUH DVVLJQHG H[DFWO\ LI DQG RQO\ LI WKH\ OLH LQ WKHLU UHVSHFWLYH DFKLHYDEOH VXEVSDFH 7KH DFKLHYDEOH VXEVSDFH DVVRFLDWHG WR WKH LWK PRGH LV GHILQHG E\ /M 0;" ';M .f% f ZKHUH K^ LV WKH PHDVXUHG HLJHQYDOXH RI WKH LWK PRGH :KHQ DOO Q FRPSRQHQWV RI WKH H[SHULPHQWDO HLJHQYHFWRUV DUH DYDLODEOH WKH LWK EHVW DFKLHYDEOH HLJHQYHFWRUV LV GHILQHG DV WKH

PAGE 59

OHDVW VTXDUH SURMHFWLRQ RI WKH LWK H[SHULPHQWDO HLJHQYHFWRU YH RQ WKH LWK DFKLHYDEOH VXEVSDFH /M 7KLV SURMHFWLRQ LV VFKHPDWLFDOO\ LOOXVWUDWHG LQ )LJXUH )LJXUH %HVW $FKLHYDEOH (LJHQYHFWRU 3URMHFWLRQ 7KLV EHVW DFKLHYDEOH HLJHQYHFWRU LV JLYHQ E\ 9 OHD / /L / O /L;HL f :KHQ RQO\ D VXEVHW V RI WKH HLJHQYHFWRU FRPSRQHQWV DUH PHDVXUHG V Q WKH OHDVW VTXDUH SURMHFWLRQ GLVFXVVHG DERYH FDQ EH XVHG WR VLPXOWDQHRXVO\ H[SDQG DQG SURMHFW WKH PHDVXUHG HLJHQYHFWRUV ,Q WKLV FDVH WKH LWK H[SDQGHG EHVW DFKLHYDEOH H[SHULPHQWDO HLJHQYHFWRU LV JLYHQ E\ f§HD /M /c /M /M
PAGE 60

6HOHFWLRQ RI %Q 7KH 6XEVSDFH 5RWDWLRQ 0HWKRG 6R IDU WKH FRQWURO LQIOXHQFH PDWUL[ KDV QRW \HW EHHQ FRPSOHWHO\ GHILQHG 7KH SUHFHGLQJ IRUPXODWLRQ VXJJHVW WKDW GLIIHUHQW %R PD\ SRVVLEO\ UHVXOW LQ GLIIHUHQW DGMXVWHG )(0 +HQFH LW LV HVVHQWLDO WR GHYHORS D SK\VLFDOO\ PHDQLQJIXO UDWLRQDOH WR VHOHFW %R =LPPHUPDQ DQG :LGHQJUHQ f SURSRVHG DQ DSSURDFK WHUPHG WKH PRGH VHOHFWLRQ PHWKRG WKDW FRQVLVWV LQ VHOHFWLQJ %R VXFK WKDW WKH XQPHDVXUHG PRGHV RI WKH VWUXFWXUH DUH QHDUO\ XQFKDQJHG ,Q RWKHU ZRUG %R LV VHOHFWHG VXFK WKDW RQO\ WKH PHDVXUHG PRGHV RI WKH VWUXFWXUH DUH FRUUHFWHG 7KLV VHOHFWLRQ WHFKQLTXH IL[HV WKH DFKLHYDEOH VXEVSDFHV LQ ZKLFK WKH HLJHQYHFWRUV PXVW OLH DQG KHQFH SODFHV D OLPLWDWLRQ RQ WKH DVVLJQPHQW SURFHVV ,Q PRVW VWXGLHG FDVHV WKH H[SHULPHQWDO HLJHQYHFWRUV ZHUH QRW DVVLJQHG H[DFWO\ VLQFH WKHLU DVVLJQPHQW fVXFFHVVf GHSHQGV RQ WKH ORFDWLRQV YLVDYLV WKH DFKLHYDEOH VXEVSDFHV VHW E\ WKH VHOHFWLRQ RI %R ,Q WKLV ZRUN D QHZ PHWKRG RI VHOHFWLQJ %T WHUPHG WKH VXEVSDFH URWDWLRQ PHWKRG LV SURSRVHG 7KH VXEVSDFH URWDWLRQ PHWKRG LV EDVHG RQ VHOHFWLQJ %R VXFK WKDW WKH PHDVXUHG HLJHQYHFWRUV OLH H[DFWO\ LQ WKH DFKLHYDEOH HLJHQYHFWRUV VXEVSDFHV 7KLV SURFHGXUH LV LOOXVWUDWHG LQ )LJXUH DQG LV DFFRPSOLVKHG E\ VHWWLQJ %T DV % EU EI EUS EL Ec E f ZKHUH EU UHDO P; 'NM .fYHM EU LPDJLQDU\ 0N 'NM .fYHM M S ZKHUH YH LV WKH HLJHQYHFWRU DVVRFLDWHG ZLWK WKH MWK H[SHULPHQWDOO\ PHDVXUHG PRGH DQG LW LV DVVXPHG WKDW DOO Q FRPSRQHQWV RI WKH H[SHULPHQWDO HLJHQYHFWRUV DUH DYDLODEOH 7KLV FRXOG EH DFFRPSOLVKHG E\ DQ\ RI WKH SURFHGXUHV GLVFXVVHG LQ &KDSWHU &OHDUO\ ZKHQ %T LV VHOHFWHG DV VKRZQ LQ (T f WKH PHDVXUHG H[SDQGHG HLJHQYHFWRUV OLH H[DFWO\ LQ WKH DFKLHYDEOH VXEVSDFHV GHILQHG LQ (T f +HQFH WKHUH LV QR QHHG IRU WKH SURMHFWLRQ RSHUDWLRQV GHILQHG LQ (T f 7KLV HOLPLQDWHV WKH UHTXLUHG S LQYHUVHV RI Q[Q PDWULFHV

PAGE 61

LQYROYHG LQ FRPSXWLQJ WKH DFKLHYDEOH VXEVSDFHV $V ZLOO EH VHHQ LQ &KDSWHUV DQG WKH HOHPHQWV RI %T DV GHILQHG E\ (T f JLYH DQ LQGLFDWLRQ WR WKH SVHXGRFRQWUROOHU DERXW WKH H[WHQW RI PRGLILFDWLRQ RI HDFK '2) LQ RUGHU IRU WKH VWUXFWXUH WR H[KLELW WKH MWK PHDVXUHG HLJHQYDOXH DQG HLJHQYHFWRU )LJXUH 5RWDWLRQ RI WKH $FKLHYDEOH 6XEVSDFH 1XPHULFDO ,OOXVWUDWLRQV ,Q WKLV VHFWLRQ WKH FKDUDFWHULVWLFV RI WKH SURSRVHG HQKDQFHPHQW WR WKH V\PPHWULF HLJHQVWUXFWXUH DVVLJQPHQW PRGHO UHILQHPHQW DOJRULWKP 6($05$f DUH HYDOXDWHG DQG FRPSDUHG WR RWKHU UHILQHPHQW WHFKQLTXHV IRU WZR H[DPSOH SUREOHPV 7KH ILUVW SUREOHP LV D ZLGHO\XVHG VSULQJPDVV FRPSXWHU VLPXODWHG H[DPSOH .DEH f ,W LV XVHG KHUH IRU WKH SXUSRVHV RI LOOXVWUDWLQJ PRGHO UHILQHPHQW IRU D ODUJH ORFDO GLVFUHSDQF\ DQDORJRXV WR D GDPDJH GHWHFWLRQ VLWXDWLRQ 7KH SKHQRPHQD RI JOREDOORFDO PRGH VZLWFKLQJ DQG ORDG SDWK SUHVHUYDWLRQ DUH H[DPLQHG LQ WKLV SUREOHP 7KH VHFRQG SUREOHP LV XVHG WR LOOXVWUDWH WKH FKDUDFWHULVWLFV RI WKH HQKDQFHG 6($05$ LQ XSGDWLQJ WKH ILQLWH HOHPHQW PRGHO )(0f RI D ODERUDWRU\ FDQWLOHYHU EHDP XVLQJ DFWXDO PHDVXUHG PRGDO SDUDPHWHUV

PAGE 62

'DPDJH 'HWHFWLRQ .DEHfV 3UREOHP .DEHfV HLJKW GHJUHH RI IUHHGRP VSULQJPDVV V\VWHP LV VKRZQ LQ )LJXUH 7KH PDVV DQG VWLIIQHVV SURSHUWLHV RI WKH V\VWHP DUH LQFOXGHG LQ WKH ILJXUH 7KLV SUREOHP SUHVHQWV D FKDOOHQJLQJ VLWXDWLRQ IRU GDPDJH GHWHFWLRQ LQ WKDW VWLIIQHVV YDOXHV RI YDULRXV PDJQLWXGHV DUH LQFOXGHG 7KH PRGHO H[KLELWV FORVHO\VSDFHG IUHTXHQFLHV DQG ERWK ORFDO DQG JOREDO PRGHV RI YLEUDWLRQ PL NM PJ PM M N N N N N )LJXUH .DEHfV 3UREOHP $ YDULDWLRQ RI .DEHfV RULJLQDO SUREOHP LV XVHG KHUH 5DWKHU WKDQ WKH VWDQGDUG LQLWLDO PRGHO FRPPRQO\ XVHG ZKLFK KDV LQFRUUHFW YDOXHV IRU DOO RI WKH FRQQHFWLQJ VSULQJV RQO\ D VLQJOH VSULQJ FRQVWDQW LV FKDQJHG 7KLV LV UHIOHFWLYH RI WKH IDFW WKDW GDPDJH PD\ RFFXU DV D ODUJH ORFDO FKDQJH LQ WKH VWLIIQHVV RI D VWUXFWXUDO PHPEHU

PAGE 63

/RFDO WR *OREDO 0RGH &KDQJH ,Q WKH ILUVW SUREOHP .DEHfV LQLWLDO PRGHO LV RQO\ LQFRUUHFW IRU WKH VSULQJ EHWZHHQ PDVVHV DQG $ YDOXH RI ILYH WLPHV WKDW RI WKH H[DFW VSULQJ LV DVVXPHG LQ WKLV SUREOHP &KDQJLQJ WKH VSULQJ YDOXH IURP WR DOVR FDXVHV D ORFDO PRGH RI YLEUDWLRQ WR EH UHSODFHG E\ D JOREDO PRGH WKXV SUHVHQWLQJ D GLIILFXOW FKDOOHQJH IRU GDPDJH GHWHFWLRQ )LJXUH SUHVHQWV HOHPHQWE\HOHPHQW VWLIIQHVV PDWUL[ UHVXOWV IRU DSSO\LQJ WKH %DUXFK DQG %DU ,W]KDFN XSGDWH f DQG WKH V\PPHWULF HLJHQVWUXFWXUH DVVLJQPHQW PRGHO UHILQHPHQW DOJRULWKP %DUXFK 'DPDJH LQGLFDWHV WKDW WKH XSGDWH ZDV PDGH XVLQJ %DUXFK DQG %DU ,W]KDFNfV DOJRULWKP 6($0 'DPDJH LQGLFDWHV WKDW WKH XSGDWH ZDV PDGH XVLQJ WKH 6($05$ ZLWK %R VHOHFWHG E\ XVLQJ WKH PRGDO 0f VHOHFWLRQ PHWKRG 6($65 'DPDJH LQGLFDWHV WKDW WKH XSGDWH ZDV PDGH E\ XVLQJ WKH 6($05$ ZLWK %R VHOHFWHG XVLQJ 6XEVSDFH 5RWDWLRQ 65f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f WKH GDPDJH YHFWRU LV JLYHQ DV D > @7 ZKHUH WKH HOHPHQW QXPEHU FRUUHVSRQGV WR WKH VWUXFWXUDO '2)V DQG D QXPEHU OHVV WKDQ LQGLFDWHV WKH

PAGE 64

$FWXDO 'DPDJH DL F nE ‘F 2 HV [ : HV F nE fF 2 Z RQ ,QGLFHV 6($0 'DPDJH ,QGLFHV F6 & nE fF 2 L IV£ RQ ‘ : RQ 6($65 'DPDJH ,QGLFHV )LJXUH 5HVXOWV IRU .DEHfV 3UREOHP XVLQJ WKH VW 0RGH )XOO (LJHQYHFWRU SRVVLELOLW\ RI GDPDJH DIIHFWLQJ WKDW '2) ,W LV REYLRXV WKDW '2)V DQG DUH DIIHFWHG E\ GDPDJH EXW WKH UHVXOWV DOVR LQGLFDWH VWURQJ GDPDJH RI '2) ,Q WKH VHFRQG FDVH DV VKRZQ LQ )LJXUH LW LV DVVXPHG WKDW WKH ILUVW WKUHH PRGHV RI YLEUDWLRQ KDYH EHHQ PHDVXUHG EXW RQO\ WKH ILUVW WKUHH FRPSRQHQWV RI WKH HLJHQYHFWRUV KDYH EHHQ PHDVXUHG 7KH HLJHQYHFWRUV FRPSRQHQWV ZHUH H[SDQGHG IRU WKH %DUXFK XSGDWH XVLQJ G\QDPLF H[SDQVLRQ %HUPDQ DQG 1DJ\ f ZLWK VXEVHTXHQW RUWKRJRQDOL]DWLRQ %DUXFK DQG %DU ,W]KDFN f 7KH OHDVW VTXDUHV H[SDQVLRQ ZDV XVHG IRU WKH 6($0 XSGDWH 7KH 6($65 XSGDWH XWLOL]HG WKH RUWKRJRQDO 3URFUXVWHV H[SDQVLRQ &KDSWHU f ,Q FRPSDULQJ )LJXUH WR )LJXUH LW LV FOHDU WKDW WKH GDPDJH GHWHFWLRQ FDSDELOLWLHV RI DOO WKUHH DOJRULWKPV KDYH EHHQ GHJUDGHG ZKHQ XVLQJ H[SDQGHG PRGH VKDSHV HYHQ WKRXJK PRUH PRGHV KDYH EHHQ PHDVXUHG +RZHYHU ERWK WKH 6($0 DQG 6($65 XSGDWHV JLYH D FOHDU

PAGE 65

$FWXDO 'DPDJH FG & nE fF 2 R FG [ : %DUXFK 'DPDJH D & nG f& 2 : RQ ,QGLFHV 6($0 'DPDJH ,QGLFHV Q6 & nG fF 2 R &4 FG F nG ‘F 2 O DL RQ Z RQ ,QGLFHV 6($65 'DPDJH ,QGLFHV )LJXUH 5HVXOWV IRU .DEHfV 3UREOHP XVLQJ 0RGHV DQG (LJHQYHFWRUV &RPSRQHQWV LQGLFDWLRQ WR ERWK WKH ORFDWLRQ DQG H[WHQW RI GDPDJH 8VLQJ /LQf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

PAGE 66

U & fE fF 2 \ RV [ : $FWXDO 'DPDJH Q F nE f& 2 Z RQ ,QGLFHV ,QGLFHV 6($0 'DPDJH Q & nE ‘& 2 ‘ & R J m %DUXFK 'DPDJH HG F fE r& R &' L Z RQ ,QGLFHV 6($65 'DPDJH ,QGLFHV )LJXUH 5HVXOWV IRU .DEHf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

PAGE 67

,QGLFHV 6($0 'DPDJH F nE : A ,QGLFHV 2 ,QGLFHV )LJXUH 5HVXOWV IRU .DEHfV 3UREOHP XVLQJ WKH VW 0RGH )XOO (LJHQYHFWRUV XVHG LQ WKH XSGDWH ,W VKRXOG EH QRWHG WKDW WKH %DUXFK DQG 6($0 XSGDWH WHQGV WR IRFXV HOHPHQWDO FKDQJHV LQ WKH IRXUWK DQG VL[WK URZV RI WKH VWLIIQHVV PDWUL[ LQGLFDWLQJ WKH SRVVLELOLW\ RI GDPDJH EHWZHHQ WKHVH GHJUHHV RI IUHHGRP EXW FHUWDLQO\ JLYLQJ QR FOHDU LQGLFDWLRQ WR WKH H[WHQW RI GDPDJH 8VLQJ /LQf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

PAGE 68

6($0 2ULJLQDO ([DFW 2ULJLQDO XQDEOH WR SUHGLFW WKH H[DFW H[WHQW 8VLQJ /LQfV DOJRULWKP WKH GDPDJH YHFWRU LV JLYHQ DV DB > @7 $JDLQ LW LV GLIILFXOW IURP LQVSHFWLRQ RI D WR GHWHUPLQH WKH ORFDWLRQ RI GDPDJH ,Q IDFW LQVSHFWLRQ RI D LQGLFDWHV WKDW '2) LV WKH PRVW OLNHO\ GDPDJHG '2) ,QGLFHV P %DUXFK 'DPDJH ,QGLFHV )LJXUH 5HVXOWV IRU .DEHfV 3UREOHP XVLQJ 0RGHV DQG (LJHQYHFWRUV &RPSRQHQWV ,W VKRXOG EH QRWHG WKDW LQ WKLV SUREOHP LW ZDV FULWLFDO WR KDYH WKH SURSHU '2)V PHDVXUHG :KHQ WKH VHFRQG WHVW FDVH ZDV UXQ ZLWK WKH ILUVW WKUHH '2)V PHDVXUHG QR DOJRULWKP ZDV DEOH WR ORFDWH GDPDJH ,Q WKLV FDVH WKH HLJHQYHFWRUV FRPSRQHQWV ZHUH UHODWLYHO\ XQDIIHFWHG E\ GDPDJH WKXV FDXVLQJ VXEVWDQWLDO HUURU LQ WKH HLJHQYHFWRU H[SDQVLRQ SURFHVV

PAGE 69

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fV ILUVW WKUHH PRGHV RI YLEUDWLRQ ZHUH LGHQWLILHG DQG DUH UHSRUWHG LQ 7DEOHV DQG )LJXUH ([SHULPHQWDO &DQWLOHYHU %HDP

PAGE 70

7DEOH 6WUXFWXUDO 3URSHUWLHV RI WKH &DQWLOHYHU %HDP /HQJWK P 0DVV/HQJWK NJP 0RPHQW RI ,QHUWLD H P
PAGE 71

*X\DQ UHGXFWLRQ f HOLPLQDWLQJ WKH URWDWLRQDO GHJUHHV RI IUHHGRP 7KHUH DUH VHYHUDO SRVVLEOH HUURUV DIIHFWLQJ WKH DFFXUDF\ RI WKLV )(0 7KH PRVW REYLRXV LV WKH IDFW WKDW D SHUIHFW FDQWLOHYHU FRQGLWLRQ LV DVVXPHG ,Q DGGLWLRQ DQ DUWLILFLDO HUURU ZDV SXUSRVHO\ LQWURGXFHG E\ VHOHFWLQJ WKH
PAGE 72

6($05$ )LJXUH ([SHULPHQWDO DQG $QDO\WLFDO )UHTXHQF\ 5HVSRQVH )XQFWLRQV RI WKH &DQWLOHYHU %HDP PRGDO SURSHUWLHV ZHUH FRUUXSWHG E\ PHDVXUHPHQW HUURUV 7KLV FDQ EH DWWULEXWHG WR WKH IDFW WKDW WKHUH LV QR V\PPHWULF XSGDWHG )(0 WKDW LV FRQVLVWHQW ZLWK WKH SUHVHQW 6($05$fV IRUPXODWLRQ 5HFDOO WKDW 6($05$fV PRGLILFDWLRQV WR WKH LQLWLDO VWLIIQHVV DQG GDPSLQJ PDWULFHV LQ LWV XSGDWLQJ SURFHVV DUH JLYHQ E\ $. %+R%ZLWK + )* $' %T+M%f ZLWK + )*@ &OHDUO\ IURP (T f WKH SHUWXUEDWLRQV LH PRGLILFDWLRQVf WR WKH LQLWLDO VWLIIQHVV DQG GDPSLQJ PDWULFHV DUH FRQVWUDLQHG E\ WKH UHODWLRQVKLS UDQJH$.f UDQJH$.7f UDQJH$'f UDQJH$'7f f

PAGE 73

7KLV UHODWLRQVKLS FDQ DOZD\V EH VDWLVILHG IRU WKH FDVHV ZKHQ WKH H[SHULPHQWDO PRGDO SURSHUWLHV DUH FRQVLVWHQW ZLWK DQ XSGDWH LQ ZKLFK Lf HLWKHU $. RU $' LV ]HUR RU LLf $. LV SURSRUWLRQDO WR $' $. $' LV D VFDODUf )RU DOO RWKHU FDVHV 6($05$ PLJKW IDLO WR SURGXFH V\PPHWULF $. DQG $' $ PRUH IOH[LEOH DQG JHQHUDO IRUPXODWLRQ WKDW DFFRXQWV IRU VXFK VKRUWFRPLQJ LV SURSRVHG LQ 6HFWLRQ RI &KDSWHU 7KH IRUPXODWLRQ DV SUHVHQWHG LQ &KDSWHU LV PRUH HOHJDQW HIILFLHQW DQG JXDUDQWHHV IRU DOO VLWXDWLRQV D V\PPHWULF XSGDWHG )(0 6XPPDU\ $ SUHYLRXVO\ GHYHORSHG PRGHO UHILQHPHQW DOJRULWKP EDVHG RQ WKH JHQHUDO PDWKHPDWLFDO IUDPHZRUN RI HLJHQVWUXFWXUH DVVLJQPHQW WKHRU\ =LPPHUPDQ DQG :LGHQJUHQ f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f ZDV WHVWHG IRU LWV VXLWDELOLW\ IRU PRGHO UHILQHPHQW DQG VWUXFWXUDO GDPDJH DVVHVVPHQW 7KH SHUIRUPDQFHV RI 6($65 LQ D GDPDJH DVVHVVPHQW SUREOHP RQ D FKDOOHQJLQJ VLPXODWHG VWUXFWXUH ZDV SUHVHQWHG DQG FRPSDUHG WR RWKHU DOJRULWKPV 7KH UHVXOWV DFTXLUHG XVLQJ WKH 6($65 ZHUH VXSHULRU

PAGE 74

&+$37(5 '$0$*( /2&$7,21 7+( 68%63$&( 527$7,21 $/*25,7+0 ,QWURGXFWLRQ ,Q WKLV FKDSWHU D FRPSXWDWLRQDOO\ DWWUDFWLYH DOJRULWKP LV SURSRVHG WR SURYLGH DQ LQVLJKW WR WKH ORFDWLRQ RI VWUXFWXUDO GDPDJH 7KH SURSRVHG DOJRULWKP LV VLPLODU WR WKH 0RGDO )RUFH (UURU &ULWHULD SURSRVHG E\ VHYHUDO UHVHDUFKHUV 2MDOYR DQG 3LOQ *\VLQ f +RZHYHU D JUHDWHU LQVLJKW RI WKH 0RGDO )RUFH (UURU FULWHULD LV SURYLGHG )XUWKHU D QHZ YLHZSRLQW ZKLFK DOORZV IRU WKH UHGXFWLRQ RI WKH HIIHFWV RI PHDVXUHPHQW HUURUV LQ WKH H[SHULPHQWDO PRGDO SDUDPHWHUV IRU D FHUWDLQ FODVV RI VWUXFWXUHV LV DOVR GLVFXVVHG $V ZLOO EH VKRZQ LQ WKH QH[W VHFWLRQV WKH SURSRVHG GDPDJH ORFDWLRQ DOJRULWKP UHTXLUHV RQO\ PDWUL[VFDODU DQG PDWUL[YHFWRU PXOWLSOLFDWLRQ 7KH 6XEVSDFH 5RWDWLRQ $OJRULWKP 7KH 'LUHFW 0HWKRG $VVXPH WKDW DQ Q'2) ILQLWH HOHPHQW PRGHO RI WKH fKHDOWK\f XQGDPDJHGf VWUXFWXUH H[LVWV $V VHHQ LQ WKH HDUOLHU FKDSWHUV WKH VWDQGDUG GLIIHUHQWLDO HTXDWLRQ JRYHUQLQJ WKH G\QDPLF PRWLRQ RI VXFK VWUXFWXUHV LV JLYHQ E\ f 0Z 'Z .Z ZKHUH 0 DQG DUH WKH Q [ Q DQDO\WLFDO PDVV GDPSLQJ DQG VWLIIQHVV PDWULFHV Z LV D Q [ YHFWRU RI SRVLWLRQV DQG WKH RYHUGRWV UHSUHVHQW GLIIHUHQWLDWLRQ ZLWK UHVSHFW WR WLPH 7KH HLJHQYDOXH SUREOHP DVVRFLDWHG ZLWK (T f LQ VHFRQG RUGHU ODPEGDf IRUP LV JLYHQ DV f

PAGE 75

ZKHUH ;K DQG YK GHQRWH WKH LWK HLJHQYDOXH DQG HLJHQYHFWRU UHVSHFWLYHO\ RI WKH SUHGDPDJHG fKHDOWK\f VWUXFWXUH ,W LV DVVXPHG WKDW (T f LV VDWLVILHG IRU DOO PHDVXUHG fKHDOWK\f HLJHQYDOXHVHLJHQYHFWRUV 7KLV FDQ EH HQIRUFHG E\ FRUUHODWLQJ WKH RULJLQDO )(0 0' DQG .f SRVVLEO\ WKURXJK WKH XVH RI D PRGHO UHILQHPHQW SURFHGXUH 1H[W FRQVLGHU WKDW WKH S HLJHQYDOXHV DQG HLJHQYHFWRUV ;Gc DQG YA RI D SRVWGDPDJH PRGDO VXUYH\ RI WKH VWUXFWXUH DUH DYDLODEOH LQ ZKLFK AG ;K YG A \K ,Q WKH SUHVHQW IRUPXODWLRQ LW LV DVVXPHG WKDW WKH GLPHQVLRQ RI WKH PHDVXUHG HLJHQYHFWRU LV WKH VDPH DV WKH DQDO\WLFDO HLJHQYHFWRU $V GLVFXVVHG LQ &KDSWHU WKLV LV WUXH Lf ZKHQ DOO )(0 '2)V DUH PHDVXUHG LLf DIWHU WKH DSSOLFDWLRQ RI DQ HLJHQYHFWRU H[SDQVLRQ DOJRULWKP RU LLLf DIWHU WKH DSSOLFDWLRQ RI D ILQLWH HOHPHQW PRGHO UHGXFWLRQ DOJRULWKP 7KH LGHDO VLWXDWLRQ ZRXOG EH WR PHDVXUH DOO )(0 '2)V VLQFH WKH HLJHQYHFWRU H[SDQVLRQ SURFHVV ZRXOG LQWURGXFH DGGLWLRQDO HUURUV LQ WKH fH[SDQGHGf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f ;G' $'Gf $.ffA f $OWKRXJK RQO\ S RI WKH Q HLJHQYDOXHVHLJHQYHFWRUV DUH DVVXPHG PHDVXUHG S m Q (T f KROGV IRU DQ\ SDUWLFXODU HLJHQYDOXH DQG HLJHQYHFWRU RI WKH GDPDJHG VWUXFWXUH EHFDXVH WKH SHUWXUEDWLRQ PDWULFHV DUH DVVXPHG WR EH H[DFW *URXSLQJ DOO SHUWXUEDWLRQ PDWULFHV RQ WKH ULJKWKDQG VLGH GHILQHV D GDPDJH YHFWRU Gc Df Ef ZKHUH =G ;A0 ;G' .

PAGE 76

$OWKRXJK (TV D Ef \LHOG WKH VDPH GDPDJH YHFWRU LW VKRXOG EH QRWHG WKDW =G DQG WKH FRHIILFLHQW PDWUL[ RI (T Ef DUH QRW HTXDO :KHQ WKH PHDVXUHG HLJHQGDWD DUH QRW FRUUXSWHG E\ QRLVH DQ LQVSHFWLRQ RI Gc LQ WHUPV RI WKH (T Ef UHYHDOV WKDW WKH MWK HOHPHQW RI GM ZLOO EH ]HUR ZKHQ WKH MWK URZV RI WKH SHUWXUEDWLRQ PDWULFHV DUH ]HUR LH WKH ILQLWH HOHPHQW PRGHO IRU WKH MWK GHJUHH RI IUHHGRP LV QRW GLUHFWO\ DIIHFWHG E\ GDPDJH &RQYHUVHO\ D GHJUHH RI IUHHGRP ZKRVH ILQLWH HOHPHQW PRGHO KDV EHHQ DIIHFWHG E\ GDPDJH ZLOO UHVXOW LQ D QRQ]HUR HQWU\ LQ Gc 7KXV WKH GHJUHHV RI IUHHGRP ZKLFK KDYH EHHQ DIIHFWHG E\ GDPDJH FDQ EH GHWHUPLQHG E\ LQVSHFWLQJ WKH HOHPHQWV RI Gc 9HFWRU Gc DV GHILQHG LQ (T Ef DOVR UHYHDOV WKDW RQO\ D VLQJOH PRGH RI YLEUDWLRQ QHHGV WR EH PHDVXUHG H[DFWO\ WR GHWHUPLQH WKH GDPDJH ORFDWLRQV H[DFWO\ 7KLV LV WUXH LQ HYHQ PXOWLSOH PHPEHU GDPDJH VLWXDWLRQV 0RUH LPSRUWDQWO\ WKH YHFWRU Gc FDQ EH GHWHUPLQHG IURP WKH RULJLQDO ILQLWH HOHPHQW PRGHO 0'.f DQG WKH PHDVXUHG HLJHQYDOXHV DQG HLJHQYHFWRUV ;G DQG \G XVLQJ (T Df 7KXV WKHUH LV QR QHHG WR XVH D PRGHO UHILQHPHQW DOJRULWKP WR DWWHPSW WR HVWLPDWH WKH H[DFW SHUWXUEDWLRQ PDWULFHV LQ RUGHU WR ORFDWH WKH GDPDJH ,I WKH GDPSLQJ WHUP LV LJQRUHG (T Df LV HVVHQWLDOO\ WKH 0RGDO )RUFH (UURU FULWHULD DV SURSRVHG E\ 2MDOYR f IRU XVH DV D GLDJQRVWLF fWRROf WR ORFDWH PRGHOOLQJ HUURUV LQ )(0V $ SK\VLFDO LQWHUSUHWDWLRQ RI (T Df SURYLGHG E\ 2MDOYR ZDV f Gc LV WKH DSSOLHG KDUPRQLF IRUFH HUURU GLVWULEXWLRQ DSSOLHG DW IUHTXHQF\ ;G ZKLFK LV QHFHVVDU\ WR FDXVH WKH DQDO\WLFDO PRGHO WR YLEUDWH ZLWK PRGH VKDSH YG f +RZHYHU (T Ef SURYLGHV D PXFK FOHDUHU LQWHUSUHWDWLRQ RI WKH GDPDJH YHFWRU Gc IRU WKH GDPDJH ORFDWLRQ SUREOHP LQ ZKLFK WKH SHUWXUEDWLRQ PDWULFHV DUH VSDUVH ,Q SUDFWLFH WKH SHUIHFW ]HURQRQ]HUR SDWWHUQ RI WKH GDPDJH YHFWRU Gc UDUHO\ RFFXUV GXH WR HUURUV SUHVHQW LQ WKH H[SHULPHQWDOO\ PHDVXUHG HLJHQYDOXHV DQG HLJHQYHFWRUV 7KLV VFHQDULR ZDV VWXGLHG DQG GLVFXVVHG IRU WKH XQGDPSHG FDVH E\ *\VLQ f LQ WKH FRQWH[W RI HLJHQYHFWRU H[SDQVLRQ WHFKQLTXHV *\VLQ f REVHUYHG WKDW LQ FHUWDLQ VSHFLILF FDVHV RI HLJHQYHFWRU H[SDQVLRQ HUURUV WKH GDPDJH YHFWRU GHILQHG E\ (T Df PD\ OHDG WR LQFRUUHFW FRQFOXVLRQV FRQFHUQLQJ WKH ORFDWLRQ RI GDPDJH 7KH QH[W VHFWLRQ SURSRVHV D QHZ

PAGE 77

YLHZSRLQW ZKLFK DOORZV IRU WKH UHGXFWLRQ RI WKH HIIHFWV RI PHDVXUHPHQW HUURUV IRU FHUWDLQ FODVVHV RI VWUXFWXUHV 7KH 6XEVSDFH 5RWDWLRQ $OJRULWKP 7KH $QJOH 3HUWXUEDWLRQ 0HWKRG ,Q RUGHU WR SURYLGH DQ DOWHUQDWLYH YLHZ RI WKH VWDWH RI GDPDJH (T Df LV UHZULWWHQ DV GM YGL __ ,, ,, YGc __ FRV-f f ZKHUH GO LV WKH MWK FRPSRQHQW RU MWK '2)f RI WKH LWK GDPDJH YHFWRU ]-G LV WKH MWK URZ RI WKH PDWUL[ = DQG  LV WKH DQJOH EHWZHHQ WKH YHFWRUV ]? DQG YU, rL L r f§Gc Xc ,Q WKH FDVH ZKHQ WKH PHDVXUHPHQWV DUH IUHH RI HUURU D ]HUR GM FRUUHVSRQGV WR D M RI QLQHW\ GHJUHHV ZKHUHDV D QRQ]HUR GM FRUUHVSRQGV WR D M GLIIHUHQW IURP QLQHW\ GHJUHHV (UURUV LQ WKH H[SHULPHQWDO PHDVXUHPHQWV RI PRGDO SDUDPHWHUV ZLOO FDXVH VOLJKW SHUWXUEDWLRQV LQ WKH DQJOHV  WKDW GHVWUR\ WKH ]HURQRQ]HUR SDWWHUQ RI WKH GDPDJH YHFWRU 2QH ZRXOG LQLWLDOO\ H[SHFW WKDW WKH FRPSRQHQWV RI G FRUUHVSRQGLQJ WR WKH GDPDJHG '2)V ZRXOG EH VXEVWDQWLDOO\ ODUJHU WKDQ WKH RWKHU HOHPHQWV +RZHYHU E\ LQVSHFWLQJ (T f D ODUJH Gc FRPSRQHQW FRXOG EH GXH WR D ]-G URZ QRUP VXEVWDQWLDOO\ ODUJHU WKDQ RWKHU URZV RI =G FRXSOHG ZLWK D VOLJKW GHYLDWLRQ RI IURP QLQHW\ GHJUHHV GXH WR PHDVXUHPHQW QRLVH +HQFH ZKHQ GHDOLQJ ZLWK D VWUXFWXUH ZKRVH )(0 UHVXOWV LQ ]URZ QRUPV RI GLIIHUHQW RUGHU RI GL PDJQLWXGH LW LV PRUH UHDVRQDEOH WR XVH WKH GHYLDWLRQ RI WKH DQJOHV M IURP QLQHW\ GHJUHHV IRU GDPDJH ORFDWLRQ DL M r ZKHUH f &26 f 7KH DQJOH  LV GHWHUPLQHG IURP (T f DQG DM LV WKH MWK FRPSRQHQW RI Dc

PAGE 78

3UDFWLFDO ,VVXHV &XPXODWLYH 'DPDJH /RFDWLRQ 9HFWRUV 7KH GLVFXVVLRQ LQ WKH SUHYLRXV VHFWLRQ VXJJHVWV WKDW IRU D JLYHQ PRGH WKH GDPDJH LV ORFDWDEOH LI WKH SHUWXUEDWLRQ WR WKH DQJOH GXH WR WKH PHDVXUHPHQW HUURU LV OHVV WKDQ WKH DQJOH SHUWXUEDWLRQ GXH WR WKH GDPDJH +HQFH PRGHV WKDW DUH KLJKO\ DIIHFWHG E\ WKH GDPDJH DUH H[SHFWHG WR SURYLGH EHWWHU DVVHVVPHQW WR WKH ORFDWLRQ RI WKH GDPDJH ZKHQ HUURUV DUH SUHVHQW LQ WKH PHDVXUHG HLJHQGDWD $V ZLOO EH VHHQ LQ H[DPSOH SUREOHPV &KDSWHU f FHUWDLQ PRGHV DUH PRUH VXVFHSWLEOH WR D JLYHQ VWDWH RI GDPDJH WKDQ RWKHUV 7KLV LV PDLQO\ GXH WR WKH IDFW WKDW GLIIHUHQW HOHPHQWV RI WKH VWUXFWXUH KDYH GLIIHUHQW OHYHOV RI FRQWULEXWLRQ WR WKH WRWDO VWUDLQ HQHUJ\ RI D JLYHQ PRGH .DVKDQJDNL f )XUWKHUPRUH D SDUWLFXODU SDUW RI WKH VWUXFWXUH XVXDOO\ KDV GLIIHUHQW RUGHUV RI VWUDLQ HQHUJ\ FRQWULEXWLRQ IRU GLIIHUHQW PRGHV 8VXDOO\ LI WKH GDPDJH RFFXUV LQ D UHJLRQ RI KLJK VWUDLQ HQHUJ\ IRU D JLYHQ PRGH WKDW PRGH ZRXOG EH KLJKO\ VXVFHSWLEOH WR WKH GDPDJH DQG KHQFH ZRXOG UHIOHFW WKH VWDWH RI GDPDJH 7R DFFRPPRGDWH WKLV W\SH RI SUREOHP ZKHQ WKH QXPEHU RI PHDVXUHG PRGHV S LV JUHDWHU WKDQ RQH WZR GLIIHUHQW FRPSRVLWH GDPDJH YHFWRUV PD\ EH GHILQHG DV 3 m _Dc_ $f L L ,Q (T f WKH GDPDJH YHFWRUV Gc DUH QRUPDOL]HG ZLWK UHVSHFW WR WKHLU FRUUHVSRQGLQJ HLJHQYHFWRUV YG 7KH UHDVRQ IRU WKLV QRUPDOL]DWLRQ LV VXFK WKDW WKH FRPSRVLWH YHFWRU LJQRUHV WKH LQKHUHQW fZHLJKWLQJf RI __ \G __ ZKLFK LV XVXDOO\ RI GLIIHUHQW RUGHUV RI PDJQLWXGHV IRU GLIIHUHQW PHDVXUHG PRGHV ,W VKRXOG EH QRWHG WKDW LQ WKH PXOWLPRGH PHDVXUHPHQW FDVH (T f LV SUHIHUDEOH ZKHQ WKH YDOXHV RI __ ]-G __ DUH RI GLIIHUHQW RUGHUV RI PDJQLWXGH IRU GLIIHUHQW PHDVXUHG PRGHV $JDLQ LQ SUDFWLFH WKH '2)V DIIHFWHG E\ WKH GDPDJH DUH H[SHFWHG

PAGE 79

WR KDYH VXEVWDQWLDOO\ ODUJHU G RU D )LQDOO\ WKH GDPDJHG DUHDV RI WKH VWUXFWXUH FDQ WKHQ EH ORFDWHG XVLQJ WKH NQRZOHGJH RI WKH fGDPDJHGf '2)V DQG WKH FRQQHFWLYLW\ RI WKH )(0 ,W LV LQWHUHVWLQJ WR QRWH WKDW (TV D Ef UHYHDO DQ LQWHUHVWLQJ UHODWLRQVKLS EHWZHHQ YDULRXV PRGHO UHILQHPHQW DOJRULWKPV 0RGHO UHILQHPHQW WHFKQLTXHV DWWHPSW WR DSSUR[LPDWH WKH H[DFW SHUWXUEDWLRQ PDWULFHV E\ XVLQJ OLPLWHG PRGDO GDWD EXW GR VR LQ GLIIHUHQW PDQQHUV +RZHYHU (T D Ef LQGLFDWHV WKDW LI WKH PRGHO UHILQHPHQW WHFKQLTXH KDV VDWLVIDFWLRQ RI WKH HLJHQSUREOHP DV DQ HTXDOLW\ FRQVWUDLQW WKH FDOFXODWHG SHUWXUEDWLRQ PDWULFHV $0 $' DQG $. DUH FRQVWUDLQHG WR EH UHODWHG WR WKH RULJLQDO ILQLWH HOHPHQW PRGHO 0 DQG DQG WKH PHDVXUHG HLJHQGDWD E\ (TV D Ef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f RU Ef DQG WKH RULJLQDO )(0 FRQQHFWLYLW\ WKH HQJLQHHU FDQ GHGXFH ZKLFK '2)V KDYH EHHQ GDPDJHG ,W LV UHDVRQDEOH WR DVVXPH WKDW QRQ]HUR HOHPHQWV LQ HDFK GDPDJH YHFWRU Gc DVVRFLDWHG ZLWK fXQGDPDJHG '2)Vf DUH GXH WR HLJHQYHFWRU HUURUV 7KHVH HOHPHQWV FDQ WKHQ EH VHW WR ]HUR ,Q DGGLWLRQ WKH PDJQLWXGH RI WKH HOHPHQWV RI Gc DW WKH fGDPDJHGf '2)V FDQ EH DGMXVWHG E\ XVLQJ NQRZOHGJH RI '2) FRQQHFWLYLW\ DQG WKH SURSHUWLHV RI WKH HOHPHQW SURSHUW\ PDWULFHV FRQQHFWLQJ WKH fGDPDJHG '2)Vf 7KH HOHPHQW SURSHUW\ PDWULFHV SURYLGH FRQVWUDLQWV UHODWLQJ WKH HIIHFW RI GDPDJH RQ HDFK HOHPHQW '2) 7KH QRLVH ILOWHULQJ DOJRULWKP FRQVLVWV VLPSO\ RI UHSODFLQJ WKH Gc YHFWRUV E\ GI ZKHUH GI LV REWDLQHG IURP Gc DV GHVFULEHG DERYH 7KH LWK ILOWHUHG HLJHQYHFWRU \GI FDQ WKHQ EH REWDLQHG IURP VROYLQJ f

PAGE 80

XVLQJ *DXVVLDQ HOLPLQDWLRQ ,Q WKLV FDOFXODWLRQ WKH EDQGHGQHVV RI W\SLFDO )(0 PDWULFHV VKRXOG EH H[SORLWHG (VVHQWLDOO\ WKH ILOWHUHG HLJHQYHFWRU LV MXVW WKH HLJHQYHFWRU WKDW LI PHDVXUHG ZRXOG KDYH SURGXFHG WKH GDPDJH YHFWRU GI ([SHULHQFH JDLQHG LQ XVLQJ WKH HLJHQYHFWRU ILOWHULQJ DOJRULWKP LQGLFDWHV WKDW LW LV EHVW WR XVH VWUXFWXUDO PDWUL[ SURSHUWLHV 0 .f WKDW Lf DUH ILQLWH HOHPHQW FRQVLVWHQW DQG LLf KDYH QRW EHHQ fFRUUXSWHGf E\ PHDVXUHPHQW QRLVH %\ ILQLWH HOHPHQW FRQVLVWHQW LW LV PHDQW WKDW WKH SURSHUW\ PDWULFHV FDQ EH DFKLHYHG E\ D ILQLWH HOHPHQW SURJUDP 1RWH WKDW PHDVXUHPHQW QRLVH FDQ EH LQWURGXFHG LQ WKH SURSHUW\ PDWUL[ WKURXJK D )(0 UHILQHPHQW DOJRULWKP +HQFH WKH SURSHUW\ PDWULFHV WKDW VKRXOG EH XVHG DUH WKH RULJLQDO SURSHUW\ PDWULFHV XQUHILQHGf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

PAGE 81

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fPRGHO UHILQHPHQW SKLORVRSK\f PLQLPXP FKDQJH PDGH WR WKH RULJLQDO )(0 7KH PLQLPXP FKDQJH FRQVWUDLQW KDV D FOHDU WHQGHQF\ WR VPHDU WKH FKDQJHV WKURXJKRXW WKH HQWLUH )(0 +RZHYHU LQ PRVW FDVHV WKLV SKLORVRSK\ LV QRW FRQVLVWHQW ZLWK WKH HIIHFW RI VWUXFWXUDO GDPDJH RQ )(0V ,Q IDFW WKH HIIHFWV RI VWUXFWXUDO GDPDJH RQ )(0V DUH XVXDOO\ fQRQPLQLPDOf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

PAGE 82

$VVXPH WKDW fS GDPDJHGf HLJHQYDOXHV DQG HLJHQYHFWRUV KDYH EHHQ PHDVXUHG DQG WKDW WKH RULJLQDO )(0 KDV EHHQ FRUUHFWHG VXFK WKDW LWV PRGDO SURSHUWLHV PDWFK WKH PHDVXUHG PRGDO SURSHUWLHV RI WKH KHDOWK\ PRGHO 7KH HLJHQYDOXH SUREOHP RI D GDPDJHG VWUXFWXUH VKRZQ LQ (T f IRU DOO S PHDVXUHG PRGHV FDQ EH ZULWWHQ LQ PDWUL[ IRUP DV 09G$M '9G$G .9G $0G9G$ $'G9G$G $.G9G % f ZKHUH % f§ G_G GS ZKHUH DOO YDULDEOHV KDYH WKH VDPH GHILQLWLRQV DV LQ WKH SUHYLRXV FKDSWHU 1RWH WKDW PDWUL[ % FDQ EH GHWHUPLQHG IURP WKH )(0 0 .f DQG WKH fSf PHDVXUHG HLJHQYDOXHV DQG HLJHQYHFWRUV $V GLVFXVVHG HDUOLHU WKH GDPDJH H[WHQW SUREOHP FRQVLVW RI ILQGLQJ WKH SHUWXUEDWLRQ PDWULFHV $0A $'M DQG $.c VXFK WKDW (T f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f LV GHULYHG 7KLV WKHRU\ ZLOO EH H[WHQVLYHO\ XVHG WKURXJKRXW WKH UHPDLQGHU RI WKLV FKDSWHU 352326,7,21 6XSSRVH WKDW ;< e 5Q[S DUH JLYHQ ZKHUH S Q DQG UDQN;f UDQN
PAGE 83

$; < ZLWK $7 $ f 7KHQ ODf ,I WKH VHW LV QRQHPSW\ WKH PLQLPXP UDQN RI DQ\ PDWUL[ $ LQ LV S 1H[W GHILQH S WR EH D VXEVHW RI FRPSULVHG RI DOO $ VXFK WKDW UDQN$f S 7KHQ OEf ,I WKH PDWUL[ <7; LV V\PPHWULF WKHQ RQH PHPEHU RI S LV JLYHQ E\ $S <+
PAGE 84

ZKHUH WKH VXSHUVFULSW M LQGLFDWHV WKH MWK IDPLO\ PHPEHU RI *3 WKH XM DUH WKH OHIW DQG ULJKW VLQJXODU YHFWRUV DQG WKH R_ DUH WKH QRQ]HUR VLQJXODU YDOXHV RI $S,Q WKH H[SDQGHG VLQJXODU YDOXH GHFRPSRVLWLRQ WKH Sf WR Q VLQJXODU YHFWRUV DUH QRW VKRZQ LQ WKH IDFWRUL]DWLRQ EHFDXVH RI WKHLU FRUUHVSRQGLQJ ]HUR VLQJXODU YDOXHV 1RWH WKDW WKH OHIW DQG ULJKW VLQJXODU YHFWRUV DUH WKH VDPH EHFDXVH $SLV UHVWULFWHG WR EH V\PPHWULF )RU (T f WR EH VDWLVILHG WKH UDQJH RI < $S$SM7 DQG 8PXVW EH HTXDO 7KXV DQ\ FROXPQ RI < FDQ EH ZULWWHQ DV D OLQHDU FRPELQDWLRQ RI WKH XMfV 7KH PDWULFHV < DQG 8DUH WKHQ UHODWHG E\ D XQLTXH S[S LQYHUWLEOH PDWUL[ 4c < 84f 6XEVWLWXWLQJ (T f LQWR (T f JLYHV $SR <4Mnf]M4Mn7f<7 <+-
PAGE 85

(T f OHDGV WR WKH FRQFOXVLRQ WKDW $S M LV WKH XQLTXH PHPEHU RI WKH VHW bS 7KLV PHPEHU LV JLYHQ E\ (T f ,OO $W WKLV SRLQW WKH 0537 DV GHILQHG LQ 3URSRVLWLRQ DVVXPHV WKDW WKH PDWULFHV ; DQG < DUH RI IXOO UDQN ,Q SUDFWLFDO XVHV DV ZLOO EH VHHQ ODWHU PDWUL[ ; LV XVXDOO\ RI IXOO UDQN 7KH UDQN UHTXLUHPHQW RQ PDWUL[ < FDQ EH RI VRPH FRQFHUQ VLQFH LW LV GLUHFWO\ UHODWHG WR WKH UDQN RI PDWUL[ $ 7KH QH[W SURSRVLWLRQ DGGUHVVHV WKH FDVH LQ ZKLFK PDWUL[ < LV UDQN GHILFLHQW 352326,7,21 6XSSRVH WKDW ; < ( 5Q[P DUH JLYHQ DQG UDQN;f P DQG UDQN
PAGE 86


PAGE 87

WKH VWUXFWXUDO GDPDJH DIIHFWV Lf RQO\ WKH PDVV SURSHUWLHV RU LLf RQO\ WKH VWLIIQHVV SURSHUWLHV RU LLLf VLPXOWDQHRXVO\ WKH PDVV DQG VWLIIQHVV SURSHUWLHV 'DPDJH ([WHQW 0DVV 3URSHUWLHV ,Q WKLV FDVH LW LV DVVXPHG WKDW WKH HIIHFW RI GDPDJH RQ WKH VWLIIQHVV SURSHUWLHV RI WKH VWUXFWXUH LV QHJOLJLEOH :LWK WKLV DVVXPSWLRQ (T f FDQ EH UHZULWWHQ DV 09G$ .9G $0G9G$ % f 1RWH WKDW WKH HLJHQYHFWRUV DUH UHDO DQG WKH HLJHQYDOXHV DUH SXUHO\ LPDJLQDU\ )XUWKHU WKH HLJHQYHFWRUV DUH OLQHDUO\ LQGHSHQGHQW ZKLFK LPSOLHV WKDW WKH PDWUL[ SURGXFW 9G$G LV RI IXOO FROXPQ UDQN LI ULJLG ERG\ PRGHV DUH QRW LQFOXGHG $VVXPH IRU WKH PRPHQW WKDW % LV RI IXOO UDQN UDQN%f Sf 7KHQ 3URSRVLWLRQ FDQ EH DSSOLHG WR GHWHUPLQH WKH SHUWXUEDWLRQ PDWUL[ $0G DV $0G %%79G$Mf 9 f E\ OHWWLQJ < % DQG ; 9G$G 1RWH WKDW WKH UHTXLUHG LQYHUVLRQ LV WKDW RI D S[S PDWUL[ ZKHUH fSf LV WKH QXPEHU RI PHDVXUHG PRGHV $V GLVFXVVHG LQ 3URSRVLWLRQ WKLV LQYHUVLRQ LV IHDVLEOH LI PDWUL[ % LV RI IXOO UDQN DQG WKH ULJLG ERG\ PRGHV RI WKH V\VWHP DUH RPLWWHG LQ WKH FRPSXWDWLRQV :KHQ PDWUL[ % LV UDQN GHILFLHQW 3URSRVLWLRQ VKRXOG EH XVHG WR UHQGHU WKH FRPSXWDWLRQ SRVVLEOH 7KH SURSHUWLHV DVVRFLDWHG ZLWK $0G DV FRPSXWHG LQ (T f DUH DV IROORZV 3523(57< 7KH SHUWXUEDWLRQ PDWUL[ $0G GHILQHG LQ (T f ZLOO EH V\PPHWULF LI WKH HLJHQYHFWRUV 9G DUH VWLIIQHVV RUWKRJRQDO LH WKH HLJHQYHFWRUV DUH RUWKRJRQDO ZLWK UHVSHFW WR WKH RULJLQDO VWLIIQHVV PDWUL[ 3URRI 3URSRVLWLRQ OFf LQ FRQMXQFWLRQ ZLWK 3URSRVLWLRQ Ef LPSOLHV WKDW WKH H[LVWHQFH RI WKH XQLTXH V\PPHWULF UDQN S $0G UHTXLUHV WKH V\PPHWU\ RI WKH PDWUL[ SURGXFW %79G$G 7KH V\PPHWULF HTXLYDOHQFH DVVRFLDWHG ZLWK WKLV PDWUL[ SURGXFW LV

PAGE 88

%79G$ HH $MYM% f 6XEVWLWXWLQJ WKH H[SUHVVLRQ IRU % IURP (T f LQWR (T f JLYHV $-Y-P9M$r 9M.9G$G 6 $9!I9G$ $AY-.9 f ZKHUH WKH V\PPHWU\ RI 0 DQG $G KDV EHHQ XVHG LQ ZULWLQJ (T f )URP (T f LW LV FOHDU WKDW WKH HTXLYDOHQFH LV WUXH LI YG.9Gf$G $GYM.9Gf f (TXDWLRQ f ZLOO REYLRXVO\ EH VDWLVILHG LI WKH PHDVXUHG HLJHQYHFWRUV DUH VWLIIQHVV RUWKRJRQDO %DUXFK f WUHDWHG RQH DSSURDFK WR PDVV RUWKRJRQDOL]H WKH PHDVXUHG HLJHQYHFWRUV $ VLPLODU DSSURDFK FDQ EH XVHG WR RUWKRJRQDOL]H WKH PHDVXUHG HLJHQYHFWRUV ZLWK UHVSHFW WR WKH VWLIIQHVV PDWUL[ ‘ 3523(57< 7KH XSGDWHG ILQLWH HOHPHQW PRGHO )(0f GHILQHG E\ WKH RULJLQDO PDVV DQG VWLIIQHVV PDWUL[ DORQJ ZLWK WKH SHUWXUEDWLRQ PDVV PDWUL[ FRPSXWHG XVLQJ (T f SUHVHUYH WKH ULJLG ERG\ FKDUDFWHULVWLFV RI WKH RULJLQDO )(0 3URRI 7KLV LV DSSDUHQW LQ WKDW WKH RULJLQDO VWLIIQHVV PDWUL[ LV XQFKDQJHG DQG WKDW WKH ULJLG ERG\ PRGHV DUH GHILQHG DV PRGHV ZKRVH HLJHQYHFWRUV OLH LQ WKH QXOO VSDFH RI WKH VWLIIQHVV PDWUL[ 'DPDJH ([WHQW 6WLIIQHVV 3URSHUWLHV +HUH LW LV DVVXPHG WKDW WKH HIIHFW RI GDPDJH RQ WKH PDVV SURSHUWLHV RI WKH VWUXFWXUH LV QHJOLJLEOH :LWK WKLV DVVXPSWLRQ (T f FDQ EH UHZULWWHQ DV 09M$M .9G $.G9G % f )RU WKLV SUREOHP WKH HLJHQYHFWRUV DUH UHDO DQG WKH HLJHQYDOXHV DUH SXUHO\ LPDJLQDU\ 7KH HLJHQYHFWRUV DUH DOVR OLQHDUO\ LQGHSHQGHQW ZKLFK LPSOLHV WKDW PDWUL[ LV RI IXOO UDQN ,I

PAGE 89

PDWUL[ % LV DVVXPHG WR EH RI IXOO UDQN UDQN%f Sf 3URSRVLWLRQ FDQ EH XVHG WR GHWHUPLQH WKH SHUWXUEDWLRQ WR WKH RULJLQDO VWLIIQHVV PDWUL[ $.G %%79GfO%7 f 7KLV H[SUHVVLRQ IRU $.A LV GHWHUPLQHG E\ VHWWLQJ < % DQG ; 9M LQ (T f 7KH SURSHUWLHV DVVRFLDWHG ZLWK $.A DV FRPSXWHG E\ (T f DUH DV IROORZ 3523(57< 7KH PDWUL[ $. ZLOO EH V\PPHWULF LI WKH HLJHQYHFWRUV DUH PDVV RUWKRJRQDO LH WKH HLJHQYHFWRUV DUH RUWKRJRQDO ZLWK UHVSHFW WR WKH RULJLQDO PDVV PDWUL[ 7KH SURRI RI 3URSHUW\ IROORZV YHU\ PXFK WKH VDPH SDWWHUQ DV WKH RQH SUHVHQWHG IRU $0M 3URSHUW\ f 3523(57< 7KH XSGDWHG )(0 GHILQHG E\ WKH RULJLQDO PDVV DQG VWLIIQHVV PDWULFHV DQG WKH SHUWXUEDWLRQ VWLIIQHVV PDWUL[ $.M SUHVHUYHV WKH ULJLG ERG\ PRGH FKDUDFWHULVWLFV LI WKH PHDVXUHG HLJHQYHFWRUV DQG WKH ULJLG ERG\ PRGHV DUH PDVV RUWKRJRQDO 3URRI 7KH RULJLQDO ULJLG ERG\ PRGHV RI DQ XQGDPSHG V\VWHP DUH GHILQHG E\ WKH HLJHQYDOXH SUREOHP .YU ;U0YU f ZKHUH WKH VXEVFULSW U GHQRWHV WKH ULJLG ERG\ PRGHVf DQG ? LV HTXDO WR ]HUR 7KXV WKH ULJLG ERG\ PRGHV OLH LQ WKH QXOO VSDFH RI WKH RULJLQDO VWLIIQHVV PDWUL[ 7KH ULJLG ERG\ PRGHV RI WKH V\VWHP ZLOO EH SUHVHUYHG LQ WKH XSGDWHG PRGHO LI WKH RULJLQDO ULJLG ERG\ PRGHV OLH LQ WKH QXOO VSDFH RI WKH XSGDWHG VWLIIQHVV PDWUL[ H f§ $.GfYU Df $.GYU Ef ZKHUH YHFWRU H LV ]HUR LI WKH WKH ULJLG ERG\ PRGHV DUH SUHVHUYHG (TXDWLRQ f KDV EHHQ XVHG WR DUULYH DW WKH H[SUHVVLRQ VKRZQ LQ (T Ef 6XEVWLWXWLQJ (T f LQWR Ef JLYHV

PAGE 90

H %%79Gf f%7YU f %\ XWLOL]LQJ WKH V\PPHWU\ RI WKH RULJLQDO PDVV DQG VWLIIQHVV PDWULFHV DORQJ ZLWK (T f (T f FDQ EH H[SDQGHG DV H %%79Gf 9M.Y $G9M0YU f 7KH ILUVW WHUP LQ WKH SDUHQWKHVLV LV ]HUR EHFDXVH WKH PDWUL[YHFWRU SURGXFW .YU LV ]HUR E\ GHILQLWLRQ 7KH VHFRQG WHUP ZLOO EH ]HUR LI WKH ULJLG ERG\ PRGHV DQG WKH PHDVXUHG PRGH VKDSHV DUH PDVV RUWKRJRQDO 'DPDJH ([WHQW 0DVV DQG 6WLIIQHVV 3URSHUWLHV ,Q WKLV FDVH LW LV DVVXPHG WKDW WKH VWUXFWXUDO GDPDJH DIIHFWV VLPXOWDQHRXVO\ WKH PDVV DQG VWLIIQHVV SURSHUWLHV RI WKH VWUXFWXUH :LWK WKLV DVVXPSWLRQ (T f FDQ EH UHZULWWHQ DV 09G$r .9G $0G9G$G $.G9G % f $SSOLFDWLRQ RI 7KH 0537 $VVXPH WKDW (T f FDQ EH GHFRXSOHG DV IROORZV $0G9G %P Df $.G9G %N Ef 7KHQ WKH 0537 DV IRUPXODWHG LQ 3URSRVLWLRQ FDQ EH DSSOLHG WR GHWHUPLQH WKH SHUWXUEDWLRQ PDWULFHV $0A DQG $.M DV $0G %P %M9MI f %M Df $.G %N %A9Nfan %I Ef 1RWH WKDW WKH PDWULFHV %P9G DQG %N 9G DUH LQYHUWLEOH LI %P DQG %A DUH RI IXOO UDQN :KHQ WKHVH UDQN UHTXLUHPHQWV DUH QRW PHW 3URSRVLWLRQ FDQ EH XVHG WR PDNH WKH FRPSXWDWLRQV SRVVLEOH

PAGE 91

'HFRPSRVLWLRQ RI 0DWUL[ % 7KH GHFRPSRVLWLRQ SUREOHP DV LOOXVWUDWHG LQ WKH SUHYLRXV VHFWLRQ LV HTXLYDOHQW WR WKH SUREOHP RI VROYLQJ IRU WKH PDWULFHV %P DQG %A 6R IDU WKH RQO\ FRQVWUDLQW WKDW WKHVH XQNQRZQ PDWULFHV PXVW VDWLVI\ LV JLYHQ E\ WKH H[SUHVVLRQ % %P %N f ZKLFK UHVXOWV IURP (TV f DQG f 1DWXUDOO\ WKHUH LV DQ LQILQLWH VHW RI VROXWLRQV %P %Nf WKDW VDWLVI\ (T f 7R DUULYH DW D XQLTXH VROXWLRQ DGGLWLRQDO SK\VLFDOO\ PHDQLQJIXO FRQVWUDLQWV FDQ EH HQIRUFHG 7KH GHFRPSRVLWLRQ SURSRVHG KHUHLQ H[SORLWV WKH FURVVRUWKRJRQDOLW\ UHODWLRQV WKDW DULVH IURP WKH V\PPHWULF QDWXUH RI WKH SURSHUW\ PDWULFHV DQG WKH XQGDPSHG DVVXPSWLRQ %\ PHDVXULQJ PDVV QRUPDOL]HG fGDPDJHGf HLJHQYHFWRUV ZKLFK LV SRVVLEOH LI D GULYLQJ SRLQW PHDVXUHPHQW LV PDGHf WKH FURVVRUWKRJRQDOLW\ UHODWLRQV DVVRFLDWHG ZLWK WKH GDPDJHG VWUXFWXUH FDQ EH ZULWWHQ DV 9M0 $0Gf9G ,S[S Df 9M. $.GfYG GLDJZG_ RfG!f 4G Ef LQ ZKLFK FRG LV WKH QDWXUDO IUHTXHQF\ RI WKH LWK PRGH RI WKH fGDPDJHGf VWUXFWXUH 0DWUL[ ,S;S LV WKH S[S LGHQWLW\ PDWUL[ $ UHDUUDQJHPHQW RI (T f \LHOGV 9M $0G 9G 9M 0 9G ,S[S 9M %P Df 9M $.G 9G 9M 9G 4G 9M %N Ef &OHDUO\ WKH PDWULFHV %P DQG %A FDQ EH FRPSXWHG IURP (TV f ,Q WKH UDUH VLWXDWLRQ WKDW WKH QXPEHU RI PHDVXUHG PRGHV LV HTXDO WR WKH QXPEHU RI '2)V LQ WKH )(0 S Qf WKHVH FDQ EH FRPSXWHG E\ VLPSO\ LQYHUWLQJ PDWUL[ 9G 8QIRUWXQDWHO\ DV GLVFXVVHG HDUOLHU WKH QXPEHU RI PHDVXUHG PRGHV LV XVXDOO\ PXFK OHVV WKDQ WKH QXPEHU RI )(0 '2)V S m Qf ,Q WKLV FDVH WKH VROXWLRQ WKDW QDWXUDOO\ FRPHV WR PLQG LV WR XVH WKH SVHXGRLQYHUVH RI PDWUL[ 9M 7KH LQFRQYHQLHQFH RI WKLV DSSURDFK LV WKDW WKH VSDUVLW\ SDWWHUQ RI PDWUL[ % ZLOO QRW EH

PAGE 92

UHIOHFWHG LQ WKH FRPSXWHG PDWULFHV %P DQG %A 5HPHPEHU WKDW WKH VSDUVLW\ SDWWHUQ RI % DV GLVFXVVHG LQ &KDSWHU LQGLFDWHV WKH ORFDWLRQ RI WKH GDPDJH DIIHFWLQJ WKH VWUXFWXUH $ PRUH SK\VLFDOO\ LQWXLWLYH DSSURDFK LV WR FRQVWUDLQ %P DQG %A WR H[KLELW WKH VDPH VSDUVLW\ SDWWHUQ DV PDWUL[ % 7KLV LV GRQH E\ FDVWLQJ % LQ DQ HTXDWLRQ VLPLODU WR WKH H[SUHVVLRQV RI (TV f 7KH SUREOHP LQ TXHVWLRQ LV WKHQ WR ILQG DQ Q[S PDWUL[ 3 WKDW VDWLVILHV 39-%f % f 0DWUL[ 3 FDQ EH FRPSXWHG DV S EY-Efn f 7KH LQYHUVH LQYROYHG LQ WKLV FRPSXWDWLRQ LV WKDW RI D S[S PDWUL[ ZKLFK LV LQYHUWLEOH LI PDWUL[ % LV RI IXOO UDQN 1RZ WKDW 3 LV FRPSXWHG %P DQG %A FDQ EH FRPSXWHG XVLQJ (T f DV %P 3 YM 0 9G ,S[Sf Df %W 3YM.9G 4Gf Ef ,W LV FOHDU IURP (T f WKDW 3 ZLOO KDYH WKH VDPH VSDUVLW\ SDWWHUQ DV PDWUL[ % +HQFH %P DQG %A ZLOO DOVR UHIOHFW WKH LPSRUWDQW VSDUVLW\ SDWWHUQ RI % 7KH FRPSXWHG PDWULFHV %P DQG %N FDQ DOVR EH XVHG WR GHWHUPLQH WKH HIIHFW RI WKH GDPDJH UHVSHFWLYHO\ RQ WKH PDVV DQG VWLIIQHVV SURSHUWLHV $V LQ &KDSWHU FXPXODWLYH YHFWRUV DVVRFLDWHG WR %P DQG %AFDQ DOVR EH GHILQHG ZKHQ PRUH WKDQ RQH PHDVXUHG PRGH LV DYDLODEOH + 9 aPLO A O\ 3W-,9G Df Ef ZKHUH GP DQG GN DUH UHVSHFWLYHO\ WKH LWK FROXPQ RI PDWUL[ %P DQG %A

PAGE 93

3523(57< 7KH SHUWXUEDWLRQ PDWULFHV $0M $.Gf FRPSXWHG IURP WKH 0537 XVLQJ WKH %P DQG UHVXOWLQJ IURP WKH GHFRPSRVLWLRQ GLVFXVVHG DERYH ZLOO EH V\PPHWULF 3URRI 7KH SHUWXUEDWLRQ PDWULFHV $0M DQG $.G ZLOO EH V\PPHWULF EHFDXVH WKH\ VDWLVI\ WKH UHODWLRQVKLSV LQ (TV f DQG WKH ULJKW KDQG VLGHV RI WKHVH HTXDWLRQV DUH V\PPHWULF 'DPDJH ([WHQW 3URSRUWLRQDOO\ 'DPSHG 6WUXFWXUHV 6LQFH PDQ\ VWUXFWXUHV KDYH QRQQHJOLJLEOH GDPSLQJ LW LV RI SUDFWLFDO LQWHUHVW WR H[WHQG WKH 0537 WR DGGUHVV GDPSHG VWUXFWXUHV ,Q WKLV DQDO\VLV WKH VWUXFWXUH XQGHU FRQVLGHUDWLRQ LV DVVXPHG WR H[KLELW SURSRUWLRQDO GDPSLQJ 'DPDJH ([WHQW 6WLIIQHVV DQG 'DPSLQJ 3URSHUWLHV ,W LV DVVXPHG WKDW WKH HIIHFW RI WKH VWUXFWXUDO GDPDJH RQ WKH PDVV SURSHUWLHV LV QHJOLJLEOH ,Q WKLV FRQWH[W (T f LV UHZULWWHQ DV 09G$ '9G$G .9G $'G9G$G $.G9G HH % f 7KH FRPSOH[ FRQMXJDWH RI (T f LV $'G9G$G $.G9G % f ZKHUH WKH RYHUEDU LQGLFDWHV WKH FRPSOH[ FRQMXJDWH RSHUDWRU DQG WKH IDFW WKDW $'M $.G DQG 9M DUH UHDO KDV EHHQ XVHG LQ ZULWLQJ (T f 6XEWUDFWLQJ (T f IURP (T f JLYHV $'G9G$G $Gf % %f f ,I % f§ %f LV DVVXPHG WR EH RI IXOO UDQN 3URSRVLWLRQ FDQ EH DSSOLHG WR GHWHUPLQH WKH SHUWXUEDWLRQ PDWUL[ $'G DV $'G % %f+G% %f7 ZLWK +G % %f79G$G $Gf f

PAGE 94

1RWH WKDW $'G DV GHILQHG E\ (T f LV UHDO 3RVWPXOWLSO\LQJ (T f E\ $G DQG (T f E\ $G DQG VXEWUDFWLQJ WKH WZR HTXDWLRQV OHDGV WR $.f9G$G $Gf %$G %$Gf f ZKHUH WKH IDFW WKDW $G DQG $G DUH GLDJRQDO PDWULFHV KDV EHHQ XVHG LQ ZULWLQJ (T f ,I %$G %$Gf LV DVVXPHG WR EH RI IXOO UDQN 3URSRVLWLRQ FDQ DOVR EH DSSOLHG WR GHWHUPLQH WKH SHUWXUEDWLRQ PDWUL[ $.G DV $.G %$G %$Gf+N%$G %$Gf7 ZLWK +N f§ ?7 %$G %$Gf 9G$G $Gf f 1RWH WKDW $.G DV GHILQHG E\ (T f LV DOVR UHDO 3523(57< 7KH SHUWXUEDWLRQ PDWULFHV $'G DQG $.M DV FRPSXWHG DERYH ZLOO EH V\PPHWULF LI WKH PHDVXUHG HLJHQYHFWRUV 9G DUH PDVV RUWKRJRQDO LH WKH HLJHQYHFWRUV DUH RUWKRJRQDO ZLWK UHVSHFW WR WKH RULJLQDO XQSHUWXUEHG PDVV PDWUL[ 3URRI 0DWUL[ $'G LV V\PPHWULF LI +G LV V\PPHWULF RU HTXLYDOHQWO\ LI + LV V\PPHWULF +HQFH WR JHW D V\PPHWULF $'G WKH IROORZLQJ HTXLYDOHQFH PXVW EH VDWLVILHG % %f79G$G $Gf $G $GfY-% %f f 6XEVWLWXWLQJ WKH H[SUHVVLRQV IRU % DQG % IURP (TV f DQG f UHVSHFWLYHO\ LQWR (T f \LHOGV $M9M0 $G9M' $YM0 $G9M'f 9G$G $Gf $G $Gf9M 09G$G '9G$G 09G$G '9G$Gf f

PAGE 95

1RWH WKDW LQ (T f WKH WHUPV LQYROYLQJ PDWUL[ $.G FDQFHOHG RXW $ IXUWKHU H[SDQVLRQ DQG VLPSOLILFDWLRQ RI (T f \LHOGV D Df 9M09G $G $Gf $G $Gf 9M09G D$Gf f ZKLFK LV FOHDUO\ VDWLVILHG LI WKH PHDVXUHG fGDPDJHGf HLJHQYHFWRUV 9G DUH PDVV RUWKRJRQDO ,OO /LNHZLVH WKH SHUWXUEDWLRQ PDWUL[ $.G DV FRPSXWHG LQ (T f LV V\PPHWULF LI LV V\PPHWULF RU HTXLYDOHQWO\ LI +Na LV V\PPHWULF 7KLV V\PPHWU\ UHTXLUHPHQW \LHOGV WKH IROORZLQJ HTXLYDOHQFH %$G %$Gf79G$G $Gf V $G $Gf9M%$G %$Gf f 6XEVWLWXWLRQ RI WKH H[SUHVVLRQV IRU % DQG % LQWR (T f \LHOGV ;M$M9M0 $G9G. $G$G9M0 $GYMLFf 9G$G $Gf f 6 $G $GfY09G$G$G .9G$G 09A$A .9G$Gf LQ ZKLFK WKH WHUPV LQYROYLQJ PDWUL[ $'M FDQFHO 0DQLSXODWLQJ DQG VLPSOLI\LQJ (T f \LHOGV Df$ $GƒGfYM09G$G $Gf V $G $GfYM09G$G$ $Gƒf f 7KLV HTXLYDOHQFH LV REYLRXVO\ VDWLVILHG LI WKH HLJHQYHFWRUV DUH PDVV RUWKRJRQDO ,OO ‘ 3523(57< 7KH XSGDWHG )(0 GHILQHG E\ WKH RULJLQDO )(0 DQG WKH SHUWXUEDWLRQ PDWULFHV $'G DQG $.G FRPSXWHG IURP (TV f DQG f SUHVHUYHV WKH RULJLQDO ULJLG ERG\ PRGHV LI WKH PHDVXUHG HLJHQYHFWRUV DQG WKH ULJLG ERG\ PRGHV DUH PDVV RUWKRJRQDO

PAGE 96

3URRI $V GLVFXVVHG HDUOLHU D ULJLG ERG\ PRGH LV GHILQHG DV D PRGH ZKRVH HLJHQYDOXH LV HTXDO WR ]HUR DQG ZKRVH HLJHQYHFWRU OLHV LQ WKH QXOO VSDFH RI WKH )(0 VWLIIQHVV PDWUL[ +HQFH WKH ULJLG ERG\ PRGHV RI WKH RULJLQDO V\VWHP DUH SUHVHUYHG LQ WKH XSGDWHG )(0 LI WKH\ OLH LQ WKH QXOO VSDFH RI WKH SHUWXUEHG VWLIIQHVV PDWUL[ &RQVLGHU WKH UHODWLRQVKLS H $.GfYU f ZKHUH YU LV D ULJLG ERG\ PRGH HLJHQYHFWRU &OHDUO\ WKH ULJLG ERG\ PRGH DVVRFLDWHG WR HLJHQYHFWRU YU LV SUHVHUYHG LI H %\ GHILQLWLRQ YU LV D ULJLG ERG\ HLJHQYHFWRU RI WKH RULJLQDO V\VWHP KHQFH (T f FDQ EH VLPSOLILHG DV H f§ $.GYU f 6XEVWLWXWLQJ WKH H[SUHVVLRQ IRU $. DV GHILQHG LQ (T f LQWR (T f JLYHV f H %$G %$Gf+N%$G %$Gf7YU 6XEVWLWXWLRQ RI WKH H[SUHVVLRQV IRU % DQG % LQWR WKLV HTXDWLRQ \LHOGV 6 %$G %$Gf+ $G$M9M0 $G9M. $G$G9G0 $G9G. YU f %\ XVLQJ WKH IDFW WKDW YU LV D ULJLG ERG\ HLJHQYHFWRU RI WKH RULJLQDO V\VWHP LH .YU f (T f FDQ EH VLPSOLILHG DV %$G %$Gf+W $G$G $G$6 fYM0\ f ,W LV FOHDU IURP (T f WKDW H LI WKH ULJLG ERG\ PRGH YU DQG WKH PHDVXUHG HLJHQYHFWRUV 9G DUH PDVV RUWKRJRQDO LH 9G0\U f P 'DPDJH ([WHQW 0DVV DQG 'DPSLQJ 3URSHUWLHV ,Q WKLV FDVH LW LV DVVXPHG WKDW WKH HIIHFW RI WKH VWUXFWXUDO GDPDJH RQ WKH VWLIIQHVV SURSHUWLHV LV QHJOLJLEOH ,Q WKLV FRQWH[W (T f LV UHZULWWHQ DV

PAGE 97

09G$r '9G$G .9G $0G9G$ $'G9G$G % f %\ XVLQJ DQ DSSURDFK VLPLODU WR RQH XVHG LQ WKH SUHFHGLQJ VHFWLRQ (T f DQG LWV FRPSOH[ FRQMXJDWH FDQ EH PDQLSXODWHG WR \LHOG WKH IROORZLQJ GHFRPSRVLWLRQ $0G9G$A$G $G$Gf %$G %$Gf f $'G9G$G$ $G$GM %$ %$GM f $JDLQ E\ DSSO\LQJ WKH 0537 WR WKH SUHFHGLQJ HTXDWLRQV $0G DQG $'G DUH GHWHUPLQHG WR EH $0G %$G %$Gf+P%$G %$Gf ZLWK +P O %$G %$Gf79GA$G$G $G$Gf f $'G %$G %$f+G%$G %$GM ZLWK +G %$G %$rf 9G$G$G $G$GM f§ O f &OHDUO\ WKH SHUWXUEDWLRQ PDWULFHV $0G DQG $'G DV GHILQHG E\ (TV f DQG f DUH UHDO 3523(57< 7KH SHUWXUEDWLRQ PDWULFHV $0G DrG $'G DV FRPSXWHG DERYH ZLOO EH V\PPHWULF LI WKH PHDVXUHG HLJHQYHFWRUV 9G DUH VWLIIQHVV RUWKRJRQDO LH WKH HLJHQYHFWRUV DUH RUWKRJRQDO ZLWK UHVSHFW WR WKH RULJLQDO XQSHUWXUEHG VWLIIQHVV PDWUL[ 3523(57< 7KH XSGDWHG )(0 GHILQHG E\ WKH RULJLQDO )(0 DQG WKH SHUWXUEDWLRQ PDWULFHV $0G DQG $'G SUHVHUYHV WKH RULJLQDO ULJLG ERG\ PRGHV 7KH SURRI RI 3URSHUW\ LV VWUDLJKWIRUZDUG VLQFH WKH RULJLQDO VWLIIQHVV PDWUL[ LV XQFKDQJHG VHH 3URSHUW\ f 7KH SURRI RI 3URSHUW\ IROORZV YHU\ PXFK WKH VDPH SDWWHUQ DV WKH SURRI RI 3URSHUW\

PAGE 98

'DPDJH ([WHQW 0DVV DQG 6WLIIQHVV 3URSHUWLHV ,Q WKLV SUREOHP LW LV DVVXPHG WKDW WKH HIIHFW RI WKH VWUXFWXUDO GDPDJH RQ WKH GDPSLQJ SURSHUWLHV LV QHJOLJLEOH )RU WKLV VLWXDWLRQ WKH JHQHUDO HLJHQYDOXH SUREOHP GHILQHG LQ (T f DVVRFLDWHG WR WKLV FDVH FDQ EH VLPSOLILHG DV 09G$ '9G$G .9G $0G9G$G $.G9G % f $OJHEUDLF PDQLSXODWLRQV RI (T f DQG LWV FRPSOH[ FRQMXJDWH \LHOG WKH IROORZLQJ GHFRPSRVLWLRQ $0G9G$ $GM % %f $.G9G$ $Mf %$G %$GM f f 7KH SHUWXUEDWLRQ PDWULFHV $0WM DQG $.M FDQ WKHQ EH FRPSXWHG XVLQJ WKH 0537 $0G % %f+P% %f7 ZLWK +P % %f79G$A $GfO $.G %$ %$A+A%D%$GM ZLWK +N 7 LL %$6 %$G_ 9G$G $Mf f f 1RWH WKDW $0M DQG $.M DV GHILQHG E\ (TV f DQG f DUH UHDO 3523(57< 7KH SHUWXUEDWLRQ PDWULFHV $0M DQG $. DV FRPSXWHG DERYH ZLOO EH V\PPHWULF LI WKH PHDVXUHG HLJHQYHFWRUV 9M DUH GDPSLQJ RUWKRJRQDO LH WKH HLJHQYHFWRUV DUH RUWKRJRQDO ZLWK UHVSHFW WR WKH RULJLQDO XQSHUWXUEHG GDPSLQJ PDWUL[ 3523(57< 7KH XSGDWHG )(0 GHILQHG E\ WKH RULJLQDO )(0 DQG WKH SHUWXUEDWLRQ PDWULFHV $0M DQG $.M SUHVHUYHV WKH RULJLQDO ULJLG ERG\ PRGHV LI WKH PHDVXUHG HLJHQYHFWRUV DQG WKH ULJLG ERG\ PRGHV DUH GDPSLQJ RUWKRJRQDO

PAGE 99

7KHVH SURRIV RI WKH DERYH WZR SURSHUWLHV DUH QRW UHSRUWHG KHUH 7KH\ IROORZ YHU\ PXFK WKH VDPH SDWWHUQ DV WKH SURRIV LQ 6HFWLRQ 'DPDJH ([WHQW 0DVV 'DPSLQJ DQG 6WLIIQHVV 3URSHUWLHV 7KH HLJHQYDOXH SUREOHP RI D SURSRUWLRQDOO\ GDPSHG V\VWHP ZLWK DOO SURSHUW\ PDWULFHV VLPXOWDQHRXVO\ DIIHFWHG E\ GDPDJH FDQ EH UHDUUDQJHG LQWR WKH IRUP 09G$ '9G$G .9G $0G9G$ $'G9G$G $.G9G V % f 7KH WKHRU\ GHYHORSHG LQ 6HFWLRQ FDQ EH H[SDQGHG WR DGGUHVV WKLV SDUWLFXODU SUREOHP 7KH FURVVRUWKRJRQDOLW\ UHODWLRQVKLSV DVVRFLDWHG ZLWK WKLV W\SH RI VWUXFWXUHV DUH 9M0 $0Gf9G ,S[S Df 9M' $'Gf9G GLDJLG_:G@ G Ef 9-. $.Gf9G GMDJRMG_ ZGf 4G Ff 1RWLFH WKDW WKH FURVVRUWKRJRQDOLW\ UHODWLRQVKLSV LQ (TV Df DQG Ff DUH H[DFWO\ WKH VDPH DV WKH RQHV DVVRFLDWHG ZLWK XQGDPSHG V\VWHPV UHSRUWHG LQ (TV Df DQG Ef $V EHIRUH WKHVH FURVVRUWKRJRQDOLW\ FRQGLWLRQV FDQ DOVR EH UHDUUDQJHG DV YM $0G 9G 9M 0 9G ,S[S ( P I 7 ,,, Df 9M$'G9G 9M'9G 9M% Ef 9M $.G 9G 9M 9G 4G V Y,%N Ff )ROORZLQJ WKH H[DFW VDPH DUJXPHQW GLVFXVVHG IRU XQGDPSHG V\VWHPV LQ 6HFWLRQ DQ Q[S PDWUL[ 3 WKDW VDWLVILHV WKH UHODWLRQ 3YM%f % f LV VRXJKW ZKHUH % LV FRPSXWHG XVLQJ (T f DQG YG%f LV D S[S PDWUL[ $OWKRXJK % LV D FRPSOH[ PDWUL[ WKH Q[S PDWUL[ 3 LV UHDO VLQFH 9G LV UHDO +HQFH IRU FRPSXWDWLRQDO HIILFLHQF\ PDWUL[ 3 FDQ FRPSXWHG IURP

PAGE 100

3 %UYM%Uf f ZKHUH %U LV WKH UHDO SDUW RI % ,Q (T f LW LV DVVXPHG WKDW PDWUL[ YG% M LV LQYHUWLEOH :LWK 3 FRPSXWHG WKH QH[W VWHS LV WR GHWHUPLQH WKH GHFRPSRVHG GDPDJH YHFWRUV WKDW LQGLFDWH WKH HIIHFWV RI WKH GDPDJH RQ WKH PDVV GDPSLQJ DQG VWLIIQHVV PDWULFHV %P $09G 3 9M 0 9G ,S[Sf Df %G $'9G 3 9M 9G Gf Ef %N $.9G 3 9M 9G 4Gf Ff 7KH PLQLPXP UDQN SHUWXUEDWLRQ WKHRU\ 0537f DV IRUPXODWHG LQ 3URSRVLWLRQ FDQ DJDLQ EH DSSOLHG WR GHWHUPLQH WKH SHUWXUEDWLRQ PDWULFHV $0M $'M DQG $.M DV $0G %P %M9GUn %O Df $'G %G %G9Gf f %M Ef $.G %N %M9Wfn %M Ff 1RWH WKDW WKH PDWULFHV %P9G %GU9G DQG %A9G DUH S[S PDWULFHV WKDW DUH LQYHUWLEOH LI %P %M DQG %N DUH RI IXOO UDQN $V LQ DOO RWKHU FDVHV DOUHDG\ VWXGLHG 3URSRVLWLRQ FDQ EH XVHG WR GHDO ZLWK WKH VLWXDWLRQ ZKHQ DQ\ RQH RI WKHVH PDWULFHV DUH UDQN GHILFLHQW 7KH FXPXODWLYH GDPDJH ORFDWLRQ YHFWRU DVVRFLDWHG WR %P DQG %N GHILQHG LQ (TV DUH DOVR DSSOLFDEOH WR WKLV SUREOHP $Q DGGLWLRQDO FXPXODWLYH GDPDJH YHFWRU DVVRFLDWHG WR WKH SHUWXUEDWLRQV LQ WKH GDPSLQJ SURSHUWLHV FDQ EH VLPLODUO\ GHILQHG DV GG f ZKHUH GG LV WKH LWK FROXPQ RI PDWUL[ %G

PAGE 101

3523(57< 7KH SHUWXUEDWLRQ PDWULFHV $0G $'G $.Gf DV FRPSXWHG DERYH ZLOO EH V\PPHWULF 3URRI 7KLV LV WUXH VLQFH WKHVH SHUWXUEDWLRQ PDWULFHV DUH FRQVWUDLQHG WR VDWLVI\ WKH UHODWLRQVKLSV LQ (TV f 1RWLFH WKDW WKH ULJKW KDQG VLGHV RI (TV f DUH V\PPHWULF 'DPDJH ([WHQW 1RQSURSRUWLRQDOOY 'DPSHG 6WUXFWXUHV $V UHSRUWHG HDUOLHU LQ WKH EHJLQQLQJ RI WKH FKDSWHU WKH HLJHQYDOXH SUREOHP DVVRFLDWHG ZLWK WKH G\QDPLF GLIIHUHQWLDO HTXDWLRQ RI PRWLRQ RI D GDPDJHG QRQSURSRUWLRQDOO\ GDPSHG V\VWHP FDQ EH ZULWWHQ DV 09G$c '9G$G .9G $0G9G$G $'G9G$G $.G9G V % f ZKHUH WKH PDWULFHV 0 $0G $'G $.A 9G DQG $G KDYH WKH VDPH GHILQLWLRQV DV LQ WKH SUHYLRXV VHFWLRQV $ QXPEHU RI FURVVRUWKRJRQDOLW\ UHODWLRQVKLSV DVVRFLDWHG WR JHQHUDO QRQSURSRUWLRQDOO\ GDPSHG V\VWHPV KDYH EHHQ UHSRUWHG LQ &KDSWHU 7KH FURVVRUWKRJRQDOLW\ FRQGLWLRQV WKDW DUH UHOHYDQW WR WKH IROORZLQJ IRUPXODWLRQV DUH n9$7 9G GHQRWHV WKH FRPSOH[ FRQMXJDWH WUDQVSRVH RSHUDWRU ([SDQVLRQ RI WKH DERYH WKUHH HTXDWLRQV \LHOG UHVSHFWLYHO\ WKH IROORZLQJ UHODWLRQV R 6 L UYG$G Rn L R > YG $'Gf B $.‘9G$Gf $.Gf YG 0 $0Gf UYG$G nRn 0 $0Gf B $'Gf / 9G f f f r $G9G0 $0Gf9G$G 9G. $.Gf9G >@ f

PAGE 102

$G9G' $'Gf9G$G $G9G. $.Gf9G 9. $.Gf9G$G >@ f 9G0 $0f9G$G $G9G0 $0Gf9G 9' $'f9G >@ f 7KHVH HTXDWLRQV FDQ EH IXUWKHU PDQLSXODWHG WR EHFRPH UHVSHFWLYHO\ : $G9G 0 9G$G 9G 9G f $A9G $0G 9G$G 96 $.G 9G : $G9G 9G$G $G9G 9G 9G 9G$G f $G9G $'G 9G$G $A9G $.G 9G $.G 9G$G : 0 9G$G $G9G 0 9G 9G 9G f 9G $0G 9G$G $!G $0G 9G 9 $'G 9G 1RWLFH WKDW : L : DQG : FDQ EH FRPSXWHG XVLQJ WKH HTXDWLRQV LQ WHUPV RI NQRZQ PDWULFHV 0 9G DQG $G 7KH DERYH FURVVRUWKRJRQDOLW\ FRQGLWLRQV DUH GHULYHG IURP WZR GLIIHUHQW VWDWH VSDFH UHSUHVHQWDWLRQV RI WKH HTXDWLRQ RI PRWLRQ VKRZQ LQ (T f VHH &KDSWHU f )URP HDFK UHSUHVHQWDWLRQ WZR FURVVRUWKRJRQDOLW\ FRQGLWLRQ ZHUH GHULYHG 7KH WZR VHWV SURYLGH H[DFWO\ WKH VDPH LQIRUPDWLRQ DERXW (T f EXW LQ D GLIIHUHQW IRUPDW +HQFH RQO\ WZR RI WKH WKUHH FURVVRUWKRJRQDOLW\ UHODWLRQV DUH OLQHDUO\ LQGHSHQGHQW ,Q IDFW (T f FDQ EH JHQHUDWHG E\ VXEWUDFWLQJ (T f IURP f $Q\ VHW RI WZR OLQHDUO\ LQGHSHQGHQW FURVVRUWKRJRQDOLW\ UHODWLRQVKLSV FDQ EH YLHZHG DV D GHFRXSOLQJ RI WKH HTXDWLRQ RI PRWLRQ (T ff +HQFH IRU WKH SUHVHQW SUREOHP WKHUH DUH RQO\ WZR GLVWLQFW OLQHDUO\ LQGHSHQGHQWf PDWUL[ HTXDWLRQV IRU WKH WKUHH XQNQRZQ SHUWXUEDWLRQ PDWULFHV $0G $'G $.Gf 1DWXUDOO\ LI WKH SUREOHP FRQVLVWV RI VROYLQJ IRU DOO WKUHH XQNQRZQ PDWULFHV WKHUH ZLOO H[LVW DQ LQILQLWH QXPEHU RI VROXWLRQV $GGLWLRQDO FRQVWUDLQWV DUH KHQFH QHHGHG WR JHW D EHWWHU GHILQHG SUREOHP ,Q WKLV IRUPXODWLRQ LW LV DVVXPHG WKDW WKH GDPDJH RQO\ DIIHFWV WZR RI WKH WKUHH SURSHUW\ PDWULFHV 0DWKHPDWLFDOO\ WKLV FDQ EH WUDQVODWHG DV D FRQVWUDLQW RQ RQH RI WKUHH SHUWXUEDWLRQ PDWULFHV HLWKHU $0G RU $'G RU $.GfWR EH HTXDO WR ]HUR

PAGE 103

'DPDJH ([WHQW 'DPSLQJ DQG 6WLIIQHVV 3URSHUWLHV ,Q WKLV FDVH LW LV DVVXPHG WKDW WKH HIIHFWV RI WKH GDPDJH RQ WKH PDVV SURSHUWLHV DUH QHJOLJLEOH $0M f 7KH FRUUHVSRQGLQJ HLJHQYDOXH HTXDWLRQ DQG WKH FURVVRUWKRJRQDOLW\ FRQGLWLRQV UHOHYDQW WR WKLV SUREOHP DUH 09G$ '9G$G .9G $'G9G$G $.G9G % f $.G 9G : f 96 $'G 9G : f 7KH DSSURDFK XVHG LQ WKLV IRUPXODWLRQ IROORZV YHU\ PXFK WKH VDPH SDWWHUQ DV WKH RQH GHYHORSHG LQ 6HFWLRQ 7KH EDVLF LGH£LV WR JHQHUDWH IURP(TV f SUREOHPV WKDW FDQ WDNH WKH DGYDQWDJH RI WKH DOUHDG\ H[LVWLQJ 0537 FRQFHSW %DVHG RQ WKH VDPH DUJXPHQW DV LQ 6HFWLRQ D PDWUL[ 3 e &Q[S LV VRXJKW VXFK WKDW 39G%f % f ,W LV $VVXPHG WKDW Y-Ef LV LQYHUWLEOH WKHQ PDWUL[ 3 LV JLYHQ E\ 3 %
PAGE 104

$. G $' 9Gf5 YGfO I3:f5 3:f YGf5 n 9GfO L3:f 3:f Df Ef ZKHUH WKH QRWDWLRQ ;f5 DQG ;f LQGLFDWH UHVSHFWLYHO\ WKH UHDO DQG LPDJLQDU\ SDUW RI PDWUL[ ; 7KH 0537 FDQ QRZ EH DSSOLHG WR GHWHUPLQH WKH SHUWXUEDWLRQ PDWULFHV $'A DQG $.M $'G )G )M8Gff )M Df $.G )N)M8Gfn )7 Ef ZKHUH 8G )N )G 9Gf5 YGf 3:f5 3:f SZfU 3:f 5HPHPEHU WKDW :M DQG : FDQ EH FRPSXWHG LQ WHUPV RI NQRZQ PDWULFHV DV VKRZQ LQ (TV f DQG f 7KH FRPSXWDWLRQV LQ (TV Df DQG Ef UHTXLUH WKDW )G8Gf DQG )G8Gf EH LQYHUWLEOH $V GLVFXVVHG LQ WKH HDUOLHU VHFWLRQV 3URSRVLWLRQ FDQ EH XVHG ZKHQ WKHVH LQYHUVH UHTXLUHPHQWV DUH QRW PHHW 'DPDJH ([WHQW 0DVV DQG 'DPSLQJ 3URSHUWLHV +HUH LW LV DVVXPHG WKDW WKH GDPDJH HIIHFWV RQ WKH VWLIIQHVV SURSHUWLHV DUH QHJOLJLEOH $.G f %\ FRQVLGHULQJ WKLV DVVXPSWLRQ WKH HLJHQYDOXH HTXDWLRQ DVVRFLDWHG ZLWK WKLV SUREOHP LV IRXQG WR EH 09G$ '9G$G .9G $0G9G$ $'G9G$G % f 7KH FURVVRUWKRJRQDOLW\ FRQGLWLRQV UHOHYDQW WR WKH SUHVHQW GHYHORSPHQW DUH REWDLQHG IURP (TV f DQG f E\ VHWWLQJ $.G WR ]HUR $G9G $0G 9G$G : M f

PAGE 105

$G9M $'G 9G$G : f 7KH VROXWLRQ SURFHGXUH WR WKH SUREOHP RI GHWHUPLQLQJ $0G DQG $'G LV H[DFWO\ WKH VDPH DV WKH RQH IRUPXODWHG LQ WKH SUHFHGLQJ VHFWLRQ )LUVW HTXDWLRQV f DQG f DUH FDVW LQ WKH IROORZLQJ IRUP $0G 9G$G 3: Df $'G 9G$G 3: Ef ZKHUH 3 % $G9G%f 7KH DERYH WZR HTXDWLRQV LQYROYH FRPSOH[ PDWULFHV 7R JXDUDQWHH WKDW WKH SHUWXUEDWLRQ PDWULFHV $0G DQG $'G DUH UHDO (TV Df DQG Ef DUH XVHG LQ FRQMXQFWLRQ ZLWK WKHLU FRUUHVSRQGLQJ FRPSOH[ FRQMXJDWHV 7KH FRPELQDWLRQV RI WKH WZR VHWV RI FRPSOH[ FRQMXJDWH SDLUV \LHOG $0 $' 9G$Gf5 9G$Gf@ >3:f5 3:f 9G$Gf5 9G$Gf_ 3:f 3:f Df Ef )LQDOO\ WKH SHUWXUEDWLRQ PDWULFHV $0G DQG $'G DUH UHVSHFWLYHO\ GHWHUPLQHG IURP (TV Df DQG Ef E\ XVLQJ WKH 0537 $0G )P )P8Gf n )O Df $'G )G )M8Gfn )M Ef 8G )P )G 9G$Gf5 9G$Gf 3:f5 3:f 3:fU 3:f@ ZKHUH

PAGE 106

$JDLQ WKH FRPSXWDWLRQV LQ (T Df DQG Ef DUH IHDVLEOH LI DQG RQO\ LI )P8Gf DQG )G8Gf DUH ERWK LQYHUWLEOH 'DPDJH ([WHQW 0DVV DQG 6WLIIQHVV 3URSHUWLHV ,Q VRPH VLWXDWLRQV VWUXFWXUDO GDPDJH LV VXFK WKDW LW RQO\ DIIHFWV WKH PDVV DQG VWLIIQHVV SURSHUWLHV RI WKH V\VWHP 7KH IROORZLQJ H[SUHVVLRQ DUH WKH UHVXOWV RI VHWWLQJ WKH GDPSLQJ SHUWXUEDWLRQ WR ]HUR $'G f LQ (TV f f DQG f 09G$ '9G$G .9G $0G9G$ $.G9G % f $G9G $.G 9G 9G $.G 9G$G : f $G9G $0G 9G 9G $0G 9G$G : f (TXDWLRQ f LV WKH UHDUUDQJHG HLJHQYDOXH SUREOHP DVVRFLDWHG ZLWK WKH SUHVHQW FDVH 7KH H[SUHVVLRQV LQ (TV f DQG f DUH WKH fUHDUUDQJHGf FURVVRUWKRJRQDOLW\ FRQGLWLRQV WKDW DUH UHOHYDQW WR WKH IRUWKFRPLQJ IRUPXODWLRQ )LUVW (TV f DQG f DUH FRQYHQLHQWO\ VLPSOLILHG WR WKH VWDQGDUG IRUPDW HQFRXQWHUHG LQ WKH SUHYLRXV WZR VHFWLRQV 6LQFH $G DQG $G DUH GLDJRQDO PDWULFHV (TV f FDQ EH ZULWWHQ DV 9G $.G 9G 4 f ZKHUH T Z 7KH YDULDEOHV T DQG Z DUH WKH FRPSRQHQWV LQ WKH LWK URZ DQG MWK FROXPQ RI PDWUL[ 4N DQG : UHVSHFWLYHO\ $G LV WKH LWK GLDJRQDO FRPSRQHQW RI WKH HLJHQYDOXH PDWUL[ $G 6LPLODUO\ (T f FDQ DOVR EH ZULWWHQ DV 9G $0G 9G 4 f ZKHUH T Z AG AGMMf (TXDWLRQV f DQG f FDQ WKHQ EH WUDQVIRUPHG DV

PAGE 107

$.G 9G 34 Df $0G 9G 34 Ef ZKHUH 3 % YG%f n 1RWLFH WKDW WKH DERYH FRPSXWDWLRQV UHTXLUH WKDW PDWUL[ YG%f EH LQYHUWLEOH 7KH VRXJKW SHUWXUEDWLRQ PDWULFHV $0G DQG $.GDUH HQVXUHG WR EH UHDO E\ VXSSOHPHQWLQJ (TV ODf DQG Ef ZLWK WKHLU FRPSOH[ FRQMXJDWHV 7KH FRPSOH[ FRQMXJDWH SDLUV DUH WKHQ FRPELQHG WRJHWKHU LQ WKH IROORZLQJ IDVKLRQ $.G 9Gf5 n YG9 >S4Df5 34Mff $0G n9Gf5 n YGf >34f S4f@ Df Ef $W WKLV SRLQW WKH 0537 LV DSSOLHG WR (TV Df DQG Ef DQG WKH SHUWXUEDWLRQ PDWULFHV $0G DQG $.G DUH GHWHUPLQHG $.G )N )M8Gf )I Df $0G )P )MXGfBO )M Ef ZKHUH XG >YGf5 9Gf )N >34fU 34Lf@ )P >34fU L 34f$JDLQ LW LV DVVXPHG WKDW 3URSRVLWLRQ KDV EHHQ DSSOLHG LI QHFHVVDU\ WR HQVXUH WKDW WKH LQYHUVHV RI WKH PDWULFHV )P8Gf DQG )A8Gf H[LVW 3523(57< ,Q WKH DERYH WKUHH GDPDJH FRQILJXUDWLRQV FRQVLGHUHG IRU QRQSURSRUWLRQDOO\ GDPSHG VWUXFWXUHV WKH FRPSXWHG SHUWXUEDWLRQ PDWULFHV $0G $'G RU $.G f IURP WKH 0537 ZLOO EH V\PPHWULF

PAGE 108

3URRI 7KLV LV WUXH VLQFH WKH GHYHORSHG IRUPXODWLRQV UHTXLUH WKH SHUWXUEDWLRQ PDWULFHV DORQJ ZLWK WKH fKHDOWK\f )(0 WR VDWLVI\ WKH FURVV RUWKRJRQDOLW\ FRQGLWLRQV LQ (TV f 7KHVH FURVVRUWKRJRQDOLW\ FRQGLWLRQV DUH VDWLVILHG LI DQG RQO\ LI WKH FRUUHVSRQGLQJ VWDWH PDWULFHV DUH V\PPHWULF WKDW LV R V n0 $0Gf R O O Z 4 R f B $'Gf $.Gf $.Gf $'Gf B $.Gf $.Gf 0 $0Gfn 0 $0Gf B $'Gf 0 $0Gf 0 $0Gfn $'Gf f 6LQFH WKH PDWULFHV 0 DUH NQRZQ WR EH V\PPHWULF WKH DERYH UHODWLRQVKLSV DUH VDWLVILHG LI DQG RQO\ LI WKH SHUWXUEDWLRQ PDWULFHV $0G $'G DQG $.G DUH V\PPHWULF 3UDFWLFDO ,VVXHV ,Q DOO FDVHV GLVFXVVHG LQ WKH SUHYLRXV VHFWLRQV RI WKLV &KDSWHU WKH GDPDJH H[WHQW SUREOHP ERLOHG GRZQ WR VROYLQJ SUREOHPV RI WKH IRUP $; < f LQ ZKLFK $ LV DQ XQNQRZQ Q[Q PDWUL[ UHSUHVHQWLQJ RQH RI WKH SHUWXUEDWLRQ PDWULFHV $0G $'M DQG $.Mf DUrG ; DQG < DUH NQRZQ Q[S FRPSOH[ PDWULFHV 0DWUL[ ; LV D IXQFWLRQ RI WKH GDPDJHG VWUXFWXUH HLJHQYHFWRU 9Gf DQG HLJHQYDOXH $Gf PDWULFHV 5HFDOO WKDW S UHSUHVHQWV WKH QXPEHU RI PRGHV DQG Q LV WKH WRWDO QXPEHU RI '2)V LQ WKH VWUXFWXUHfV )(0 $V GLVFXVVHG LQ &KDSWHU WKH FROXPQV RI PDWUL[ < SURYLGH XVHIXO LQIRUPDWLRQ LQ GHWHUPLQLQJ WKH ORFDWLRQ

PAGE 109

RI WKH VWUXFWXUDO GDPDJH )RU HDFK SUREOHP D FXPXODWLYH GDPDJH YHFWRU DVVRFLDWHG WR WKH FROXPQV RI PDWUL[ < FDQ EH GHILQHG DV S O\ 9 GL £f3 = f L O GL 5HFDOO WKDW WKH '2)V RI WKH )(0 DIIHFWHG E\ WKH GDPDJH DUH DVVRFLDWHG ZLWK WKH FRPSRQHQWV RI YHFWRU G ZKLFK DUH VXEVWDQWLDOO\ ODUJHU WKDQ WKH RWKHU FRPSRQHQWV RI G &RQVLGHULQJ WKH SUHVHQW JHQHULF SUREOHP WKH H[WHQW SUREOHP FRQVLVWV RI GHWHUPLQLQJ D V\PPHWULF UHDO PDWUL[ $ WKDW VDWLVILHV (T f $ VROXWLRQ WR WKLV SUREOHP EDVHG RQ WKH 0LQLPXP 5DQN 3HUWXUEDWLRQ 7KHRU\ 0537f ZDV GHYHORSHG LQ 3URSRVLWLRQ 7KH VROXWLRQ WHFKQLTXH SUHVHQWHG LQ 3URSRVLWLRQ UHTXLUHV WKH S[S PDWUL[ <7;f WR EH LQYHUWLEOH 3URSRVLWLRQ ZDV WKHQ SUHVHQWHG WR GHDO ZLWK WKH FDVH ZKHQ PDWUL[ <7;f LV QRW LQYHUWLEOH ,Q WKH IRUWKFRPLQJ GLVFXVVLRQV WZR VWUDWHJLHV WR LPSURYH WKH H[WHQW FDOFXODWLRQ DUH SUHVHQWHG 7KH &RQFHSW RI f%HVWf 0RGHV 1RWH WKDW IRU WKH VDNH RI JHQHUDOLW\ WKH IRUWKFRPLQJ GLVFXVVLRQ LV EDVHG RQ WKH JHQHULF SUREOHP VKRZQ LQ (T f KRZHYHU LW VKRXOG EH YLHZHG DV UHSUHVHQWDWLYH RI DOO H[WHQW SUREOHPV GLVFXVVHG LQ WKH HDUOLHU VHFWLRQV RI WKLV FKDSWHU $OWKRXJK 3URSRVLWLRQ GHDOV ZLWK QRLVH IUHH PHDVXUHPHQWV LWV SUDFWLFDO XVHIXOQHVV LV IRU WKH FDVH ZKHQ WKH HLJHQYDOXH HLJHQYHFWRU PHDVXUHPHQWV DUH FRUUXSWHG E\ QRLVH HLWKHU WKURXJK DFWXDO PHDVXUHPHQW HUURU RU HLJHQYHFWRU H[SDQVLRQ LQGXFHG HUURUVf ,Q JHQHUDO PDWUL[ < ZLOO W\SLFDOO\ EH RI IXOO FROXPQ UDQN GXH WR WKH SUHVHQFH RI QRLVH UDQN
PAGE 110

HLJHQYDOXHHLJHQYHFWRU PHDVXUHPHQW HUURUV 7KHUH DUH WZR SRVVLEOH WHFKQLTXHV IRU HVWLPDWLQJ WKH fWUXHf UDQN RI < 7KH ILUVW WHFKQLTXH ZKLFK DPRXQWV WR D EUXWHIRUFH QXPHULFDO FRPSXWDWLRQ LV WR FDOFXODWH WKH 6LQJXODU 9DOXH 'HFRPSRVLWLRQ 69'f RI < 9DULRXV UDQN FULWHULD PHDVXUHV DV IRUPXODWHG E\ -XDQJ DQG 3DSSD f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f LQGLFDWHV ZKLFK VWUXFWXUDO '2)V KDYH EHHQ GLUHFWO\ DIIHFWHG E\ WKH GDPDJH 8VLQJ FRQQHFWLYLW\ LQIRUPDWLRQ IURP WKH RULJLQDO )(0 LQ FRQMXQFWLRQ ZLWK NQRZOHGJH RI WKH GDPDJHG '2)V DOORZV IRU WKH GHWHUPLQDWLRQ RI ZKLFK HOHPHQWV RI WKH )(0 KDYH EHHQ GDPDJHG 7KH fWUXHf UDQN RI < FDQ EH HVWLPDWHG E\ DGGLQJ XS WKH UDQNV RI WKH HOHPHQW PDWULFHV DVVRFLDWHG ZLWK HDFK GDPDJHG HOHPHQW 2QFH WKH WUXH UDQN RI < KDV EHHQ HVWLPDWHG FRUUHVSRQGLQJ FROXPQV RI ; DQG < DUH HOLPLQDWHG VXFK WKDW WKH DFWXDO UDQN RI % LV VHW WR WKH HVWLPDWHG WUXH UDQN RI < ,Q 3URSRVLWLRQ LW ZDV VKRZQ WKDW ZLWK QRLVH IUHH PHDVXUHPHQWV LW GRHV QRW PDWWHU ZKLFK U PRGHV IURP WKH VHW RI S PHDVXUHG PRGHV DUH XVHG LQ WKH H[WHQW FDOFXODWLRQ +RZHYHU ZKHQ WKH PHDVXUHPHQWV DUH FRUUXSWHG E\ QRLVH LW LV EHVW WR XVH WKRVH PRGHV ZKLFK EHVW UHIOHFW WKH GDPDJH VWDWH DV LQGLFDWHG E\ WKH FXPXODWLYH GDPDJH YHFWRU (T f $V ZLOO EH LOOXVWUDWHG LQ WKH H[DPSOH SUREOHPV SUHVHQWHG LQ &KDSWHU WKH FRQFHSW GLVFXVVHG KHUH LPSURYHV WKH GDPDJH H[WHQW HVWLPDWLRQ 7KLV LV HVSHFLDOO\ WUXH ZKHQ WKH UDQN RI WKH SHUWXUEDWLRQ PDWUL[ Uf GXH WR WKH GDPDJH LV OHVV WKDQ WKH QXPEHU RI PHDVXUHG PRGHV Sf 7KH UHDVRQ LV WZRIROG )LUVW WKH UDQN RI WKH FRPSXWHG SHUWXUEDWLRQ PDWUL[ LV FRQVLVWHQW ZLWK WKH UDQN RI WKH DFWXDO SHUWXUEDWLRQ PDWUL[ 6HFRQG WKH PHDVXUHPHQW HUURUV LQ WKH H[SHULPHQWDO PRGHV WKDW GR QRW UHIOHFW WKH DFWXDO VWDWH RI WKH GDPDJH DUH QRW LQFOXGHG LQ WKH FRPSXWDWLRQ

PAGE 111

$SSOLFDWLRQ RI WKH (LJHQYHFWRU )LOWHULQJ $OJRULWKP $QRWKHU WHFKQLTXH WKDW FDQ EH XVHG WR LPSURYHG WKH GDPDJH H[WHQW HVWLPDWLRQ LV WKH HLJHQYHFWRU IGWHULQJ DOJRULWKP GHYHORSHG LQ &KDSWHU 7KH PRGHV WKDW VKRXOG EH ILOWHUHG DQG XVHG LQ WKH H[WHQW FRPSXWDWLRQ DUH WKH RQHV WKDW UHIOHFW WKH RYHUDOO VWUXFWXUH FXPXODWLYH GDPDJH YHFWRU GHILQHG LQ (TV f RU f RI &KDSWHU 7KH FKDUDFWHULVWLFV RI WKH HLJHQYHFWRU ILOWHULQJ DOJRULWKP DUH LOOXVWUDWHG LQ &KDSWHU ,W VKRXOG EH SRLQWHG RXW WKDW LW LV QRW D UHTXLUHPHQW RI WKH H[WHQW DOJRULWKP WR XVH WKH HLJHQYHFWRU ILOWHULQJ DOJRULWKP EXW FHUWDLQO\ LW GRHV DOORZ IRU PRUH HQJLQHHULQJ MXGJHPHQW WR HQWHU LQWR WKH H[WHQW FDOFXODWLRQ 7KH FKRLFH RI ZKHWKHU RU QRW WR XVH WKH ILOWHULQJ DOJRULWKP LV GHSHQGHQW RQ Lf WKH HUURUV LQ WKH HLJHQYHFWRUV DV VHHQ IURP WKH GDPDJH YHFWRUV LLf WKH UHTXLUHPHQW RQ DFFXUDF\ RI WKH H[WHQW FDOFXODWLRQ DQG LLLf WKH VL]H RI WKH )(0 ZKLFK GHILQHV WKH DGGLWLRQDO FRPSXWDWLRQDO EXUGHQ 6XPPDU\ $ FRPSXWDWLRQDOO\ DWWUDFWLYH DOJRULWKP ZDV GHYHORSHG WR SURYLGH DQ LQVLJKW WR WKH H[WHQW RI VWUXFWXUDO GDPDJH 7KH GHYHORSHG WKHRU\ FDOOHG WKH 0LQLPXP 5DQN 3HUWXUEDWLRQ 7KHRU\ 0537f PDNHV XVH RI DQ H[LVWLQJ ILQLWH HOHPHQW PRGHO RI WKH fKHDOWK\f VWUXFWXUH DQG D VXEVHW RI H[SHULPHQWDOO\ PHDVXUHG PRGDO SURSHUWLHV RI WKH fGDPDJHGf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

PAGE 112

&+$37(5 9$/,'$7,21 $1' $66(660(17 2) 7+( 68%63$&( 527$7,21 $/*25,7+0 $1' 7+( 0,1,080 5$1. 3(5785%$7,21 7+(25< ,QWURGXFWLRQ ,Q WKLV FKDSWHU WKH FKDUDFWHULVWLFV RI WKH VXEVSDFH URWDWLRQ DOJRULWKP &KDSWHU f DQG WKH PLQLPXP UDQN SHUWXUEDWLRQ WKHRU\ &KDSWHU f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fUHDOOLIHf V\VWHPV .DEHfV 3UREOHP 3UREOHP 'HVFULSWLRQ .DEHfV HLJKW GHJUHH RI IUHHGRP VSULQJPDVV SUREOHP LV VKRZQ LQ )LJXUH ZKLFK LQFOXGHV WKH VWLIIQHVV DQG PDVV YDOXHV IRU WKH H[DFW PRGHO 7KH PDLQ REMHFWLYH RI WKLV SUREOHP LV WR LOOXVWUDWH WKH GDPDJH ORFDWLRQ DOJRULWKP IRUPXODWHG LQ &KDSWHU 7KH 6XEVSDFH 5RWDWLRQ DOJRULWKP GLUHFW PHWKRG LV FRPSDUHG WR WKH $QJOH 3HUWXUEDWLRQ 0HWKRG .H\ SRLQWV GLVFXVVHG LQ WKH IRUPXODWLRQ ZLOO EH KLJKOLJKWHG $SSOLFDWLRQ RI WKH GDPDJH

PAGE 113

H[WHQW DOJRULWKP RI &KDSWHU LV DOVR DGGUHVVHG %RWK DOJRULWKPV ZLOO EH HYDOXDWHG XVLQJ QRLV\ PRGDO PHDVXUHPHQWV LQ DQ DWWHPSW WR UHSURGXFH D SUDFWLFDO WHVWLQJ VLWXDWLRQ PL PJ PM M NL N N N N N )LJXUH .DEHfV 3UREOHP $ YDULDWLRQ RI .DEHfV RULJLQDO SUREOHP LV XVHG KHUH 5DWKHU WKDQ WKH VWDQGDUG PRGHO FRPPRQO\ XVHG ZKLFK KDV LQFRUUHFW YDOXHV IRU DOO RI WKH FRQQHFWLQJ VSULQJV RQO\ D VLQJOH VSULQJ FRQVWDQW LV FKDQJHG 7KLV LV UHIOHFWLYH RI WKH IDFW WKDW GDPDJH PD\ RFFXU DV D ODUJH ORFDO FKDQJH LQ WKH VWLIIQHVV RI D VWUXFWXUDO PHPEHU ,Q WKLV SUREOHP WKH LQLWLDO PRGHO LV LQFRUUHFW IRU WKH VSULQJ EHWZHHQ PDVVHV VHYHQ f DQG HLJKW f 'DPDJH ZLOO EH PRGHOHG DV D FKDQJH LQ VSULQJ FRQVWDQW IURP WR &KDQJLQJ WKH VSULQJ LQ WKLV IDVKLRQ FDXVHV WKH GDPDJH WR RFFXU DW GHJUHHV RI IUHHGRP ZKLFK KDYH D VPDOO __ =G __ LQ FRPSDULVRQ WR RWKHU '2)V 7KXV LW ZRXOG EH H[SHFWHG WKDW GDPDJH ZRXOG EHVW EH ORFDWHG XVLQJ WKH $QJOH 3HUWXUEDWLRQ PHWKRG DV GHILQHG LQ (T f ,Q WKLV SUREOHP LW LV DVVXPHG WKDW RQO\ WKH ILUVW WZR PRGHV RI YLEUDWLRQ DUH PHDVXUHG 1RWH WKDW WKH GDPDJH GRHV QRW DIIHFW WKH PDVV SURSHUWLHV RI WKH VWUXFWXUH

PAGE 114

'DPDJH /RFDWLRQ )LJXUHV DQG VKRZ UHVSHFWLYHO\ WKH GDPDJH ORFDWLRQ UHVXOWV RI XVLQJ WKH 6XEVSDFH 5RWDWLRQ GLUHFW Gcf PHWKRG DQG WKH $QJOH 3HUWXUEDWLRQ 0HWKRG Df )LJXUH SURYLGHV WKH GDPDJH YHFWRU DV GHILQHG E\ /LQfV DOJRULWKP f 7KHVH UHVXOWV ZHUH JHQHUDWHG XVLQJ RQO\ WKH IXQGDPHQWDO PRGH RI YLEUDWLRQ HLJHQGDWD ,Q WKHVH ILJXUHV WKH [FRRUGLQDWH UHSUHVHQWV WKH HOHPHQWV RI WKH GDPDJH YHFWRU Gc RU Dcf ZKHUH WKH LWK HOHPHQW RI Gc Dcf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f DQG f WKH ORFDWLRQ DOJRULWKP LV DEOH WR H[DFWO\ ORFDWH WKH FRUUHFW GDPDJH ZKHQ SURYLGHG ZLWK RQO\ D VLQJOH FRUUHFW HLJHQYDOXHHLJHQYHFWRU SDLU DV JXDUDQWHHG E\ (T f )URP WKH ORZHU SORWV LW LV VHHQ WKDW WKH DOJRULWKP H[SHULHQFHV D GHJUDGDWLRQ LQ GDPDJH GHWHFWLRQ FDSDELOLW\ DV WKH HUURU LQ WKH HLJHQYHFWRU LV LQFUHDVHG ,Q IDFW ZKHQ SUHVHQWHG ZLWK b HLJHQYHFWRU QRLVH WKH GLUHFW VXEVSDFH URWDWLRQ PHWKRG LQFRUUHFWO\ LGHQWLILHV WKH GDPDJH +RZHYHU ZKHQ XVLQJ WKH DQJOH SHUWXUEDWLRQ PHWKRG WKH ORFDWLRQ DOJRULWKP LV DEOH WR GLVFHUQ GDPDJH ZLWK D b QRLVH OHYHO $V VKRZQ LQ )LJXUH /LQfV DOJRULWKP LV XQDEOH WR GLVFHUQ GDPDJH LQ WKLV SDUWLFXODU WHVW SUREOHP )LJXUHV DQG VKRZ WKH UHVXOWV RI XVLQJ WKH DQJOH SHUWXUEDWLRQ PHWKRG DQG /LQfV DOJRULWKP ZKHQ SURYLGHG ZLWK FRPSOHWH ILUVW DQG VHFRQG PRGH HLJHQYDOXHHLJHQYHFWRU LQIRUPDWLRQ $JDLQ WKH DQJOH SHUWXUEDWLRQ PHWKRG LV DEOH WR FOHDQO\ GHWHFW GDPDJH HYHQ ZKHQ SUHVHQWHG ZLWK HLJHQYHFWRU LQIRUPDWLRQ FRUUXSWHG ZLWK WHQ SHUFHQW QRLVH $JDLQ /LQfV DOJRULWKP LV XQDEOH WR GLVFHUQ GDPDJH

PAGE 115

([DFW GDPDJH K 2n ‘ '2) VW PRGH L r ‘ '2) VW PRGH b HUURU VW PRGH b HUURU n f§n rf§ rf§ '2) )LJXUH .DEHfV 3UREOHP 'DPDJH /RFDWLRQ 5HVXOWV XVLQJ WKH 6XEVSDFH 5RWDWLRQ 'LUHFW 0HWKRG ZLWK WKH (LJHQGDWD RI WKH VW 0RGH ([DFW GDPDJH D } '2) )LJXUH .DEHfV 3UREOHP 'DPDJH /RFDWLRQ 5HVXOWV XVLQJ WKH $QJOH 3HUWXUEDWLRQ 0HWKRG ZLWK WKH (LJHQGDWD RI WKH VW 0RGH

PAGE 116

([DFW 'DPDJH 2n f r f'2) VW PRGH b HUURU VW PRGH )LJXUH .DEHfV 3UREOHP 'DPDJH /RFDWLRQ 5HVXOWV XVLQJ /LQfV $OJRULWKP ZLWK WKH (LJHQGDWD RI WKH VW 0RGH ([DFW GDPDJH D )LJXUH .DEHfV 3UREOHP 'DPDJH /RFDWLRQ 5HVXOWV XVLQJ WKH $QJOH 3HUWXUEDWLRQ 0HWKRG ZLWK WKH (LJHQGDWD RI WKH VW DQG QG 0RGHV

PAGE 117

([DFW 'DPDJH PRGHV t '2) PRGHV t b HUURU )LJXUH .DEHfV 3UREOHP 'DPDJH /RFDWLRQ 5HVXOWV XVLQJ /LQfV $OJRULWKP ZLWK WKH (LJHQGDWD RI WKH VW DQG QG 0RGHV 'DPDJH ([WHQW ,Q WKH SUHYLRXV WZRPRGH QRLV\ WHVW FDVH LQVSHFWLRQ RI WKH FXPXODWLYH GDPDJH YHFWRU FOHDUO\ LQGLFDWHV WKDW WKH WK DQG WK GHJUHHV RI IUHHGRP KDYH H[SHULHQFHG GDPDJH ,QVSHFWLQJ WKH VWUXFWXUDO FRQQHFWLYLW\ LW FDQ EH GHGXFHG WKDW WKH VSULQJ EHWZHHQ PDVV DQG KDV EHHQ GDPDJHG 7KH UDQN RI WKH VLQJOH VSULQJ fHOHPHQWf PDWUL[ LV RQH 7KXV WKH UDQN RI WKH fWUXHf SHUWXUEDWLRQ WR WKH VWLIIQHVV PDWUL[ GXH WR GDPDJH LV RQH )URP 3URSRVLWLRQ DQG LW LV FOHDU WKDW RQO\ H[SHULPHQWDO GDWD IURP RQH PRGH RI YLEUDWLRQ VKRXOG EH XVHG WR FRPSXWH WKH H[WHQW RI GDPDJH ,Q WKH QRLV\ FDVHV DV GLVFXVVHG LQ 6HFWLRQ WKH PRGH WKDW VKRXOG EH XVHG LQ WKH H[WHQW FDOFXODWLRQ LV WKH RQH DVVRFLDWHG ZLWK D Gc WKDW PRVW FOHDQO\ GHPRQVWUDWHV WKH GDPDJH ORFDWLRQ VKRZQ LQ )LJXUH $ VLPSOH LQVSHFWLRQ RI WKH LQGLYLGXDO Gc RI WKH WZR PHDVXUHG PRGHV LQGLFDWHV WKDW PRGH SURYLGHV WKH EHVW LQVLJKW LQWR WKH VWDWH RI GDPDJH )LJXUH SUHVHQWV HOHPHQWE\HOHPHQW VWLIIQHVV SHUWXUEDWLRQ PDWUL[ UHVXOWV IURP WKH DSSOLFDWLRQ RI WKH H[WHQW DOJRULWKP IRUPXODWHG LQ 6HFWLRQ XVLQJ PRGH

PAGE 118

HLJHQGDWD 7KH [FRRUGLQDWHV RQ HDFK SORW DUH WKH LQGLFHV RI D FROXPQ YHFWRU FRQVWUXFWHG E\ VWRULQJ WKH XSSHU WULDQJXODU SRUWLRQ RI WKH SHUWXUEDWLRQ VWLIIQHVV PDWUL[ $.Gf LQ D FROXPQ YHFWRU 7KH \FRRUGLQDWH RQ DOO SORWV FRQVLVWV RI WKH GLIIHUHQFH EHWZHHQ WKH XSGDWHG VWLIIQHVV PDWUL[ DQG WKH RULJLQDO VWLIIQHVV PDWUL[ LH $.Af ,Q WKH QRLVH IUHH FDVH LW VKRXOG EH QRWHG WKDW WKH H[DFW GDPDJH ZRXOG EH FRPSXWHG XVLQJ DQ\ VLQJOH VHW RI HLJHQYDOXHHLJHQYHFWRU PHDVXUHPHQW 7KH H[WHQW DOJRULWKP H[SHULHQFHV D PRGHVW GHJUDGDWLRQ DV WKH PHDVXUHPHQWV DUH FRUUXSWHG E\ QRLVH ([DFW GDPDJH ,QGLFHV QG PRGH ,QGLFHV QG PRGH b HUURU ,QGLFHV )LJXUH .DEHfV 3UREOHP 'DPDJH ([WHQW 5HVXOWV XVLQJ WKH 0537 ZLWK WKH (LJHQGDWD RI 0RGH 7R FRQWUDVW WKH SURSRVHG PLQLPXP UDQN SHUWXUEDWLRQ DSSURDFK WR WKH RSWLPDO PDWUL[ XSGDWH DSSURDFKHV FRQVLGHU WKH DSSOLFDWLRQ WR WKH FXUUHQW SUREOHP RI WKH FRPPRQO\ XVHG PHWKRG IRUPXODWHG E\ %DUXFK DQG %DU ,W]KDFN f $OWKRXJK WKLV DOJRULWKP ZDV GHYHORSHG IURP D PRGHO UHILQHPHQW YLHZSRLQW LW KDV EHHQ LQYHVWLJDWHG IRU SRVVLEOH XVHV LQ GDPDJH GHWHFWLRQ 6PLWK f ,W LV HYLGHQW IURP )LJXUH WKDW WKLV DOJRULWKP LV XQDEOH WR GLVFHUQ WKH GDPDJH IRU WKLV SDUWLFXODU H[DPSOH ZKHQ SURYLGHG ZLWK Lf WKH VHFRQG PRGHfV H[DFW HLJHQGDWD RU LLf WKH ILUVW VL[ PRGHVf H[DFW HLJHQGDWD ,Q IDFW LW FDQ EH VHHQ LQ WKH XSSHU

PAGE 119

([DFW GDPDJH O f§ ,QGLFHV PRGHV WR Q ’ , OBB ,QGLFHV QG PRGH LU \ ,7 ,QGLFHV PRGHV WR f§ , ,QGLFHV )LJXUH .DEHfV 3UREOHP 'DPDJH ([WHQW 5HVXOWV XVLQJ %DUXFKfV 0HWKRG ULJKW SORW RI )LJXUH WKDW FRQVLVWHQW ZLWK WKH PLQLPXP )UREHQLXV QRUP FKDQJH IRUPXODWLRQ WKH DSSURDFK WHQGV WR fVPHDUf WKH FKDQJHV WKURXJKRXW WKH HQWLUH VWLIIQHVV PDWUL[ LQVWHDG RI ORFDOL]LQJ WKHP DW WKH GDPDJH ORFDWLRQ ,Q WKLV SUREOHP WKLV PHWKRG LV RQO\ DEOH WR FOHDQO\ DVFHUWDLQ WKH GDPDJH ZKHQ SURYLGHG ZLWK WKH H[DFW HLJHQGDWD RI WKH ILUVW VHYHQ PRGHV 'DPDJH 'HWHFWLRQ )LIW\%D\ 7ZR'LPHQVLRQDO 7UXVV 8QGDPSHG )(0 3UREOHP 'HVFULSWLRQ 7KH ED\ GLPHQVLRQDO WUXVV XVHG LQ WKLV H[DPSOH LV VKRZQ LQ )LJXUH 7KH JHRPHWULF DQG PDWHULDO SURSHUWLHV RI WKH WUXVV DUH DOVR JLYHQ LQ WKH ILJXUH (DFK WUXVV PHPEHU ZDV PRGHOHG DV D URG HOHPHQW 7KH ILQLWH HOHPHQW PRGHO RI WKH VWUXFWXUH KDV GHJUHHV RI IUHHGRP DQG LV XQGDPSHG ,Q WKLV H[DPSOH GDPDJH LV VLPXODWHG E\ UHGXFLQJ WKH
PAGE 120

ED\ ,Q WKLV PHPEHU WKH
PAGE 121

H[SHULHQFHV D GHJUDGDWLRQ LQ GHWHFWLQJ GDPDJH DV WKH HUURU LQ WKH HLJHQYHFWRUV LV LQFUHDVHG KRZHYHU WKH GDPDJH LV VWLOO ORFDWDEOH ([DFW 'DPDJH 0RGHV WR FG W F :f D FG 4 '2) 0RGHV WR b HUURU '2) 0RGHV WR b HUURU )LJXUH )LIW\%D\ 7UXVV 'DPDJH /RFDWLRQ 5HVXOWV XVLQJ WKH 6XEVSDFH 5RWDWLRQ $OJRULWKP ZLWK WKH (LJHQGDWD RI WKH )LUVW 7HQ 0RGHV 'DPDJH ([WHQW :LWK NQRZOHGJH RI WKH ORFDWLRQ RI GDPDJH IURP WKH SUHYLRXV DQDO\VLV WKH UDQN RI WKH fWUXHf SHUWXUEDWLRQ WR WKH VWLIIQHVV PDWUL[ $,4 FDQ EH IRXQG E\ DGGLQJ WKH UDQN RI WKH HOHPHQW VWLIIQHVV PDWUL[ RI WKH GDPDJHG PHPEHUV +HQFH WKH UDQN RI WKH SHUWXUEDWLRQ WR WKH VWLIIQHVV PDWUL[ GXH WR GDPDJH LV WZR EHFDXVH WZR PHPEHUV KDYLQJ UDQN RQH HOHPHQW PDWULFHV DUH GDPDJHG )URP 3URSRVLWLRQ DQG LW LV FOHDU WKDW RQO\ H[SHULPHQWDO HLJHQGDWD IURP WZR PRGHV RI YLEUDWLRQ VKRXOG EH XVHG WR FRPSXWH WKH H[WHQW RI WKH GDPDJH ,Q WKH QRLV\ FDVHV WKH WZR PRGHV WKDW VKRXOG EH XVHG DUH WKH RQHV ZLWK GcfV WKDW PRVW FOHDQO\ GHPRQVWUDWH WKH GDPDJH VKRZQ LQ )LJXUH 7KHVH PRGHV FDQ EH GHWHUPLQHG E\

PAGE 122

LQVSHFWLQJ WKH LQGLYLGXDO GDPDJH YHFWRUV Gc DVVRFLDWHG ZLWK WKH PHDVXUHG PRGHV ,Q UHYLHZLQJ WKH LQGLYLGXDO Gc LW ZDV GHWHUPLQHG WKDW PRGHV DQG SURYLGH WKH EHVW LQVLJKW LQWR WKH VWDWH RI WKH GDPDJH 7KH UHVXOWV RI DSSO\LQJ WKH H[WHQW DOJRULWKP WR GHWHUPLQH WKH SHUWXUEDWLRQV WR WKH VWLIIQHVV PDWUL[ GXH WR WKH GDPDJH $.A DUH VKRZQ LQ )LJXUH 7KH PHVK SORWV DUH GLPHQVLRQDO UHSUHVHQWDWLRQV RI WKH SHUWXUEDWLRQ PDWULFHV 7KH URZV DQG FROXPQV RI WKH PHVK SORWV FRUUHVSRQG WR WKH URZV DQG FROXPQV RI WKH SHUWXUEDWLRQ PDWULFHV 7KH fKHLJKWf RI HDFK SHDN UHSUHVHQWV WKH PDJQLWXGH RI WKH SHUWXUEDWLRQ PDGH WR HDFK PDWUL[ HOHPHQW ,Q )LJXUH WKH XSSHU OHIW PHVK SORW UHSUHVHQWV WKH H[DFW GDPDJH 7KH XSSHU ULJKW SORW FRUUHVSRQGV WR WKH FDVH ZKHUH H[DFW HLJHQGDWD DUH XVHG LQ FRPSXWLQJ WKH H[WHQW RI GDPDJH 1RWH WKDW ZLWK RQO\ WZR QRLVH IUHH HLJHQYDOXHV HLJHQYHFWRUV PRGHV DQG f WKH DOJRULWKP LV DEOH WR UHSURGXFH WKH H[DFW GDPDJH $V VWDWHG DQG SURYHG LQ 3URSRVLWLRQ WKH H[DFW GDPDJH FDQ EH FRPSXWHG XVLQJ DQ\ WZR QRLVH IUHH PRGHV WKDW KDYH D FRUUHVSRQGLQJ % GHILQHG LQ 6HFWLRQ f RI UDQN WZR 7KH OHIW DQG ULJKW PHVK SORWV RI WKH VHFRQG URZ RI )LJXUH FRUUHVSRQG WR WKH FDVHV ZKHUH WKH HLJHQYHFWRUV KDYH EHHQ FRUUXSWHG ZLWK b DQG b UDQGRP QRLVH UHVSHFWLYHO\ 7KH DOJRULWKP DJDLQ GHPRQVWUDWHV JRRG SHUIRUPDQFH ZKHQ IDFHG ZLWK QRLV\ HLJHQGDWD PLGGOH SORWVf 7KH UHVXOWV RI DSSO\LQJ WKH HLJHQYHFWRU ILOWHULQJ DOJRULWKP GHVFULEHG LQ 6HFWLRQ WR SUHGLFW WKH H[WHQW XVLQJ PRGHV DQG DUH VKRZQ LQ WKH OHIW DQG ULJKW ORZHU SORWV RI )LJXUH ,W LV REYLRXV WKDW WKH ILOWHULQJ SURFHVV JUHDWO\ HQKDQFHV WKH DFFXUDF\ RI WKH H[WHQW DOJRULWKP )LJXUH GLVSOD\V WKH 0537fV DVVHVVPHQW RI WKH GDPDJH H[WHQW IRU WKH QRLV\ HLJHQYHFWRU FDVHV ZKHQ WKH HLJHQGDWD RI WKH ILUVW WHQ PRGHV RI YLEUDWLRQ DUH XVHG $ FRPSDULVRQ RI WKH UHVXOWV LQ )LJXUH DQG WKRVH LQ WKH VHFRQG URZ RI )LJXUH VKRZV WKDW WKH DERYH SURFHGXUH ZKLFK UHVXOWHG LQ WKH XVH RI RQO\ PRGHV DQG WR DVVHVV WKH H[WHQW SURYLGHG FOHDQHU UHVXOWV ,Q WKH PHVK SORWV RI )LJXUH DGGLWLRQDO QXPHULFDO YDOXHV DSSHDU DW WKH fXQGDPDJHGf '2)V ZKLFK FDQ EH DWWULEXWHG WR WKH QRLVH DGGHG E\ WKH DGGLWLRQ RI WKH H[WUD PRGHV 7KH UHVXOWDQW SHUWXUEDWLRQ PDWULFHV ZKHQ XVLQJ DOO WHQ PRGHV DUH RI UDQN ZKLFK LV QRW FRQVLVWHQW ZLWK WKH fDFWXDOf GDPDJH SHUWXUEDWLRQ ZKLFK LV RI UDQN )XUWKHUPRUH WKLV fWHQ PRGHV SURFHGXUHf LV OHVV

PAGE 123

0RGHV t b HUURU 0RGHV t b HUURU 0RGHV t b HUURU )LOWHUHG )LOWHUHG )LJXUH )LIW\%D\ 7UXVV 'DPDJH ([WHQW 5HVXOWV XVLQJ WKH 0537 ZLWK WKH (LJHQGDWD RI 0RGHV DQG HIILFLHQW VLQFH LW UHTXLUHV WKH LQYHUVLRQ RI D E\ PDWUL[ FRQWUDVWHG WR WKH E\ LQYHUVH XVHG LQ WKH fWZR PRGHV SURFHGXUHf 7KH SHUFHQWDJH HUURUV RI WKH SUHGLFWHG GDPDJH H[WHQW ZLWK UHVSHFW WR WKH H[DFW VWLIIQHVV GDPDJH IRU DOO VWXGLHG FDVHV ZLWK DQG ZLWKRXW ILOWHULQJ

PAGE 124

DUH OLVWHG LQ 7DEOH ,W FDQ EH VHHQ IURP WKH WDEOH WKH UHVXOWV REWDLQHG ZKHQ WKH HLJHQYHFWRU ILOWHULQJ SURFHVV LV XVHG DUH PXFK PRUH DFFXUDWH 7KH UHVXOWV IURP WKH fWZR PRGHV SURFHGXUHf DUH LQ DOO FDVHV EXW RQH EHWWHU WKDQ WKH fWHQ PRGHV SURFHGXUHf 0RGHV WR b HUURU 0RGHV WR b HUURU )LJXUH )LIW\%D\ 7UXVV 'DPDJH ([WHQW 5HVXOWV XVLQJ WKH 0537 :LWK WKH (LJHQGDWD RI WKH )LUVW 7HQ 0RGHV 0RGHV t 0RGHV WR )LJXUH )LIW\%D\ 7UXVV 'DPDJH ([WHQW 5HVXOWV XVLQJ %DUXFKfV $OJRULWKP )RU WKH VDNH RI FRPSDULVRQ UHVXOWV IURP WKH DSSOLFDWLRQ RI %DUXFK DQG %DU ,W]KDFNfV DOJRULWKP f DUH SURYLGHG LQ )LJXUH 7KHVH UHVXOWV ZHUH REWDLQHG XVLQJ Lf QRLVH IUHH HLJHQGDWD IURP PRGHV DQG DQG LLf QRLVH IUHH HLJHQGDWD IURP WKH ILUVW WHQ PRGHV ,Q ERWK FDVHV %DUXFKfV DOJRULWKP IDLOV WR DVFHUWDLQ WKH H[WHQW RI WKH GDPDJH ,Q WKH ILUVW FDVH WKH DSSURDFK FRPSOHWHO\ fVPHDUHGf WKH FKDQJHV WKURXJKRXW WKH HQWLUH VWLIIQHVV PDWUL[ ,Q WKH VHFRQG FDVH WKH FKDQJHV DUH PRUH ORFDOL]HG DURXQG WKH GDPDJHG DUHDV

PAGE 125

7DEOH )LIW\%D\ 7UXVV 6XPPDU\ RI 'DPDJH ([WHQW 5HVXOWV XVLQJ WKH 0537 3HUFHQWDJH (UURU ZLWK 5HVSHFW WR WKH ([DFW 'DPDJH :LWKRXW )LOWHULQJ 0RGHV WR :LWKRXW )LOWHULQJ 0RGHV t :LWK )LOWHULQJ 0RGHV t (LJHQYHFWRUV (UURU 8SSHU /RQJHURQ RI %D\ 7KUHH /RZHU /RQJHURQ RI %D\ )RUW\ 8SSHU /RQJHURQ RI %D\ 7KUHH /RZHU /RQJHURQ RI %D\ )RUW\ 8SSHU /RQJHURQ RI %D\ 7KUHH /RZHU /RQJHURQ RI %D\ )RUW\ b 1RW $SSOLFDEOH 1RW $SSOLFDEOH b b 1RW $SSOLFDEOH 1RW $SSOLFDEOH b b b b b b b b b b b b b b

PAGE 126

([SHULPHQWDO 6WXG\ 7KH 1$6$ EDY 7UXVV 3UREOHP 'HVFULSWLRQ 7KH HLJKW ED\ K\EULGVFDOHG WUXVV XVHG LQ WKLV LQYHVWLJDWLRQ LV SDUW RI D VHULHV RI VWUXFWXUHV GHVLJQHG IRU UHVHDUFK LQ G\QDPLF VFDOH PRGHO JURXQG WHVWLQJ RI ODUJH VWUXFWXUHV DW WKH 1$6$ /DQJOH\ 5HVHDUFK &HQWHU $PRQJ RWKHU VWXGLHV D FRPSOHWH DQDO\WLFDO DQG H[SHULPHQWDO DQDO\VLV RI WKLV WUXVV ZDV SHUIRUPHG WR JHQHUDWH D UHDOLVWLF WHVWEHG IRU VWUXFWXUDO GDPDJH ORFDWLRQH[WHQW DOJRULWKPV .DVKDQJDNL HW DO .DVKDQJDNL f 7KH WUXVV FRQILJXUDWLRQ XVHG LQ WKLV DQDO\VLV ZDV FDQWLOHYHUHG DQG LQVWUXPHQWHG ZLWK QLQHW\VL[ DFFHOHURPHWHUV WR PHDVXUH DOO WKUHH WUDQVODWLRQDO GHJUHHV RI IUHHGRP DW HDFK RI WKH WKLUW\WZR XQFRQVWUDLQHG QRGHV $ VFKHPDWLF RI WKLV WUXVV LV VKRZQ LQ )LJXUH )LJXUH VKRZV D FORVHXS RI WKH WUXVV ODFLQJ SDWWHUQ DORQJ ZLWK VWUXW GHILQLWLRQV WKDW ZLOO EH XVHG LQ WKH UHPDLQGHU RI WKLV VWXG\ )LJXUH 7KH 1$6$ (LJKW%D\ +\EULG6FDOHG 7UXVV 'DPDJH &DVHV

PAGE 127

)LJXUH 7KH 1$6$ %D\ 7UXVV /DFLQJ 3DWWHUQ $ QLQHW\VL[ GHJUHH RI IUHHGRP XQGDPSHG ILQLWH HOHPHQW PRGHO )(0f RI WKH RULJLQDO fKHDOWK\f WUXVV ZDV JHQHUDWHG XVLQJ 06&1$675$1 ,Q WKLV )(0 HDFK WUXVV VWUXW ZDV PRGHOHG DV D URG HOHPHQW &RQFHQWUDWHG PDVVHV ZHUH DGGHG DW HDFK QRGH WR DFFRXQW IRU WKH MRLQW DQG LQVWUXPHQWDWLRQ PDVV SURSHUWLHV .DVKDQJDNL f 7KH PDVV DQG VWUXW SURSHUWLHV RI WKH WUXVV DUH VXPPDUL]HG LQ 7DEOHV DQG 7DEOH 0DVV 3URSHUWLHV RI WKH (LJKW %D\ 7UXVV :HLJKW 3RXQGVf 7RWDO 1XPEHU LQ (LJKW %D\ 7UXVV 7RWDO :HLJKW 3RXQGVf 1RGH %DOO /RQJHURQ 6WUXW 'LDJRQDO 6WUXW -RLQW $VVHPEO\ 7ULD[ %ORFN 7RWDO :HLJKW RI WKH 7UXVV

PAGE 128

7DEOH 6WUXW 3URSHUWLHV RI WKH (LJKW %D\ 7UXVV /RQJHURQ 'LDJRQDO 6WUXW /HQJWK LQ LQ 6WUXW 6WLIIQHVV ($/fHII OELQ ,ELQ ([SHULPHQWDO PRGDO DQDO\VLV RI WKH WUXVV ZDV SHUIRUPHG IRU WKH fQR GDPDJHf DQG VL[WHHQ GDPDJH FDVHV ,Q WKH WHVWLQJ WKH H[FLWDWLRQ VRXUFH ZDV SURYLGHG E\ WZR VKDNHUV 0RGDO SDUDPHWHUV ZHUH LGHQWLILHG IURP WKH PHDVXUHG IUHTXHQF\ UHVSRQVH IXQFWLRQV XVLQJ WKH 3RO\UHIHUHQFH FRPSOH[ H[SRQHQWLDO WHFKQLTXH )RU HDFK FDVH ILYH PRGHV RI YLEUDWLRQ ZHUH LGHQWLILHG (DFK PHDVXUHG PRGH FRQVLVWV RI D QDWXUDO IUHTXHQF\ DQG LWV FRUUHVSRQGLQJ PRGH VKDSH ZLWK PHDVXUHPHQWV DW DOO QLQHW\VL[ )(0 GHJUHHV RI IUHHGRP )RU WKH fQR GDPDJHf FDVH WKH ILUVW VHFRQG IRXUWK DQG ILIWK PRGH DUH EHQGLQJ PRGHV DQG WKH WKLUG PRGH LV WKH ILUVW WRUVLRQDO PRGH ,Q WKH GDPDJH FDVHV WKH VDPH ILYH PRGHV ZHUH PHDVXUHG KRZHYHU WKHLU RUGHU YDULHV VLQFH LQ VRPH FDVHV WKH GDPDJH FRXOG FDXVH PRGH VZLWFKLQJ ,W VKRXOG EH QRWHG WKDW GXULQJ WKH WHVWLQJ SURFHVV WZR DFFHOHURPHWHUV DW QRGH ILIWHHQ IDLOHG 7KHVH GHIHFWLYH DFFHOHURPHWHUV ZRXOG DIIHFW WKH FRPSRQHQW RI WKH PHDVXUHG PRGHV DVVRFLDWHG ZLWK '2)V DQG 'DWD IRU ILIWHHQ RI WKH VL[WHHQ GDPDJH FDVHV WHVWHG ZHUH UHFHLYHG IURP 1$6$ 7KH ILIWHHQ GDPDJH FDVHV WKDW ZHUH UHFHLYHG DUH LGHQWLILHG LQ WKH VFKHPDWLF RI WKH WUXVV VKRZQ LQ )LJXUH )RU HDFK GDPDJH FDVH WKH W\SH RI HOHPHQW DQG WKH )(0 GHJUHHV RI IUHHGRP DIIHFWHG E\ GDPDJH DUH VKRZQ LQ 7DEOH ,Q GDPDJH FDVHV $ WR 1 VWUXFWXUDO GDPDJH FRQVLVWHG RI WKH IXOO UHPRYDO RI RQH VWUXW IURP WKH WUXVV &DVH 2 GDPDJH FRQVLVWV RI WKH IXOO UHPRYDO RI WZR VWUXWV ,Q &DVH 3 RQH RI WKH VWUXWV ZDV EXFNOHG WR LOOXVWUDWH D SDUWLDO GDPDJH VFHQDULR 1RWH WKDW WKH IRUWKFRPLQJ DQDO\VLV RI WKH ILIWHHQ GDPDJH FDVHV ZDV SHUIRUPHG ZLWK QR D SULRUL NQRZOHGJH RI WKH DFWXDO GDPDJH ORFDWLRQV

PAGE 129

7DEOH 1$6$ %D\ 7UXVV 'DPDJH &DVH 'HILQLWLRQV 'DPDJH &DVH (OHPHQW 7\SH 'DPDJHG '2)V $ /RQJHURQ & /RQJHURQ 8SSHU 'LDJRQDO ( ;%DWWHQ ) )DFH 'LDJRQDO /RQJHURQ + /RQJHURQ /HIW 'LDJRQDO /RQJHURQ 5LJKW 'LDJRQDO / /RQJHURQ 0 /RQJHURQ 1 /RQJHURQ r /RQJHURQ t 'LDJRQDO =%DWWHQ r 7ZR VWUXWV UHPRYHG rr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

PAGE 130

LWV IUHTXHQF\ UHVSRQVH IXQFWLRQV WR WKRVH PHDVXUHG H[SHULPHQWDOO\ 2QH VXFK FRPSDULVRQ LV VKRZQ LQ )LJXUH ,W LV FOHDU IURP WKLV W\SLFDO IUHTXHQF\ UHVSRQVH FRPSDULVRQ WKDW WKH RULJLQDO )(0 LV ODFNLQJ LQ DFFXUDF\ 7DEOH &RPSDULVRQ RI $QDO\WLFDO DQG ([SHULPHQWDO )UHTXHQFLHV 1DWXUDO )UHTXHQFLHV +]f 02'( $QDO\WLFDO ([SHULPHQWDO +HQFH WKH ILUVW VWHS LQ WKLV VWXG\ ZDV WR UHILQH WKH RULJLQDO )(0 )RU WKH UHILQHPHQW SURFHVV LW ZDV DVVXPHG WKDW WKH RULJLQDO PDVV PDWUL[ LV DQ DFFXUDWH UHSUHVHQWDWLRQ RI WKH WUXVVf PDVV SURSHUWLHV 7KH LQDFFXUDF\ RI WKH RULJLQDO )(0 ZDV EHOLHYHG WR EH VROHO\ GXH WR PRGHOLQJ HUURUV LQ WKH VWLIIQHVV SURSHUWLHV 7KH PLQLPXP UDQN SHUWXUEDWLRQ WKHRU\ GLVFXVVHG LQ &KDSWHU 6HFWLRQ f ZDV XVHG WR FRUUHFW WKH RULJLQDO VWLIIQHVV PDWUL[ ,Q RUGHU WR JHW D V\PPHWULF XSGDWHG VWLIIQHVV PDWUL[ WKH PHDVXUHG PRGH VKDSHV HLJHQYHFWRUVf ZHUH PDVV RUWKRJRQDOL]HG XVLQJ WKH 2SWLPXP :HLJKWHG 2UWKRJRQDOL]DWLRQ WHFKQLTXH %DUXFK f $ PHVK SORW RI WKH FKDQJHV PDGH WR WKH VWLIIQHVV PDWUL[ $.f IURP WKH UHILQHPHQW SURFHVV LV VKRZQ LQ )LJXUH $V FDQ EH VHHQ LQ WKH ILJXUH WKHUH DUH WKUHH DUHDV RI PDMRU FKDQJH PDGH WR WKH RULJLQDO VWLIIQHVV PDWUL[ 2QH DUHD RI FKDQJH FRUUHVSRQGV WR WKH FDQWLOHYHU HQG RI WKH WUXVV 7KLV FKDQJH ZDV H[SHFWHG VLQFH D SHUIHFW FDQWLOHYHU FRQGLWLRQ ZDV DVVXPHG LQ JHQHUDWLQJ WKH RULJLQDO )(0 1RWH WKDW LW LV FRPPRQ NQRZOHGJH WKDW SHUIHFW FDQWLOHYHU FRQGLWLRQV FDQQRW EH SURGXFH LQ SUDFWLFH 2WKHU FKDQJHV FDQ EH VHHQ LQ WKH PLGGOH RI WKH PHVK SORW 7KHVH FKDQJHV RFFXU DW DQG DURXQG '2)V DQG ZKLFK DUH WKH ORFDWLRQ RI WKH EDG VHQVRUV +HQFH WKHVH FKDQJHV ZHUH DWWULEXWHG WR WKH HIIHFW RI WKH EDG VHQVRUV
PAGE 131

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

PAGE 132

5227 )LJXUH 1$6$ ED\ 7UXVV 3HUWXUEDWLRQ WR WKH 2ULJLQDO 6WLIIQHVV 0DWUL[ WKDW 5HVXOWHG IURP WKH 5HILQHPHQW 3URFHVV 'DPDJH /RFDWLRQ )RU HDFK GDPDJH FDVH WKH PRGDO SDUDPHWHUV RI WKH ILYH PHDVXUHG PRGHV ZHUH XVHG WR FRPSXWH D FXPXODWLYH GDPDJH ORFDWLRQ YHFWRU &'/9f WR GHWHUPLQH WKH ORFDWLRQ RI WKH VWUXFWXUDO GDPDJH 6LQFH WKH YDOXHV RI __ ]-G __ DV GHILQHG LQ 6HFWLRQ DUH RI WKH VDPH RUGHUV RI PDJQLWXGH IRU DOO '2)V DQG DOO ILYH PHDVXUHG PRGHV WKH FXPXODWLYH GDPDJH ORFDWLRQ YHFWRU &'/9f FRXOG EH FRPSXWHG XVLQJ HLWKHU (T f RU (T f 7KH DQJOH SHUWXUEDWLRQ &'/9 (T ff DVVRFLDWHG WR GDPDJH FDVH ) LV VKRZQ LQ )LJXUH f 7KH XSSHU OHIW FRUQHUV RI )LJXUHV f WR f GLVSOD\ SORWV RI WKH FXPXODWLYH GDPDJH ORFDWLRQ YHFWRUV DV FDOFXODWHG E\ (T f IRU DOO FDVHV ZKHUH WKH GDPDJH ZDV VXFFHVVIXOO\ ORFDWHG ,Q WKHVH SORWV WKH &'/9 ZHUH XQLW QRUPDOL]HG DQG WKHLU HOHPHQWV ZHUH SORWWHG YHUVXV WKH )(0 '2)V 7KH SHUIRUPDQFH RI WKH ORFDWLRQ DOJRULWKP IRU HDFK GDPDJH FDVH LV VXPPDUL]HG LQ WKH VHFRQG FROXPQ RI 7DEOH $ VXPPDU\ RI WKH SURFHGXUHV XVHG LQ WKH LQWHUSUHWDWLRQ RI WKH &'/9 RI HDFK GDPDJH FDVH LV DV IROORZV

PAGE 133

f ,QLWLDOO\ D FRPSDUDWLYH VWXG\ RI WKH &'/9 IRU DOO FDVHV ZDV SHUIRUPHG WR GHGXFH WKH HIIHFWV RI WKH EDG DFFHOHURPHWHUV $ IDLUO\ ODUJH QXPHULFDO FRPSRQHQW DW '2) ZDV GHWHFWHG LQ PRVW RI WKH ILIWHHQ &'/9 7KLV FRPSRQHQW ZDV EHOLHYHG WR EH GXH WR WKH fEDGf VHQVRU ORFDWHG DW '2) +HQFH IRU DOO FDVHV LW ZDV GHFLGHG WR LJQRUH WKH FRPSRQHQW RI WKH &'/9 DW '2) f )RU GDPDJH FDVHV $ & + 0 1 DQG 2 WKH ORFDWLRQ RI WKH GDPDJH ZDV GHWHUPLQHG E\ VLPSO\ FRQVLGHULQJ WKH '2)V DVVRFLDWHG ZLWK WKH VXEVWDQWLDOO\ ODUJHU QXPHULFDO FRPSRQHQWV RI WKH &'/9fV DV EHLQJ WKH GDPDJHG '2)V 7KH VPDOOHU QXPHULFDO FRPSRQHQWV RI WKH &'/9fV DW WKH RWKHU '2)V FDQ EH DWWULEXWHG WR PHDVXUHPHQW HUURUV 1RWLFH WKDW IRU FDVHV 0 DQG 1 WKH FRPSRQHQW RI WKH &'/9 DW '2) LV IDLUO\ VPDOO DQG KHQFH QHJOLJLEOH ,W VKRXOG EH DOVR QRWHG WKDW WKH GDPDJH LQ &DVH $ DIIHFWV RQO\ RQH GHJUHH RI IUHHGRP f VLQFH WKH VWUXW LQYROYHG LV D ORQJHURQ FRQQHFWHG WR WKH FDQWLOHYHUHG HQG f )RU GDPDJH FDVH ( WKH '2)V DVVRFLDWHG ZLWK WKH &'/9 FRPSRQHQWV RI JUHDWHU RUGHU RI PDJQLWXGH DUH DQG %\ XWLOL]LQJ WKH FRQQHFWLYLW\ RI WKH RULJLQDO )(0 WKH FRPELQDWLRQV RI WKHVH '2)V WKDW DUH SK\VLFDOO\ PHDQLQJIXO LH ERXQG D VWUXW DUH f DQG f %DVHG RQ WKHVH UHVXOWV LW ZDV GHFLGHG WKDW WZR VWUXWV FRQQHFWLQJ '2) f DQG f ZHUH GDPDJHG +RZHYHU LQ DFWXDOLW\ WKH RQO\ GDPDJHG VWUXW LV WKH RQH FRQQHFWLQJ '2)V f f )RU GDPDJH FDVH / E\ LJQRULQJ '2) WKH fEDGf VHQVRUf WKH RQO\ '2) WKDW LV FOHDUO\ GDPDJHG LV 1R VWUXW ZLWK '2) LV FRQQHFWHG WR WKH ZDOO 7KH GDPDJH RI D VWUXW QRW FRQQHFWHG WR WKH FDQWLOHYHUHG HQG PXVW DIIHFW DW OHDVW WZR '2)V 6LQFH '2) LV LQ WKH \GLUHFWLRQ LW ZDV GHGXFHG WKDW WKH PRVW SUREDEOH GDPDJHG VWUXW LV D ORQJHURQ FRQQHFWHG DW RQH HQG WR '2) 7KH WZR FDQGLGDWH VWUXWV DUH WKH RQHV ERXQGHG E\ '2)V f DQG f +RZHYHU WKH FRPSRQHQW RI WKH &'/9 DVVRFLDWHG ZLWK

PAGE 134

'2) LV VPDOO DQG LV LQ QR ZD\ DIIHFWHG E\ GDPDJH 7KLV SURPSWHG WKH GHGXFWLRQ WKDW WKH VWUXW FRQQHFWLQJ '2)V DQG LV WKH GDPDJHG PHPEHU LQ &DVH / f )RU GDPDJH FDVH 3 WKH VXEVWDQWLDOO\ ODUJHU FRPSRQHQWV RI WKH &'/9 RFFXU DW '2)V DQG '2) LJQRUHGf $JDLQ ZLWK WKH XVH RI WKH FRQQHFWLYLW\ RI WKH RULJLQDO )(0 LW LV GHGXFHG WKDW WKH RQO\ FRPELQDWLRQ RI WKHVH WKUHH '2)V WKDW DUH FRQQHFWHG E\ D VWUXW LV f 7KHVH '2)V DUH H[DFWO\ WKH GDPDJHG '2)V RI &DVH 3 f 'DPDJH &DVH ) LV WKH RQO\ FDVH ZKHUH WKH ORFDWLRQ DOJRULWKP IDLOHG WR ORFDWH WKH GDPDJH 7KH FXPXODWLYH GDPDJH YHFWRU IRU WKLV FDVH LV VKRZQ LQ )LJXUH 7KH UHDVRQ WKH GDPDJH ZDV QRW ORFDWHG LQ WKLV FDVH LV WKDW LW LQYROYHV D IDFH GLDJRQDO VWUXW )DFH GLDJRQDO VWUXWV LQ JHQHUDO DIIHFW RQO\ WKH D[LDO PRGHV DQG KDYH VOLJKW RU QR HIIHFWV RQ WKH EHQGLQJ DQG WRUVLRQDO PRGHV $V UHSRUWHG HDUOLHU WKH RQO\ DYDLODEOH PHDVXUHGf PRGHV DUH EHQGLQJ DQG WRUVLRQDO PRGHV ,W ZDV GHWHUPLQHG DQDO\WLFDOO\ WKDW WKH ILUVW D[LDO PRGH RI WKH WUXVV RFFXUV DW WKH VL[WK PRGH .DVKDQJDNL HW DO f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f QHJOLJLEOH RQ WKH RYHUDOO LQHUWLDO SURSHUWLHV DQG LLf D VXEVWDQWLDO VWLIIQHVV ORVV 8VLQJ WKH H[WHQW DOJRULWKP GLVFXVVHG LQ 6HFWLRQ WKH SHUWXUEDWLRQ WR WKH VWLIIQHVV PDWUL[ $.Mf GXH WR WKH GDPDJH LV FDOFXODWHG IRU HDFK GDPDJH FDVH ZKHUH WKH GDPDJH ZDV VXFFHVVIXOO\ ORFDWHG 7R LQVXUH WKH V\PPHWU\ RI $.A WKH

PAGE 135

HLJHQYHFWRUV XVHG LQ WKHVH FDOFXODWLRQV ZHUH ILUVW PDVV RUWKRJRQDOL]HG XVLQJ WKH 2SWLPXP :HLJKWHG 2UWKRJRQDOL]DWLRQ WHFKQLTXH %DUXFK DQG %DU ,W]KDFN f 7KH %UXWH )RUFH 0HWKRG 7KH XSSHU ULJKW FRPHUV RI )LJXUHV WR GLVSOD\ PHVK SORWV RI WKH FDOFXODWHG $ ,4f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fDFWXDOf $,4 7KH UDQN RI WKH fDFWXDOf $. FDQ EH HVWLPDWHG E\ DGGLQJ WKH UDQN RI DOO HOHPHQW VWLIIQHVV PDWULFHV ZKLFK FRQQHFW GDPDJHG '2)V 6LQFH WKH HOHPHQWDO VWLIIQHVV PDWUL[ RI WKH WUXVV XQGHU LQYHVWLJDWLRQ LV UDQN RQH WKH UDQN RI WKH fDFWXDOf $.M IRU HDFK GDPDJH FDVH LV HTXDO WR WKH QXPEHU RI GDPDJHG VWUXWV +HQFH RQO\ PRGDO SDUDPHWHUV IURP RQH PRGH DUH QHHGHG WR FRPSXWH WKH GDPDJH H[WHQW RI DOO FDVHV IHDWXULQJ RQH GDPDJHG VWUXW )RU &DVH 2 VLQFH WZR VWUXWV DUH GDPDJHG GDWD IURP WZR PRGHV DUH QHHGHG )RU HDFK FDVH WKH PRGHV WKDW VKRXOG EH XVHG DUH WKH RQHV WKDW PRVW FOHDQO\ GHPRQVWUDWH WKH VWDWH RI GDPDJH RI WKDW FDVH 7KHVH PRGHV FDQ EH VLPSO\ GHWHUPLQHG E\ LQVSHFWLQJ WKH LQGLYLGXDO GDPDJH ORFDWLRQ YHFWRU Gc 7KH PRGHV WKDW SURYLGH WKH EHVW LQVLJKW LQWR WKH VWDWH RI WKH GDPDJH IRU HDFK GDPDJH FDVH DUH UHSRUWHG LQ WKH WKLUG FROXPQ RI 7DEOH 7KH XQLW QRUPDOL]HG GDPDJH ORFDWLRQ YHFWRUV DVVRFLDWHG ZLWK WKH fEHVWf PRGHV IRU DOO FDVHV DUH

PAGE 136

SORWWHG YHUVXV WKH )(0 '2)V LQ WKH PLGGOH OHIW RI )LJXUHV WR 7KH PHVK SORWV RI WKH $.GfV FRPSXWHG XVLQJ RQO\ WKH fEHVWf PRGHVf IRU DOO GDPDJH FDVHV DUH VKRZQ LQ WKH PLGGOH ULJKW FRPHUV RI )LJXUHV WR ,W LV FOHDU WKDW WKH SHUIRUPDQFH RI WKH H[WHQW DOJRULWKP IRU DOO FDVHV KDV EHHQ JUHDWO\ LPSURYHG LQ FRPSDULVRQ WR LWV SHUIRUPDQFH ZKHQ DOO ILYH PHDVXUHG PRGHV ZHUH XVHG 7KH HVWLPDWHG HIIHFWLYH VWLIIQHVV ORVV OELQf IRU DOO GDPDJH FDVHV FRPSXWHG IURP WKLV SURFHVV DUH UHSRUWHG LQ WKH IRXUWK FROXPQ RI 7DEOH )RU HDFK GDPDJH FDVH WKH HIIHFWLYH VWLIIQHVV ORVV ZDV FDOFXODWHG E\ DYHUDJLQJ WKH VWLIIQHVV FKDQJHV $.Mf DVVRFLDWHG ZLWK WKH GDPDJHG '2)V ,Q WKH RULJLQDO )(0 WKH ORQJHURQV ;EDWWHQV DQG =EDWWHQV KDYH DQ HIIHFWLYH VWLIIQHVV RI OELQ ZKLOH WKH HIIHFWLYH VWLIIQHVV RI WKH GLDJRQDO VWUXWV LV OELQ .DVKDQJDNL f 7KH UHPRYDO RI DQ\ VWUXW UHVXOWV LQ WKH FRPSOHWH ORVV RI WKH VWLIIQHVV RI WKDW VWUXW 7KH ILIWK FROXPQ RI 7DEOH UHSRUWV WKH SHUFHQWDJH HUURU RI WKH HVWLPDWHG VWLIIQHVV ORVVHV ZLWK UHVSHFW WR WKH RULJLQDO )(0 7KH SHUFHQWDJH HUURU IRU &DVH 3 EXFNOHG VWUXWf ZDV QRW FRPSXWHG VLQFH WKH GDPDJH H[WHQW LV XQNQRZQ $SSOLFDWLRQ RI WKH (LJHQYHFWRU )LOWHULQJ $OJRULWKP 7KH GDPDJH H[WHQW DVVHVVPHQW FDQ EH IXUWKHU LPSURYHG E\ XVLQJ WKH HLJHQYHFWRU ILOWHULQJ DOJRULWKP GLVFXVVHG LQ 6HFWLRQ )RU FRQVLVWHQF\ WKH PRGHVf XVHG LQ WKH ILOWHULQJ SURFHVV DUH WKH fEHVWf PRGHVf DV GHWHUPLQHG DERYH DQG UHSRUWHG LQ WKH WKLUG FROXPQ RI 7DEOH )RU HDFK FDVH WKH fXQGDPDJHGf FRPSRQHQWV RI WKH GDPDJH YHFWRUV Gc DUH ILUVW VHW WR ]HUR 7KH FRPSRQHQWV RI Gc DVVRFLDWHG WR WKH fGDPDJHGf '2)V DUH WKHQ FRQVWUDLQHG WR EH FRQVLVWHQW ZLWK WKH DFWXDO HIIHFW RI WKH GDPDJH DV GLFWDWHG E\ WKH HOHPHQW VWLIIQHVV PDWUL[ RI WKH GDPDJH VWUXWV 7KH GDPDJH LQ ;EDWWHQV ORQJHURQV RU =EDWWHQV DIIHFWV WZR '2)V RQH '2) DW HDFK RI WKH WZR QRGHV RI WKH VWUXWf ,Q RUGHU IRU WKH GDPDJH WR EH ILQLWH HOHPHQW FRQVLVWHQW WKH FRPSRQHQW DVVRFLDWHG ZLWK WKHVH WZR '2)V VKRXOG EH HTXDO LQ PDJQLWXGH DQG RI RSSRVLWH VLJQ 'DPDJH LQ XSSHU IDFH OHIW RU ULJKW GLDJRQDO VWUXWV DIIHFWV IRXU '2)V WZR DW HDFK QRGH RI WKH VWUXWf 7KHVH IRXU '2)V FDQ EH FODVVLILHG LQWR WZR VHWV RI WZR '2)V LQ WKH VDPH GLUHFWLRQ [ \ RU ]f 7KH '2)V RI D JLYHQ VHW DUH LQ JHQHUDO HTXDO LQ PDJQLWXGH DQG RI

PAGE 137

RSSRVLWH VLJQV ,Q WKH ED\ WUXVV DOO XSSHU IDFH OHIW RU ULJKW GLDJRQDOV DUH HLWKHU DW D r RU r DQJOH ZLWK UHVSHFW WR WKHLU LQSODQH JOREDO FRRUGLQDWH D[LV 7KLV FRQVWUDLQV WKH '2)V RI WKH WZR VHWV WR KDYH WKH VDPH PDJQLWXGH LQ RUGHU WR EH FRQVLVWHQW ZLWK WKH )(0 7KH '2)V DVVRFLDWHG ZLWK D JLYHQ QRGH KDYH WKH VDPH VLJQ ZKHQ WKH DQJOH LV r WKH\ DUH RI RSSRVLWH VLJQ ZKHQ WKH DQJOH LV r )RU WKH FDVHV ZKHUH D VLQJOH VWUXW LV GDPDJHG WKH PDJQLWXGH RI D '2) FRXOG EH VHW DUELWUDULO\ 7KH UHDVRQ LV WKDW HLJHQYHFWRUV DUH XQLTXH LQ D UHODWLYH VHQVH LH LI 8M LV DQ HLJHQYHFWRU DVVRFLDWHG WR D JLYHQ PRGH WKHQ DXc LV DOVR DQ HLJHQYHFWRU DVVRFLDWHG WR WKDW VDPH PRGH D EHLQJ DQ\ VFDODU 7KXV WKH FRPSRQHQWV RI WKH GDPDJH YHFWRU DUH DOVR D IXQFWLRQ RI WKH VFDODU D 7KHVH FRPSRQHQWV FDQ EH VHW WR DQ\ YDOXHV E\ YDU\LQJ D )RU WKH VLQJOH PHPEHU GDPDJH FDVHV XQGHU VWXG\ WKLV SURFHVV LV VXPPDUL]HG LQ 7DEOH 7DEOH 6XPPDU\ RI WKH )LOWHULQJ 3URFHVV IRU 6LQJOH 0HPEHU 'DPDJH &DVHV 1RGH 1RGH '2) ; < ] ; < ] ;%DWWHQ 3 3 /RQJHURQ 3 S =%DWWHQ S S &DVH rf S S S S &DVH rf S S S S &DVH rf S S S S ZKHUH LV DQ\ VFDODU 1RWH WKDW FDVHV $ & ( + / 0 1 DQG 3 LQYROYH WKH GDPDJH RI HLWKHU DQ ;EDWWHQ ORQJHURQ RU =EDWWHQ $V GLVFXVVHG HDUOLHU DOO '2)V RI WKH IGWHUHG GDPDJH YHFWRU GI DVVRFLDWHG WR D JLYHQ GDPDJH PHPEHU KDYH WKH VDPH PDJQLWXGH ([SHULHQFH JDLQHG E\ XVLQJ WKH HLJHQYHFWRU ILOWHULQJ DOJRULWKP LQGLFDWHV WKDW LQ WKH PXOWLSOH PHPEHU GDPDJH VFHQDULR FDVH 2f WKH UDWLR EHWZHHQ WKH fV IURP GLIIHUHQW GDPDJHG VWUXWV FRQWDLQV LPSRUWDQW

PAGE 138

LQIRUPDWLRQ DERXW WKH UHODWLYH GDPDJH H[WHQW DPRQJ WKH GDPDJHG VWUXWV +HQFH LQ FDVH 2 D JLYHQ 3 DVVRFLDWHG ZLWK D JLYHQ GDPDJHG VWUXW FDQQRW EH FKRVHQ DUELWUDU\ 7KLV S VKRXOG EH HVWLPDWHG E\ DYHUDJLQJ WKH FRPSRQHQWV RI WKH XQILOWHUHG GDPDJH YHFWRU GI FRUUHVSRQGLQJ WR WKH GDPDJHG VWUXW LQ TXHVWLRQ 7KH DEVROXWH YDOXH RI WKH XQLW QRUPDOL]HG ILOWHUHG GDPDJH ORFDWLRQ YHFWRUV DVVRFLDWHG ZLWK DOO FDVHV DUH SORWWHG YHUVXV WKH )(0 '2)V LQ WKH ERWWRP OHIW FRUQHUV RI )LJXUHV WR 1RZ WKDW WKH fILOWHUHGf GDPDJH YHFWRUV DUH JHQHUDWHG WKH QH[W VWHS LV WR FRPSXWH WKH FRUUHVSRQGLQJ ILOWHUHG GDPDJH HLJHQYHFWRU XVLQJ (T 7KH SURFHVV RI FRPSXWLQJ WKH SHUWXUEDWLRQ GXH WR WKH GDPDJH XVLQJ WKH ILOWHUHG HLJHQYHFWRUV LV DV LQ WKH SUHYLRXV VHFWLRQ )RU DOO GDPDJH FDVHV WKH PHVK SORWV RI WKH $.AfV FRPSXWHG XVLQJ WKH fILOWHUHGf PRGHVf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

PAGE 139

'DPDJH 9HFWRU 'DPDJH 9HFWRU 'DPDJH 9HFWRU 'DPDJH /RFDWLRQ 0RGHV 'DPDJH /RFDWLRQ 0RGH 'DPDJH /RFDWLRQ 0RGH )LOWHUHG '2) 'DPDJH ([WHQW 0RGHV 'DPDJH ([WHQW 0RGH 'DPDJH ([WHQW 0RGH )LOWHUHG )LJXUH 1$6$ ED\ 7UXVV 'DPDJH $VVHVVPHQW RI &DVH $

PAGE 140

'DPDJH 9HFWRU 'DPDJH 9HFWRU 'DPDJH 9HFWRU 'DPDJH /RFDWLRQ 0RGHV 'DPDJH /RFDWLRQ 0RGH '2) 'DPDJH ([WHQW 0RGHV 'DPDJH ([WHQW 0RGH 'DPDJH ([WHQW 0RGH )LOWHUHG )LJXUH 1$6$ ED\ 7UXVV 'DPDJH $VVHVVPHQW RI &DVH &

PAGE 141

'DPDJH 9HFWRU 'DPDJH 9HFWRU 'DPDJH 9HFWRU 'DPDJH /RFDWLRQ 0RGHV 'DPDJH /RFDWLRQ 0RGH 'DPDJH /RFDWLRQ 0RGH )LOWHUHG P P 2n rf§ '2) 'DPDJH ([WHQW 0RGHV 'DPDJH ([WHQW 0RGH 'DPDJH ([WHQW 0RGH )LOWHUHG )LJXUH 1$6$ ED\ 7UXVV 'DPDJH $VVHVVPHQW RI &DVH '

PAGE 142

'DPDJH 9HFWRU 'DPDJH 9HFWRU 'DPDJH 9HFWRU 'DPDJH /RFDWLRQ 0RGHV 'DPDJH /RFDWLRQ 0RGH 'DPDJH /RFDWLRQ 0RGH )LOWHUHG 'DPDJH ([WHQW 0RGHV 'DPDJH ([WHQW 0RGH 'DPDJH ([WHQW 0RGH )LOWHUHG )LJXUH 1$6$ ED\ 7UXVV 'DPDJH $VVHVVPHQW RI &DVH (

PAGE 143

'DPDJH 9HFWRU 'DPDJH 9HFWRU 'DPDJH 9HFWRU 'DPDJH /RFDWLRQ 0RGHV 'DPDJH /RFDWLRQ 0RGH 'DPDJH /RFDWLRQ 0RGH )LOWHUHG '2) 'DPDJH ([WHQW 0RGHV 'DPDJH ([WHQW 0RGH 'DPDJH ([WHQW 0RGH )LOWHUHG )LJXUH 1$6$ ED\ 7UXVV 'DPDJH $VVHVVPHQW RI &DVH *

PAGE 144

'DPDJH 9HFWRU 'DPDJH 9HFWRU 'DPDJH 9HFWRU @ 'DPDJH /RFDWLRQ 0RGH )LOWHUHG 'DPDJH ([WHQW 0RGHV 'DPDJH ([WHQW 0RGH 'DPDJH ([WHQW 0RGH )LOWHUHG )LJXUH 1$6$ ED\ 7UXVV 'DPDJH $VVHVVPHQW RI &DVH +

PAGE 145

'DPDJH 9HFWRU 'DPDJH 9HFWRU 'DPDJH 9HFWRU 'DPDJH /RFDWLRQ 0RGHV 21 12 f OLIOIOOP OOOL M L ZW8LLZY '2) 'DPDJH /RFDWLRQ 0RGH 'DPDJH /RFDWLRQ 0RGH )LOWHUHG '2) 'DPDJH ([WHQW 0RGHV 'DPDJH ([WHQW 0RGH 'DPDJH ([WHQW 0RGH )LOWHUHG )LJXUH 1$6$ ED\ 7UXVV 'DPDJH $VVHVVPHQW RI &DVH ,

PAGE 146

'DPDJH 9HFWRU 'DPDJH 9HFWRU 'DPDJH 9HFWRU 'DPDJH /RFDWLRQ 0RGHV OO:Q8, /ZL QLO :OIOOI0+ MO £LLOOX/ '2) 'DPDJH /RFDWLRQ 0RGH 'DPDJH /RFDWLRQ 0RGH )LOWHUHG 'DPDJH ([WHQW 0RGHV 'DPDJH ([WHQW 0RGH 'DPDJH ([WHQW 0RGH )LOWHUHG )LJXUH 1$6$ ED\ 7UXVV 'DPDJH $VVHVVPHQW RI &DVH -

PAGE 147

'DPDJH 9HFWRU 'DPDJH 9HFWRU 'DPDJH 9HFWRU 'DPDJH /RFDWLRQ 0RGHV Q KPL,/, LWLLLWI L LO PL IOO LNIOO0/ '2) 'DPDJH /RFDWLRQ 0RGH )LOWHUHG '2) 'DPDJH ([WHQW 0RGHV 'DPDJH ([WHQW 0RGH 'DPDJH ([WHQW 0RGH )LOWHUHG )LJXUH 1$6$ ED\ 7UXVV 'DPDJH $VVHVVPHQW RI &DVH .

PAGE 148

'DPDJH 9HFWRU 'DPDJH 9HFWRU 'DPDJH 9HFWRU 'DPDJH /RFDWLRQ 0RGH 'DPDJH /RFDWLRQ 0RGH )LOWHUHG '2) 'DPDJH ([WHQW 0RGHV 'DPDJH ([WHQW 0RGH 'DPDJH ([WHQW 0RGH )LOWHUHG )LJXUH 1$6$ ED\ 7UXVV 'DPDJH $VVHVVPHQW RI &DVH /

PAGE 149

'DPDJH 9HFWRU 'DPDJH 9HFWRU 'DPDJH 9HFWRU 'DPDJH /RFDWLRQ 0RGHV n ;RQO/LIOLLLL-L UIEO/ID O WQ77-OUYOO7I!7IIOQ:IKYI7M7QO8'QB '2) 'DPDJH /RFDWLRQ 0RGH )LOWHUHG 'DPDJH ([WHQW 0RGHV 'DPDJH ([WHQW 0RGH 'DPDJH ([WHQW 0RGH )LOWHUHG )LJXUH 1$6$ ED\ 7UXVV 'DPDJH $VVHVVPHQW RI &DVH 0

PAGE 150

'DPDJH 9HFWRU 'DPDJH 9HFWRU 'DPDJH 9HFWRU '2) 'DPDJH ([WHQW 0RGHV 'DPDJH ([WHQW 0RGH 'DPDJH ([WHQW 0RGH )LOWHUHG )LJXUH 1$6$ ED\ 7UXVV 'DPDJH $VVHVVPHQW RI &DVH 1

PAGE 151

'DPDJH ([WHQW 0RGHV 'DPDJH ([WHQW 0RGHV t 'DPDJH /RFDWLRQ 0RGHV t )LOWHUHG 'DPDJH ([WHQW 0RGHV t )LOWHUHG '2) )LJXUH 1$6$ ED\ 7UXVV 'DPDJH $VVHVVPHQW RI &DVH 2

PAGE 152

'DPDJH 9HFWRU 'DPDJH 9HFWRU 'DPDJH 9HFWRU 'DPDJH /RFDWLRQ 0RGH )LOWHUHG '2) 'DPDJH ([WHQW 0RGHV 'DPDJH ([WHQW 0RGH 'DPDJH ([WHQW 0RGH )LOWHUHG )LJXUH 1$6$ ED\ 7UXVV 'DPDJH $VVHVVPHQW RI &DVH 3

PAGE 153

7DEOH 1$6$ %D\ 7UXVV 6XPPDU\ RI WKH 'DPDJH $VVHVVPHQW 5HVXOWV :LWKRXW )LOWHULQJ :LWK )LOWHULQJ 'DPDJH &DVH /RFDWLRQ 3HUIRUPDQFH %HVW PRGH ([WHQW OELQf b HUURU ([WHQW OELQf b HUURU $ & ( R ) /RFDWLRQ QRW GHWHFWHG + / 0 1 2 /f 'f /f 'f 3 QRW GHI QRW GHI 'DPDJH FOHDUO\ ORFDWHG 'DPDJH ORFDWHG ZLWK IXUWKHU DQDO\VLV 'DPDJH QRW ORFDWHG R ORFDWLRQ QDUURZHG WR ZLWKLQ WZR PHPEHUV /f /RQJHURQ 'f GLDJRQDO

PAGE 154

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a P %HDP
PAGE 155

)(0 ZDV JHQHUDWHG XVLQJ WKH HLJKW HTXDO OHQJWK HOHPHQW GLVFUHWL]DWLRQ VKRZQ LQ )LJXUH 7KH HIIHFW RI WKH QRQ VWUXFWXUDO PDVV ZDV FRQVLGHUHG QRQVWLIIHQLQJ DQG FRQFHQWUDWHG DW QRGH 7KLV PDVV ZDV PRGHOOHG E\ DGGLQJ LWV PDVV DQG PRPHQW RI LQHUWLD WR QRGH EHQGLQJ DQG URWDWLRQ '2)V UHVSHFWLYHO\ 7KH PDVV ORDGHG EHDP DV GHVFULEHG DERYH ZDV FRQVLGHUHG WKH fKHDOWK\f FRQILJXUDWLRQ 6WUXFWXUDO GDPDJH FRQVLVWHG RI WKH UHPRYDO RI WKH QRQVWUXFWXUDO PDVV IURP WKH EHDP ([SHULPHQWDO PRGDO DQDO\VLV RI WKH EHDP ZDV SHUIRUPHG RQ WKH fKHDOWK\ DQG fGDPDJHGf EHDP FRQILJXUDWLRQV 0RGDO SDUDPHWHUV ZHUH LGHQWLILHG XVLQJ IUHTXHQF\ GRPDLQ WHFKQLTXHV DQG WKH 5DWLRQDO )UDFWLRQ /HDVW 6TXDUH VLQJOH GHJUHH RI IUHHGRP FXUYH ILWWLQJ DOJRULWKP 7KH H[FLWDWLRQ VRXUFH XVHG ZDV DQ LPSDFW KDPPHU DQG WKH GULYLQJ SRLQW PHDVXUHPHQW ZDV DQ DFFHOHURPHWHU PRXQWHG DW WKH IUHH HQG RI WKH EHDP )RU HDFK FRQILJXUDWLRQ KHDOWK\ DQG GDPDJHGf IRXU PRGHV RI YLEUDWLRQ ZHUH PHDVXUHG (DFK PRGH FRQVLVWHG RI D QDWXUDO IUHTXHQF\ DQG LWV FRUUHVSRQGLQJ PRGH VKDSH ZLWK PHDVXUHPHQWV DW RQO\ WKH HLJKW )(0 EHQGLQJ GHJUHHV RI IUHHGRP $QDO\WLFDO DQG ([SHULPHQWDO 0RGHOV 'LPHQVLRQ &RUUHODWLRQ 7KH QXPEHU RI PHDVXUHG HLJHQYHFWRU FRPSRQHQWV f LV OHVV WKDQ WKH QXPEHU RI '2)V LQ WKH )(0 f ,Q IDFW RQO\ WKH EHQGLQJ '2)V RI WKH EHDP ZHUH PHDVXUHG H[SHULPHQWDOO\ $V GLVFXVVHG LQ &KDSWHU WZR DSSURDFKHV DUH DYDLODEOH WR FRUUHODWH WKHVH GLPHQVLRQV Lf H[SDQVLRQ RI WKH PHDVXUHG HLJHQYHFWRUV RU LLf UHGXFWLRQ RI WKH )(0 ,W ZDV IRXQG WKDW D )(0 UHGXFWLRQ LV EHWWHU VXLWHG IRU WKLV DSSOLFDWLRQ 7KXV WKH )(0 ZDV UHGXFHG XVLQJ WKH LPSURYHG UHGXFWLRQ V\VWHP ,56f PHWKRG 2f&DOODKDQ f 5HILQHPHQW RI WKH 2ULJLQDO )(0 7DEOH VKRZV D FRPSDULVRQ EHWZHHQ WKH DQDO\WLFDOO\ FRPSXWHG DQG H[SHULPHQWDOO\ PHDVXUHG QDWXUDO IUHTXHQFLHV )URP WKLV FRPSDULVRQ LW LV FOHDU WKDW WKH RULJLQDO UHGXFHG )(0 GRHV QRW DFFXUDWHO\ SUHGLFW WKH G\QDPLF EHKDYLRU RI WKH fKHDOWK\f EHDP 7KH ILUVW VWHS LQ WKLV VWXG\ ZDV WR UHILQH WKH RULJLQDO )(0 )RU WKH UHILQHPHQW SURFHVV LW ZDV DVVXPHG WKDW

PAGE 156

WKH RULJLQDO PDVV PDWUL[ LV DQ DFFXUDWH UHSUHVHQWDWLRQ RI WKH VWUXFWXUHfV PDVV SURSHUWLHV 7KH LQDFFXUDF\ RI WKH RULJLQDO )(0 ZDV EHOLHYHG WR EH VROHO\ GXH WR PRGHOLQJ HUURUV LQ WKH VWLIIQHVV SURSHUWLHV 7KH DOJRULWKP GLVFXVVHG LQ 6HFWLRQ ZDV XVHG WR FRUUHFW WKH RULJLQDO UHGXFHG VWLIIQHVV PDWUL[ ,Q RUGHU WR JHW D V\PPHWULF XSGDWHG VWLIIQHVV PDWUL[ WKH PHDVXUHG PRGH VKDSHV HLJHQYHFWRUVf ZHUH PDVV RUWKRJRQDOL]HG XVLQJ WKH 2SWLPXP :HLJKWHG 2UWKRJRQDOL]DWLRQ WHFKQLTXH %DUXFK DQG %DU ,W]KDFN f 7DEOH $QDO\WLFDO DQG ([SHULPHQWDO )UHTXHQFLHV RI WKH f+HDOWK\f 6WUXFWXUH 0RGH $QDO\WLFDO )UHTXHQF\ +]f ([SHULPHQWDO )UHTXHQF\ +]f 'DPDJH /RFDWLRQ 7KH QH[W VWHS RI WKLV DQDO\VLV LV WR GHWHUPLQH WKH ORFDWLRQ RI WKH VWUXFWXUDO GDPDJH 8VLQJ WKH UHILQHG )(0 DQG WKH PRGDO SDUDPHWHUV RI WKH IRXU PRGHV RI YLEUDWLRQ PHDVXUHG IURP WKH fGDPDJHGf EHDP D FXPXODWLYH GDPDJH ORFDWLRQ YHFWRU ZDV FDOFXODWHG &KDSWHU f 6LQFH WKH YDOXHV RI __]-G __ DV GHILQHG LQ 6HFWLRQ DUH RI GLIIHUHQW RUGHUV RI PDJQLWXGH WKH FXPXODWLYH GDPDJH ORFDWLRQ YHFWRU VKRXOG EH FRPSXWHG XVLQJ (T f 7KH XSSHU OHIW FRPHU RI )LJXUH GLVSOD\V WKH SORW RI WKH XQLW QRUPDOL]HG FXPXODWLYH GDPDJH ORFDWLRQ YHFWRU DV FDOFXODWHG E\ (T f )URP WKLV SORW LW LV FOHDU WKDW '2) KDV EHHQ DIIHFWHG E\ GDPDJH 7KLV LV H[DFWO\ WKH EHQGLQJ '2) ZKHUH WKH QRQVWUXFWXUDO PDVV ZDV PRXQWHG 7KH

PAGE 157

VPDOO QXPHULFDO HOHPHQWV DW DOO RWKHU '2)V FDQ EH DWWULEXWHG WR H[SHULPHQWDO PHDVXUHPHQW QRLVH '2) EW! V '2) 'DPDJH ([WHQW 0RGH )LJXUH 0DVV /RDGHG &DQWLOHYHUHG %HDP 'DPDJH $VVHVVPHQW 'DPDJH ([WHQW 7KH ILQDO VWHS RI WKLV DQDO\VLV LV WR GHWHUPLQH WKH H[WHQW RI VWUXFWXUDO GDPDJH :LWK NQRZOHGJH RI WKH GDPDJH ORFDWLRQ WKH UDQN RI WKH fWUXHf PDVV SHUWXUEDWLRQ PDWUL[ $0G LV RQH VLQFH WKHUH LV RQO\ RQH '2) DIIHFWHG E\ WKH GDPDJH DQG WKDW '2) LV QRW FRQQHFWHG WR WKH FDQWLOHYHUHG HQG +HQFH LQ RUGHU WR FRPSXWH D UDQN RQH $0G RQO\ RQH PRGH RI YLEUDWLRQ LV QHHGHG $V GLVFXVVHG HDUOLHU WKH PRGH WKDW VKRXOG EH XVHG LV WKH RQH WKDW PRVW FOHDQO\ GHPRQVWUDWHV WKH GDPDJH GHWHFWHG E\ WKH FXPXODWLYH GDPDJH ORFDWLRQ YHFWRU 7KH GDPDJH

PAGE 158

YHFWRU œ ZDV GHWHUPLQHG WR SURYLGH WKH EHVW LQVLJKW LQWR WKH VWDWH RI GDPDJH 7KH GDPDJH YHFWRU  LV VKRZQ LQ WKH XSSHU ULJKW FRUQHU RI )LJXUH 7KH FDOFXODWHG $0M XVLQJ PRGH GDWD LV VKRZQ LQ WKH ORZHU SORW RI )LJXUH ,W LV FOHDU WKDW WKH H[WHQW FDOFXODWLRQ KDV FRQFHQWUDWHG WKH PDMRU FKDQJHV DW WKH '2) DIIHFWHG E\ WKH GDPDJH )URP WKH H[WHQW FDOFXODWLRQ WKH PDVV ORVV ZDV HVWLPDWHG WR EH NLORJUDPV ZKLFK LV ZLWKLQ b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fKHDOWK\f PRGHO LV SURSRUWLRQDO DQG HTXDO WR [ WLPHV WKH fKHDOWK\f PDVV PDWUL[ SOXV [O WLPHV WKH fKHDOWK\f VWLIIQHVV PDWUL[ $V LQ WKH SUREOHP RI 6HFWLRQ GDPDJH LV VLPXODWHG E\ UHGXFLQJ WKH
PAGE 159

SUDFWLFH WKH QRLVH LQ HLJHQYHFWRU LQIRUPDWLRQ FRXOG EH GXH WR ERWK PHDVXUHPHQW DQGRU H[SDQVLRQ HUURUV ( 2[O2 [ SVLf S [O NJP OELQf $ [ + P LQf / P LQf )LJXUH %D\ 'LPHQVLRQDO 7UXVV 'DPDJH /RFDWLRQ &XPXODWLYH GDPDJH ORFDWLRQ YHFWRUV DV GHILQHG LQ &KDSWHU DUH ILUVW FRPSXWHG IRU DOO WKUHH VFHQDULRV XVLQJ WKH PRGDO SURSHUWLHV RI WKH WHQ fGDPDJHGf PRGHV ,Q WKLV SUREOHP DOO URZV RI PDWUL[ =G GHILQHG LQ &KDSWHU f DUH RI WKH VDPH RUGHU RI PDJQLWXGH KHQFH HLWKHU (T f RU (T f FDQ EH XVHG WR FRPSXWH WKH FXPXODWLYH GDPDJH YHFWRUV 7KH XSSHU OHIW SORW RI )LJXUH UHSUHVHQWV WKH H[DFW GDPDJH FRPSXWHG IURP WKH H[DFW GDPDJH SHUWXUEDWLRQ PDWULFHV $,4 DQG $'Mf 7KH XSSHU ULJKW SORW FRUUHVSRQGV WR WKH FDVH ZKHUH WKH H[DFW fGDPDJHGf HLJHQYHFWRU LQIRUPDWLRQ LV SURYLGHG WR WKH VXEVSDFH URWDWLRQ GDPDJH ORFDWLRQ DOJRULWKP 7KH ORZHU OHIW DQG ULJKW SORWV FRUUHVSRQG WR WKH FDVHV ZKHUH WKH H[DFW fGDPDJHGf HLJHQYHFWRUV KDYH EHHQ FRUUXSWHG ZLWK b DQG b UDQGRP QRLVH UHVSHFWLYHO\ $V VKRZQ LQ )LJXUH WKH ORFDWLRQ DOJRULWKP LV DEOH WR H[DFWO\ ORFDWH WKH GDPDJH ZKHQ SUHVHQWHG ZLWK QRLVH IUHH LQIRUPDWLRQ $OWKRXJK QRW DV FOHDQ WKH GDPDJH FDQ VWLOO EH FOHDUO\ ORFDWHG LQ WKH QRLV\ HLJHQYHFWRUV FDVHV

PAGE 160

([DFW 'DPDJH 0RGHV WR '2) R +f§n 2 f S+ 7 & RD F 4 0RGHV WR b HUURU 7 8X!QLL '2) 0RGHV WR b HUURU '2) )LJXUH %D\ 'LPHQVLRQDO 7UXVV 'DPDJH /RFDWLRQ 'DPDJH ([WHQW 7KH GDPDJH ORFDWLRQ DVVHVVHG LQ WKH SUHYLRXV VHFWLRQ FDQ EH XVHG LQ FRQMXQFWLRQ ZLWK WKH WUXVV ILQLWH HOHPHQW FRQQHFWLYLW\ WR GHWHUPLQH WKH GDPDJHG WUXVV VWUXWV 7KH UDQN RI WKH fWUXHf SHUWXUEDWLRQ PDWUL[ $.A RU $'Mf FDQ EH IRXQG E\ DGGLQJ WKH UDQN RI WKH HOHPHQW VWLIIQHVV RU GDPSLQJf PDWUL[ RI WKH GDPDJHG VWUXWV +HQFH WKH UDQN RI WKH SHUWXUEDWLRQ WR WKH VWLIIQHVV PDWUL[ GXH WR GDPDJH LV WZR EHFDXVH WZR VWUXWV RI UDQN RQH HOHPHQW PDWULFHV DUH GDPDJHG 1RWH WKDW WKH HOHPHQW VWLIIQHVV PDWUL[ RI D VWUXW LV RQH VLQFH LW LV PRGHOOHG DV D URG HOHPHQW %HFDXVH WKH GDPSLQJ RI WKH VWUXFWXUH LV SURSRUWLRQDO DQG WKH PDVV SURSHUWLHV DUH XQDIIHFWHG E\ WKH GDPDJH LW LV GHGXFHG WKDW WKH UDQN RI WKH SHUWXUEDWLRQ WR WKH GDPSLQJ PDWUL[ LV WKH VDPH DV WKH UDQN RI WKH SHUWXUEDWLRQ WR WKH VWLIIQHVV PDWUL[ )URP SURSRVLWLRQV

PAGE 161

DQG LW LV FOHDU WKDW RQO\ H[SHULPHQWDO GDWD IURP WZR PRGHV RI YLEUDWLRQ DUH QHHGHG WR FRPSXWH WKH H[WHQW RI WKH GDPDJH ,Q WKH QRLV\ VLWXDWLRQV WKH WZR PRGHV WKDW VKRXOG EH XVHG DUH WKH RQHV WKDW PRVW FOHDQO\ GHPRQVWUDWH WKH GDPDJH VKRZQ LQ )LJXUH 7KHVH PRGHV FDQ EH GHWHUPLQHG E\ LQVSHFWLQJ WKH LQGLYLGXDO GDPDJH YHFWRUV $Q LQVSHFWLRQ RI WKH LQGLYLGXDO GDPDJH YHFWRUV DVVRFLDWHG ZLWK HDFK fQRLV\f HLJHQYHFWRU VXJJHVWV WKDW PRGHV DQG SURYLGH WKH EHVW LQVLJKW LQWR WKH VWDWH RI WKH GDPDJH 7KH UHVXOWV RI DSSO\LQJ WKH 0537 6HFWLRQ (T ff WR GHWHUPLQH WKH SHUWXUEDWLRQV WR WKH VWLIIQHVV PDWUL[ GXH WR WKH GDPDJH DUH VKRZQ LQ )LJXUH &OHDUO\ IURP MXGJLQJ WKH XSSHU ULJKW PHVK SORW RI )LJXUH WKH 0537 LV DEOH WR UHSURGXFH WKH H[DFW GDPDJH ZLWK RQO\ WZR QRLVH IUHH PRGHV 7KH DOJRULWKP GHPRQVWUDWHV JRRG SHUIRUPDQFH ZKHQ IDFHG ZLWK QRLV\ HLJHQGDWD ORZHU SORWVf 7KH SHUFHQWDJH HUURUV ZLWK UHVSHFW WR WKH H[DFW VWLIIQHVV GDPDJH IRU DOO VWXGLHG FDVHV DUH OLVWHG LQ 7DEOH 7KH SHUWXUEDWLRQV WR WKH GDPSLQJ PDWUL[ $'M GXH WR WKH GDPDJH DUH HVWLPDWHG E\ WKH H[WHQW DOJRULWKP 6HFWLRQ (T ff ZLWK H[DFWO\ WKH VDPH DFFXUDF\ DV IRU $. ,Q IDFW WKH XQVHDOHG PHVK SORWV RI WKH FRPSXWHG $'MfV DUH WKH VDPH DV WKH RQHV VKRZQ LQ )LJXUH 7KLV ZRXOG EH H[SHFWHG VLQFH DV UHSRUWHG HDUOLHU WKH GDPDJH LQ WKH GDPSLQJ SURSHUWLHV LV SURSRUWLRQDO WR WKH GDPDJH LQ WKH VWLIIQHVV SURSHUWLHV 7DEOH %D\ 'LPHQVLRQDO 7UXVV 6XPPDU\ RI 3HUFHQWDJH (UURU :LWK 5HVSHFW WR WKH ([DFW 'DPDJH 3HUFHQWDJH (UURU ZLWK UHVSHFW WR H[DFW VWLIIQHVV RU GDPSLQJf (LJHQYHFWRUV (UURU 8SSHU /RQJHURQ 2I %D\ 7KUHH /RZHU /RQJHURQ RI %D\ )RUW\ b b b

PAGE 162

([DFW 'DPDJH 0RGHV t )LJXUH %D\ 'LPHQVLRQDO 7UXVV 'DPDJH ([WHQW (LJKW%D\ 7ZR'LPHQVLRQDO 0DVV/RDGHG &DQWLOHYHUHG 7UXVV 3UREOHP 'HVFULSWLRQ 7KH VWUXFWXUH XQGHU LQYHVWLJDWLRQ LV WKH HLJKWED\ WZRGLPHQVLRQDO PDVVORDGHG FDQWLOHYHUHG WUXVV VKRZQ LQ )LJXUH 7KLV VWUXFWXUH ZDV GHVLJQHG 5LGHV f WR HPXODWH W\SLFDO SURSHUWLHV RI VSDFH VWUXFWXUHV ORZ IUHTXHQF\ PRGHV ZLWK ODUJH QRQVWUXFWXUDO PDVV 7KH JHRPHWULF DQG PDWHULDO SURSHUWLHV RI WKH WUXVV DUH JLYHQ LQ )LJXUH 7KH WUXVV FRQVLVWV RI VWUXWV DQG QRGHV )RXUWHHQ RI WKH QRGHV DUH ORDGHG ZLWK FRQFHQWUDWHG QRQVWUXFWXUDO PDVV RI PDJQLWXGH OEVVHFLQ DQG WKH UHPDLQLQJ WZR QRGHV KDYH ODUJH OXPSHG PDVVHV RI PDJQLWXGH OEVVHFLQ (DFK WUXVV VWUXW ZDV PRGHOHG DV D

PAGE 163

URG HOHPHQW ZLWK QHJOLJLEOH PDVV 7KH ILQLWH HOHPHQW PRGHO KDV WUDQVODWLRQDO '2)V '2)V SHU QRGHf 7KH WUXVV DV GHVFULEHG DERYH LV FRQVLGHUHG WKH KHDOWK\ XQGDPDJHGf FRQILJXUDWLRQ LQ RXU VWXG\ 7ZR SUREOHPV EDVHG RQ WKLV VWUXFWXUH DUH SUHVHQWHG WR LOOXVWUDWH WKH FKDUDFWHULVWLFV RI WKH SURFHGXUH RI DSSO\LQJ WKH 0537 WR VLPXOWDQHRXVO\ GHWHUPLQH WKH GDPDJH H[WHQW LQ DOO SURSHUW\ PDWULFHV RI XQGDPSHG 0 .f DQG SURSRUWLRQHG GDPSHG 0 .f VWUXFWXUHV )LJXUH 7KH (LJKW%D\ 7ZR'LPHQVLRQDO 0DVV/RDGHG &DQWLOHYHUHG 7UXVV 3URSRUWLRQDOO\ 'DPSHG &RQILJXUDWLRQ 'DPDJH RI 6PDOO 2UGHU RI 0DJQLWXGH ,Q WKLV ILUVW SUREOHP LW LV DVVXPHG WKDW WKH GDPSLQJ RI WKH WUXVV LV SURSRUWLRQDO DQG HTXDO WR O[O2 WLPHV WKH fKHDOWK\f PDVV PDWUL[ SOXV O[O2 WLPHV WKH fKHDOWK\f VWLIIQHVV PDWUL[ ,Q WKLV H[DPSOH WKH GDPDJH LV VLPXODWHG E\ D b VWLIIQHVV UHGXFWLRQ RI WKH GDUNHQHG VWUXW DQG D b UHGXFWLRQ RI PDVV 0f 7KH GDPSLQJ RI WKH fGDPDJHGf PRGHO LV DOVR DVVXPHG SURSRUWLRQDO DQG HTXDO WR O[O2 WLPHV WKH fGDPDJHGf PDVV PDWUL[ SOXV O[O2 WLPHV WKH fGDPDJHGf VWLIIQHVV PDWUL[ )RU WKH GDPDJH DQDO\VLV LW LV DVVXPHG WKDW RQO\ WKH ILUVW IRXU PRGHV RI YLEUDWLRQ DUH DYDLODEOH (DFK DYDLODEOH PRGH FRQVLVWV RI DQ HLJHQYDOXH DQG DQ HLJHQYHFWRU ZLWK HQWULHV DW DOO )(0 '2)V 7KLV SUHVHQW SUREOHP LV VLPLODU WR D SUREOHP

PAGE 164

LQYHVWLJDWHG E\ 5LFOHV f ZLWK WKH H[FHSWLRQ WKDW LQ 5LFOHV f WKH WUXVV ZDV DVVXPHG XQGDPSHG 'DPDJH /RFDWLRQ 7KH ILUVW VWHS LQ WKH DQDO\VLV LV WR XVH WKH GDPDJH ORFDWLRQ DOJRULWKP DV GLVFXVVHG LQ &KDSWHU WR GHWHUPLQH WKH ORFDWLRQ RI GDPDJH 7KH OHIW SORW RI )LJXUH UHSUHVHQWV WKH XQLW QRUPDOL]HG H[DFW FXPXODWLYH GDPDJH ORFDWLRQ YHFWRU &'/9f (TV Ef DQG ff FRPSXWHG IURP WKH H[DFW SHUWXUEDWLRQ PDWULFHV $0M $'M DQG $.A 7KH XQLW QRUPDOL]HG &'/9 FRPSXWHG IURP WKH VXEVSDFH GDPDJH ORFDWLRQ DOJRULWKP (TV Ef DQG ff LV VKRZQ LQ WKH ULJKW SORW RI )LJXUH $V SUHYLRXVO\ SURYHQ WKH ORFDWLRQ DOJRULWKP LV DEOH WR H[DFWO\ ORFDWH WKH GDPDJH ZKHQ SUHVHQWHG ZLWK QRLVH IUHH GDWD ([DFW 'DPDJH 0RGHV WR )LJXUH 3UREOHP &XPXODWLYH 'DPDJH /RFDWLRQ 9HFWRU )LUVW )RXU 0RGHV 'HFRPSRVLWLRQ RI 0DWUL[ % 7KH QH[W VWHS LV WR GHFRPSRVH PDWUL[ % DV GHILQHG E\ (T f LQWR %P %A DQG %A $V GLVFXVVHG LQ 6HFWLRQ WKH SURSRVHG GHFRPSRVLWLRQ UHTXLUHV WKDW PDWUL[ % EH RI IXOO UDQN $ VLPSOH VLQJXODU YDOXH GHFRPSRVLWLRQ RI PDWUL[ % FRPSXWHG XVLQJ DOO DYDLODEOH PRGHV VKRZV WKDW LWV UDQN LV 7KHUHIRUH RQO\ WKUHH RI WKH IRXU DYDLODEOH PRGHV VKRXOG EH XVHG $ IXUWKHU LQYHVWLJDWLRQ UHYHDOV WKDW DQ\ WKUHH RI WKH IRXU PRGHV UHVXOW LQ D IXOO UDQN %

PAGE 165

+HQFH DQ\ FRPELQDWLRQ RI WKUHH PRGHV FDQ EH XVHG LQ WKH GHFRPSRVLWLRQ 7KH WKUHH SORWV LQ WKH XSSHU KDOI RI )LJXUH VKRZ WKH XQLW QRUPDOL]HG H[DFW FXPXODWLYH YHFWRU (T f 6HFWLRQ f FRPSXWHG XVLQJ WKH H[DFW SHUWXUEDWLRQ PDWULFHV $0M $' DQG $.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f F 0O '2) '2) '2) )LJXUH 3UREOHP &XPXODWLYH 9HFWRUV $VVRFLDWHG :LWK WKH ([DFW DQG &RPSXWHG %P %G %A )LUVW 7KUHH 0RGHV

PAGE 166

'DPDJH ([WHQW 1RZ WKDW %P %M DQG %A KDYH EHHQ FRPSXWHG WKH ILQDO VWHS LV WR FRPSXWH WKH SHUWXUEDWLRQ WR WKH SURSHUW\ PDWULFHV GXH WR WKH GDPDJH $0G $'G $.Gf XVLQJ WKH PLQLPXP UDQN SHUWXUEDWLRQ WKHRU\ 0537f DV GHILQHG LQ 6HFWLRQ 8VLQJ WKH &RPSXWHG $0G &RPSXWHG $'G $Q\ 7ZR 0RGHV $Q\ 7KUHH 0RGHV &RPSXWHG $.G $Q\ 2QH 0RGH )LJXUH 3UREOHP ([DFW DQG &RPSXWHG $0G $'G $.G FRQQHFWLYLW\ LQIRUPDWLRQ RI WKH RULJLQDO )(0 DORQJ ZLWK WKH LQIRUPDWLRQ RI WKH GDPDJH HIIHFW RQ WKH WKUHH SURSHUW\ PDWULFHV VKRZQ LQ )LJXUH ff LW FDQ EH GHGXFHG WKDW WKH UDQN RI WKH fWUXHf $0G LV WZR DQG WKH UDQN RI WKH WUXH $.G LV RQH %HFDXVH RI WKH IDFW WKDW WKH GDPDJH LQ WKH GDPSLQJ LV SURSRUWLRQDO WR WKH GDPDJH LQ WKH PDVV DQG VWLIIQHVV LW FDQ DOVR EH GHGXFHG WKDW WKH UDQN RI WKH fWUXHf $'G LV WKUHH 7KH UDQN LQIRUPDWLRQ LV DVFHUWDLQHG E\ LGHQWLI\LQJ ZKLFK VWUXFWXUDO HOHPHQWV FRQQHFW WKH GDPDJHG '2)V DQG WKHQ DGGLQJ XS WKH UDQN RI HDFK GDPDJHG VWUXFWXUDO HOHPHQW 7KH UDQN HVWLPDWLRQ FRXOG DOVR EH REWDLQHG E\ SHUIRUPLQJ D VLQJXODU YDOXH GHFRPSRVLWLRQ RQ %P %G DQG %A )URP WKH UDQN LQIRUPDWLRQ

PAGE 167

DQG SURSRVLWLRQV DQG LW LV FOHDU WKDW RQO\ WZR PRGHV DUH QHHGHG WR FRPSXWHG $0G XVLQJ WKH 0537 /LNHZLVH RQO\ WKUHH PRGHV DUH QHHGHG WR FRPSXWHG $'A DQG RQO\ RQH PRGH LV QHHGHG WR FRPSXWH $. 6LQFH ZH DUH GHDOLQJ ZLWK QRLVH IUHH HLJHQGDWD DQ\ WZR WKUHH PRGHV RU DQ\ RQH PRGH FDQ EH XVHG LQ WKH FRPSXWDWLRQ $0M $'M RU $.A UHVSHFWLYHO\ 0HVK SORWV RI WKH H[DFW DQG FRPSXWHG $0 $' DQG $.M PDWULFHV DUH VKRZQ LQ )LJXUHV $JDLQ D FRPSDULVRQ EHWZHHQ H[DFW DQG FRPSXWHG $0G $' DQG $.G VKRZV WKDW WKH 0537 LV DEOH WR UHSURGXFH WKH H[DFW GDPDJH HIIHFW ,Q WKLV SUREOHP DV VKRZQ LQ WKLV LQYHVWLJDWLRQ WKH SURSRVHG GDPDJH GHWHFWLRQ WHFKQLTXH LV DEOH WR DVVHVV WKH GDPDJH H[DFWO\ ZKHQ WKUHH QRLVH IUHH HLJHQYDOXHVHLJHQYHFWRUV DUH XVHG 8QIRUWXQDWHO\ LQ SUDFWLFH WKH PHDVXUHG HLJHQGDWD DUH DOZD\V FRUUXSWHG E\ QRLVH 7R VLPXODWH VRPH NLQG RI D SUDFWLFDO VLWXDWLRQ UDQGRP QRLVH ZDV DGGHG WR WKH HLJHQGDWD :KHQ XVLQJ QRLV\ HLJHQGDWD ILUVW IRXU PRGHVf WKH SUHVHQW DOJRULWKP ZDV XQDEOH WR ORFDWH WKH GDPDJH 7KH UHDVRQ LV WKDW WKH GDPDJH DV VLPXODWHG LV RI VPDOO RUGHU RI PDJQLWXGH ZKLFK UHVXOWV LQ RQO\ D VPDOO FKDQJH LQ WKH HLJHQGDWD RI WKH KHDOWK\ PRGHO ,Q WKLV FDVH WKH QRLVH WRWDOO\ PDVNV WKH GDPDJH DV UHIOHFWHG LQ WKH fPHDVXUHGf QRLV\ HLJHQGDWD 8QGDPSHG &RQILJXUDWLRQ 'DPDJH RI /DUJH 2UGHU RI 0DJQLWXGH ,Q WKLV SUREOHP WKH WUXVV LV DVVXPHG XQGDPSHG 7KH GDPDJH LV VLPXODWHG E\ D b VWLIIQHVV UHGXFWLRQ RI WKH GDUNHQHG VWUXW DQG D b UHGXFWLRQ RI PDVV 0M )RU WKH DQDO\VLV LW LV DVVXPHG WKDW RQO\ WKH ILUVW ILYH PRGHV RI YLEUDWLRQ DUH DYDLODEOH (DFK DYDLODEOH PRGH FRQVLVW RI DQ HLJHQYDOXH DQG DQ HLJHQYHFWRU ZLWK HQWULHV DW DOO )(0 '2)V 7KLV SUREOHP LV LQYHVWLJDWHG IRU IRXU GLIIHUHQW FDVHV (DFK FDVH UHSUHVHQWV D GLIIHUHQW DPRXQW RI UDQGRP QRLVH DGGHG WR WKH HLJHQYHFWRUV RI WKH DYDLODEOH PRGHV b b b DQG b 1RLVH )UHH (LJHQGDWD 7KH SURFHGXUHV XVHG LQ WKH LQYHVWLJDWLRQ RI WKLV FDVH DUH VLPLODU WR WKH RQH XVHG LQ WKH SUHYLRXV SUREOHP 7KH FRPSXWHGH[DFW FXPXODWLYH GDPDJH ORFDWLRQ YHFWRU DQG WKH

PAGE 168

FXPXODWLYH YHFWRU DVVRFLDWHG ZLWK %P DQG %N DUH GLVSOD\HG LQ WKH ILUVW URZ RI SORWV RI )LJXUH /LNHZLVH WKH ILUVW URZ RI SORWV RI )LJXUH VKRZ WKH FRPSXWHGH[DFW $0M DQG $.M ZKHQ XVLQJ QRLVH IUHH GDWD $JDLQ DV SUHYLRXVO\ VKRZQ H[DFW GDWD SURYLGHV H[DFW UHVXOWV 1RLV\ (LJHQGDWD 7KH HIIHFW RI LQWURGXFLQJ QRLVH LQWR WKH fPHDVXUHGf HLJHQYHFWRUV LV VKRZQ LQ WKH ORZHU WKUHH URZV RI SORWV LQ )LJXUHV DQG ,Q )LJXUH LW LV REYLRXV WKDW DV QRLVH LV LQFUHDVHG WKH FXPXODWLYH GDPDJH YHFWRU % ILUVW FROXPQf EHFRPHV FRUUXSWHG WR WKH SRLQW WKDW ZKHQ b QRLVH LV DGGHG LW LV GLIILFXOW WR DVFHUWDLQ LQIRUPDWLRQ FRQFHUQLQJ WKH VWDWH RI GDPDJH ,Q SHUIRUPLQJ WKH GHFRPSRVLWLRQ RI % LQWR %P DQG %N RQH VKRXOG RQO\ XVH WKRVH PRGHV RI YLEUDWLRQV ZKRVH GDPDJH YHFWRU G UHIOHFWV WKH VDPH fQDWXUH RI GDPDJHf DV WKH FXPXODWLYH GDPDJH YHFWRU % )RU WKLV SDUWLFXODU H[DPSOH RQO\ PRGHV DQG PHHW WKLV FULWHULRQ IRU WKH FDVHV RI b DQG b UDQGRP QRLVH )RU WKH b QRLVH FDVH WKH FXPXODWLYH GDPDJH YHFWRU DQG WKH LQGLYLGXDO PRGH GDPDJH YHFWRUV GLG QRW FOHDUO\ LQGLFDWH GDPDJH 7R PDLQWDLQ D OHYHO RI FRQVLVWHQF\ WKH VDPH PRGH VHW f ZDV XVHG IRU WKH FDVH RI b QRLVH 7KH GHFRPSRVLWLRQ RI % LQWR %P DQG %N IRU WKH FDVH RI QRLV\ PHDVXUHPHQWV LV VKRZQ LQ WKH UHPDLQLQJ URZV DQG FROXPQV RI )LJXUH $V VKRZQ LQ )LJXUH WKH DGGHG QRLVH KDV WZR HIIHFWV LQ WKH GHFRPSRVLWLRQ SURFHVV 7KH ILUVW ZKLFK LV UDWKHU REYLRXV LV WKDW QRLVH FDXVHV WKH DSSHDUDQFH RI VPDOO QXPHULFDO FRPSRQHQW DW DOO '2)V DOWKRXJK LQ WKH ORZ OHYHO QRLVH FDVHV b DQG bf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f UDQN FDOFXODWLRQ RI %P DQG %N DQG LLf FRPSDULVRQ RI LQGLYLGXDO FROXPQV RI %P DQG %N WR WKHLU DVVRFLDWHG FXPXODWLYH YHFWRU )RU WKH FDVHV RI b DQG b

PAGE 169

'DPDJH ,QGLFDWRU 'DPDJH ,QGLFDWRU 'DPDJH ,QGLFDWRU 'DPDJH ,QGLFDWRU 1RLVH )UHH (LJHQGDWD &RPSXWHG 4 / -/PA-O 0 LQ ,, QU\MW b 1RLVH $GGHG WR WKH (LJHQYHFWRUV b 1RLVH $GGHG WR WKH (LJHQYHFWRUV b 1RLVH $GGHG WR WKH (LJHQYHFWRUV )LJXUH 3UREOHP &XPXODWLYH 9HFWRUV $VVRFLDWHG :LWK % %P %G %A % &RPSXWHG 8VLQJ 0RGHV %P %G %N &RPSXWHG 8VLQJ PRGHV t

PAGE 170

QRLVH WKH H[WHQW FDOFXODWLRQ VWLOOV SURYLGHV D UHDVRQDEOH LQGLFDWLRQ RI GDPDJH DOWKRXJK WKH PDJQLWXGH RI GDPDJH LV LQ HUURU 7KH SHUFHQWDJH HUURUV RI WKH HVWLPDWHG PDVV DQG VWLIIQHVV GDPDJH ZLWK UHVSHFW WR WKH H[DFW GDPDJH DUH OLVWHG LQ 7DEOH )LJXUH VKRZV WKH H[WHQW FDOFXODWLRQ ZLWK QR HQKDQFHPHQWV )RU WKH FDVH LQ ZKLFK b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b b b 6XPPDU\ $ WHFKQLTXH WKDW DSSURDFKHV WKH VWUXFWXUDO GDPDJH DVVHVVPHQW SUREOHP LQ D GHFRXSOHG IDVKLRQ ZDV GHPRQVWUDWHG DQG HYDOXDWHG XVLQJ QXPHULFDO DQG DFWXDO H[SHULPHQWDO WHVW GDWD ,Q DOO SUHVHQWHG H[DPSOHV WKH VWUXFWXUDO GDPDJH ZDV ILUVW ORFDWHG E\ XVLQJ WKH VXEVSDFH URWDWLRQ GDPDJH ORFDWLRQ DOJRULWKP IRUPXODWHG LQ &KDSWHU :LWK ORFDWLRQ GHWHUPLQHG WKH

PAGE 171

([DFW &RPSXWHG $0G $Q\ WZR 0RGHV ([DFW &RPSXWHG $.A $Q\ 2QH 0RGH &RPSXWHG $0G 0RGHV t &RPSXWHG $.M 0RGH b 1RLVH $GGHG WR WKH (LJHQYHFWRUV &RPSXWHG $0G 0RGHV t &RPSXWHG $.r 0RGH &RPSXWHG $0G 0RGHV t &RPSXWHG $.A 0RGH b 1RLVH $GGHG WR WKH (LJHQYHFWRUV )LJXUH 3UREOHP ([DFW DQG &RPSXWHG $0G $'G $.M

PAGE 172

PLQLPXP UDQN SHUWXUEDWLRQ WKHRU\ 0537f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

PAGE 173

&+$37(5 &21&/86,21 $1' 68**(67,216 )25 )8785( :25. 7KLV VWXG\ LQYHVWLJDWHG WKH GHYHORSPHQW RI IRXU DOJRULWKPV UHOHYDQW WR WKH DUHDV RI ILQLWH HOHPHQW PRGHO UHILQHPHQW DQG VWUXFWXUDO GDPDJH DVVHVVPHQW $OO IRXU DOJRULWKPV PDNH XVH RI DQ RULJLQDO ILQLWH HOHPHQW PRGHO )(0f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f 1H[W D VXEVSDFH URWDWLRQ WHFKQLTXH WR LPSURYH WKH SHUIRUPDQFH RI DQ H[LVWLQJ PRGHO UHILQHPHQW DOJRULWKP WHUPHG 6($05$ =LPPHUPDQ DQG :LGHQJUHQ f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

PAGE 174

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f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fGDPDJHGf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fUHDO OLIHf VWUXFWXUHV 7KLV ZDV GHPRQVWUDWH LQ WKH LQYHVWLJDWLRQ RI WKH H[SHULPHQWDO GDWD DQG LQ SDUWLFXODU WKH 1$6$ ED\ WUXVV H[DPSOHV

PAGE 175

7R IXOO\ GHPRQVWUDWH WKH SUDFWLFDOLW\ RI WKH 0537 EDVHG GDPDJH H[WHQW DOJRULWKPV DQG WKH VXEVSDFH URWDWLRQ GDPDJH ORFDWLRQ DOJRULWKP LQ KDQGOLQJ fUHDO OLIHf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fROGf VWUXFWXUHV WKDW KDYH QR DQDO\WLFDO PRGHO

PAGE 176

5()(5(1&(6 $GHOPDQ + 0 DQG +DIWND 5 7 f f6HQVLWLYLW\ $QDO\VLV RI 'LVFUHWH 6WUXFWXUDO 6\VWHPVf $0$ -RXUQDO 9RO 1R SS $QGU\ $ 1 6KDSLUR ( < DQG &KXQJ & f f(LJHQVWUXFWXUH $VVLJQPHQW )RU /LQHDU 6\VWHPVf ,((( 7UDQVDFWLRQV RQ $HURVSDFH DQG (OHFWURQLF 6\VWHPV 9RO $(6 1R SS %DUXFK 0 f f2SWLPXP :HLJKWHG 2UWKRJRQDOL]DWLRQ RI 0HDVXUHG 0RGHVf $,$$ -RXUQDO 9RO 1R SS %DUXFK 0 DQG %DU ,W]KDFN < f f2SWLPXP :HLJKWHG 2UWKRJRQDOL]DWLRQ RI 0HDVXUHG 0RGHVf $,$$ -RXUQDO 9RO 1R SS %HUPDQ $ DQG 1DJ\ ( f f,PSURYHPHQWV RI D /DUJH $QDO\WLFDO 0RGHO 8VLQJ 7HVW 'DWDf $,$$ -RXUQDO 9RO 1R SS %URFN ( f f2SWLPDO 0DWULFHV 'HVFULELQJ /LQHDU 6\VWHPVf $0$ -RXUQDO 9RO 1R SS &KHQ & DQG *DUED $ f f$QDO\WLFDO 0RGHO ,PSURYHPHQW 8VLQJ 0RGDO 7HVW 5HVXOWVf $,$$ -RXUQDO 9RO 1R SS &KHQ & DQG *DUED $ f f2Q2UELW 'DPDJH $VVHVVPHQW IRU /DUJH 6SDFH 6WUXFWXUHVf $,$$ -RXUQDO 9RO 1R SS &ROOLQV +DUW & +DVVHOPDQ 7 DQG .HQQHG\ % f f6WDWLVWLFDO ,GHQWLILFDWLRQ RI 6WUXFWXUHVf $,$$ -RXUQDO 9RO 1R SS &UHDPHU 1 DQG +HQGULFNV 6 / f f6WUXFWXUDO 3DUDPHWHU ,GHQWLILFDWLRQ 8VLQJ 0RGDO 5HVSRQVH 'DWDf 3URFHHGLQJV RI WKH WK 93,t68$,$$ 6\PSRVLXP RQ '\QDPLFV DQG &RQWUROV IRU /DUJH 6WUXFWXUHV %ODFNVEXUJ 9$ SS (ZLQV f 0RGDO 7HVWLQJ 7KHRU\ DQG 3UDFWLFH %UXHO t .MDHU /HWFKZRUWK +HUWIRUGVKLUH (QJODQG )ODQLJDQ & & f f&RUUHFWLRQ RI )LQLWH (OHPHQW 0RGHOV 8VLQJ 0RGH 6KDSH 'HVLJQ 6HQVLWLYLW\f 3URFHHGLQJV RI WKH WK ,QWHUQDWLRQDO 0RGDO $QDO\VLV &RQIHUHQFH )LUHQ]D ,WDO\ SS

PAGE 177

)UHHG $ 0 DQG )ODQLJDQ & & f f$ &RPSDULVRQ RI 7HVW$QDO\VLV 0RGHO 5HGXFWLRQ 0HWKRGVf 6RXQG DQG 9LEUDWLRQ 0DUFK SS )XK &KHQ 6 DQG %HUPDQ $ f f6\VWHP ,GHQWLILFDWLRQ RI $QDO\WLFDO 0RGHOV RI 'DPSHG 6WUXFWXUHVf 3URFHHGLQJV RI WKH WK $,$$ 6WUXFWXUHV 6WUXFWXUDO '\QDPLFV DQG 0DWHULDOV &RQIHUHQFH 3DOP 6SULQJV &$ SS *ROXE + DQG 9DQ /RDQ & ) f 0DWUL[ &RPSXWDWLRQV 7KH -RKQV +RSNLQV 8QLYHUVLW\ 3UHVV %DOWLPRUH 0' *X\DQ 5 f f5HGXFWLRQ RI 6WLIIQHVV DQG 0DVV 0DWULFHVf $,$$ -RXUQDO 9RO 1R S *\VLQ + f f&RPSDULVRQ RI ([SDQVLRQ 0HWKRGV IRU )( 0RGHOLQJ (UURU /RFDOL]DWLRQf 3URFHHGLQJV RI WKH WK ,QWHUQDWLRQDO 0RGDO $QDO\VLV &RQIHUHQFH .LVVLPPHH )/ SS +DMHOD 3 DQG 6RHLUR ) f f5HFHQW 'HYHORSPHQWV LQ 'DPDJH 'HWHFWLRQ %DVHG RQ 6\VWHP ,GHQWLILFDWLRQ 0HWKRGVf 6WUXFWXUDO 2SWLPL]DWLRQ 9RO SS +DQDJXG 6 0H\\DSSD 0 &KHQJ < 5 DQG *UDLJ f f,GHQWLILFDWLRQ RI 6WUXFWXUDO '\QDPLF 6\VWHPV ZLWK 1RQSURSRUWLRQDO 'DPSLQJf 3URFHHGLQJV RI WKH WK $,$$ 6WUXFWXUHV 6WUXFWXUDO '\QDPLFV DQG 0DWHULDOV &RQIHUHQFH 3DOP 6SULQJV &$ SS +H\OHQ : DQG 6DDV 3 f f&RUUHODWLRQ RI $QDO\VLV DQG 7HVW LQ 0RGHOLQJ RI 6WUXFWXUHV $VVHVVPHQW DQG 5HYLHZf 3URFHHGLQJV RI WKH WK ,0$& /RQGRQ (QJODQG SS +XJKHV 7 5 f 7KH )LQLWH (OHPHQW 0HWKRG 3UHQWLFH +DOO (QJOHZRRG &OLIIV 1,EUDKLP 6 5 DQG 6DDIDQ $ $ f f&RUUHODWLRQ RI $QDO\VLV DQG 7HVW LQ 0RGHOLQJ RI 6WUXFWXUHV $VVHVVPHQW DQG 5HYLHZf 3URFHHGLQJV RI WKH WK ,0$& /RQGRQ (QJODQG SS ,QPDQ f 9LEUDWLRQ :LWK &RQWURO 0HDVXUHPHQW DQG 6WDELOLW\ 3UHQWLFH +DOO (QJOHZRRG &OLIIV 1,QPDQ DQG 0LQDV & f f0DWFKLQJ $QDO\WLFDO 0RGHOV ZLWK ([SHULPHQWDO 0RGDO 'DWD LQ 0HFKDQLFDO 6\VWHPVf &RQWURO DQG '\QDPLFV 6\VWHPV 9RO SS -XDQJ -1 DQG 3DSSD 5 6 f f$Q (LJHQV\VWHP 5HDOL]DWLRQ $OJRULWKP IRU 0RGDO 3DUDPHWHU ,GHQWLILFDWLRQ DQG 0RGHO 5HGXFWLRQf -RXUQDO RI *XLGDQFH &RQWURO DQG '\QDPLFV 9RO 1R SS .DEH $ 0 f f6WLIIQHVV 0DWUL[ $GMXVWPHQW 8VLQJ 0RGH 'DWDf $,$$ -RXUQDO 9RO 1R SS

PAGE 178

.DPPHU & f f7HVW$QDO\VLV 0RGHO 'HYHORSPHQW 8VLQJ DQ ([DFW 0RGDO 5HGXFWLRQf ,QWHUQDWLRQDO -RXUQDO RI $QDO\WLFDO DQG ([SHULPHQWDO 0RGDO $QDO\VLV 9RO 1R SS .DPPHU & f f2SWLPXP $SSUR[LPDWLRQ IRU 5HVLGXDO 6WLIIQHVV LQ /LQHDU 6\VWHP ,GHQWLILFDWLRQf $,$$ -RXUQDO 9RO 1R SS .DRXN 0 DQG =LPPHUPDQ & Df f(YDOXDWLRQ RI WKH 0LQLPXP 5DQN 8SGDWH LQ 'DPDJH 'HWHFWLRQ $Q ([SHULPHQWDO 6WXG\f 3URFHHGLQJV RI WKH WK ,0$& .LVVLPPHH )/ SS .DRXN 0 DQG =LPPHUPDQ & Ef f6WUXFWXUDO 'DPDJH $VVHVVPHQW 8VLQJ D *HQHUDOL]HG 0LQLPXP 5DQN 3HUWXUEDWLRQ 7KHRU\f 3URFHHGLQJV RI WKH WK $,$$ 6WUXFWXUHV 6WUXFWXUDO '\QDPLFV DQG 0DWHULDOV &RQIHUHQFH /D -ROOD &$ SS .DVKDQJDNL 7 $/ f f*URXQG 9LEUDWLRQ 7HVWV RI D +LJK )LGHOLW\ 7UXVV )RU 9HULILFDWLRQ RI RQ 2UELW 'DPDJH /RFDWLRQ 7HFKQLTXHVf 1$6$ 7HFKQLFDO 0HPRUDQGXP .DVKDQJDNL 7 $/ 6PLWK 6 : DQG /LP 7 : f f8QGHUO\LQJ 0RGDO 'DWD ,VVXHV IRU 'HWHFWLQJ 'DPDJH LQ 7UXVV 6WUXFWXUHVf 3URFHHGLQJV RI WKH UG $,$$ 6WUXFWXUHV 6WUXFWXUDO '\QDPLFV DQG 0DWHULDOV &RQIHUHQFH 'DOODV 7; SS .LGGHU 5 / f f5HGXFWLRQ RI 6WUXFWXUDO )UHTXHQF\ (TXDWLRQVf $,$$ -RXUQDO 9RO 1R SS /LP 7 : f f6XEPDWUL[ $SSURDFK WR 6WLIIQHVV 0DWUL[ &RUUHFWLRQ 8VLQJ 0RGDO 7HVW 'DWDf $,$$ -RXUQDO 9RO 1R SS /LQ & 6 f f/RFDWLRQ RI 0RGHOLQJ (UURUV 8VLQJ 0RGDO 7HVW 'DWDf $,$$ -RXUQDO 9RO 1R SS 0DUWHQVVRQ f f2Q WKH 0DWUL[ 5LFFDWL (TXDWLRQf ,QIRUPDWLRQ 6FLHQFHV 9RO SS f§ 0DUWLQH] 5HG+RUVH DQG $OOHQ f f6\VWHP ,GHQWLILFDWLRQ 0HWKRGV IRU '\QDPLF 6WUXFWXUDO 0RGHOV RI (OHFWURQLF 3DFNDJHVf 3URFHHGLQJV RI WKH QG $,$$ 6WUXFWXUHV 6WUXFWXUDO '\QDPLFV DQG 0DWHULDOV &RQIHUHQFH %DOWLPRUH 0' SS 0F*RZDQ 3 ( f '\QDPLF 7HVW$QDO\VLV &RUUHODWLRQ 8VLQJ 5HGXFHG $QDO\WLFDO 0RGHOV 0 6 7KHVLV (QJLQHHULQJ 0HFKDQLFV 2OG 'RPLQLRQ 8QLYHUVLW\ 1RUIRON 9$ 0HLURYLWFK / f &RPSXWDWLRQDO 0HWKRGV LQ 6WUXFWXUDO '\QDPLFV 6LMKRII t 1RRU'KRII $OSKHQ DDQ GHQ 5LMQ 7KH 1HWKHUODQGV

PAGE 179

0HLURYLWFK / f (OHPHQWV RI 9LEUDWLRQ $QDO\VLV 0F*UDZ+LOO %RRN &RPSDQ\ 1HZ
PAGE 180

6PLWK 6 : DQG %HDWWLH & $ f f6HFDQW0HWKRG $GMXVWPHQW IRU 6WUXFWXUDO 0RGHOVf $,$$ -RXUQDO 9RO 1R SS 6PLWK 6 : DQG +HQGULFNV 6 / f f(YDOXDWLRQ RI 7ZR ,GHQWLILFDWLRQ 0HWKRGV IRU 'DPDJH 'HWHFWLRQ LQ /DUJH 6SDFH 7UXVVHVf 3URFHHGLQJV RI WKH WK 93,t68$,$$ 6\PSRVLXP RQ '\QDPLFV DQG &RQWUROV IRU /DUJH 6SDFH 6WUXFWXUHV 9LUJLQLD 3RO\WHFKQLF ,QVWLWXWH DQG 6WDWH 8QLYHUVLW\ %ODFNVEXUJ 9$ SS 6RHLUR ) f 6WUXFWXUDO 'DPDJH $VVHVVPHQW 8VLQJ ,GHQWLILFDWLRQ 7HFKQLTXHV 3K' 'LVVHUWDWLRQ 'HSDUWPHQW RI $HURVSDFH (QJLQHHULQJ 0HFKDQLFV DQG (QJLQHHULQJ 6FLHQFH 8QLYHUVLW\ RI )ORULGD *DLQHVYLOOH )/ 6ULQDWKNXPDU 6 f f(LJHQYDOXHV(LJHQYHFWRUV $VVLJQPHQW 8VLQJ 2XWSXW )HHGEDFNf ,((( 7UDQVDFWLRQV RQ $XWRPDWLF &RQWURO 9RO $& SS :LGHQJUHQ 0 f $Q $QDO\WLFDO 0HWKRG IRU WKH 6\PPHWULF &RUUHFWLRQ RI 0DWKHPDWLFDO 0RGHOV RI 9LEUDWLQJ 6\VWHPV 8VLQJ (LJHQVWUXFWXUH $VVLJQPHQW 06 7KHVLV 'HSDUWPHQW RI 0HFKDQLFV 5R\DO ,QVWLWXWH RI 7HFKQRORJ\ 6WRFNKROP 6ZHGHQ :KLWH & : DQG 0D\WXP % f f(LJHQVROXWLRQ 6HQVLWLYLW\ WR 3DUDPHWULF 0RGHO 3HUWXUEDWLRQf 6KRFN DQG 9LEUDWLRQ %XOOHWLQ 9RO 3DW SS =LPPHUPDQ & DQG .DRXN 0 Df f(LJHQVWUXFWXUH $VVLJQPHQW $SSURDFK IRU 6WUXFWXUDO 'DPDJH 'HWHFWLRQf $,$$ -RXUQDO 9RO 1R SS =LPPHUPDQ & DQG .DRXN 0 Ef f6WUXFWXUDO 'DPDJH 'HWHFWLRQ 8VLQJ D 6XEVSDFH 5RWDWLRQ $OJRULWKPf 3URFHHGLQJV RI WKH UG $,$$ 6WUXFWXUHV 6WUXFWXUDO '\QDPLFV DQG 0DWHULDOV &RQIHUHQFH 'DOODV 7; SS =LPPHUPDQ & DQG 6PLWK 6 : f f0RGHO 5HILQHPHQW DQG 'DPDJH /RFDWLRQ IRU ,QWHOOLJHQW 6WUXFWXUHVf FKDSWHU LQ ,QWHOOLJHQW 6WUXFWXUDO 6\VWHPV +6 7]RX HGLWRUf .OXZHU $FDGHPLF 3XEOLVKHUV $PVWHUGDP 7KH 1HWKHUODQGV =LPPHUPDQ & DQG :LGHQJUHQ : f f(TXLYDOHQFH 5HODWLRQV IRU 0RGHO &RUUHFWLRQ RI 1RQSURSRUWLRQDOO\ 'DPSHG /LQHDU 6\VWHPVf 3URFHHGLQJV RI WKH 6HYHQWK 93,t68 6\PSRVLXP RQ WKH '\QDPLFV DQG &RQWURO RI /DUJH 6WUXFWXUHV %ODFNVEXUJ 9$ SS =LPPHUPDQ '& DQG :LGHQJUHQ : f f0RGHO &RUUHFWLRQ 8VLQJ D 6\PPHWULF (LJHQVWUXFWXUH $VVLJQPHQW 7HFKQLTXHf $,$$ -RXUQDO 9RO 1R SS

PAGE 181

%,2*5$3+,&$/ 6.(7&+ 0RKDPHG .DRXN UHFHLYHG D %DFKHORU RI 6FLHQFH LQ DHURVSDFH HQJLQHHULQJ IURP WKH 'HSDUWPHQW RI $HURVSDFH (QJLQHHULQJ 0HFKDQLFV DQG (QJLQHHULQJ 6FLHQFH DW WKH 8QLYHUVLW\ RI )ORULGD LQ 'HFHPEHU RI )URP WKH VDPH GHSDUWPHQW KH WKHQ UHFHLYHG D 0DVWHU RI 6FLHQFH LQ DHURVSDFH HQJLQHHULQJ LQ $XJXVW RI

PAGE 182

, FHUWLI\ WKDW KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWDEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ 'DYLG & 6PPHUPDQ &KDLU $VVRFLDWH 3URIHVVRU RI $HURVSDFH (QJLQHHULQJ 0HFKDQLFV DQG (QJLQHHULQJ 6FLHQFH FHUWLI\ WKDW KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWDEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ /DZUHQFH ( 0DOYHUQ 3URIHVVRU (PHULWXV RI $HURVSDFH (QJLQHHULQJ 0HFKDQLFV DQG (QJLQHHULQJ 6FLHQFH FHUWLI\ WKDW KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWDEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ %KDYDQL 6DQNDU $VVRFLDWH 3URIHVVRU RI $HURVSDFH (QJLQHHULQJ 0HFKDQLFV DQG (QJLQHHULQJ 6FLHQFH FHUWLI\ WKDW KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWDEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ 6 1RUPDQ )LW]&R\ $VVLVWDQW 3URIHVVRU RI $HURVSDFH (QJLQHHULQJ 0HFKDQLFV DQG (QJLQHHULQJ 6FLHQFH

PAGE 183

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

PAGE 184

81,9(56,7< 2) )/25,'$