Vibration testing by design

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Vibration testing by design excitation and sensor placement using genetic algorithms
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xiii, 113 leaves : ill. ; 29 cm.
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English
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Larson, Cinnamon Buckels
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Subjects / Keywords:
Structural analysis (Engineering) -- Testing   ( lcsh )
Modal analysis -- Research   ( lcsh )
Aerospace Engineering, Mechanics and Engineering Science thesis, Ph. D
Dissertations, Academic -- Aerospace Engineering, Mechanics and Engineering Science -- UF
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bibliography   ( marcgt )
non-fiction   ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1996.
Bibliography:
Includes bibliographical references (leaves 108-112).
Statement of Responsibility:
by Cinnamon Buckels Larson.
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Typescript.
General Note:
Vita.

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University of Florida
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VIBRATION TESTING BY DESIGN:
EXCITATION AND SENSOR PLACEMENT
USING GENETIC ALGORITHMS














By

CINNAMON BUCKELS LARSON


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


1996


































To Jim and Audrey











ACKNOWLEDGEMENTS


I would like to sincerely thank my advisor, Dr. David Zimmerman, for all of the support,

advice, and knowledge he has given to me. He has worked hard on my behalf, obtaining

funding for my support and providing me with several research opportunities. He gave me

the guidance to learn and the room to grow. I will forever be indebted to him.

I would like to thank my husband, Jim, his parents, and my daughter, Audrey. Without

their love and support I never would have made it. I would like to thank my sisters, Beth,

Kim, Cynthia, and Erin, my brother Laing, and my best friend Leslie for their never-ending

encouragement and love. I am truly blessed.

I would like thank my dear friends Mohamed Kaouk and William Leath for their advice,

support, and companionship through graduate school. I would also like to thank the entire

staff and faculty of the Aerospace Engineering, Mechanics, and Engineering Science

department. Specifically, I would like to acknowledge my committee, Dr. Norman Fitz-Coy,

Dr. Daniel Drucker, Dr. Marc Hoit, Dr. Peter Ifju, and Dr. Bavani Sankar for their advice.

I would like to acknowledge Sandia National Laboratories for their financial support

and for the research opportunities they have given me. Specifically, I would like to thank Ed

Marek, Clay Fulcher, and Scott Klenke. I would also like to acknowledge General Motors

for providing data for my studies.

I would like to thank the Florida/NASA Space Grant Consortium whose financial

support made my graduate studies possible.













TABLE OF CONTENTS

page

ACKNOWLEDGEMENTS ........................................... iii

LIST OF TABLES ................................................ vii

LIST OF FIGURES ............................................... viii

KEY TO ABBREVIATIONS ......................................... xi

ABSTRACT ....................................................... xii

CHAPTERS


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

1.1 Finite Element Model Refinement ............................. 2
1.2 Modal Testing: Sensor and Actuator Placement .................. 5
1.3 Current Study Objective .................................... 7

2 GENETIC ALGORITHMS: THEORY AND APPLICATION .......... 9

3 FINITE ELEMENT MODEL REFINEMENT USING
GENETIC ALGORITHMS ..................................... 14

3.1 Introduction ........................................... 14
3.2 Model Refinement Problem Formulation ....................... 14
3.3 Genetic Algorithm Application ............................... 15
3.4 Numerical Example: Six Bay Truss FEM ...................... 17
3.4.1 M odel Refinement .................................... 18
3.4.2 Effect of Noise .................. ..................... 20
3.5 Conclusions ........................................... 22

4 MODAL TEST EXCITATION AND SENSOR PLACEMENT:
CURRENT TECHNIQUES ..................................... 25








4.1 Introduction ......................................

4.2 Effective Independence .............................

4.3 Kinetic Energy ....................................

4.4 Eigenvector Product ..............................

4.5 Driving Point Residue ..............................

5 MODAL TEST EXCITATION AND SENSOR PLACEMENT:
NEW TECHNIQUES ..................................


5.1 Introduction ................

5.2 Mode Indicator Function ........

5.2.1 Excitation Placement ......

5.2.2 Sensor Placement ........

5.3 Observability and Controllability

5.3.1 Excitation Placement ......

5.3.2 Sensor Placement .........


6 PRE-MODAL TEST PLANNING ALGORITHM APPLICATION:
NASA EIGHT-BAY TRUSS ..............................


.... 49


6.1 Introduction ......................

6.2 NASA Eight-Bay Test Bed ..........

6.2.1 Excitation Placement ..........

6.2.2 Sensor Placement .............

6.2.3 Results: Random Sensor Location

6.3 Computational Efficiency ...........

6.4 Conclusion .....................


7 PRE-MODAL TEST PLANNING ALGORITHM APPLICATION:
MICRO-PRECISION INTERFEROMETER TRUSS .................

7.1 Introduction ................... ....................

7.2 Micro-Precision Interferometer Test Bed .......................

7.3 Excitation Placement ......................................

7.4 Sensor Placement .........................................

7.4.1 Unconstrained Sensor Placement ........................


....... 25

....... 25

....... 29

....... 31

....... 3 1



....... 35








7.4.2 Triaxially Constrained Sensor Placement .................. 76
7.4.3 Unconstrained vs. Triaxially-Constrained Sensor Sets ........ 80
7.5 Effect of Model Error ....................................... 80
7.5.1 Excitation Placement with Model Error ................... 82
7.5.2 Sensor Placement with Model Error ...................... 83
7.6 Computational Cost ....................................... 88
7.7 Conclusions ............................... ............. 89

8 PRE-MODAL TEST PLANNING APPLICATION:
CAR BODY ................................... ............ 5 95

8.1 Introduction ..................................... ....... 95
8.2 Excitation Placement ....................................... 95
8.3 Sensor Placement ................... ........ ........... 102

9 CONCLUSIONS AND FUTURE WORK .......................... 105

REFERENCES ....................................... ........... 108

BIOGRAPHICAL SKETCH .......................................... 112













LIST OF TABLES


Table pa

5.1 GMIF Design Variable Description .................................. 38

6.1 Eight-Bay Truss Frequencies and Mode Description .................... 50

6.2 Percent Difference in FE and Identified Frequencies .................... 57

6.3 Total Floating Point Calculations for Each Placement Technique .......... 60

7.1 Reduced MPI FEM Frequencies Compared with MPI
Modal Test Frequencies .................................... ...... 64

7.2 Original, GMIF, and GCON Excitation Locations MIF Values ............ 66

7.3 Difference Between Pre- and Post-Corrupted Model
Frequencies and Mode Shapes ..................................... 82

7.4 Number of Sensor or Triax Sets That Change
W hen M odel Error Is Added ...................................... 87

7.5 Floating Point Calculations for MPI Sensor and Excitation Placement ...... 89

8.1 Car Body Excitation Location and Orientation ........................ 96

8.2 Mode Indicator Function Values for Various Excitation Placements ........ 97

8.3 Controllability Angles (in degrees) for Excitation Placements ............. 101

8.4 Excitation Placement Techniques Floating Point Calculations ............. 101

8.5 Triaxial Sensor Placement Techniques Floating Point Calculations ......... 104













LIST OF FIGURES


Figure page

1.1 Finite Element Modeling ........................................ 2

1.2 Finite Element Model Refinement ................................. 3

1.3 M odal Testing ........................................... ..... 6

2.1 Genetic Algorithms as Robust Problem Solvers ....................... 9

2.2 Coding of a Four Design Variable Problem .......................... 11

2.3 Cross-Over Examples ........................................... 12

2.4 Genetic Algorithm Flow Chart .................................... 13

3.1 Six Bay Truss with 25 DOFs ..................................... 17

3.2 Generational Data, Measured Modes with No Noise ................... 19

3.3 Generational Eigensolution Data, Measured Modes with No Noise ....... 21

3.4 FRF after 0, 5, 10, and 20 Generations, Measured Modes with No Noise ... 21

3.5 Generational Data, Measured Modes with 15% Noise .................. 23

3.6 Generational Eigensolution Data, Measured Modes with 15% Noise ...... 23

3.7 FRF After 20 Generations, Measured Modes with 15% Noise ........... 24

4.1 Typical Driving Point Residue (NASA 8-Bay Truss) ................... 34

5.1 Typical M IF Plot ................. ...... .... ...... ............ 37

5.2 Excitation Selection by GM IF ..................................... 38

5.3 State Space Variable Description .................................. 42

5.4 Controllability and Observability ................................. 44

6.1 NASA 8-Bay Truss ............................................. 50

6.2 Eight-Bay Excitation Locations ........................... ........ 51








6.3 Eight-Bay Excitation Locations Frequency Response
of Time Domain Data ............................................. 53

6.4 Eight-Bay Excitation Placement Cross-Orthogonality of
Identified and FEM Modes 1 to 5 .................................. 53

6.5 Eight-Bay Sensor Locations ...................................... 55

6.6 Eight-Bay Sensor Placement Cross-Orthogonality of
Identified and FEM Modes 1 to 5 ......................... ......... 56

6.7 Eight-Bay Cross-Orthogonalities of Five Techniques Compared to 300
Random Sensor Sets ................. ...... ....... .... ....... ... 59

7.1 MPI Structure .................... .......... .... ............. 62

7.2 Excitation Placement on MPI Structure ............................. 63

7.3 Typical Frequency Response for MPI Structure ....................... 65

7.4 Comparison of Selected Excitation and Random Excitation MIF Values ... 68

7.5 Comparison of Selected Excitation and Random Excitation
Controllability Angles .......................................... 69

7.6 Cross-Orthogonality Between FE Modes and Identified Modes .......... 70

7.7 Unconstrained MPI Sensor Sets ................................... 72

7.8 Cross-Orthogonality Between MPI FE and Identified Modes,
18 Unconstrained Sensors ...................................... 74

7.9 Triaxially Constrained MPI Sensor Sets ................ ....... 77

7.10 Cross-Orthogonality Between MPI FE and Identified Modes,
6 Triaxially Constrained Sensors .................................. 78

7.11 Model Error Added to MPI FEM ................................. 81

7.12 True vs. Corrupted MPI Mode Shapes .............. ............. .. 81

7.13 GMIF Derived Excitation Locations ................................ 83

7.14 GCON Derived Excitation Locations ............................... 84

7.15 Model Error Effect on Unconstrained MPI Sensor Sets ................. 85

7.16 Model Error Effect on Triaxially Constrained MPI Sensor Sets ........... 86

7.17 Cross-Orthogonality Between MPI Identified Modes Using Corrupted Model
and Uncorrupted FEM Modes (18 Unconstrained Sensors) .............. 91








7.18 Cross-Orthogonality Between MPI Identified Modes Using Corrupted
Model and Uncorrupted FEM Modes (6 Triaxially Constrained Sensors) ... 93

8.1 Car Body Shaker Locations ....................................... 96

8.2 MIF Values for Excitation Placements Compared to 500 Randomly
Located 3-Point Excitations ...................................... 99

8.3 Controllability Values for Excitation Placements Compared to 500
Randomly Located 3-Point Excitations .................. .......... 100

8.4 Triaxially Located Car Body Sensor Sets ............... ............. 102














KEY TO ABBREVIATIONS


AKE .....

ARS ......

DOF ......

DPR ......

El ........

ERA ......

EVP .....

FEM .....

FRF ......

GA .......

GCON ....

GMIF .....

GMRA ....

KE .....

MIF ......

MPI ......

OBS ......


average kinetic energy

average random sample

degree of freedom

driving-point residue

effective independence

eigensystem realization algorithm

eigenvector product

finite element model

frequency response function

genetic algorithm

genetic controllability algorithm

genetic mode indicator function algorithm

genetic model refinement algorithm

kinetic energy

mode indicator function

micro-precision interferometer

observability












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

VIBRATION TESTING BY DESIGN:
EXCITATION AND SENSOR PLACEMENT
USING GENETIC ALGORITHMS

By

Cinnamon Buckels Larson

May 1996


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

This dissertation is an investigation of the use of genetic algorithms for the purposes of

finite element model refinement and pre-modal test planning. The objective of a model

refinement technique is to use information, about a structure, obtained during a vibration test

to update the analytical model. The product of this process is an updated model of a structure

which possesses dynamic properties closer to the dynamics obtained from the modal test of

the structure. A genetic algorithm is used to vary finite element structural parameters to

obtain an updated model with measured modal properties.

Although one purpose of a modal test is to use the information to update finite element

models, the information obtained may be used for other purposes such as damage

assessment, critical loads and frequency determination, and vibration control design. The

type of information to be realized from a vibration test may well govern how and where the

structure is excited and observed. The principal purpose of the current work is to explore the

subject of pre-modal test planning for excitation and sensor placement. An overview of

several existing sensor and excitation placement techniques is presented as a platform for the

current study. The sensor placement techniques include effective independence, kinetic








energy, and eigenvector product and the excitation placement techniques include

eigenvector product, kinetic energy, and driving-point residue. Two new sensor and two new

excitation placement techniques are developed using normal mode indicator functions, and

the concept of modal controllability and observability along with genetic algorithms. The

new and existing techniques are compared using three finite element models: the NASA

eight-bay truss, the Jet Propulsion Laboratory Micro-Precision Interferometer test bed, and a

car body.














CHAPTER 1
INTRODUCTION


The area of structural engineering encompasses the design, manufacture, and test of a

wide variety of systems. These basic steps are all used in the design of any structure, whether

it is a household appliance, an automobile, a bridge, an aircraft, or a spacecraft. In the past,

systems were over-engineered and over-built resulting in an increase in the time and material

required to build them. Often times the steps in the engineering process were repeated

several times until the designed system performed satisfactorily. With the advent of the

computer, tools have been and are continuing to be developed which enable the structural

engineer to improve on each step of the engineering process. These tools not only help to

limit the time, material, and cost that it takes to complete the engineering steps but they also

help to limit the repetition of these steps.

Structural computer modelling and vibration testing are two tools that have been

developed with the aid of computers. Knowledge about material properties and structural

dynamics may be used to create computer models of a system, which in turn may be used to

predict dynamic performance and limitations. Once the system has been built, modal testing

may be used to gain a greater insight into the dynamics of the structure and to update the

computer model. An updated computer model may be used as a health monitoring tool for

the structure after it goes into use.








1.1 Finite Element Model Refinement


One of the most common modelling techniques is finite element modelling (FEM)

which can be used to represent the continuous medium of a structure as a connection of finite

elements. This enables a system with distributed mass, damping, and stiffness properties to

be represented as a lumped parameter system with discrete mass damping and stiffness

properties (Figure 1.1). In other words, an infinite degree of freedom system is represented

as a finite degree of freedom system.


continuous -
medium


I I
FINITE ELEMENT MODEL


Figure 1.1 Finite Element Modeling


A wide variety of finite element modelling software is available to the designer. The basic

steps involved in the modelling process are as follows:

1. Divide continuum into a finite number of elements.

2. Select node points where equilibrium conditions are enforced.

3. Determine element types and properties.

elemental type (rod, plate, etc.) and location

elemental displacement, stiffness, stress-strain, node-point lodes

4. Assemble elemental matrices (mass,stiffness, and damping).


5. Develop equilibrium equations for node point location.


lumped parameters




\__________








6. Create global mass damping and stiffness matrices from which frequencies and

mode shapes are calculated.

The resulting FEM may be used to evaluate the efficacy of a design before it is built.

Critical loads, resonant frequencies, and mode shapes may be predicted using the FEM and

appropriate changes to the design based on these values may be made. Once the structure has

been built, the FEM may be used to predict the performance of the structure under working

conditions as well as serve as a damage assessment tool. However, while the original FEM is

a predicted representation of a particular structure, the dynamic performance of the FEM

very rarely matches the performance of the as-built structure. In order to correlate the

dynamics of the structure to those of the FEM, the model must undergo a refinement process.

The basic steps of FEM refinement are shown in Figure 1.2. The dynamic properties of the

FEM are compared to the dynamic properties extracted from a vibration test of the structure.

The resulting information is used to refine the model so that the modal properties of the FEM

agree with modal properties from the vibration test.


Figure 1.2 Finite Element Model Refinement








Some of the earliest work done in model refinement was proposed by Rodden (1967),

who explored using modal test data to generate analytical mass and stiffness properties of the

structure being tested. The early work of Rodden has broadened into the modern FEM

refinement techniques. Algorithms used to address the FEM refinement can be broadly

classified as falling into one of four different approaches: optimal-matrix updates, sensitivity

methods, eigenstructure assignment techniques, and minimum-rank perturbation methods.

Survey papers providing an overview of these techniques are provided in papers by Ibrahim

and Saafan, 1987; Heylen and Sas, 1987; and Zimmerman and Kaouk, 1992.

In the optimal matrix update formulation, perturbation matrices for the mass, stiffness,

and/or damping matrices are determined which minimize a given cost function subject to

various constraints. Typical constraints may include satisfaction of the eigenproblem for all

measured modes, definiteness of the updated property matrices and preservation of the

original sparsity pattern of the property matrices. Baruch and Bar Itzhack (1978) worked on

an optimal update of the global stiffness matrix with a cost function that minimized the

Frobenius norm of the perturbation matrix. Their work was expanded to look at updating

mass, damping, and stiffness matrices (Berman and Nagy, 1983; Fuh et al., 1984; Hanagudet

al., 1984). Kabe (1985) and Kammer (1987) expanded on this work further by looking at

matrix updating while preserving the sparsity pattern of the original FE global matrices.

Sensitivity methods for model refinement make use of sensitivity derivatives of modal

parameters with respect to physical design variables (Martinez et al., 1991) or with respect to

matrix element variables (Matzen, 1987). When varying physical parameters, the updated

model is consistent within the original FE program framework. A variety of derivatives and

optimization techniques have been used (Collins et al., 1974; Chen and Garba, 1980;

Adelman and Haftka, 1986; Creamer and Hendricks, 1987; Flanigan, 1991). In the current

work a physical parameter update technique using a genetic algorithm is developed.

Inman and Minas (1990) proposed designing pseudo-controllers to be applied to the

FEM in an iterative fashion resulting in a match between measured and FE modal properties.








These controllers were then translated into matrix updates. These techniques, known as

control-based eigenstructure assignment techniques, are based on work done in

eigenstructure control (Andry et al., 1983). Zimmerman and Widengren (1990) proposed a

non-iterative eigenstructure assignment formulation using an algebraic Riccati equation.

Finally, the development of a minimum rank update theory has been recently proposed

as a computationally attractive approach for model refinement and damage detection

(Zimmerman and Kaouk, 1992). The update to each property matrix is of minimum rank and

is equal to the number of experimentally measured modes which the modified model is to

match.



1.2 Modal Testing: Sensor and Actuator Placement


Regardless of the method used to perform FEM refinement, a modal test must be

performed on a structure or its components in order to obtain the experimental information to

correlate with the analytical information contained in the FEM. Finite element model

refinement is only one use for modal data. Modal analysis is also a tool for damage

assessment and force reconstruction. Several issues may govern the use of modal tests for

these purposes. The final use of modal test data governs the pretest planning associated with

modal testing. The placement of actuators for structural excitation purposes and the

placement of sensors for structural response observations may well depend on whether the

data will be used for modal parameter estimation, mode orthogonality for FEM correlation,

identification of uncertain parameters in FEMs, structural health monitoring, or force

reconstruction.

The science of modal testing is thoroughly discussed in D.J Ewins' book Modal Testing:

Theory and Practice. The basic steps involved in a modal test are discussed here for

completeness and are pictured in Figure 1.3. The structure being tested must be dynamically

excited and the response of the structure to this input must be measured. The excitation may

be accomplished using an impact hammer, a shaker, or a release from an initial structural








displacement. The response of the structure is generally measured using piezoelectric

accelerometers, which are mounted on the structure in various locations. The force and

response signals are sent to a processor or analyzer, after being filtered and amplified, from

which a frequency response function (FRF) of the structure is obtained. The modal

properties (mode shapes, damping ratios, and frequencies) may be obtained from the

analysis of the FRF.


FORCE TRANSDUCER



shaker impact hammer

conditioning
amplifiers






frequencies
structure mode shapes

RESPONSE TRANSDUCER:

signal processor
triax single DOF


Figure 1.3 Modal Testing


Modal testing of a structure can be a costly venture in terms of time and money.

Pre-modal test planning can be essential in saving cost by determining ahead of time the

appropriate transducers and analyzers to use for the job at hand. Another aspect of pre-test

planning, the one associated with the current study, is the optimal selection of the location of

the force and response transducers. It is desired to obtain the most information at the least

cost, which involves minimizing the number of transducers to be used. The selection of the








transducers will also depend on what the modal data will be used for (i.e., model refinement

or damage detection).

A majority of the sensor placement research that has been done may be broadly

classified into two areas, system identification and optimal control. Yu and Seinfield (1973),

Le Pourhiet and Le Letty (1978), Omatu et al. (1978), Sawaragi et al. (1978), and Qureshi et

al. (1980) have all done work in the area of sensor placement for system identification. Shah

and Udwadia (1978) and Udwadia and Garba (1985) have done work in sensor placement for

structural parametric identification. Kammer (1991) approached the system identification

sensor placement problem for the purpose of FEM validation. Goodson and Polis (1978)

researched the selection of sensors for optimal structural control. Juang and Rodriguez

(1979) looked at sensor placement for identification and control purposes.

Information about the FEM has been used by modal test designers to place sensors and

actuators on a structure for the purpose of modal testing. Finite element DOFs which have

high kinetic energy are good choices for sensor or actuator placement because more

information may be extracted about and more energy may be input to the structure at these

points (Kammer, 1991; Flanigan and Hunt, 1993; Lim, 1991). Jarvis (1991) proposes using

FE mode shape products to find sensor and actuator locations for modal testing. Kientzy et

al. (1989) used driving point residues or modal participation factors to determine modal test

excitation locations. Mode indicator functions have been used by the modal test engineers to

tune modes during a modal test (Hunt et al., 1984).



1.3 Current Study Objective



Proper modal test planning is needed in order to obtain the largest amount of

information about a structure relative to the task of the data at the smallest cost. The

refinement of finite element models is a task which benefits from modal test planning. The

objective of the current studies is to explore the areas of finite element model refinement and








pre-modal test planning. Specifically, the use of the optimization technique, genetic

algorithms (GAs), in these two areas will be examined.

The genetic algorithm is an optimization tool which has been developed in the past 20

years. An overview of the theory and applications of the GA is given in Chapter 2. A

structural-parameter update model-refinement algorithm is developed using a genetic

algorithm in Chapter 3 and is applied using a FEM of a two-dimensional truss structure.

The topics of pre-modal test planning actuator and sensor placement are examined in

Chapters 4 and 5. In Chapter 4 several techniques which have been developed in existing

literature are outlined. The excitation placement techniques of kinetic energy eigenvector

product, and driving point residue and the sensor placement techniques of effective

independence, kinetic energy, and eigenvector product are reviewed. In an effort to improve

on the current sensor and actuator placement technologies, two new actuator and sensor

placement techniques are developed in Chapter 5. The normal mode indicator function

(MIF) of a FEM is used with a GA to optimally find excitation locations. In addition the MIF

is used as a tool to locate sensors. The second sensor and actuator placement algorithms use a

degree of controllability and observability calculated using the FEM information.

The effectiveness of the current sensor and actuator placement techniques and those

developed in Chapter 5 are explored using several structural FEMs in Chapters 6,7, and 8. In

Chapter 6 NASA's Eight-Bay truss, in Chapter 7 the micro-precision interferometer truss,

and in Chapter 8 a FEM of a General Motors car body are used as examples to explore the

effectiveness of all of the sensor and actuator placement algorithms for the purpose of

pre-modal test planning. Concluding remarks and a discussion of future work in the areas

outlined above are given in Chapter 9.














CHAPTER 2
GENETIC ALGORITHMS:
THEORY AND APPLICATION


The motivation behind the development of GAs is that they are robust problem solvers

for a wide class of problems, as depicted in Figure 2.1. However, it should be noted that they

are not as efficient as nonlinear optimization techniques over the class of problems which are

ideally suited for nonlinear optimization; namely continuous design variables with a

continuous differentiable unimodal design space. Genetic Algorithms have the capability to

solve continuous, discrete and a combination of continuous and discrete optimization

problems.

Nonlinear Optimization

Genetic Algorithms




Random Walk

Problem Class

Figure 2.1 Genetic Algorithms as Robust Problem Solvers

Genetic algorithms are an optimization method which is based on Charles Darwin's

survival of the fittest theories (Holland, 1975). The basic concept of the GA is that a

population of designs is allowed to evolve over a period of time. The most fit members of

that population are most likely to survive thus enabling their genetic code (or design

information) to be passed down to future generations. More than just the information

contained in the initial population may be passed down to future generations. As in nature as

the evolutionary process progresses, mutations may occur in the offspring which may or may








not result in more fit population members. Ideally, this evolutionary process will result in a

population of members more fit than the original initial population. Genetic Algorithms may

therefore be described as a directed random search or as a compromise between determinism

and chance.

Genetic algorithms are radically different from the more traditional design optimization

techniques. Genetic algorithms work with a coding of the design variables, as opposed to

working with the design variables directly. The search is conducted from a population of

designs (i.e., from a large number of points in the design space), unlike the traditional

algorithms which search from a single design point. The GA requires only objective

function information, as opposed to gradient or other auxiliary information, which is usually

required in other optimization techniques. The GA is based on probabilistic transition rules,

as opposed to deterministic rules. There are five main operations in a basic GA: coding,

evaluation, selection, crossover and mutation.

Coding is the process in which each design variable is coded as a q-bit binary number.

Discrete variables would each be assigned a unique binary string. A continuous design

variable Bi is approximated by 2q discrete numbers between lower and upper bounds for the

design variable,


B = Bn + inary# (B B n) (2.1)
i imin 2q 1 imax imin


where Bimin and Bimax are the lower and upper bounds on the ith continuous design variable

and binary# is an integer number between zero and 2q -1. The continuous to discrete coding

is like that of an analog to digital converter used in control systems. A population member

is obtained by concatenating all design variables to obtain a single string of ones and zeros.

Thus, a population member contains all information to completely specify the total design.

For example, consider a design which has three continuous variables B1, B2 and B3

represented by 5-bit, 6-bit, and 4-bit numbers and a discrete design variable B4 which can

take on four different values. An example of a population of members containing this design








information is pictured in Figure 2.2. A population is defined to be a grouping of npop

members, where npop is the number of members in the population.


Design Variables
5 bit 6 bit 4 bit 2 bit
population continuous continuous continuous discrete
member
member B1 B2 B3 B4

1 10111 010001 0011' 11

2 0 001 1 0 1 1 1 0 1 0 1 101 1 0





npop 1 0 1 0 1 10 00 00 11 0 1



Figure 2.2 Coding of a Four Design Variable Problem


Evaluation is the process of assigning a fitness measure to each member of the current

population. The fitness measure is typically chosen to be related to the objective function

which is to be minimized or maximized. No gradient or auxiliary information is used; only

the value of the fitness function is needed. Therefore, GAs are less likely than traditional

"hill climbing" algorithms to become "trapped" at a local minima or maxima. Additionally,

because no gradient information is required, the design space is allowed to be discontinuous.

Selection is the operation of choosing members of the current generation to produce the

prodigy of the next generation. Selection is biased toward the most fit members of the

population. Therefore, designs which are better as viewed from the fitness function, and

therefore the objective function, are more likely to be chosen as parents.

Crossover is the process in which design information is transferred to the prodigy from

the parents. Crossover amounts to a swapping of various strings of ones and zeroes between

the two parents to obtain two children. Two possible types of cross-over are illustrated in

Figure 2.3.








Point Cross-over Pattern Cross-over

parent 1 1 0 1000 1 1 010001
parent 2 1 1 1 0 1 0 0 1 10 0 1 0 0

swap: point swap x x x
pattern
child 1 0 10o 1 0 0 1 1 1 0 1 0 0
child 1 1 1:0 0 0 1 1 0 0 0 0 0 1


Figure 2.3 Cross-Over Examples



Mutation is a low probability random operation that may perturb the design represented

by the prodigy. The mutation operator is used to retain design information over the entire

domain of the design space during the evolutionary process.

Holland (1975) developed the concept of schema, which for a very simple GA

implementation explains why GAs work. Schema are a similarity template defined by O's,

l's and x's, where x's are the don't care symbol. Thus, for a design coded with a total of

8-bits, one schema would be 10xxxxxx. All designs which have a 1 in the most significant

bit and a 0 in the 2nd most significant bit would be said to contain this schema. Holland's

Schema Convergence Theorem states that under certain combinations of selection,

crossover, and mutation, the expected number of schema H at generation k+1, n(H,k+l), is

given as

fitH
n(H,k + 1) = (1 e) n(H, k) (2.2)
tavg

where E is a number much less than 1 and fitH and fitavg are the average fitness of all designs

containing schema H and of the population as a whole, respectively. Thus, if those designs

which contain schema H have on the whole a higher average fitness than the overall general

population, the expected number of schema H in the next generation will be greater than or

equal to the number of schema H in the current generation.







The proof is valid only for a specific combination of selection, crossover and mutation.
It should be noted that a general proof for more complex GAs has not been developed.
However, there exists a wide body of literature which demonstrates the power and capability
of advanced GAs (Schaffer 1989, Grefenstette 1987). A flow chart summarizing the GA
process is shown in Figure 2.4.



CODING
Create Initial Population

EVALUATION
Evalute Population Fitness
(objective function calculation)

SSELECTION
Apply Selection Criteria NO
(which members reproduce?)

-F
CROSSOVER and MUTATION evaluate
create new members stopping New Design
criteria


Figure 2.4 Genetic Algorithm Flow Chart














CHAPTER 3
FINITE ELEMENT MODEL REFINEMENT
USING GENETIC ALGORITHMS




3.1 Introduction

As discussed in Chapter 1, an important tool in the design of engineering structures is the

finite element model (FEM). Recall that FEM refinement techniques may be classified as

optimal matrix updates, sensitivity updates, eigenstructure assignment updates, and

minimum-rank perturbation updates. Sensitivity methods use sensitivity derivatives of

modal parameters with respect to physical design variables or with respect to matrix element

variables. These derivatives are used in order to determine what changes to make to the

physical parameters or elemental matrices of the FEM in order to obtain in a refined model

with measured modal properties. A GA-driven model refinement technique is developed to

update FE structural parameters to provide an updated model with the measured modal

characteristics. This model refinement technique is illustrated using a numerical example.


3.2 Model Refinement Problem Formulation

For a given undamped structure it is assumed that an n-DOF FEM is developed and

results in the second-order linear differential equation of motion,

Mx + Kx = 0 (3.2.1)

where M and K are the original analytic (nxn) mass and stiffness matrices, and x is the (nxl)

position vector. The over-dots represent differentiation with respect to time. The eigenvalue

problem associated with Eq. (3.2.1) can be written as








?2Mr + Kr = 0 (3.2.2)

where Xr is the rth eigenvalue and r is the rth mass orthogonal eigenvector of the original

analytical system. It is assumed that the original analytic model of Eq. (3.2.1) does not

satisfy the eigenvalues (Xmr) and the mass orthogonal eigenvectors (Omr) of the

experimentally measured system,

X2mrMm(r + Kmmr = 0 (3.2.3)

where Mm and Km are the experimentally derived mass and stiffness matrices of the

structure. Therefore, a discrepancy between the original analytic and measured modal

information will result in an eigenvalue problem of the form

2,[M + AM(p)]r + [K + AK(p)]!kmr = 0 (3.2.4)

where AM(p) and AK(p) are perturbation matrices which are functions of the structural

parameters vector p. These perturbation matrices represent the mismatch between the

original analytic mass and stiffness matrices and the experimentally derived mass and

stiffness matrices. In order to develop an updated analytical model which is in agreement

with measured modal data, a structural parameters vector p must be found which satisfies

Eq. (3.2.4) for all measured modes. In the following sections, a model refinement technique

is developed which employes the use of a GA to find the structural parameters vector which

will result in an updated FEM whose modal properties match the measured modal properties

of the as-built structure.



3.3 Genetic Algorithm Application

As explained in Chapter 1, a FEM is a lumped parameter representation of a continuous

structure. Information about the geometry and material of a structure are used to estimate its

properties and to create the FEM. Mass, density, Young's modulus, cross-sectional area, and

moment of inertia are some example of the properties which are used to develop a FEM.








Since these properties are estimated for the structure, they may be perturbed to give an

updated FEM with the same modal properties as experimentally measured from the true

structure. A Genetic Model Refinement Algorithm (GMRA) is developed which uses a GA

to search for updated structural parameters which will result in an updated FEM with

corresponding measured modal properties. An outline of the steps of GMRA, which follow

those for a GA given in Chapter 2, follows.

Coding. The design variables used in GMRA are continuous design variables which

represent the structural parameters to be changed. Limits may be set on the amount of

perturbation allowed for each design variable or structural parameter. This enables the user

to allow small perturbations to variables about which they are certain, such as cross-sectional

area or moment of inertia and larger perturbations to variables about which they are less

certain, such as density or Young's modulus.

Evaluation. The most fit members of a population are those which minimize a chosen

objective function. An objective function has been formulated which states, the most fit

structural parameters vector p is one which results in an updated analytical model which

gives the smallest value for the objective function


obj = xur -+ m1 Qu 1mr hr (3.3.1)
r= 1 r=1

where Xur and (ur are the rth eigenvalue and eigenvector of the updated analytical model.

The first summation of Eq. (3.3.1) provides for the minimization between measured

eigenvalues and updated analytic eigenvalues. As the updated eigenvalue approaches the

measured eigenvalue the first summation approaches zero. The absolute value of the

difference in each measured and updated eigenvalue is divided by the corresponding

measured eigenvalue to insure that each frequency contributes equally to the objective

function. The second summation of Eq. (3.3.1) is the 2-norm between the difference in

measured and updated eigenvectors and provides for the minimization between measured

and updated mode shapes. As the updated eigenvector approaches the measured eigenvector








the second summation approaches zero. The variables w, and hr are weights which can be

changed in order to emphasize agreement between specific measured and updated

eigenvalues/eigenvectors. By changing the weights, emphasis can be placed either on

updated eigenvalue or updated eigenvector agreement with measured data.




3.4 Numerical Example: Six Bay Truss FEM



A finite element model of a six bay truss with 25-DOFs was developed to test GMRA. A

picture of the truss is given in Figure 3.1.

A= llcm2
E = 7.03x109 kg/m2
p = 2685 kg/m3
l = 0.75 m



Figure 3.1 Six Bay Truss With 25 DOFs


In Figure 3.1, A is the cross sectional area, E is the modulus of elasticity, p is the density

of all of the members, and I is the length of each bay. It is assumed that the analytic model is

incorrect and needs to be changed in order to facilitate agreement in analytic and measured

modal properties. To obtain experimental modal information, the FEM of the six bay truss is

altered and the "experimental" modal information is calculated. It is assumed that the

dimension of the measured eigenvector is the same as that of the analytic eigenvector. This

can be accomplished using an eigenvector expansion algorithm (Berman and Nagy, 1971;

Smith and Beatie,1990; Zimmerman and Smith, 1992). An alternate formulation would be

to let the vector norm calculation of Eq. (3.3.1) take place over only those components of the

eigenvector that are measured, therefore, eliminating any error that may be introduced by

expanding the measured eigenvectors.








3.4.1 Model Refinement

As a first example, it is assumed that all of the structural parameters are known to be

correct except for E, the modulus of elasticity. In addition, the properties of all of the

diagonal members, all of the horizontal members, and all of the vertical members are linked.

GMRA is used to find the updated structural parameters vector, Pu, which minimizes Eq.

(3.3.1),

u = {Ed Eh Ev} (3.4.1)

The components of pu (Ed, Eh, and Ev ) are the moduli of elasticity of the diagonal,

horizontal, and vertical truss members. The original analytic model has the same modulus

of elasticity for all members, which will be referred to as Enom. The resulting analytical

parameters vector is


PA = Enom Enom Enom} (3.4.2)

The value for Enom is 7.03x109 kg/m2. The structural parameters vector which is used to

generate the experimentally measured model is


pm = {0.95Enom 0.90Enom 0.92Enom} (3.4.3)

The genetic algorithm is instructed to search for three design variables (components of

the structural parameters vector) which are in the range of 0.7Enom to 1.3Enom. An initial

population representing the horizontal, vertical, and diagonal moduli of elasticity is

randomly generated within the limits of 0.7Enom to 1.3Enom. A member is added to this

initial population which represents the original analytic model parameters vector. Since the

original population is randomly generated, there is a good possibility that one of those initial

population members will be more fit than the member representing the original analytic

model. Therefore, the improvement in the updated eigensolution would in part be due to a

random search. Even though this random chance is a benefit of genetics, in order to show the

true improvement to the original analytic model by the genetic algorithm, an initial





19



population is chosen with all members less fit than the member representing the original

analytic model. In order to facilitate frequency matching, the weights of Eq. (3.3.1) are set to

emphasize minimization of the eigenvalue portion of the cost function. Five measured

modes are supplied and GMRA is instructed to run for twenty generations (Figure 3.2).

Minimum and Average Objective Functions Diagonal Truss Members

0 1.2
o0
S0.8


0
-^ | 0.8-
0--------- ------------------ I~ 0.
0 10 20 0 10 20
Generation Generation
Horizontal Truss Members Vertical Truss Members

S 1.2 1.2-
o 0

0 0

0.8- 0.8

0 10 20 0 10 20
Generation Generation

Figure 3.2 Generational Data, Measured Modes with No Noise


The top left graph of Figure 3.2 shows the average fitness (dashed line) and maximum

fitness (solid line) of the current population at each generation. As the generations increase

the average and maximum fitness improves, which corresponds to a decrease in the value of

the objective function. After the first generation there is substantial improvement in the

updated objective function over the original analytic objective function value. Since the

members of the population which are randomly generated are all less fit than the original

analytical model member, the decrease in objective function over the first generation is due

to the children of the initial population. Since it is the goal to minimize the objective

function, this is a desirable trend. Also as the generation increases, the average fitness

approaches the maximum fitness. This is due to the fact that as generations evolve, the








overall population tends toward the most fit member. After a certain number of generations,

the diversity in the population members decreases.

The other three graphs of Figure 3.2 show how the actual design variables modulii of

elasticity) are varying over the generations. The straight lines on these graphs correspond to

the "experimental" diagonal, horizontal, and vertical moduli of elasticity which were used to

generate the measured modes. It is seen that within the first few generations, the parameters

have quickly converged near their "experimental" values. In later generations it is seen that

Ev, the vertical member's elastic modulus, varies widely. Physically, this is due to the fact the

lower modes are fairly insensitive to the stiffness of the vertical members.

The generational trends of the eigensolution of the updated model are shown in Figure

3.3. The top left graph is the norm of the difference in the measured and updated

eigenvectors. It can be seen that over some of the generations the value of this norm increases

instead of decreasing. This is due to the fact that the eigenvalue portion of the cost function is

weighted more heavily in order to insure modal frequency agreement. It can be seen that an

increase in the norm in Figure 3.3 corresponds to the overall updated eigenvalues moving

closer to the measured eigenvalues resulting in a decrease in the value of the objective

function. A comparison of the first three frequencies of the updated model with respect to the

measured frequencies over 20 generations is shown in the other three graphs of Figure 3.3.

The FRFs of the pre-genetics model and the FRFs of the post-genetic models after 0, 5,

10, and 20 generations are pictured in Figure 3.4. An immediate improvement in the FRF of

the system can be seen after five generations.


3.4.2 Effect of Noise


The example presented would be expected to behave differently when there is noise

present in the measured modal information. To simulate noise in the measured data, 5% and

15% random noise were added to the measured eigenvectors. It is assumed that the modal

frequencies are measured accurately. As in the case with no noise, an initial population is










Norm (Measured-Updated) Eigenvectors


3 -

2-




0
0 10 20 30
Generation


ona Frequency: Measured(..) Upoate

67

66

65

64

63


rst Frequency: Measured(..) Updated(-)


Thi


124

122

120

118


1 10 20
Generation

rd Frequency: Measured(..) Updatec


0 10 20 0 10 20
Generation Generation

Figure 3.3 Generational Eigensolution Data, Measured Modes with No Noise


10-5




. 10-8
o



10-11


Frequency (Hz)


10-s




10-~

ai


0 100 200 34
Frequency (Hz)

After 20 Generations


0 100 200 300 0 100 200 300
Frequency (Hz) Frequency (Hz)

(...) Updated Model (-) Measured Model


Figure 3.4 FRF After 0, 5, 10, and 20 Generations, Measured Modes with No Noise


10-5




0 10-8
g-io-


10-5




S10-8

Cr:


d(-)








generated with one member which represents the original analytic model and with other

members randomly generated which have cost function values greater than that for the

original analytic model member. This is to show how the cost function is minimized due to

genetics and not just due to a random selection of a more fit design. GMRA was run using 5%

and 15% noise with 5 measured modes. The generational results for 15% noise are pictured

in Figures 3.5 and 3.6, and the FRF is pictured in Figure 3.7. Graphically a similar trend was

observed for 5% noise.

An immediate improvement in the cost function can be seen after 5 generations, as

shown in the top left graph of Figure 3.5. The generational updated eigensolution data of

Figure 3.6 shows a similar trend to that of the example with no noise. The norm of the

difference in measured eigenvectors with 15% noise and the updated eigenvectors is given in

the top left graph of Figure 3.6. The straight dotted line in this figure corresponds to the norm

of the difference in measured eigenvectors with and without noise.

The FRF pictured in Figure 3.7 shows how the model which was updated using noisy

data compares with measured model without noise and with the original analytic model.

Because the cost function was weighted to be more heavily affected by the frequencies, the

effect that noisy modes may have had on the update was minimized.



3.5 Conclusions


One of the benefits of using a GA to search for an updated parameters vector is that the

search is conducted from several points in the design space whereas conventional gradient

sensitivity methods conduct the search from a single point. This helps enable GMRA to

avoid getting stuck in a local minimum in addition to completing the search faster. Based on

the evaluation of the data of this example, GMRA was successful in identifying an updated

parameters vector which resulted in an updated FEM with measured modal properties. One,

draw back to GMRA is that it requires an eigensolution of the FEM in order to calculate the

objective function of Eq. (3.3.1). For large FEMs this objective function evaluation is a









computationally expensive calculation, and would need to be redesigned to make GMRA a

feasible model refinement tool.


Minimum and Averaee Obiective Functions


10 20
Generation

Horizontal Truss Members


Diaeonal Truss Members


10 20
Generation

Vertical Truss Members


0 10 20 0 10 20
Generation Generation

Figure 3.5 Generational Data, Measured Modes with 15% Noise

Norrm (Measured-Updated) Eigenvectors st Frequency: Measured(..) Updated(-)


34


33


32


0 10 20 30
Generation

Scond Frequency: Measured(..) Updated(-)


0


Thi
124

122

120

118


10 20
Generation

rd Frequency: Measured(..) Update<



l__


0 10 20 0 10 20
Generation Generation

Figure 3.6 Generational Eigensolution Data, Measured Modes with 15% Noise











10-5



10-6



10-7



Q. 10-S



10-9



10-10


Analytic Model(--), Measured Model No Noise(-), and Updated Model (...)


10-1 1 I1I
0 50 100 150 200 250 300

Frequency (Hz)


Figure 3.7 FRF After 20 Generations, Measured Modes with 15% Noise













CHAPTER 4
MODAL TEST EXCITATION AND SENSOR PLACEMENT:
CURRENT TECHNIQUES



4.1 Introduction

In the current literature various techniques for excitation and sensor selection for modal

testing exist. These techniques vary in computational complexity, cost, and accuracy.

Several of these techniques were explored in the current study as a basis for comparison for

the excitation and sensor placement techniques developed in the next chapter. An overview

of the excitation placement techniques of kinetic energy, driving point residues, and

eigenvector products and of the sensor placement techniques of effective independence,

kinetic energy, and eigenvector product is given in the following sections.


4.2 Effective Independence

Effective independence (El) is a technique developed to place sensors for the purpose of

obtaining structural information for FEM verification for large space structures (Kammer,

1991). It follows from the work done by Shah and Udwadia (1978) and Udwadia and Garba

(1985). The sensor locations are chosen such that the trace and determinant of the Fisher

information matrix (corresponding to the target modal partitions) are maximized and the

condition number minimized. By maximizing the determinant of the Fisher information

matrix, the covariance matrix of the estimate error would be minimized, thus giving the best

estimate of the structural response. A reduced sensor set is obtained in an iterative fashion

from an initial candidate set by removing sensors from those DOFs (i.e., removing rows from

the Fisher information matrix) which contribute least to the linear independence of the target

modes.








In order to perform test analysis mode shape correlation using a cross-orthogonality

criterion, the measured modes obtained during the modal test must be linearly independent.

A summary of the derivation given in Kammer's paper (1991) follows. The output of the

sensors can be expressed as the product between the FEM target mode matrix partitioned to a

candidate set of sensors, Os, and the modal coordinates q

Us = Osq + W2 = H + I2 (4.2.1)

with Gaussian white noise Yo2 added. It is assumed that the FEM mode shapes are linearly

independent. The sensors are sampled and an estimate of the state of the system is calculated

as


q = TI: s] sus (4.2.2)

In order to obtain the best estimate of the state of the structure, the covariance matrix of the

estimate error must be a minimum. The covariance matrix is given by


P E[(q )(q q)T] = 2] (4.2.3)


Assuming that the sensors measure displacement (acceleration may also be considered), the

covariance matrix may be rewritten as

S4TT2-1
P= ~~~0 s =Q-1 (4.2.4)

where Q is the Fisher information matrix and can be rewritten as

Q = s = 2o (4.2.5)
0 0
Ao will now be referred to as the Fisher information matrix. In order to minimize the

covariance error P, Q must be maximized; therefore Ao must be maximized. Kammer (1991)

states that the determinant of the Fisher information matrix for the best linear estimate is a

maximum for all linear unbiased estimators. Therefore, one wishes to maximize the








determinant of the Fisher information matrix. From Eq. (4.2.5), the Fisher information

matrix is calculated to be the product of the transpose of the target mode matrix times the

target mode matrix,

Ao = ITs (4.2.6)

The first step is to calculate the eigenvalues XA and eigenvectors yA of the Fisher

information matrix. Since it is assumed that the original FEM mode shapes are linearly

independent, Ao will be positive definite, the eigenvalues will be real and positive and the

eigenvectors will be orthonormal. The next step is to form the matrix product,

G = ['FsWA] [sIA] (4.2.7)

where & is an element by element multiplication. The columns of G sum up to be the

eigenvalues of A0. Next the G matrix is scaled by the inverse of the eigenvalues of A0,


FE= G kA1 (4.2.8)


The effective independence vector is then calculated by summing the rows of the FE matrix,

n
FE1j
j=1
n
n FE2j
ED= 1. (4.2.9)

n
Z FE(nDOF)j
j=l

where n is the number of target modes. The ith term in the ED vector is hypothesized to be

the contribution of the ith sensor to the linear independence of the FEM modes. A value of
1.0 in the ED vector corresponds to a DOF that is essential to the linear independence of the
target modes (i.e., that DOF must be retained as a measurement location). The DOF which








contributes least to the linear independence (i.e., lowest ED value) is removed from the FEM

target mode matrix. The Fisher information matrix Ao, the G and FE matrices, and effective

independence vector ED are recalculated and the next sensor location is deleted from the

target set. This iterative process is performed until the desired number of sensors remains.

The minimum number of sensors required for identification corresponds to the number of

target modes supplied.

The previously described technique chooses single DOFs to place sensors. Often times

a modal tester uses triaxial sensors instead of single DOF sensors. Assume the FEM being

used to place sensors has 3 DOF per node. The El algorithm is modified to choose 3 DOFs at

a time (corresponding to a node point) which contributed least to the linear independence of

the target modes were eliminated over each iteration. The El value for each node is calculated

as a sum of the El of each DOF of that node. The ED values for the 3 DOFs at each node are

summed as,

ED(1) + ED(2) + ED(3)
EDti = ED(4) + ED(5) + ED(6) (4.2.10)
ED(s 2) + ED(s 1) + ED(S)


The 3 DOFs which contributes least to the linear independence (i.e., lowest EDtriax value)

are removed from the FEM target mode matrix. However, if 1 of the DOFs for a particular

node had an EI value of 1.0, meaning that that DOF was essential to the linear independence

of the target modes, that node point would be retained, regardless of the ranking of its node

point El sum rating compared to the other node points. The Fisher information matrix Ao,

the G and FE matrices, and effective independence vector EDtriax are recalculated and the

next triax sensor location is deleted from the target set. This iterative process is performed

until the desired number of sensors remains.

It is suggested that to increase computational efficiency for large FEM, the original

FEM should be reduced down to a candidate set of measurement locations larger than the

number of sensors to be placed before performing the effective independence calculations.








One suggested technique for this reduction is modal kinetic energy, which is discussed in the

next section.


4.3 Kinetic Energy

The use of kinetic energy for optimal sensor placement as well as target mode

identification has been discussed in several papers (Salama et al., 1987; Kammer, 1991). The

modal kinetic energy is calculated using the FEM mass matrix and target modes. The kinetic

energy of the ith DOF of the jth mode is given as

nDOF
KEij = ij Mik kj (4.3.1)
k=l

where nij is the ijth entry of the FEM modal matrix 4, Mik is the ikth entry of the FEM mass

matrix M, and nDOF is the total number of DOFs of the mass and modal matrices. The

kinetic energy matrix, KE, can be expressed as the matrix product

DOF1
KE = (D M = DOF2 (4.3.2)
SKEDOFn

where @ denotes an element by element multiplication of the matrix D and the matrix

resulting from the product of M and 0. The rows of the KE matrix correspond to the DOFs

of the model and the columns correspond to the modes of the FEM. Locations for actuation

or sensing are chosen as those DOFs with a maximum value of kinetic energy for a given

mode. For example, assume the 4 contains FEM modes 1 through 10 for a given structure

and one wishes to sense or excite the third mode. The DOF (row) with a maximum kinetic

energy value for the third mode or column of the KE matrix would be selected. It is assumed

that by placing the sensors at points of maximum kinetic energy, the sensors will have the

maximum observability of the structural parameters of interest.

If the modal test designer wishes to place triaxially constrained sensors, then a KE

matrix may be calculated by summing the rows of Eq. (4.3.2) corresponding to DOFs for








each node. Then the node points with maximum KE over the modes of interest may be

chosen as locations for actuators or sensors.

KEDOF1 + KEDOF2 + KEDOF3
SKEDOF + KEDOF2 + KDOF3 (4.3.3)
KEtriax (4.3.3)
KEDOFn-2 + KEDOFn-1 + KEDOFn

The kinetic energy objective function precludes placing any sensors or actuators at

nodal points since there is no motion and zero kinetic energy at these points (i.e., the Q entry

would be zero resulting in a zero product). This could be a limiting factor in the pretest

planning. To combat this problem, sensors can also be placed using maximum average

kinetic energy (AKE) technique. A sensor is placed at a DOF with a maximum average

kinetic energy over a range of modes of interest. In using an average kinetic energy, a DOF is

not necessarily excluded if it is a node point of a particular mode. The average kinetic energy

vector is calculated as

N
SKE1k
k=l
N
KE2k
AKE = k= 1 /N (4.3.4)

N
SKE(ndof)k
k=1

where N is the number of modes in the mode shape matrix 0 (i.e., the number of columns

of the KE matrix). The sensor or actuator locations are found by finding the DOFs of the

maximum average kinetic energies. Triaxially constrained sensors may be placed by taking

the sum of the average kinetic energy for the DOFs for each node and choosing the nodes

with maximum average kinetic energy.

In addition, it should be noted that the mass weighting inherent to the kinetic energy and

average kinetic energy approaches causes the sensor or excitation placement to become

dependent on the finite element discretization of the structure. There is an inherent bias








against the placement of sensors in the areas of the structure in which a fine mesh size (and

thus small mass) is used.


4.4 Eigenvector Product

This technique uses modal products from the reduced FEM eigenvectors to identify

possible locations for sensors or excitation. By choosing a frequency range of interest and

the corresponding FEM eigenvectors (or modes) in that range, the eigenvector product is

calculated as


EVP = 21 (0 2 0 N (4.4.1)


where ( represents an element by element multiplication of the mode shape vectors 4. The

ith entry of the EVP is given as

EVPi = (i1)i2i3 .iN (4.4.2)

This product is calculated for all candidate DOF sensor or actuator locations. A

maximum value of this product corresponds to a candidate location of reference or excitation

(Jarvis, 1991). This technique also precludes the placement of sensors at nodal points which

result in zero eigenvector products. If this presents a problem for a given test case, the

eigenvector product can be replaced equivalently by an absolute value eigenvector sum, over

the FE target modes of interest.

The eigenvector product may be used to place triaxially constrained sensors by

summing up the entries of Eq. (4.4.2) which correspond to a particular node point. The node

points with the maximum eigenvector product sum are then chosen as points of reference.


4.5 Driving Point Residue

A FEM can be used to identify the best locations and directions for exciting a structure

by an evaluation of driving point residues (DPRs) or modal participation factors (Kientzy et







al, 1989). A DPR is a measure of how much a particular mode is excited at a particular DOE

The point and direction of excitation are chosen where the DPRs are maximized (to excite a

given mode) or minimized (to avoid exciting a given mode). An equation of motion in

Laplace domain for a structure may be written as

[Ms2 + Cs + K]X(s) = F(s) or B(s)X(s) = F(s) (4.5.1)

where M, C, and K are the (nxn) mass, damping and stiffness matrices, s is the complex

Laplace variable (s = ( + ico), and F(s) is the transformed excitation forces. Equation (4.5.1)

may be solved for the transformed displacement responses, X(s),

X(s) = H(s)F(s) where H(s) = B(s)-1 (4.5.2)

and H(s) is referred to as the transfer matrix. The system transfer matrix for a structure with

damping can be expressed in the form

H(s) = + (4.5.3)
k=lS-k S k

where Rk and Rk* are the modal residues and Xk and Xk* are the complex conjugate pairs of

eigenvalues of the transfer matrix. The residues can be written in terms of the mode shapes

Ok as
kN Tk
H(s)= s Ak kX sT A -1 (4.5.4)
k= S- k S-k

where Ak is the mode shape scaling constant. For a structure which is lightly damped, the

following two inequalities are true:

Ok < (Wk and Imaginaryl{k} < Real{(k} for k = 1 to N (4.5.5)

When these conditions are imposed, the mode shape scaling constants can be written in the

form

A = for k = 1 to N (4.5.6)
Ak -mk(Ok








and the residues become


Rk(a, b) = [k(a)k(b)] for k=1 to N (4.5.7)
(mkwk)

where Rk(a,b) is the residue between DOF a and DOF b, Ok(a) is the kth mode shape

component at DOF a,
are scaled such that they are mass orthonormal, (i.e., TM4=I, where
columns are the mode shapes 4k (for k=1 to N), M is the FEM mass matrix, and I is the

identity matrix) then the residues (in terms of the displacements) may be written as


Rk(a,a) k(a) for k=1 to N (4.5.8)


or equivalently in terms of acceleration

Rk(a, a) = k(a)2wk for k=1 to N (4.5.9)

The easiest way to evaluate several residues at once is to display them graphically. The

DPRs that were calculated for the NASA 8-bay truss are shown in Figure 4.1. The DPRs are

graphed in order of weighted average residue in order to discriminate against zero DPRs.

The weighted average residue is calculated as

waDPR = average DPR x minimum DPR (4.5.10)

Each vertical line on the graph represents the range of DPRs from maximum to

minimum over all the modes of interest for a single candidate DOF The highest weighted

average which is the best driving point is displayed first. The residues in the top graph are the

square root of the sum of the squares of the residues in the x, y, and z direction plotted on a log

scale. The bottom graphs are the residues for the x, y, and z direction plotted on unit

normalized linear scales. The top graph is used to choose the node at which to place the

excitation device. The bottom graphs are used to find the x, y, or z direction of the excitation.

In order to insure that an excitation location will give uniform participation of as many

target modes as possible, it is desired to find a high average residue for a given DOF as well as









Weighted Average dprs


co "-ttff" t t tt tI" I :

c 4-
cn-
0
0-
2 3 4 1 5 8 6 7 13161514191820171110129 21242322252827262933130
-2- node
0 5 10 15 20 25 30
node
X,Y,Z dof dprs
3 I I I z


"02
a, y
N




0 5 10 15 20 25 30
node

Figure 4.1 Typical Driving Point Residue (NASA 8-Bay Truss)


a small residue range over all the modes of interest. For this example the highest weighted

average DPRs are at nodes 2, 3, 4, and 1 as seen in the top portion of Figure 4.1. The bottom

portion of this figure shows that the optimal directions for excitation at these node points

would be in the x and/or z direction, because the larger residues are in these directions.














CHAPTER 5
MODAL TEST EXCITATION AND SENSOR PLACEMENT:
NEW TECHNIQUES



5.1 Introduction

In an effort to improve on the existing sensor and excitation placement techniques, two

new sensor placement techniques and two new excitation techniques are developed in the

current work. The first excitation and sensor placement techniques are based on the FEM

normal Mode Indicator Function (MIF) calculation. The second excitation and sensor

placement techniques are based on the observability and controllability calculations of the

modes of the FEM. The effectiveness of these techniques, along with the techniques

discussed in Chapter 4, will be explored in subsequent chapters using several different

structural test-beds.


5.2 Mode Indicator Function

The mode indicator function (MIF) was first developed to detect the presence of real

normal modes in sine dwell modal testing (Hunt et al, 1994 and Williams et al, 1985). This

function also serves as a useful metric for pre-test analysis. While it is somewhat useful for

assessing the efficacy of sensor layout, its true utility lies in assessing the effectiveness of a

particular input in exciting the system modes.

The first step in calculating the MIF is the calculation of an acceleration frequency

response function using the FEM mode shapes and frequencies,


Hk i k (5.2.1)
= msr(w (02 + j2Sr(0)
r= 1 Mr(( r







where m number of modes in frequency range of interest

or rth mode

Okr- force input point k of the rth mode
Oir- response point i of the rth mode
o discrete frequency at which to calculate Hik

Or frequency of the rth mode
Sr viscous damping ratio of rth mode

msr modal mass of the rth mode
Next, the normal MIF is calculated using Hik as


-([Real(Hik(o)) x H(ik())
MIF((o) = i= (5.2.2)
(kH,k(W)2)
i=1
where L is the total number of response points. The MIF is nearly 1.0 except near a normal

mode, at which point it drops off considerably since the frequency response becomes mostly

imaginary at that point (i.e., Real(Hik(o)) is very small). A plot of a typical MIF is given

in Figure 5.1.

5.2.1 Excitation Placement

In pre-test planning, an excitation is desired which exhibits a sharp drop in the MIF at

each mode of interest, indicating that the mode is well excited. The Genetic Mode Indicator

Function (GMIF) excitation selection algorithm uses a genetic algorithm (Holland, 1975) to

find excitation locations and their orientations on a structure to optimally excite a given

mode or range of modes. The success of the excitation is based on the MIFs of the chosen

excitation locations. If more than one excitation is sought then a MIF must be calculated for

each. A single excitation need not exhibit a sharp MIF drop for all modes as long as the union
of the MIFs for all of the excitation sources exhibits a large drop for each target mode. Two

algorithms have been developed. The first is an unconstrained version which searches for
























0 10 20 30 40 50 60 70
frequency (Hz)


Figure 5.1 Typical MIF Plot


node point excitation locations with forces being applied in any direction at the node points.

The second algorithm is a constrained version in that the direction of the excitation is

constrained to be 0, 30, 45, 60, or 90 degrees in each x, y, and z plane. The constrained

algorithm was developed to provide an improvement in algorithm speed by reducing the

number of search points in the design space. In addition, the attachment of the excitation

hardware on the structure during the modal test would be easier if the angles of orientation

are limited. An outline of the GMIF algorithms follows.

Coding. The GA chooses an initial population of node points and directions for

excitation location. The node points and the directions are referred to as design variables.

The design variables are represented differently for the constrained and unconstrained

versions of the GMIF algorithm. In both the constrained and unconstrained cases the node

point locations of the excitations are treated as discrete design variables. Discrete design

variables represent a finite number of variables to search over, and for this application they

represent all of the node points in a FEM that are being considered as possible excitation

locations. The direction design variables are two angles in spherical coordinates, a and 3,

which are used to calculate the direction of the force as seen in Figure 5.2. For the








z
F force
P *(r,La,) x = r cosa sinp
y = r sina sin3
r z= r cospl
node
F = cosasinP3i + sinasinlj + cospk

x


Figure 5.2 Excitation Selection by GMIF

constrained algorithm the orientation of the excitation is considered discrete in the sense that

there are a finite number of angles (i.e., 0, 30, 45, 60, or 90 degrees) from which the GA

selects a and p. For the unconstrained case, the angles of orientation are considered to be

continuous in that the GA searches over all possible angles. For both cases the force is

assigned a unit magnitude in order to only evaluate the angle of orientation of the force and

not the magnitude. Table 5.1 presents a list of variables used in the GMIF selection

algorithm.



Table 5.1 GMIF Design Variable Description


Unconstrained Constrained
Design Variable Type Design Variable Type
node discrete node discrete
a (any angle) continuous a (0,30,45,60,90 degrees) discrete

P (any angle) continuous p (0,30,45,60,90 degrees) discrete


Evaluation. The next step in the GA is to evaluate the fitness of each population member

or excitation. The fitness of a member is based on the calculation of the MIF corresponding

to each force that makes up a single member. All of the MIFs for a single member are

assembled into a MIF matrix,








miffl(Wo) miffl(W2) miffl(omm Force 1
MIFm= if.(W1) miff(W2) mif2(Om+- Force 2 (5.2.3)
miffni()m oif(2) mf 2 m.ifn(o m Force nf

1st 2nd mth
natural frequencies of interest

Next the minimum of each column of MIFm is taken to find the maximum drop-off values

of the union of the MIFs of each force resulting in a minimum MIF vector,

column
MIF = minimum (MIFm) (5.2.4)

The objective function is calculated as a weighted sum of the elements of MIFv,

m
Jobj = wMIFvi (5.2.5)
i=l
The weights may be used to emphasize the drop-off values of particular modes. The

objective function of Eq. (5.2.5) is designed to find excitation sources which exhibit sharp

MIF drop-offs for as many modes as possible.

Selection, crossover, and mutation. Once the fitness of the initial population is

established the population is allowed to evolve over a fixed number of generations. The

information contained in the initial population is crossed over between members and sent to

the next generations. Members of new generations which are more fit than the previous

generation (i.e., have better drop-off values) replace the less fit members in the evolving

population. Mutations that occur in the population allow for the population to remain diverse

during the evolutionary process, keeping the design search space open.

5.2.2 Sensor Placement

Once an excitation source has been selected, the MIF corresponding to the chosen

excitation source may be used along with a GA to locate a sensor set. First, the FRF matrix is

calculated for the FEM under consideration using the chosen excitations. When the MIF is

calculated to evaluate an excitation source, all DOFs of the mode shape matrix are used to








calculate the frequency response matrix, H. When the MIF is used to evaluate a sensor

placement, only the sensor candidate DOF or three DOFs in case of a triax sensor set, is used

to calculate the frequency response matrix, H. A MIF must be calculated for each force for a

candidate sensor and the minimum MIF value for each mode is taken. The MIF values for the

target modes for the ith DOF are taken as the minimum MIF values for all of the forces in an

excitation set,

MIFi(forcel)-
column MIFi(force2) -
MIFji = minimum (5.2.6)
MIFi(forcenf) -


The MIF vector of Eq. (5.2.6) is calculated for all candidate sensor DOFs. A weighted sum

of the MIF values for each DOF is made and assembled into the MIF vector,


Z wMIFi

MIFv = wMIF2 (5.2.7)
I wMIFn


where w is an (lxm) weight vector used to emphasize MIF drop-off values. The variable

n is the total number of candidate sensor DOFs for unconstrained sensor placement or the

total number of candidate sensor nodes for triaxially-constrained sensor placement. Once

MIFv has been calculated, the node or dof with minimum MIFv sum is retained as the first

sensor. The MIFv vector is recalculated using all remaining DOFs plus the single sensor

chosen, and the next dof or node is chosen with minimum MIFy value. This iterative process

is performed until the desired number of sensors is chosen.


5.3 Observability and Controllability

Consider the set of discrete linear second-order differential equations of motion

corresponding to a particular nDOF FEM of a structure,


Mx(t) + Dx(t) + Kx(t) = Bu(t)


(5.3.1)








y(t) = Cx(t) (5.3.2)

where M, D, and K are the (nxn) mass, damping, and stiffness matrices, x(t) is the (nxl)

displacement vector, and u(t) is the (nxl) input function of the system. The over dots
represent differentiation with respect to time. By choosing


z(t x(t)] (5.3.3)
z(t) = [x(t)J


Eq. (5.3.1) can be rewritten in state space form as


z(t) = M-K -M-1D z(t) + M- ] (5.3.4)


or equivalently


z(t) = Az(t) + Bu(t) (5.3.5)

where A is the (2nx2n) state matrix, B is the (2nxo) input influence matrix, and u(t) is the

(ox 1) input function vector. The output of the system defined by Eq. (5.3.2) may be rewritten
as


y(t) = Cz(t) where C = [C 0] (5.3.6)

The vector y(t) is the (lxl) system output, and C is the (lx2n) output influence matrix. Figure
5.3 is a pictorial representation of the matrices and vectors of a state space system of
equations and describes the purpose of each.

An important consideration in the control of the system described by Eq. (5.3.4) is if the
system is controllable and observable. Another consideration is the observability and
controllability of the modes of the system defined by Eq. (5.3.4). Several techniques for
calculating the observability and controllability of modes have been explored. One of the
most common tests for controllability and observability is the Popov-Belevitch-Hautus
(PBH) test (Kailath, 1980). For the purpose of vibration control, it is most common to




42



Input Space / Output Space
u(t) y (t0

map B map C
how and where energy what information is
is injected into system State Space extracted from system
x(t)

Smap A

how system transforms
and dissipates energy


Figure 5.3 State Space Variable Description

overstep the conversion of Eq. (5.3.1) into state space form and to calculate the observability

and controllability directly from Eq. (5.3.1). The PBH eigenvector test for a second-order

system (Laub and Arnold, 1984) states that given the system defined in Eqs. (5.3.1) and

(5.3.2):

1. The ith mode will not be controllable from the jth input if and only if there

exist a left eigenvector qi such that

qi[X2M + XiD + K] = 0T (5.3.7)


qTbj = 0T (5.3.8)

2. The ith mode will not be observable from the kth output if and only if there

exist a right eigenvector pi such that

[XM + XiD + K]pi = 0 (5.3.9)


cki = 0 (5.3.10)

where bj is the jth column vector of the input influence matrix, B, and gk is the kth column

vector of the output influence matrix, C.








This evaluation of controllability and observability tells whether or not the modes are

completely observable or controllable; it does not address the issue of degree of observability

and controllability. The issue of degree of controllability and observability is explored in a

paper by Hamdan and Nayfeh (1989). In this work the matrices QTB and CP are used to

evaluate the degree of controllability and observability of the modes of a system. The matrix

QT is the transpose of the matrix whose columns are the m left eigenvectors of Eq. (5.3.7) and

B is the output influence matrix whose columns are the o output influence vectors.

Ti qT I I
QTB = b2 q b b2 b (5.3.11)
n I I I



The matrix C is the output influence matrix whose I rows contain the output influence vectors

and P is the matrix whose columns are the m right eigenvectors of Eq. (5.3.9).

-c I -
CP = 2 2. (5.3.12)


The (mxo) matrix QTB contains information about the controllability of the modes and

the (lxm) matrix CP contains information about the observability of the modes. If the ijth

entry of QTB is 0 then the ith mode is uncontrollable from the jth input. Similarly, if the kith

entry of CP is 0 then the ith mode is unobservable from the kth output. If the ijth entry of the

controllability matrix is nonzero, then what information may be gained about the degree of

controllability of the ith mode from the jth input? The ijth element of the QTB matrix is the

vector dot product of qi and bj. If the two sub-spaces spanned by these vectors are parallel

then the ith mode is completely controllable from the jth input, and if the two sub-spaces are

orthogonal then the ith mode is completely uncontrollable from the jth input. If the two

sub-spaces are neither orthogonal or parallel then the angle between the two is an indication

of the degree of controllability of the ith mode from the jth input. This relationship is

illustrated in Figure 5.4 and the magnitude of the vector dot product is,








(5.3.13)


A similar argument may be made for the observability of the ith mode from the kth output

using the magnitude of the vector dot product,


IckPil = II Pi I cos (ki


(5.3.14)


The angle 0ij is a direct measure of the degree of controllability of the ith mode and )ki is a

direct measure of the degree of observability of ith mode. The degree of controllability and

observability decrease as ij and Oki go from 0 to t/2 as shown in Figure 5.4.


CONTROLLABILITY


OBSERVABILITY


bj Qi ck Pi
completely controllable completely observable
completely controllable completely observable


Oij = 7c/2


complete uncontrollable

complete3 uncontrollable


Oki = 7C/2

completely


uncontrollable


Figure 5.4 Controllability and Observability


The above argument has been made from a dynamic controls perspective. The same

argument may be used to gain information about actuator and sensor placement during

modal tests of a structure. Using the FEM of a particular structure the degree of


IqbjI = 1 q ill bj II cos Oi








controllability of a modal test excitation layout may be used to optimally select an excitation

location. Similarly, the degree of observability of a modal test sensor layout may be used to

select a sensor configuration which will result in an increase in the amount of modal

information obtained.


5.3.1 Excitation Placement


The degree of controllability based on the calculation of the angle between the sub-space

spanned by a mode shape of the system and the sub-space spanned by the input influence

vectors of the matrix of Eq. (5.3.1) is used to evaluate how effective the input u(t) may be in

controlling the modes of the system. Consider that Eq. (5.3.1) is the equation of motion for a

particular structure and that the right hand side, Bu(t), is the force that will be applied to

excite the structure for modal testing. In order to gain the most information from the modal

test, an excitation location which will excite a chosen range of target modes well is required.

The measure of modal controllability is an indication of how great an effect a particular

input, bj, may have on the mode shapes of the system. An input with a higher degree of

controllability over a mode will be more successful in exciting that mode than an input with a

lower degree of controllability over that mode. Therefore, it is proposed that the angle Oij of

Eq. (5.3.13) may be used as a measure of how successful the input excitation bj will be in

exciting mode qi. Since there are an infinite number of possible input influence vector

values, an optimization technique is needed to search for an input influence vector which

maximizes the controllability of the target modes. A genetic algorithm is employed as the

optimization tool for this purpose.

Coding. The coding of the Genetic Controllability (GCON) Algorithm is identical to the

coding of the GMIF algorithm. One design variable represents the node point locations of

the forces, the other design variables represent the angle orientations of the forces in

spherical coordinates as described in Figure 5.2. The difference between the GMIF and the

GCON algorithms is in the fitness evaluation of the population.








Evaluation. The fitness of each population member is based on the calculation of the

controllability vector. The location and orientations of each force in a population member is

used to calculate an input influence vector. The j th force of a member is used to calculate a

portion of the input influence vector as,

cos ctf sin Pfj'
b = sin sin ifj (5.3.15)
cosf, (J


The unit magnitude vector bfj is calculated for all j forces of a member and assembled into

the global input influence vector. The global input influence vector, b, is initially an n DOF

vector of zeros. Once the individual force unit input influences are calculated, they are

placed in the global input influence vector, b, at the DOFs of each corresponding force node

point,

'0
bfl
0
b = f2 (5.3.16)
bf3
0
hfnf
0


where nf is the total number of excitation devices represented in a population member. Since

the magnitude of the big's affect the controllability of the system, each bfj is scaled to unit

magnitude so as to compare the directions of the forces as apposed to the magnitudes. The

unit input influence vector is used in conjunction with the left-hand eigenvectors of Eq.

(5.3.7) to calculate the (mx1) degree of controllability vector from Equation 5.3.13,



S= 0 2 (5.3.17)
0m








The ith entry of the degree of controllability vector is


Scos qTb (5.3.18)
Oi = COS-1 --
II 9i II


The entries of the (mx1) controllability vector, 0, represents the controllability of each of

the m modes of the system from the locations and directions defined by b. The algorithm

is designed to find excitation location which exhibit the highest degree of controllability for

the modes of interest. Therefore, the most fit members of a population are forces which

minimize the entries of the vector Q. Doing so minimizes the angle between the input vector

sub-space and the mode shape sub-spaces thus increasing the amount of controllability and

the amount of power input into the modes. The objective function is calculated as a weighted

sum of the entries of 9,

m
Jobj = i0i (5.3.19)
i=l

The weight may be used to emphasize the controllability of particular modes over other

modes.

Selection, cross-over, and mutation. The population is allowed to evolve over a fixed

number of generations as in the GMIF algorithm. The most fit members are those that

minimize the objective function of Eq. (5.3.19).

5.3.2 Sensor Placement

The degree of observability of the modes of the system in Eq. (5.3.1) is based on the

calculation of the angle between the modes of the system and the output influence matrix.

When performing a modal test of structure, it is not likely that all DOFs in the FEM will be

instrumented during the test due to cost constraints. In order to get the most information

about the modes of the system, a reduced sensor set which has the greatest degree of target

mode observability should be chosen. Therefore, the angle 0 of Eq. (5.3.14) will be used as a




48


measure of how successful a sensor configuration is in measuring a group of chosen target

modes.

There are a finite number of DOFs or sensor possibilities represented in a FEM,

therefore, an optimization technique is not needed. In order to evaluate each DOF location

individually, the output influence matrix, C, is set equal to an (nxn) identity matrix.

Therefore, the observability of the kth DOF of the ith mode is obtained from Equation 5.3.15

as


ki = kil (5.3.20)
II Pi II

where Pki is the kith entry of the right eigenvector matrix of Eq. (5.3.12) and pi is the ith

column of the right eigenvector matrix, P. If this value is calculated for all candidate DOFs

and all target modes the resulting (nxm) observability matrix,


Il 12 m
(P21 (22. 2m
( = (5.3.21)

4nl Sn2- nm

The rows of the observability matrix represent DOFs and the columns represent the

modes. Once the observability matrix has been calculated, the DOFs for sensor location

must be evaluated. The observability column corresponding to each mode is sorted, and the

DOFs with the minimum 0 values (i.e., greatest observability) for each mode are selected as

sensor locations.













CHAPTER 6
PRE-MODAL TEST PLANNING ALGORITHM APPLICATION:
NASA EIGHT-BAY TRUSS



6.1 Introduction

The NASA 8-bay truss is used to compare the techniques discussed in Chapters 4 and 5

in placing sensors and actuators for modal testing and system identification purposes. The

kinetic energy, average kinetic energy, eigenvector product, driving point residue, and

controllability techniques are used to place three excitation devices on the truss. A numerical

simulation is performed to evaluate the effectiveness of each technique to excite the first five

target modes of the structure. A cross-orthogonality check between the identified and finite

element modes is performed in addition to a frequency match comparison. Effective

independence, kinetic energy, average kinetic energy, eigenvector product, and

observability techniques are used to place sensors on the 8-bay truss, in order to best identify

the first five modes of vibration. The structural response of the truss is numerically simulated

and measured at those DOFs corresponding to the sensor locations obtained using the various

techniques. The eigensystem realization algorithm (ERA) is then used to evaluate the

effectiveness of each sensor set with respect to modal parameter identification (Juang and

Pappa 1985). A set of three hundred random sensor locations are compared to the five sensor

location techniques. The cost effectiveness of each of the excitation and sensor selection

techniques is evaluated.


6.2 NASA Eight-Bay Test Bed

The NASA eight-bay truss, pictured in Figure 6.1, is modeled with 96 DOFs and is

considered to be lightly damped. Using the FE mass and stiffness matrices supplied by








NASA, the FE mode shapes and frequencies were calculated. When the true modal tests

were performed on the truss, it was assumed that the first five modes were successfully

identified (Kashangaki, 1992). Table 6.1 list the first five frequencies and mode

descriptions.


29
25
21

13 17 18 22 7 3 2
14 28
9 f 10^ 19 /'24

1 ~ 1 20
7 1 12 16 y x

3 8


Figure 6.1 NASA 8-Bay Truss


Table 6.1 Eight-Bay Truss Frequencies and Mode Description

Mode Frequency (Hz) Description
1 13.925 1st y-x bending
2 14.441 1st y-z bending
3 46.745 1St torsional
4 66.007 2nd y-x bending
5 71.142 2nd y-z bending


6.2.1 Excitation Placement

Kinetic energy, average kinetic energy, eigenvector product, driving point residue, and

controllability techniques are each used to place three excitation devices on the 8-bay truss to

best excite the first five modes of the structure. The excitation locations for the five

techniques are pictured in Figure 6.2. The kinetic energy technique placed two excitation













y

Z
Kinetic
Energy

z
Average Kinetic Energy
and
Controllability



Eigenvector x
Product


S, x
Driving Point x x Y
Residue z

z


Figure 6.2 Eight-Bay Excitation Locations

sources towards the cantilevered end of the truss and one towards the center of the truss. The

other four techniques clustered all of the excitation sources at the cantilevered end of the

truss. It is interesting to note that the kinetic energy technique put all excitation sources in the

z-direction. The remaining four techniques placed excitation sources in both the x and z

directions, in addition to clustering two of the sources at a single node. The average kinetic

energy and controllability techniques placed the excitation devices in the same location as

seen in Figure 6.2. In a true modal test, the two excitation sources which were placed at a

single node could be combined into one excitation source at that particular node in the

xz-direction, thus reducing the number of excitation sources needed. The kinetic energy

placement could not result in this option.







Once the excitation sources have been determined using the five techniques, the truss's

response to an impulse at the chosen excitation locations is numerically simulated for the first

five modes and 5% noise is added to the response data. The response, measured at all 96

DOFs to fully evaluate the effectiveness of the each excitation placement, is sampled for a

length of 2 seconds at 200 Hz. Five percent noise is added to the response which is sent to

ERA for system identification. A comparison of the measured frequencies and

cross-orthogonalities of the identified and FE models is calculated for each excitation

placement.

The five excitation placement techniques were all successful in exciting the structure at

the target frequencies based on the comparison of the original FE and identified frequencies

and mode shape. All of the techniques had excellent matching between identified and FEM

frequencies with differences much lower than the industry standard of 5% (Flanigan and

Hunt, 1993). A Frequency Response Function (FRF) is plotted for each of the excitation

devices in Figure 6.3. The cross-orthogonalities between identified and FEM mode shapes

for the five excitation placement techniques are pictured in Figure 6.4. All off-diagonal

terms for each of the excitation sets are less than or equal to 0.02, which is well within the

industry standard of < 0.02 off-diagonals for primary modes (Flanigan and Hunt, 1993).

Of the five techniques, the cross-orthogonality of kinetic energy was worst although it

was well within the acceptable range of off-diagonal values. The cross-orthogonality of

average kinetic energy, controllability, eigenvector product, and driving point residue

techniques had similar values; all of the off-diagonal elements for each of the techniques

were : 0.01. Recall that kinetic energy placed all excitation sources in the z-direction at

three separate nodes, and that the other three location techniques placed excitation sources in

both the x and z directions and collocated two sources at one node. The similar placement

configurations at the cantilevered tip of the truss resulted in similar cross-orthogonalities.

Based on the frequency matching and cross-orthogonality between identified and FEM

frequencies and mode shapes, and on the FRFs, each of the five excitation location






















0 lo 20 30 410 0 w o 70 80 0 10 20 30 40 0 60 70 80
Frequency(Hz) Frequency(Hz)
S Eigenvector Product 1 Driving Point Residue
10to' 10
10d 10o







10 10
Iio

0 10 20 30 4 50 860 70 80 0 10 28 3 40 50 0 70 0
Frequency(Hz) Frequency(Hz)

Figure 6.3 Eight-Bay Excitation Location Frequency Response of Time Domain Data



Kinetic Enereg Average Kinetic Energy and Controllability


Mode'j -- Mode


Figure 6.4 Eight-Bay Excitation Placement Cross-orthogonality of Identified and
FEM Modes 1 to 5








techniques identified an acceptable three point excitation location for exciting the first 5

modes of the 8-bay truss.


6.2.2 Sensor Placement


Using the FE modes and frequencies, the five sensor placement techniques, effective

independence, kinetic energy, average kinetic energy, eigenvector product, and

observability were assigned the task of best identifying the first 5 modes of vibration by

placing 15 sensing devices on the truss. The five sensor set configurations are pictured in

Figure 6.5. Each of the techniques clustered the sensors in two locations on the truss at the

cantilevered end and at the midspan. Effective independence, average kinetic energy, and

eigenvector product techniques collocated fourteen of the fifteen sensors at seven node

points. The kinetic energy technique collocated twelve of the fifteen sensors at six node

points and observability collocated eight sensors at four node points. From a cost standpoint,

the collocation of as many sensors as possible is desired. None of the five techniques placed

sensors in the y DOF This is to be expected since the first five modes do not include

significant motion in the y-direction.

The simulated response, with 5% noise added, obtained using the excitation

configuration determined from the average kinetic energy technique was used to test the

sensor sets. The exact same response was used to test each sensor configuration, by taking

from the 96 DOF response only those DOFs corresponding to the sensor locations to be

evaluated. The response data of each sensor set were sent to ERA for frequency and mode

shape identification.

The excitation placement found using the average kinetic energy technique is used to

excite the structure to test the various sensor locations. Each of the five sensor placement

techniques measured the target frequencies well as can be seen from the percent difference in

the FE and identified frequencies given in Table 6.2. The 96 DOF FEM mass matrix was

reduced to a 15 DOF mass matrix using exact reduction and the cross-orthogonality between













y
Z


Effective
Independence



Kinetic
Energy




Average Kinetic
Energy



Eigenvector
Product



Observability


xz T
xz


Figure 6.5 Eight-Bay Sensor Locations


identified and FEM modes 1 through 5 was calculated using the reduced mass matrix. The

cross-orthogonality for each of the sensor sets is given in Figure 6.6.

For each of the five cases the cross-orthogonality terms were within acceptable limits.

The off-diagonal terms corresponding to the primary modes were all 4 0.02. Effective

independence, kinetic energy, and eigenvector product techniques resulted in similar

cross-orthogonalities (all off-diagonals are 4 0.01). Observability technique gave the worst

cross-orthogonality results of the five techniques even though the off-diagonal elements








remained within acceptable values. The improved performance of the effective

independence, kinetic energy, observability, and eigenvector product techniques over the

average kinetic energy technique can clearly be seen in the next section when the five

techniques are compared to the random sensor sets.


Kinetic Energy


Average Kinetic Energy Eigenvector Product


Observability


- 1.00
a 0.02
m 0.01
m<0.01


1.00



0.02
0.00


Figure 6.6 Eight-Bay Sensor Placement Cross-orthogonality
of Identified and FEM Modes 1 to 5








Table 6.2 Percent Difference in FE and Identified Frequencies


MODE El KE AKE EVP CON
1 0.042 0.052 0.066 0.042 0.17
2 0.018 0.004 0.012 0.018 0.19
3 0.141 0.036 0.013 0.142 0.30
4 1.140 0.032 0.455 1.140 0.79
5 0.021 0.008 0.037 0.021 0.15


6.2.3 Results: Random Sensor Location

Three hundred random sets of 15 sensors each were generated and evaluated in order to

assess the level of increased performance of the various sensor placement algorithms against

pure chance. The same time domain response with 5% noise used in the previous section was

partitioned to the random sensor configurations. For the time domain data of each sensor set,

ERA is used to identify the first five frequencies and mode shapes. The cross-orthogonalities

and frequency differences between the identified and FEM modes and frequencies were

calculated. Figure 6.7 is a comparison of cross-orthogonalities for the 300 random sensor

sets and for the five sensor placement techniques (EI,KE, AKE, EVP, and OBS). The bar

portion of each graph corresponds to each random sensor set value and the straight lines

correspond to the five evaluated sensor configurations (El, KE, AKE, EVP, and OBS) and to

the average value of all the random sensor sets (ARS). The top graph of Figure 6.7 is a plot of

the maximum off-diagonal elements of the cross-orthogonality matrix, the center graph is

the average off-diagonal of the cross-orthogonality matrix, and the bottom graph is the two

norm of the cross-orthogonality matrix minus the identity matrix.

In general, the average random sensor set was within the acceptable limits on frequency

matching and cross-orthogonality. This is due to the fact that a large number of sensors were

placed on the truss and approximately 1/3 of the trusses node points would be instrumented

by the random sensor sets. Statistically, the random sets would have good chances of

capturing pertinent modal information. Of the five techniques evaluated, kinetic energy,








effective independence, eigenvector product, and observability gave better results than 97%

of the random sets as can be seen in Figure 6.7. The maximum off-diagonal and the average

off-diagonal of the cross-orthogonality matrices of the three techniques are less than those

for the average random set. In addition the two norm of the difference between the

cross-orthogonality and the identity matrix for the three techniques is lower than that of the

average random set. However, the average kinetic energy gave results similar to the average

random configuration, and showed little to no improvement over the purely random

placement of fifteen sensors. The maximum off-diagonal of the AKE set was larger than that

for the average random set and the average off-diagonal and two norm were approximately

equal to those of the average random set.




6.3 Computational Efficiency



For this particular example, the results for each of the excitation and sensor placement

techniques are all relatively comparable. The targeted modes and frequencies are excited by

all of the excitation placements and properly identified by all of the sensor sets evaluated.

This is illustrated by the acceptable differences in FEM and identified frequencies and

cross-orthogonality values. The agreement between all the techniques can be partially

contributed to the fact that the "the modal test" was a numerical simulation. The differences

in the results for excitation evaluation and identification may be greater for a true modal test.

However, an important issue that must be considered when using the discussed techniques

for excitation and sensor location is the computational cost of each evaluation versus the

accuracy of the modal test results. As can be seen from Table 6.3, the most efficient

technique for excitation and sensor location is the eigenvector product technique. It may

well be that on a more complicated example, the more computationally efficient techniques

may result in modal test configurations which give less accurate modal information than the

more complex placement techniques. A larger system model with more DOFs may make the









Cross-Orthogonality: Maximum Off-Diagonal


Cross-Orthogonalitv: Average Off-Diagonal


Iw i


0 50 100 150 200 250 3

Two Norm: Cross-Orthogonality Identity


0.08 I


1.1,I


AKE
ARS
OBS
EVP
El
KE












ARS
AKE
EVP
El
OBS
KE


00











ARS
AKE
B OBS
a EVP

KE


0 50 100 150 200 250 300


Figure 6.7 Eight-Bay Cross-Orthogonalities of Five Techniques
Compared to 300 Random Sensor Sets


-Z 0.01




00.006

J l
2.)rt


0.07
0.06
Z 0.05
(q 0.04
"' !


4111








tradeoffs between the computational cost of placement and the accuracy of the modal

identification more apparent.


Table 6.3 Total Floating Point Calculations for Each Placement Technique

Technique Total flop count Placement
El 812,000 sensors
KE 92,700 sensors & excitation
AKE 93,200 sensors & excitation
EVP 480 sensor & excitation
DPR 7,600 excitation
CON 7,600 excitation
OBS 7,600 sensor


6.4 Conclusion

Based on the evaluation of the numerical simulation, each of the five excitation

techniques successfully placed three excitation sources on the structure which would excite

the first five modes of vibration. The sensor placement techniques of effective

independence, kinetic energy, eigenvector product, and observability found sensor locations

which showed better frequency matching and cross-orthogonality than 97% of the random

sensor sets. The sensor set obtained using average kinetic energy showed no improvement in

cross-orthogonality or frequency matching over those of the random sensor sets. Based on

the similar results of the placement techniques for sensors and actuators, a more complex

structure will now be used to compare the techniques discussed in this chapter as well as other

techniques outlined in Chapter 5.















CHAPTER 7
PRE-MODAL TEST PLANNING ALGORITHM APPLICATION:
MICRO-PRECISION INTERFEROMETER TRUSS




7.1 Introduction

A comparative study of several pre-modal test planning techniques is presented using

the Jet Propulsion Laboratories' Micro-Precision Interferometer (MPI) testbed. Mode

indicator functions calculated using a reduced FEM of the structure and degrees of target

mode controllability are used in conjunction with genetic algorithms to find location and

orientation of two excitation sources in order to optimally excite a chosen range of target

modes during a modal test. Effective independence, kinetic energy, eigenvector product,

observability, and mode indicator function techniques are used to place a combination of

sensors on the structure for the purpose of modal identification. The sensors are placed in

two ways: independent sensor placement and triaxially constrained placement. A numerical

simulation of the response of the structure is used to evaluate the effectiveness of each of the

placement techniques to identify the target modal parameters of the structure. The effect of

FEM error on the various placement techniques is evaluated.


7.2 Micro-Precision Interferometer Test Bed

The MPI, shown pictorially in Figure 7.1, is a tested that has been built in order to study

structural control systems in the development of space interferometers. Modal tests were

performed on the MPI structure by two independent groups (Sandia National Laboratories

and the Jet Propulsion Laboratories (Red-Horse et al., 1993; Carne et al., 1993; Levine-West

et al., 1994).



















left extending boom
z

x


right extending boom


Figure 7.1 MPI Structure


The FEM used to evaluate the placement techniques in the current work was obtained

from Sandia National Laboratories (Red-Horse et al., 1993). The model used is a 240 DOF

Guyan-reduced FEM which has been updated using the data obtained from the modal test of

the structure. The 240 DOFs correspond to three DOFs (x,y,z) at each of the 80 node balls.

The frequencies from the Guyan-reduced FEM corresponding to the first 12 non-rigid-body

modes are given in Table 7.1 and are compared to actual frequencies obtained during the

modal test.


7.3 Excitation Placement

During the original modal test of the MPI structure, two excitation sources were used as

pictured in the top portion of Figure 7.2. The lower portion of this figure is the excitation

configurations that were obtained by optimizing the MIFs of the FEM using a GA (GMIF)

and by optimizing the modal controllability of the FEM using a GA (GCON). Both the

original and the GMIF excitation locations have an exciter on the two extending booms

although they are oriented differently. The GMIF set-up moves the excitation of the right

extending boom to an interior point in comparison to the original configuration. The GCON








technique placed an exciter at the mid-point of the left extending boom and an exciter at the

top of the main boom. Figure 7.3 gives typical frequency responses for the two excitations of

the three techniques shown in Figure 7.2; the responses are measured at the sensor location

shown in Figure 7.2 in the y-directions. The straight line corresponds to the first force

located by each technique and the dotted line corresponds to the second.


Original






node 79
0.000 x
0.707 y
0.707 z


node 41
0.707 x
0.000 y
0.707 z


GMIF derived excitation


node 77
0.5225 x
0.8323 y
0.1538 z


z

X-y
x


node 19
0.2471 x
0.9618 y
0.1178 z


GCON derived excitation
node 6
sensor 0.8660 x
-0.5000 y
0.0000 z


node 67
0.000 x
0.000 y
-1.000 z


Figure 7.2 Excitation Placement on MPI Structure








Table 7.1 Reduced MPI FEM Frequencies Compared with MPI Modal Test Frequencies

Mode Frequency (Hz) Frequency (Hz)
FEM Modal Test
1 7.82 7.75
2 11.66 11.65
3 12.75 12.67
4 29.52 29.36
5 34.45 34.06
6 37.76 37.34
7 42.81 42.25
8 47.30 46.04
9 51.14 50.69
10 52.36 53.00
11 55.41 56.82
12 61.40 60.04



The excitation devices placed by the GMIF algorithm were selected to minimize an

objective function which was dependent on the MIF of each of the two excitation locations.

The MIF will be nearly 1.0 except near normal modes, at which point it drops off

considerably. This drop-off indicates that the mode is well excited. Therefore, it is desirable

to find two excitation sources (location and orientation) which exhibit a sharp drop at all of

the normal frequencies. The GMIF objective function was designed to find an excitation

sources) which exhibits sharp drop offs at normal frequencies as discussed in Chapter 5.

The GCON algorithm, as discussed in Chapter 5, was designed to find excitation sources

which minimize the angles between the input influence vector subspace and the sub-spaces

spanned by the modes of the system thus maximizing the controllability of the modes of the

structure. By choosing excitations with maximum FE modal controllability, the amount of

energy being imparted to the FE modes of the system by the excitation is theoretically

maximized. In order to compare the three excitation sources, the MIF drop off values for the

first twelve modes of each of the excitations is calculated and shown in Table 7.2. The

sharpest drop-off value for each of the modes is highlighted in bold in the table.








Original Modal Test Excitation


frequency (Hz)

GMIF Excitation


frequency (Hz)


node 41

node 79













Snode 19

- node 77














-node 6
node 67


frequency (Hz)

Figure 7.3 Typical Frequency Response for MPI Structure








Table 7.2 Original, GMIF, and GCON Excitation Locations MIF Values

Original GMIF GCON
MODE 41 79 19 77 6 67
1 0.0014 0.0006 0.0003 0.0106 0.0345 0.0129
2 0.0063 0.2601 0.0105 0.0048 0.0095 0.0079
3 0.0394 0.0022 0.4106 0.0428 0.0211 0.0256
4 0.0854 0.2017 0.0228 0.0706 0.4023 0.0287
5 0.0236 0.0420 0.0836 0.0184 0.0174 0.0408
6 0.4636 0.0609 0.0393 0.6855 0.2495 0.0862
7 0.0484 0.0489 0.0911 0.7126 0.6753 0.0372
8 0.0755 0.3263 0.3318 0.0644 0.0996 0.1947
9 0.1537 0.7790 0.7521 0.0824 0.0550 0.5615
10 0.7228 0.8957 0.1674 0.8050 0.7625 0.2260
11 0.4077 0.0407 0.0826 0.2747 0.4372 0.0747
12 0.0724 0.8088 0.4408 0.0591 0.0703 0.2013
min MIF 2 / 12 7 / 12 3 / 12



The GMIF excitations exhibit a sharper drop-off than the original excitations' MIFs for

10 of the 12 target modes. The GCON exhibit a sharper drop-off than the original excitation

for 6 of the 12 modes. Comparing all three techniques, the GMIF had the most minimum

drop-off values at 7 followed by the GCON technique at 3, and the original excitation at 2.

This is not a surprise since the GMIF is designed specifically to find excitation sources which

exhibit the greatest drop-offs. An improvement in the drop-off of the GMIF excitation over

the original excitation can especially be seen for the tenth mode.

It is interesting to note that even though the GMIF has the best overall MIF drop-off

values, the minimum drop-off values of the GCON excitations are well within acceptable

levels. The highest minimum value for the GCON technique is 0.25 for the fifth mode.

However, in the next chapter it will be shown that an excitation with good controllability

values does not necessarily have acceptable MIF values. In order to evaluate the

performance of the genetic algorithm for excitation placement, a set of 500 random 2-point








excitations is generated. The number of random excitations, 500, was chosen because the

GMIF algorithm evaluated approximately 400 population members in the search for the

chosen GMIF excitation. The MIF values for each of the random excitations are calculated,

sorted and graphed in Figure 7.4 and the controllability angles for the random excitations are

calculated, sorted, and graphed in Figure 7.5.

The top graph of Figure 7.4 shows the maximum MIF value for the original, the GMIF

derived, and the GCON derived excitations superimposed on the graph of 500 random

designs in order to compare the values. The top portion shows the GMIF excitation has a

smaller maximum MIF than 97% of the random population. The bottom portion of Figure

7.4 is a graph of the genetic MIF excitation placement algorithm objective function values of

the random and selected excitations. An optimal excitation according to the GMIF objective

function is one which has as small as possible maximum MIF drop off value. The bottom

portion of the figure shows that the GMIF excitation outperformed 100% of the random

population. This shows that the genetic algorithm was successful in finding a good

excitation based on the objective function designed in a more computationally efficient

manner than an exhaustive search.

The same evaluation was performed for the controllability angle calculations as pictured

in Figure 7.5. The controllability excitation placement algorithm objective function is

designed to find excitations which have a minimum controllability angle sum over the target

modes to be excited. Even though the original and GMIF excitations have a smaller

maximum controllability angle as seen in the middle graph of Figure 7.5, the GCON

excitation has a better angle sum as seen in the bottom graph. In fact, the GCON excitation

and the original excitations outperform 100% of the randomly generated excitations.

Numerical simulations of the MPI structural response to simultaneous impulses applied

at the two excitation locations were calculated within the MATLAB environment. Five

percent noise was added to the simulated time responses of the structure. These time

responses were used along with the Eigensystem Realization Algorithm (ERA) to identify









EXCITATION


Maximum MIF


100 150 200 250 300 350 400 4E

GMIF Objective Function
S I I


SI I i I


50 100 150 200 250 300 350 400 450 500
Random Excitations


random
original



GCON
GMIF


random



original

GCON
GMIF


Figure 7.4 Comparison of Selected Excitation and Random Excitation
MIF Values




the twelve target mode shapes and frequencies (Juang and Pappa, 1985). The evaluation of

the success of ERA to identify the frequencies and mode shapes was based on a frequency

percent difference comparison between identified and FE frequencies and on a

cross-orthogonality check between FE and identified mode shapes using an exactly reduced

mass matrix. The reduction is exact in the sense that the frequencies and mode shapes of the

reduced system match exactly their counterparts in the unreduced model (O'Callahan et al.,

1989).

When ERA was used to identify system mode shapes and frequencies, it missed the fifth

and tenth frequencies and mode shapes when the original 41/79 DOF excitation locations

were used to numerically simulate structural excitation. To illustrate this, the

cross-orthogonality between FEM and ERA identified mode shapes was calculated, and is


2

$1.5

1

0.5
0


-









Minimum Controllability Angle


O U I ---- I ---- I ----- I ---- I ----- I ---- I ---- I --------

85




75

70
0 50 100 150 200 250 300 350 400 450 500


90 Maximum Controllability Angle


89-

88 -

87

86


I uou

'1060
0


0 50 100 150 200 250 300 350

oo
GCON Objective Function


400 450 500


-o
01020

<1000


98C1 I I j


980' II i -I
0 50 100 150 200 250 300 350 400 450 500
Random Excitations


random

GMIF



original
GCON


Figure 7.5 Comparison of Selected Excitation and Random Excitation
Controllability Angles


pictured in Figure 7.6. For this example, the 240 DOF simulated response was partitioned to

the sensor configuration obtained using the El technique as discussed in the following

section. These poor cross-orthogonality results are corroborated by the frequency response

function shown in Figure 7.3 in which poor excitation can be seen for modes 5 and 10. The

GMIF and GCON derived excitations resulted in successful ERA identification of all 12

mode shapes and frequencies of the original FEM.


-


lfl


fe\


EXCITATION


random

GMIF

original
GCON






random
GCON


GMIF
original












1.00


0.50
0.25




modes [] 1.00
Effective Independence <0.25
Unconstrained Sensor Set E <0.02

Figure 7.6 Cross-Orthogonality Between FE Modes and Identified Modes



7.4 Sensor Placement

Six sensor selection techniques, effective independence, kinetic energy, average kinetic

energy, eigenvector product, observability, and MIF were used to place sensors on the MPI

structure. Most of these techniques were previously evaluated for sensor placement using

the NASA eight-bay testbed in Chapter 6. In that study, the five techniques of effective

independence, kinetic energy, average kinetic energy, eigenvector product, and

observability, performed equally well. This could be due to two reasons: (i) the structure

lacked significant dynamic complexity required to distinguish between the methods or (ii)

the methods were actually so similar that they led to similar results regardless of structural

dynamics. One purpose of this study is to again evaluate the five techniques on a more

complex dynamic system in addition to evaluating the efficacy of using the MIF for sensor

placement. The second purpose is to investigate the suitability of the techniques when the

sensors are constrained to be placed in a triaxial fashion.

Eighteen sensors were placed in two different studies using the six techniques in order to

best identify the 12 target FEM mode shapes and frequencies. First, the techniques were used

to choose 18 of the 240 DOFs as sensor locations. In the second study, the techniques were








constrained to choose 18 triaxially constrained sensors (i.e., 6 triax-sensor sets). The

excitations selected using the GMIF discussed in the previous section were used to excite the

MPI structure numerically in order to test the various sensor configurations.


7.4.1 Unconstrained Sensor Placement


The first placement study evaluated the six techniques' placement of 18 sensors on the

MPI structure at any of the 240 DOFs (x,y,z of the 80 node balls). The first 12 flexible modes

of vibration were chosen as the target modes for each technique. The locations of the sensors

obtained using each of the techniques are pictured in Figure 7.7.

All of the techniques evaluated placed a majority of the 18 sensors at the ends of the three

booms. In addition all of the techniques except MIF placed sensors in the two DOFs for each

boom which exhibited the greatest range of motion (i.e., xy for the primary boom, xz for the

extending right boom and yz for the extending left boom). The EVP technique clustered all

18 sensors at the boom tips, and the AKE techniques clustered 17 of the 18 sensors at the

boom tips, with one sensor being placed near the mid-span of an extending boom. The KE

technique placed 15 sensors at the boom tips with 3 sensors near the mid-span of the two

extending booms. The El technique placed 13 sensors at the boom tips and at least one sensor

near the mid-span of the main and extending booms. The observability technique placed 17

of the 18 sensors at the boom tips and 1 sensor at the mid-span of an extending boom. The

MIF technique resulted in the most unusual sensor configuration with several sensor being

placed in the z-direction on the main boom. The z-direction is not the primary direction of

motion for this portion of the truss.

Of the twelve target modes shapes, modes 2 through 11 exhibit a bending mode similar

to that of second-mode-cantilevered-beam bending in at least one of the main or extending

booms. Second-mode bending is clearly exhibited by the two extending booms for all target

modes and is exhibited by the main boom for some of the twelve target modes. The sensor

configurations chosen by the El and KE techniques are particularly suited to capture this








second-bending mode shape due to their placement of some sensors at mid-spans of the three

booms.

The FEM of the MPI structure was used with MATLAB to simulate a time response of

the structure to an impact applied at the GMIF chosen excitation locations for all 240 DOFs.

Five percent uncorrelated noise was added to the time response. The noisy response was then

partitioned to each of the six sensor sets and was sent to ERA for mode shape and frequency

identification.

All the techniques resulted in percent frequency difference between FEM and identified

frequencies of much less than 1% which is well within industry accepted standards (Flanigan

and Hunt, 1993). Cross-orthogonalities between FEM and identified mode shapes were

calculated for each of the techniques and are pictured in Figure 7.8. In order to calculate the

cross-orthogonalities, the 240 DOF FE mass matrix was reduced to 12 DOFs using exact

reduction. For this size model, exact reduction was computationally acceptable, therefore, it

was used to get the best cross-orthogonality comparison.

All of the off-diagonal elements of the cross-orthogonality matrix for the EI technique

are within the industry accepted standards of <0.02 for primary modes (Flanigan and Hunt,

1993). This can be seen graphically in Figure 7.8. For the KE, AKE, and MIF techniques,

almost all of the off-diagonal elements are <0.02. Some of the entries are between 0.02 and

0.1 which is within the industry standard for secondary modes (<0.1). The OBS technique

resulted in the most modes having cross-orthogonalities <0.10 which is acceptable for

secondary modes. The 8th mode was not successfully identified by the OBS sensor set. The

cross-orthogonality for the EVP technique was poor for all target modes. The

cross-orthogonality of the EVP technique was evaluated with no noise, 1%, 2%, and 5%

noise added to the time response. Of the four time responses evaluated, only the response

with no noise gave acceptable cross-orthogonality values. Based on these calculation, the

EVP technique was unsuccessful in finding an acceptable sensor set.









Effective Independence


Kinetic Energy Average Kinetic Energy


Observability


z

y
x


MIF


*z
My
Ox


Figure 7.7 Unconstrained MPI Sensor Sets


Eigenvector Product








a) Effective Independence


1.00
<0.10
<0.02


b) Kinetic Energy


1.00
<0.10
<0.02


Figure 7.8 Cross-Orthogonality Between MPI FE and Identified Modes
18 Unconstrained Sensors


1.00




0.10
0.02









1.00




0.10
0.02









1.00




0.10
0.02









d) Eigenvector Product


1.00L
<0.75
<0.50
<0.25
<0.02


e) Observability


J 1.00
< <0.50
* <0.25
* <0.10
* <0.02

e) MIF


1.00
<0.10
<0.02


Figure 7.8 continued


1.00
0.75
0.50
0.25
0.02









1.00


0.50
0.25
0.10
0.02









1.00




0.10
0.02







7.4.2 Triaxially Constrained Sensor Placement


The six sensor placement techniques were modified to place 6 triaxially constrained

sensor sets (18 total sensors) at any of the 80 node balls of the MPI structure. The resulting 6

triax-sensor sets placed using the six placement techniques are pictured in Figure 7.9. The

EI, KE, and AKE techniques grouped two sensor sets at or near the end of each boom, the

OBS technique put one, two, and three sensors at the end of each boom, and the EVP

technique placed three triax-sets at the ends of only two of the booms. It should be noted that

if the EVP placement task were extended to placing 7 triax-sets, the seventh set would be

placed at the end of the main boom using the EVP technique. The MIF sensor placement

technique put two sensors at the end of the right boom, one sensor at the end of the left boom,

one sensor at the base of the left boom, and two sensors on the main boom. As in the case of

the unconstrained set, the MIF technique did not place any sensors on the tip of the main

boom.

The time response of the MPI structure excited by the GMIF actuator locations was

partitioned to those DOFs corresponding to the six triax sensor locations chosen by the six

placement techniques. The partitioned numerical data with noise added was sent to ERA in

order to evaluate the effectiveness of each of the triax-sensor sets in identifying the system

mode shapes and frequencies. All the techniques resulted in percent frequency difference

between FEM and identified frequencies of much less than 1% which is well within industry

accepted standards. The cross-orthogonality calculations between the FEM target modes

and the ERA identified modes were performed using an exactly reduced mass matrix as in the

previous section, and are shown in Figure 7.10. All of the off-diagonal cross-orthogonality

values for the KE techniques, shown in Figure 7.10, were within the industry standard of

<0.02 for off-diagonal elements for primary modes.

The EI, AKE, and MIF techniques resulted in cross-orthogonalities which were within

this standard for most of the modes, but which were slightly above the off-diagonal standard

for a few modes. The OBS technique resulted in acceptable cross-orthogonalities for most








modes and acceptable secondary cross-orthogonalities for modes 7 through 10. The EVP

technique resulted in poor off-diagonal cross-orthogonality values for all modes.


Effective Independence


Eigenvector Product


z


x


Energy


MTF


* triaxial sensor group


Figure 7.9 Triaxially Constrained MPI Sensor Sets








a) Effective Independence


1.00




0.10
0.02
E 1.00 modest

N <0.10 3 2 3 4
E <0.02 1

b) Kinetic Energy


1.00




0.10
0.02
2 1 1.1

* <0.10 modes 6
* <0.02

c) Average Kinetic Energy


1.00




0.10
0.02
El 1.00
S<0.10 modes
* <0.02
Figure 7.10 Cross-Orthogonality Between MPI FE Modes and Identified Modes
6 Triaxially Constrained Sensors









d) Eigenvector Product


1.00
<0.75
<0.50
<0.25
<0.02


e) Observability


1.00
<0.10
<0.02


Figure 7.10 continued


1.00
0.75
0.50
0.25
0.20









1.00




0.10
0.02









1.00




0.10
0.02








7.4.3 Unconstrained vs. Triaxially-Constrained Sensor Sets


Based on the cross-orthogonalities and frequency differences between FE and identified

mode shapes and frequencies the EI, KE, AKE, and MIF techniques located sensor sets for

the unconstrained and constrained examples which were successful in identifying the target

mode set. Only a few of the cross-orthogonalities were slightly above acceptable primary

mode values, and all values were within secondary mode standards. For the unconstrained

sensor set, the El sensor set resulted in identified modes with the best cross-orthogonality

with the FE target modes. However, for the triaxially-constrained example, the KE sensor

set resulted in identified modes with the best cross-orthogonality with FE target modes. In

both constrained and unconstrained cases the EVP technique resulted in identified modes

with poor cross-orthogonalities with FE mode shapes. However, the EVP and OBS triaxially

constrained sensor sets showed an improvement over the unconstrained sets. This is unusual

because as a rule, the constrained sets do not perform as well as the unconstrained sets.


7.5 Effect of Model Error

In order to investigate the effect that model error has on the various placement

techniques, error was added to the original Guyan-reduced FEM of the MPI structure as seen

in Figure 7.11. Specifically, 1/3 of the struts' cross-sectional areas were decreased by 20%,

1/3 of the struts' cross-sectional areas were increased by 20%, and the remaining 1/3 of the

struts were unchanged.

The resulting differences in pre-corrupted and post-corrupted model frequencies and

mode shapes are listed in Table 7.3. The second column represents the percent differences in

the frequencies of the two models. The third column represents the root mean squared

(RMS) values of the absolute differences in the mode shapes of the two models. The

differences between the pre- and post-corrupted model mode shapes are shown pictorially in

Figure 7.12. The true modes are plotted along the horizontal axis and the corrupted modes

are plotted along the vertical axis.





81






AA = change in cross-sectional area


-20% AA






0% AA


+20% AA


Figure 7.11 Model Error Added to MPI FEM


0.2





2 -0.1 0 0.1 0.2
mode I




0

-0.



2 -0.1 0 0.1 0.2
mode 5

.02 -0.1 0 0.1 0.2
0





01..2 -0.1 0 0.1 0.2


- 2 -0.1 0 0.1 0.2

0 mode

0:2 -0.1 0 0.1 0.2
0,


S.2 -0.1 0 0.1 0.2



-0.2 -0.1 0 0.1 0.2


-6.2 -0.1 0 0.1 0.2



.01 2 -0.1 0 0.1 0.2


Figure 7.12 True vs. Corrupted MPI Mode Shapes








Table 7.3 Difference Between Pre- and Post-Corrupted Model Frequencies and Mode
Shapes

MODE Frequency % difference Mode Shape RMS values
1 3.15 0.90 e-3
2 3.23 5.20 e-3
3 2.01 5.20 e-3
4 3.27 4.70 e-3
5 1.27 5.60 e-3
6 1.27 10.9 e-3
7 1.76 13.9 e-3
8 0.45 11.8 e-3
9 2.23 9.00 e-3
10 3.47 10.4 e -3
11 0.12 14.7 e-3
12 1.96 14.1 e-3


7.5.1 Excitation Placement with Model Error

Once error was introduced into the MPI FEM, the GMIF and GCON excitation

placement techniques using the corrupted FEM were run. The GMIF derived excitations for

the uncorrupted and corrupted models are pictured in Figure 7.13, and the GCON derived

excitations for the uncorrupted and corrupted models are pictured in Figure 7.14. The GMIF

excitation placement changed slightly when the model error was added; only node 77

switched to node 76 when model error was added. The GCON excitation moved the shaker

from the mid span to the tip of the left boom. The directions for all of the exciters were

changed when model error was added.

In order to evaluate the excitations obtained using the corrupted FEM, the time response

of the MPI structure to impacts at the new excitations was numerically simulated using the

original uncorrupted model. This time response was then partitioned to the uncorrupted

unconstrained El sensor set of section 7.4.1 and was sent to ERA for identification. As in the

case of the uncorrupted model excitations, the target frequencies and mode shapes were










node 19
0.2471 x
node 77 0.9618 y
0.5225 x 0.1178z
0.8323 y
0.1538 z

Uncorrupted FEM

sensor
z
y ~node 19
x -0.5036 x
0.6800 y
0.5329 z
node 76
-0.5890 x
-0.5714 y
0.5714 z

Corrupted FEM

Figure 7.13 GMIF Derived Excitation Locations

successfully identified based on percent difference and cross-orthogonality calculations.

Based on these results, the error added to the FEM had little to no effect on the excitation

placement configurations' success in exciting the uncorrupted target mode shapes of the

structure for both the GMIF and GCON excitation placement techniques.

7.5.2 Sensor Placement with Model Error

Both the unconstrained and triaxially constrained sensor placement problems were

evaluated after error was added to the FEM using the six placement techniques previously

discussed. The changes in sensor set configurations for the unconstrained and constrained

sets are shown pictorially in Figure 7.15 and Figure 7.16. The original sensors placed using

the uncorrupted FEM are represented by the boxes. Any sensors that were removed from the

original sensor set after model error was introduced are represented by circles and any








node 6
0.8660 x
-0.5000 y
0.0000 z


node 67
0.000 x
0.000 y
-1.000 z
Uncorrupted FEM

z sensor node 53
|-0.5000 x
) Y 0.8660 y
0.0000 z
node 79
0.4330 x
0.2500 y
-0.8660 z


Corrupted FEM



Figure 7.14 GCON Derived Excitation Locations

sensors that were added to the original set after model error was introduced are represented

by triangles.

The total numbers of sensors that changed for the unconstrained and triaxially

constrained sensor sets after model error was introduced are listed in Table 7.4. The general

distribution of the sensors was mostly maintained after model error was added for all of the

unconstrained sensor sets except for the MIF sensor set. For the constrained sensor sets, five

of the six placement techniques resulted in a changed sensor set after model error was added.

The El technique moved one triax-set from the main boom tip to mid-boom. The KE

technique moved one triax-set from the left extending boom tip to mid-extending-boom, and

the EVP technique moved a triax-set from the left extending boom to the main boom. The

AKE technique resulted in no sensor change after model error was added. The OBS and MIF




85





m Y
l] X







Effective Independence Eigenvector Product





Z

x




Kinetic Energy Average Kinetic Energy



Observability z MIF

y














0 original sensor set O removed using model error A added using model error

Figure 7.15 Model Error Effect on Unconstrained MPI Sensor Sets







Effective Independence


Kinetic Energy Average Kinetic Energy


z

_y


Observability


MIF


* original triaxial sensors A added using model error
removed using model error
Figure 7.16 Model Error Effect on Constrained MPI Sensor Sets


Eigenvector Product








technique changed over half the triaxially constrained sensors. For both the unconstrained

and triaxially constrained cases, the MIF sensor placement technique was particularly

sensitive to FE model error.

The original uncorrupted FEM response to the GMIF derived excitation was used to

evaluate the new sensor sets obtained with the corrupted FEM. The time response discussed

in the previous section was partitioned to the new sensor configurations and ERA was used to

identify mode shapes and frequencies. Both the unconstrained and constrained sensor sets

obtained using the corrupted FEM were successful in identifying the target frequencies

within 1%, for all six techniques evaluated. The resulting cross-orthogonalities between

identified (using error sensor sets) and original FEM mode shapes were calculated and are

pictured in Fig. 7.17 and Fig. 7.18.


Table 7.4 Number of sensors or triax sets that change when model error is added

MPI Sensor Set El KE AKE EVP OBS MIF
unconstrained 2 of 18 2 of 18 1 of 18 6 of 18 2 of 18 16 of 18
constrained 1 of 6 1 of 6 0 of 6 2 of 6 3 of 6 4 of 6



For the unconstrained sensor sets, the EI, KE, AKE, and MIF techniques resulted in

generally acceptable cross-orthogonality values for the twelve target modes shown in Fig.

7.17. Only a few off-diagonal entries of the cross-orthogonalities resulting from these sensor

configuration were above the acceptable limit of <0.02 for primary modes, but were still

within the acceptable limit of <0.10 for secondary modes. The error added to the model in

this example had little effect on the placement techniques' success in identifying sensor

configurations which resulted in successful modal information identification. The EVP and

OBS techniques resulted in poor cross-orthogonalities as was the case when the uncorrupted

model was used.

For the triaxially constrained sensor configuration, the model error did not greatly affect

the uncorrupted cross-orthogonality results for the EI, KE, AKE, and MIF techniques, as




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