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STRUCTURAL DAMAGE ASSESSMENT AND FINITE ELEMENT MODEL REFINEMENT USING MEASURED MODAL DATA By MOHAMED KAOUK A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1993 A Mes Parents Tout le m6rite de ce travail, s'il en est, vous revient. ACKNOWLEDGEMENTS I would like to express my sincere gratitude toward my advisor, Dr. David Zimmerman, for his expert guidance, friendship, endless encouragement and support. I will forever be indebted to him for inspiring me in my research and for the priceless education I acquired from him. I am also grateful for the financial support he provided me during the course of my graduate studies. Words are not enough to express my deepest gratitude to my parents for their love, moral and financial support; it is these things that have made this work possible. I also wish to acknowledge my sisters and brothers for their continuous encouragement. I would like to thank the members of my supervisory committee, Drs. Norman FitzCoy, Lawrence Malvern, Bhavani Sankar, and Kermit Sigmon, for their helpful advice. I wish to thank my colleagues of the Dynamic Systems and Control Laboratory for their consideration and entertaining discussions. In particular, William Leath and Cinnamon Larson for their friendship and encouragement. I would like to thank the staff of the Aerospace Engineering, Mechanics, and Engineering Science department for their assistance throughout the years, especially Shirley Robinson for making my life easier during registration, John Young for his prompt responses in fixing my computer problems, and Ronald Brown for his assistance in the machine shop and for many stimulating discussions. I would like to acknowledge the financial support received from Harris Corporation, NASA/Florida Space Grant Consortium and Florida High Technology Council. I would like to thank Dr. T. A. Kashangaki of the NASA Langley Research Center and Dr. S. W. Smith of the University of Kentucky for providing the data of the EightBay Truss used in this study. Last, but not least, I am grateful to my good friends Joel Payabyab and Fadel Abdallah for their continuous support. TABLE OF CONTENTS paga ACKNOWLEDGEMENTS ................................................................................ iii LIST OF TABLES ................................................................................................. ix LIST OF FIGURES ............................................................................................... x A B ST R A C T ............................................................................................................ xiii CHAPTERS 1 INTRODUCTION ................................................................................. 1 1.1 Finite Element Model Refinement ................................................. 1 1.1.1 O verview ........................................................................... 1 1.1.2 Literature Survey ............................................................. 3 1.2 Structural Damage Assessment ....................................................... 6 1.3 Objective of the Present Study ....................................................... 8 2 MATHEMATICAL PRELIMINARIES AND PRACTICAL ISSUES TO THE PROBLEMS OF MODEL REFINEMENT AND DAMAGE DETECTION ............................... 11 2.1 Introduction ..................................................................................... 11 2.2 The Eigenvalue Problem of Discrete Systems ................................ 11 2.2.1 Undamped Models .............................................................. 15 2.2.2 Proportional Damped Models ............................................ 16 2.3 Experimental Modal Analysis ....................................................... 16 2.4 Analytical/Experimental Model Dimensions Correlation ............. 19 2.4.1 Model Reduction Methods .................................................. 19 2.4.1.1 Static Reduction ................................................. 21 2.4.1.2 IRS Reduction ....................................................... 22 2.4.1.3 Dynamic Reduction ............................................... 22 2.4.2 Eigenvector Expansion Methods ........................................ 23 2.4.2.1 Dynamic Expansion ............................................ 23 2.4.2.2 Orthogonal Procrustes Expansion ........................ 24 2.5 Eigenvector Orthogonalization ..................................................... 26 2.5.1 Optimal Weighted Orthogonalization .................................. 27 2.5.2 Selective Optimal Orthogonalization .................................. 27 2.6 Load Path Preservation .................................................................. 28 3 INVERSE / HYBRID PROBLEM APPROACH FOR FINITE ELEMENT MODEL REFINEMENT ................................................ 31 3.1 Introduction ................................................................................... 31 3.2 Theoretical Formulation ................................................................. 31 3.3 Numerical Illustrations .................................................................. 35 3.3 Sum m ary ......................................................................................... 37 4 SYMMETRIC EIGENSTRUCTURES ASSIGNMENT MODEL REFINEMENT ALGORITHM ....................... 39 4.1 Introduction .................................................................................... 39 4.2 Problem Formulation ..................................................................... 39 4.2.1 Standard Eigenstructure Assignment Formulation ............. 40 4.2.2 Symmetric Eigenstructure Assignment Formulation ......... 42 4.2.3 Best Achievable Eigenvectors ............................................. 44 4.2.4 Selection of Bo : The Subspace Rotation Method .............. 46 4.3 Numerical Illustrations ................................................................. 47 4.3.1 Damage Detection: Kabe's Problem ................................... 48 4.3.1.1 Local to Global Mode Change ............................ 49 4.3.1.2 Consistent Modes ................................................. 52 4.3.2 Model Refinement of a Cantilever Beam: Experimental Study .......................................................... 55 4.3.2.1 Modal Test Description ......................................... 55 4.3.2.2 Finite Element Model Description ........................ 56 4.3.2.3 Application of SEAMRA ...................................... 57 4.4 Discussion of the SEAMRA's Formulation ................................... 57 4.5 Sum m ary ......................................................................................... 59 5 DAMAGE LOCATION: THE SUBSPACE ROTATION ALG O RITH M ................................................................................. 60 5.1 Introduction .................................................................................. 60 5.2 The Subspace Rotation Algorithm: The Direct Method ................ 60 5.3 The Subspace Rotation Algorithm: The Angle Perturbation Method ................................................................. 63 5.4 Practical Issues .............................................................................. 64 5.4.1 Cumulative Damage Vectors ............................................... 64 5.4.2 Eigenvector Filtering Algorithm ......................................... 65 5.5 Sum m ary ......................................................................................... 66 6 THE MINIMUM RANK PERTURBATION THEORY............... 67 6.1 B background .................................................................................. 67 6.2 The Minimum Rank Perturbation Theory: Theoretical Background ............................................................ 68 6.3 Damage Extent: Undamped Structures ......................................... 72 6.3.1 Damage Extent: Mass Properties ........................................ 73 6.3.2 Damage Extent: Stiffness Properties .................................. 74 6.3.3 Damage Extent: Mass and Stiffness Properties ................... 76 6.3.3.1 Application of the MRPT ...................................... 76 6.3.3.2 Decomposition of Matrix B ................................. 77 6.4 Damage Extent: Proportionally Damped Structures ..................... 79 6.4.1 Damage Extent: Stiffness and Damping Properties ............ 79 6.4.2 Damage Extent: Mass and Damping Properties .................. 82 6.4.3 Damage Extent: Mass and Stiffness Properties ................... 84 6.4.4 Damage Extent: Mass, Damping and Stiffness Properties .... 85 6.5 Damage Extent: Nonproportionally Damped Structures .............. 87 6.5.1 Damage Extent: Damping and Stiffness Properties ........... 89 6.5.2 Damage Extent: Mass and Damping Properties .................. 90 6.5.3 Damage Extent: Mass and Stiffness Properties .................. 92 6.6 Practical Issues .............................................................................. 94 6.6.1 The Concept of "Best" Modes ............................................. 95 6.6.2 Application of the Eigenvector Filtering Algorithm ............ 97 6.7 Sum m ary ....................................................................................... 97 7 VALIDATION AND ASSESSMENT OF THE SUBSPACE ROTATION ALGORITHM AND THE MINIMUM RANK PERTURBATION THEORY ............................................. 98 7.1 Introduction .................................................................................. 98 7.2 Kabe's Problem .......................................................................... 98 7.2.1 Problem Description ........................................................... 98 7.2.2 Damage Location ............................................................... 100 7.2.3 Damage Extent ............................................................... 103 7.3 Damage Detection: FiftyBay TwoDimensional Truss: Undamped FEM ........................................................ 105 7.3.1 Problem Description .......................................................... 105 7.3.2 Damage Location ............................................................... 106 7.3.3 Damage Extent .................................................................. 107 7.4 Experimental Study: The NASA 8Bay Truss ............................ 112 7.4.1 Problem Description .......................................................... 112 7.4.2 Refinement of the Original FEM ....................................... 115 7.4.3 Damage Location ............................................................... 118 7.4.4 Damage Extent .................................................................. 120 7.4.4.1 The Brute Force Method ..................................... 121 7.4.4.2 The Damage Consistent Method ........................ 121 7.4.4.3 Application of the Eigenvector Filtering Algorithm ........................................................ 122 7.5 Experimental Study: Mass Loaded Cantilevered Beam .............. 140 7.5.1 Problem Description .......................................................... 140 7.5.2 Analytical and Experimental Models Dimension C orrelation ................................................................... 141 7.5.3 Refinement of the Original FEM ...................................... 141 7.5.4 Damage Location .............................................................. 142 7.5.5 Damage Extent .................................................................. 143 7.6 FiftyBay TwoDimensional Truss: Proportionally D am ped FEM ............................................................................. 144 7.6.1 Problem Description .......................................................... 144 7.6.2 Damage Location ............................................................... 145 7.6.3 Damage Extent ................................................................... 146 7.7 EightBay TwoDimensional MassLoaded Cantilevered Truss ................................................................... 148 7.7.1 Problem Description .......................................................... 148 7.7.2 Proportionally Damped Configuration: Damage of Small Order of Magnitude ............................................... 149 7.7.2.1 Damage Location ................................................. 150 7.7.2.2 Decomposition of Matrix B ................................. 150 7.7.2.3 Damage Extent .................................................... 152 7.7.3 Undamped Configuration: Damage of Large Order of Magnitude ......................................................... 153 7.7.3.1 Noise Free Eigendata ........................................... 153 7.7.3.2 Noisy Eigendata .................................................. 154 7.8 Sum m ary ........................................................................................ 156 8 CONCLUSION AND SUGGESTIONS FOR FUTURE WORK ......... 159 REFEREN CES .................................................................................................... 162 BIOGRAPHICAL SKETCH ............................................................................... 167 LIST OF TABLES Table Pa 3.1 Kabe's Problem: Elemental Stiffness Components ................................. 38 4.1 Properties of the Cantilever Beam ............................................................ 56 4.2 Measured Natural Frequencies and Damping Ratios of the Cantilever Beam ............................................................................ 56 4.3 Measured Mode Shapes of the Cantilevered Beam ................................... 56 7.1 FiftyBay Truss: Summary of Damage Detection Results using the M RPT ........................................................................ Ill 7.2 Mass Properties of the Eight Bay Truss .................................................... 113 7.3 Strut Properties of the Eight Bay Truss ..................................................... 114 7.4 NASA 8Bay: Truss Damage Case Definitions ..................................... 115 7.5 Comparison of Analytical and Experimental Frequencies ....................... 116 7.6 Summary of the Filtering Process for Single Member Damage Cases ......... 123 7.7 NASA 8Bay Truss: Summary of the Damage Assessment Results ........... 139 7.8 Mass Loaded Cantilevered Beam Properties ......................................... 140 7.9 Analytical and Experimental Frequencies of the "Healthy" Structure ...... 142 7.10 50Bay 2Dimensional Truss: Summary of the Percentage Error with Respect to the Exact Damage .......................................... 147 7.11 Problem 7.7: Percentage Error of Damage Estimate with Respect to Exact Dam age ...................................................................... 156 LIST OF FIGURES Figure Page 1.1 Overview of Finite Element Model Refinement ...................................... 2 1.2 Overview of FEM Model Refinement Process Used for D am age A ssessm ent ............................................................................ 7 2.1 Components of a Vibration Measurement System for M odal A analysis .................................................................................. 17 2.2 A simple Experimental Modal Analysis Setup ...................................... 18 2.3 Flow Chart of the Iterative Load Preservation Path Algorithm ............. 29 3.1 Kabe's Problem: Analytical Test Structure .............................................. 36 4.1 Best Achievable Eigenvector Projection ................................................ 45 4.2 Rotation of the Achievable Subspace ..................................................... 47 4.3 K abe's Problem ..................................................................................... 48 4.4 Results for Kabe's Problem using the 1st Mode, Full Eigenvector ............................................................................... 50 4.5 Results for Kabe's Problem Modes 1, 2, 3 and Eigenvectors Components 1, 2, 3 ......................................................... 51 4.6 Results for Kabe's Problem using Load Path Preservation, Modes 1, 2, 3 and Eigenvectors Components 1, 2, 3 ............................. 52 4.7 Results for Kabe's Problem using the 1 st Mode, Full Eigenvector ................................................................................. 53 4.8 Results for Kabe's Problem Modes 1,2,3 and Eigenvectors Components 1, 2, 6 ........................................................ 54 4.9 Experimental Cantilever Beam ............................................................. 55 4.10 Experimental and Analytical Frequency Response Function of the Cantilever Beam ......................................................................... 58 7.1 K abe's Problem ...................................................................................... Y' 7.2 Kabe's Problem: Damage Location Results using the Subspace Rotation Direct Method with the Eigendata of the 1st Mode ............... 101 7.3 Kabe's Problem: Damage Location Results using the Angle Perturbation Method with the Eigendata of the 1st Mode ................... 101 7.4 Kabe's Problem: Damage Location Results using Lin's Algorithm with the Eigendata of the 1st Mode .................................... 102 7.5 Kabe's Problem: Damage Location Results using the Angle Perturbation Method with the Eigendata of the 1st and 2nd Modes ....... 102 7.6 Kabe's Problem: Damage Location Results using Lin's Algorithm with the Eigendata of the 1st and 2nd Modes .................... 103 7.7 Kabe's Problem: Damage Extent Results using the MRPT with the Eigendata of M ode 2 ............................................................... 104 7.8 Kabe's Problem: Damage Extent Results using Baruch's Method ........... 105 7.9 FiftyBay TwoDimensional Truss ...................................................... 106 7.10 FiftyBay Truss: Damage Location Results using the Subspace Rotation Algorithm with the Eigendata of the First Ten Modes .......... 107 7.11 FiftyBay Truss: Damage Extent Results using the MRPT with the Eigendata of Modes 8 and 9 ..................................... 109 7.12 FiftyBay Truss: Damage Extent Results using the MRPT with the Eigendata of the First Ten Modes ............................. 110 7.13 FiftyBay Truss: Damage Extent Results using Baruch's Algorithm ....... 110 7.14 The NASA EightBay HybridScaled Truss: Damage Cases ................ 112 7.15 The NASA 8Bay Truss: Lacing Pattern ................................................ 113 7.16 NASA 8Bay Truss: Typical Frequency Response Comparison .............. 117 7.17 NASA 8Bay Truss: Perturbation to the Original Stiffness Matrix that Resulted From the Refinement Process .............................. 118 7.18 NASA 8Bay Truss: Cumulative Damage Vector Associated w ith C ase F ......................................................................................... 124 7.19 NASA 8Bay Truss: Damage Assessment of Case A .............................. 125 7.20 NASA 8Bay Truss: Damage Assessment of Case C .............................. 126 7.21 NASA 8Bay Truss: Damage Assessment of Case D .................................. z/ 7.22 NASA 8Bay Truss: Damage Assessment of Case E ............................... 128 7.23 NASA 8Bay Truss: Damage Assessment of Case G ............................... 129 7.24 NASA 8Bay Truss: Damage Assessment of Case H ............................... 130 7.25 NASA 8Bay Truss: Damage Assessment of Case I ............................ 131 7.26 NASA 8Bay Truss: Damage Assessment of Case J ............................ 132 7.27 NASA 8Bay Truss: Damage Assessment of Case K ............................ 133 7.28 NASA 8Bay Truss: Damage Assessment of Case L ............................ 134 7.29 NASA 8Bay Truss: Damage Assessment of Case M ............................ 135 7.30 NASA 8Bay Truss: Damage Assessment of Case N ............................ 136 7.31 NASA 8Bay Truss: Damage Assessment of Case 0 ............................ 137 7.32 NASA 8Bay Truss: Damage Assessment of Case P ............................ 138 7.33 The Mass Loaded Cantilevered Beam .................................................... 140 7.34 Mass Loaded Cantilevered Beam: Damage Assessment ....................... 143 7.35 50Bay 2Dimensional Truss .................................................................. 145 7.36 50Bay 2Dimensional Truss: Damage Location .................................... 146 7.37 50Bay 2Dimensional Truss: Damage Extent ....................................... 148 7.38 The EightBay TwoDimensional MassLoaded Cantilevered Truss ....... 149 7.39 Problem 7.7: Cumulative Damage Location Vector: First Four Modes ...... 150 7.40 Problem 7.7: Cumulative Vectors Associated with the Exact and Computed Bin, Bd, Bk: First Three Modes .................................... 151 7.41 Problem 7.7: Exact and Computed AMd, ADd, AKd ............................... 152 7.42 Problem 7.7: Cumulative Vector Associated with B, Bm, Bd, Bk B Computed using Modes 14 Bm, Bd, Bk Computed using Modes 3,4 & 5 ..................................... 155 7.43 Problem 7.7: Exact and Computed AMd, ADd, AKd ............................... 157 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirement for the Degree of Doctor of Philosophy STRUCTURAL DAMAGE ASSESSMENT AND FINITE ELEMENT MODEL REFINEMENT USING MEASURED MODAL DATA By Mohamed Kaouk August 1993 Chairperson: Dr. David C. Zimmerman Major Department: Aerospace Engineering, Mechanics and Engineering Science This study investigates the problems of model refinement and structural damage assessment. The essence of the model refinement problem is to adjust finite element models (FEMs) of structures with the intent of producing a correlation between experimental and analytical modal properties. Recently, the framework of model refinement has been adopted to determine the location and extent of structural damage. Damage will result in changes to the modal properties of the healthy structure. A further refinement of an accurate FEM of the structure using damaged modal parameters is expected to generate adjustments to the FEM at locations associated with damage. Analysis of these adjustments can then be used to assess damage. In this investigation, four algorithms relevant to the subjects of model refinement and damage detection are presented. The development of a model refinement algorithm with its basis in inverse theory is first presented. The algorithm, termed the inverse/hybrid approach, is illustrated in a comparative computer simulated study. Next, an existing eigenstructure assignment model refinement algorithm is improved to better approach the damage assessment problem. The enhanced algorithm is evaluated and compared to other techniques using simulated and experimental data. The algorithm is shown to perform well in assessing damage and refining FEMs. A damage location algorithm that bypasses the general framework of model refinement is discussed. The damage location algorithm, termed the subspace rotation, is similar to the modal force error criteria proposed by several researchers. Greater insight to the modal force error criteria, along with a new viewpoint that reduces the effects of measurement noise, is discussed. Furthermore, an efficient damage extent algorithm based on a minimum rank perturbation theory (MRPT) is developed. The formulation of the MRPT is consistent with the effect of most structural damage on FEMs. The characteristics of the subspace rotation algorithm and the minimum rank perturbation theory are illustrated using simulated and experimental testbeds. The decomposition of the damage assessment problem into location and extent subproblems is shown to be advantageous from both for computational efficiency and for engineering insight. CHAPTER 1 INTRODUCTION 1.1 Finite Element Model Refinement 1.1.1 Overview An important aspect in the design process of a structure is the evaluation of its performance under expected dynamic loading conditions. Dynamic performance can be analyzed by using either analytical or experimental techniques. Experimental analyses are generally very costly, time consuming and can encounter technical difficulties. One example of a commonly encountered technical barrier is the ground base reproduction of a weightless environment for the testing of space structures. This shortcoming and the advent of computers have sparked a growing interest in the uses of analytical techniques. This type of analysis utilizes a mathematical model of the actual structure for computer simulated evaluation of structural performance. Presently, most analytical models used in industry are finite element models generated by using Finite Element Methods. The accuracy of a finite element model (FEM) can be improved to some extent by increasing the number of degrees of freedom (DOFs) included in the model. However, accuracy of the FEM may still be lacking due to uncertainties in material properties, geometric layout and fabrication induced errors. Thus, it is essential to "validate" these FEMs prior to their acceptance as a basis for analysis. One way to validate them is to compare their modal properties (eigenvalues/ eigenvectors) with the measured modal parameters of the actual structure as obtained from experimental modal analysis (EMA). A FEM is acceptable when these two sets of modal properties are in agreement. Unfortunately, this agreement rarely occurs. As a result, the FEM must be adjusted to produce a correlation of analytical and experimental modal properties. An overview of this procedure is schematically illustrated in Figure 1.1 (Zimmerman and Smith, 1992). Experimental Analytical Frequency Response Mass Functions Damping or Stiffness Hankel Matrices / Matrices Modal Parameters j Modal Parameters Yes Done (Confidence in FEM established) Figure 1.1 Overview of Finite Element Model Refinement (Zimmerman and Smith, 1992) In the past, the process of adjusting FEMs was performed on an adhoc basis aided by engineering experience. This practice was naturally time consuming and in most cases inadequate for largeorder, complex FEMs that are typically needed for accurate dynamic modelling. In the past years, growing interest has been focused on developing systematic procedures to produce correlated FEMs. These efforts have resulted in the development of a large number of algorithms. Finite element model adjustment procedures have been commonly referred to in the literature as, FEM refinement, FEM adjustment, FEM correction, FEM correlation, and FEM identification. 1.1.2 Literature Survey Comprehensive literature surveys covering a large portion of the work that addresses the model refinement problem can be found in the book chapter by Zimmerman and Smith (1992), and in the papers by Ibrahim and Saafan (1987) and Heylen and Saas (1987). For completeness and to properly underline the objective of the current study, it is appropriate to include a brief survey of the development made in this area. The concept of using experimental modal data in analytical studies was initiated by an early work presented by Rodden (1967). In his work, Rodden explored the possibilities of generating mass and stiffness property matrices by using experimentally measured modal data. The experimentally generated mass and stiffness matrices were nonsymmetric. Brock (1968) improved the work of Rodden by proposing a strategy to insure symmetry of property matrices. The essence of the model refinement concept, as adopted by most researchers, is to modify the finite element model (FEM) of systems with the intention of producing a correlation of test and analytical modal parameters. In general, this task has been approached through two different philosophies. The first amounts to modifying globally the finite element property matrices of the system. This approach is commonly referred to as the property matrix update. The alternative approach is to individually correct parameters of each element of the finite element model. This will be addressed here as the physical parameter update. Property matrix update algorithms can be classified into two subclasses of methodologies: optimalmatrix update and controlbased eigenstructure assignment techniques. The basic philosophy of the optimalmatrix update is to minimize the correction to the FEM property matrices to accomplish the analytical/test modal correlation. The pioneering work in this area can be credited to Baruch and Bar Itzhack (1978). In their formulation, by assuming that the mass matrix is correct, the refinement of the stiffness matrix of an undamped FEM is cast as a constrained minimization problem. The objective of their formulation is to determine the minimal Frobeniusnorm symmetric stiffness adjustment that satisfies the eigenvalue problem in terms of the experimentally measured eigenvalues and eigenvectors. A computationally efficient closed form solution was developed for the updated stiffness matrix. Berman and Nagy (1983) extended the Baruch and Bar Itzhack approach to the refinement of both the mass and stiffness matrices. This same methodology was further extended by Fuh et al. (1984) to update the mass, damping and stiffness matrices of damped FEMs. Fuh and his colleagues used crossorthogonality relationships to correct the mass and damping matrices and a constrained minimization problem similar to the one proposed by Baruch and Bar Itzhack to determine the refined stiffness matrix. The problem of adjusting the mass, damping and stiffness matrices was also attempted by Hanagud et al. (1984). In their approach, all three property matrices of a nonproportionally damped FEM are incorporated in the constrained minimization problem. The previously stated algorithms do not preserve the load path (sparsity pattern) of the original analytical property matrices. Kabe (1985) proposed a reformulation of the Baruch and Bar Itzhack algorithm that constrains the updated stiffness matrix to preserve the load path of the original stiffness matrix. In addition, in his reformulation, he adopted a percent weighting on the stiffness changes instead of mass matrix weighting as used in the Baruch and Bar Itzhack algorithm. Kammer (1987) proposed an alternative solution approach to the problem defined by Kabe that uses projection matrix theory and the MoorePenrose generalized inverse. The algorithms formulated by Kabe and Kammer involve an excessive amount of computational effort. Another alternative and more efficient formulation of the Kabe problem that utilized a generalization of MarwillToint updates was developed by Smith and Beattie (1991). The other subclass of property matrix updates is based on the general framework of controlbased eigenstructure assignment algorithms. The essence of this approach is to determine pseudocontrollers that would assign the experimentally measured modal parameters to the original analytical FEM. The pseudocontrollers are then translated into matrix adjustment to the original property matrices. This approach was initially formulated by Inman and Minas (1990) to adjust the damping and stiffness matrices of the original FEM. In their formulation, the symmetry of the adjusted property matrices was enforced through an iterative process that involves a numerical nonlinear optimization process. Zimmerman and Widengren (1989, 1990) proposed a formulation that replaces the iterative process of Inman and Minas with a generalized algebraic Riccati equation. Further details about this approach are treated in Chapter 4. The alternative philosophy to the model refinement problem is the physical parameter update. The basic idea of this alternative approach consists of adjusting some or all structural physical parameters to produce test/analysis modal correlation. Structural parameters are design variables such as material densities, modulus of elasticity, crosssectional areas, element lengths and others. This type of procedure results in corrected FEMs that are consistent with the framework of the original FEM. Most methodologies that have adopted this philosophy have used the sensitivity derivatives of the system eigenvalues and/or eigenvectors with respect to the structural parameters. Generally, the refinement process amounts to solving for the corrected structural parameters through an iterative optimization problem directed by the sensitivity derivatives. Some notable work in this area was accomplished by Collins et al. (1974), Chen and Garba (1980), Adelman and Haftka (1986), Creamer and Hendricks (1987), Flanigan (1991), Martinez et al. (1991), to name only a few. In these studies, different sensitivity formulations and iterative optimization strategies are proposed. A direct approach that bypasses the use of closedform sensitivity derivatives was presented in the papers of Hajela and Soeiro (1990) and Soeiro (1990). White and Maytum (1976) set forth another alternative methodology that can be classified in the physical parameter update group. In their approach, adjustments to the stiffness matrix are viewed as a collection of known submatrices, elements or group of elements. Each submatrix is multiplied by an unknown scaling factor. Correction of the original stiffness matrix amounts to determining the scaling factors that would improve the correlation of analytical and experimental modal properties. The scaling factors can be physically viewed as functions of the stiffness structural parameters. A large underdetermined system of equations in terms of the scaling factor is generated. The scaling factors are then computed by using a pseudoinverse. An improvement to the White and Maytum approach is presented in the paper by Lim (1990). 1.2 Structural Damage Assessment Structures, in general, are prone to structural damage during their service lives, that could affect their overall performance and could result in catastrophic failures. This is of critical concern when expensive systems and/or human lives are at stake. On this basis, it is highly desirable to monitor the structural health of systems such as airplanes, space systems, bridges, buildings, oil platforms, among others, to prevent such catastrophic events. Currently, the most common structural health monitoring methods involve visual inspections supplemented with Xrays, acoustic emission, magnetic resonance and ultrasonic testing. These approaches can be time consuming, costly and difficult to perform on inaccessible structural components. Additionally, none of these approaches provide a quantitative assessment of the magnitude of the damage. In response to these shortcomings, structural health monitoring approaches based on the framework of existing model refinement techniques have been recently proposed by several researchers (Smith and Hendricks, 1987; Chen and Garba, 1988; Ricles and Kosmatka, 1992; Zimmerman and Kaouk, 1992a,b; Kaouk and Zimmerman, 1993a,b). These techniques utilize the vibration signature of the pre and post damaged structure, in conjunction with an analytical model of the original structure, to assess both the location and extent of the structural damage. The predamage modal parameters are used to correct (refine) the original finite element model (FEM) to determine an "accurate" reference baseline. Once damaged, the postdamage modal properties of the structure are used to further "refine" the refined analytical model. This results in perturbations to the refined analytical model. Analysis of the perturbations could indicate the damage location and extent. An overview of the application of model refinement algorithms in assessing structural damage is summarized in Figure 1.2. Undamaged Model Experimental (Correlated) Frequency Response Mass Functions Damping or Stiffness Hankel Matrices Matrices No Modal Parameters Modal Parameters Yes Done Structure Healthy Figure 1.2 Overview of Finite Element Model Refinement Process Used for Damage Assessment Notable exceptions to the direct use of FEM refinement algorithms to the damage detection problem are the work of Lin (1990), Ojalvo and Pilon (1988), and Gysin (1990). In the work of Lin, a flexibility matrix is determined using experimental data. This matrix is then multiplied by the original stiffness matrix, with those rows and/or columns that differ significantly from a row and/or column of the identity matrix indicating which degrees of freedom have been most affected by the damage. It is then assumed that damage has occurred in structural elements connecting those degrees of freedom. An overview of the work by Ojalvo and Pilon (1988), and Gysin (1990) is given in Chapter 5. Although the problems of damage assessment and model refinement are related to one another, they have quite different characteristics. In essence, the model refinement concept is based on the philosophy of the minimum change to the original FEM. Naturally, this minimum change constraint has a tendency of smearing the changes throughout the entire FEM. This aspect is inconsistent with the effects of structural damage on FEMs which are usually localized perturbations of possibly large magnitude. In fact, structural damage often occurs at discrete locations and only affects a few elements of the FEM. 1.3 Objective of the Present Study The present study investigates the development of new and promising model refinement and damage detection methodologies. Although considerable research has been done in these areas, no methodology has been fully successful in dealing with the refinement or the damage detection problem of "real life" systems. The main objective of this investigation is to formulate efficient model refinement algorithms that are consistent with the effect of structural damage. In chapter 2, practical concepts and issues related to the general areas of model refinement are presented. First, the concept of the eigenvalue problem of discrete structural models is reviewed with emphasis on the associated crossorthogonality conditions. A brief discussion of modal analysis follows. The problem associated with incomplete eigenvector measurements is then investigated and already existing techniques to deal with this problem are presented. Furthermore, eigenvector orthogonalization techniques useful to a large number of model refinement algorithms are reported. Finally, an iterative approach to the problem of load path preservation encountered in a large number of property matrix update algorithms is discussed. In chapter 3, the author proposes and formulates a new model refinement algorithm. The newly developed algorithm, named the inverse/hybrid method, is naturally based on the inverse problem. The model type under consideration is proportionally damped and the inaccuracies of the analytical model is assumed to be due to errors in the modelling of the damping and stiffness properties. A complete hybrid set of modal data is generated by approximating the unmeasured modal information by the corresponding analytical modes. Symmetry of the corrected stiffness and damping matrices is enforced by mass orthogonalization the complete hybrid eigenvector matrix. An orthogonalization procedure that assigns more confidence on the measured eigenvectors is proposed. A comparative study of the inverse/hybrid method and the algorithm proposed by Baruch and Bar Itzhack (1978) shows that both algorithms give similar results. However, it is shown that the inverse/hybrid approach is less computationally efficient. Chapter 4 proposes improvements to the symmetric eigenstructure assignment model refinement algorithm (SEAMRA) formulated by Zimmerman and Widengren (1989). The author develops a technique to enhance eigenvector assignability. The technique, termed the subspace rotation method, is based on rotating the achievable eigenvector subspaces into the experimental eigenvectors. The subspace rotation method results in both a decrease in the computational burden and an increase in the accuracy of the assigned eigenvector. The enhanced SEAMRA is then evaluated and compared to other algorithms using both computer simulated and experimental testbeds. It is shown that the enhanced algorithm is suitable for damage detection applications. In Chapter 5, an efficient damage location algorithm that utilizes modal data information but bypasses the general framework of the model refinement problem is presented. This location algorithm is an outgrowth of the subspace rotation method used to enhance eigenvector assignability of the SEAMRA. The proposed location algorithm is similar to the modal force error criteria presented by several researchers (Ojalvo and Pilon, 1988; Gysin, 1990). Further interpretation of the algorithm operation is given. Additionally, the author proposes and formulates a new viewpoint that reduces the effect of measurement noise for certain types of structures. Based on this formulation, an eigenvector filtering algorithm is also developed. Chapter 6 presents the formulation of computationally attractive damage extent algorithms that are based on a minimum rank perturbation theory (MRPT) developed by the author. The formulation of the MRPT is consistent with the effect of many classes of structural damage on FEMs. Several MRPT based algorithms are formulated to accommodate structures with undamped, proportionally damped, and nonproportionally damped FEMs. For each type of FEM, several damage scenarios are considered. Discussions of the characteristics and properties of the developed algorithms are presented along with practical issues that can be used to improve their performance. In Chapter 7, the algorithms developed in Chapters 5 and 6 are demonstrated and evaluated using both computer simulated and actual experimental data. The main objective of these example problems is to illustrate the potential of these algorithms in assessing structural damage. In all examples, the location of damage is first determined by using the algorithm presented in Chapter 5. An MRPT based algorithm is then utilized to assess the extent of the damage. Several key points made throughout the formulation in Chapters 5 and 6 are emphasized. In particular, it is shown that the damage extent calculations can be greatly enhanced by making use of the damage location algorithm. In Chapter 8, summaries and conclusions of the issues discussed in this study are presented along with some suggestions for future work. CHAPTER 2 MATHEMATICAL PRELIMINARIES AND PRACTICAL ISSUES RELATED TO THE PROBLEMS OF MODEL REFINEMENT AND DAMAGE DETECTION 2.1 Introduction In this chapter, general mathematical preliminaries and practical issues relevant to the areas of model refinement and damage detection are discussed. In Section 2.2, an overview of the general eigenvalue problem for discrete models is given. Further, the concept of modal analysis is introduced in Section 2.3. Section 2.4 deals with the concept of incomplete degrees of freedom measurement. Two alternative approaches are discussed as possible solutions to the incomplete measurement problem. Two eigenvector orthogonalization techniques are discussed in Section 2.5. Finally, in Section 2.6, an algorithm to preserve sparsity in updated property matrices is presented. The concepts discussed in this chapter will be frequently referred to in the course of the forthcoming chapters. 2.2 The Eigenvalue Problem of Discrete Systems In practice, most engineering structures are continuous systems with spatially distributed material properties. The vibration motion of these structures in terms of their distributed properties is usually governed by one or several partial differential equations. With complex systems, both the development and the analysis of partial differential equations of motion are tedious and in many cases impossible. These shortcomings and the advent of digital computers have motivated the development of approximate modelling of continuous systems in order to simplify the equations of motion. The general idea behind these approximations is to represent the exact distributed model of a system by a discrete one. This concept is known as spatial discretization which eliminates the continuous spatial dependence of the distributed properties. The discrete model is of finite order and is described by a finite number of variables known as degrees of freedom (DOFs). The number of DOFs used in the discrete model depends on the desired accuracy in representing the continuous model. Commonly, the vibration motion of systems in terms of their discrete models is described by a set of simultaneous ordinary differential equations that are usually simpler to develop and analyze than the partial differential equations of the continuous system. One of the most commonly used approximate discrete modelling techniques is the finite element method. The model generated by a finite element method is called a finite element model (FEM). More elaborate discussions of the concepts of continuous and discrete models as well as finite element methods are covered in detail in the books of Meirovitch (1986, 1980), Inman (1989), and Hughes (1987). Commonly, the free vibration motion of a structure in terms of an nth order discrete model is represented by the following set of simultaneous ordinary differential equations Mw(t) + Dw(t) + Kw(t) = 0 (2.2.1) where M, D and K are termed, respectively, the mass, damping and stiffness matrices. They are models of the mass, damping and stiffness properties of the structure. Since the system model is order n, these matrices are of dimension nxn and are generally real. The variable w(t) represents the n displacements of the nDOF model of the structure. The overdots represent differentiation with respect to time. The mass matrix, M, is always symmetric positive definite. The stiffness (K) and damping (D) matrices of nongyroscopic and noncirculatory systems are symmetric. In general, the modelling of the mass and stiffness properties of the structure is simpler and more accurate than the modelling of the damping properties. In the forthcoming discussion, it is assumed that the system under consideration is nongyroscopic and noncirculatory. The standard solution to Eq. (2.2.1) is of the form w(t) = vext (2.2.2) where v is a constant nxl vector and k is a constant scalar. Substituting Eq. (2.2.2) into Eq. (2.2.1) and dividing the resultant equation by ext yields the condition My X2 + Dv X + Kv = 0 (2.2.3) There are n sets of nontrivial complex conjugate solutions (Xi, y,) to Eq. (2.2.3). Note that since the property matrices (M, D, K) are real, if (iQ, yi) is a solution set to Eq. (2.2.3), the complex conjugate of that set is also a solution. The problem of solving for these solutions is commonly known as the eigenvalue problem and is sometimes referred to as the characteristic value problem. The scalar Xi and the vector vi are known, respectively, as the eigenvalue and eigenvector of the ith mode of vibration of the structure. For a general damped system, eigenvalues and eigenvectors are both complex. Note that Eq. (2.2.3) can be easily rearranged in the more general mathematical eigenvalue problem format as [ 0 Inxn i [ ] M'K M'D [Y = ki yi (2.2.4) where Inxn is the nxn identity matrix. Equation (2.2.4) is called a state space representation of Eq. (2.2.3). The eigenvalue and eigenvector can be related to some physical characteristic properties of structures. For that matter, the ith eigenvalue is written as Xi =  ii + j (Ii  (2.2.5) where j = 1 Note that in writing this equation it is assumed that the system is underdamped. The real scalar oi and ti are the natural frequency and damping ratio (or damping factor), respectively, of the ith mode of the structure. The eigenvector yi indicates the "shape" of vibration of the ith mode of the structure. The sets of frequencies, damping ratios and mode shapes are sometimes referred to as modal parameters. The symmetric nature of the property matrices (M, D, K) constrains the eigenvectors to satisfy some crossorthogonality relationships. For the purpose of discussing these crossorthogonality relationships, consider the following alternative state space representation of the eigenvalue problem in Eq. (2.2.3) [M O][VA VA ][A 0]+ [D K VA VA] = [ (2.2.6) 0 K V V 0 ?A K 0 V V 0 where V = [ v n ] A = diag(X1, X2, ..., Xn) where the overbar denotes the matrix complex conjugate operator. Based on this particular representation and the fact that the state matrices of Eq. (2.2.6) are symmetric, by proper normalization of the eigenvectors, the crossorthogonality relationship associated with the system are given by [VA VA M 0][VA = 2nx2n (2.2.7) V V 0 K V V 2nx2n VA VAN D K "VA VA] [A 0] V V K 0 V V 0 X Equations (2.2.7) and (2.2.8), respectively, clearly imply the following relations VA] [M _][VA] =[0] (2.2.9) V 0 K V 0 [VA]'[D K VA (2.2.10) V K O V 0 T where [ ] denotes the complex conjugate transpose operator ( [ ] = ). Note that, contrary to Eqs. (2.2.78), no particular normalization of the eigenvector is needed in order for Eqs. (2.2.910) to be satisfied. Another state space representation of Eq. (2.2.3) is given by 0 M VA VA A 0 M 0] VA VA [0 (2 [M DN V V 0 ] 0 K[ V V 0jj Based on the same argument discussed earlier, the crossorthogonality conditions that arise from this representation are M ]T[ ][ ] = 2nx2n (2.2.12) V v M Dr V Vx] VA VA T[ M ][ VA VA] [A 0] V V 0 K V V 0O Again, the following two relationships follow, VA][ [0 M][VA] = [0] (2.2.14) V M D V 0 VAI M ][VA] = [] (2.2.15) V [ 0 K V 0 2.2.1 Undamped Models In the modelling of structures, it is often assumed that the damping is negligible and hence is set to zero. For these type of system models, the eigenvalue problem in matrix form is given by MVA2 + KV = [0] (2.2.16) The matrices V and A are as defined earlier. For undamped systems, the eigenvalues X, are pure imaginary and the eigenvectors yi are real. Note that the eigenvalues are related to the system natural frequency by X2 = co? (2.2.17) Furthermore, by proper normalization of the eigenvectors, the crossorthogonality relations associated with this type of systems are VTMV = Inxn (2.2.18) VTKV = diag(wO, o, .2) (2.2.19) 2.2.2 Proportionally Damped Models When the damping of the structure is accounted for, it is sometimes modelled to be proportional, D = aM + p3K (2.2.20) where a and 03 are real scalars. The eigenvectors of a proportionally damped system are the same as the eigenvectors associated with the corresponding undamped system. The crossorthogonality relationships associated with proportionally damped models are VTMV = Inxn (2.2.21) VTDV = diag(2(o1, 22W2, 2ae0n) (2.2.22) VTKV = diag(ao, o Wn) (2.2.23) where all variables have the same definitions as in the previous discussion. A more detailed development of the eigenvalue problem can be found in the books of Meirovitch (1986, 1980) and Inman (1989). 2.3 Experimental Modal Analysis The vibration characteristics of structures can also be measured experimentally. Recall that the structural vibration characteristics are given by natural frequencies, damping ratios and mode shapes (eigenvectors). The process of measuring the modal parameters is known as modal testing or experimental modal analysis. An introductory treatment of the concept of experimental modal analysis can be found in the book of Inman (1989); a more rigorous coverage is treated in the book of Ewins (1986). The hardware components needed in a modal analysis experiment are identified in Figure 2.1 (Inman, 1989). A schematic of a simple modal vibration measurement test setup is shown in Figure 2.2. Brief descriptions of some of the components and their functions are given below. T := Transducer SC := Signal Conditioner Figure 2.1 Components of a Vibration Measurement System For Modal Analysis (Inman, 1989). The exciter is used to deliver the driving force that puts the structure in motion. The two most commonly used exciters are the shaker and the impulse hammer. Transducers are devices that measure the driving force as well as the response of the structure. They sense mechanical forces or motions and, then, convert them into electrical signals. Mechanical forces are usually recorded by a force transducers. Commonly, the vibration response of structures is measured by accelerometers. Accelerometers are mounted directly on the structure and, naturally, record the local accelerations. Signal conditioners are used to match the signal, received from the transducers, to the specification of the analyzer. These devices can also be used to amplify the signals. Accelerometer Force Transducer Signal Conditioner Accelerometer Signal Conditioner Signal Analyzer Figure 2.2 A Simple Experimental Modal Analysis Setup. The signal analyzer processes the electrical signal received from the signal conditioners. The standard type of analyzer allows time domain signals to be viewed in the frequency domain via a Fast Fourier Transform (FFT) algorithm. In an FFT, the signals are first filtered, digitized and then transformed into discrete frequency spectra. The frequency spectra can then be manipulated to compute the modal properties of the structure. It is important to note that the experimentally identified modal parameters are usually affected by unpredictable measurement noise. Typically, natural frequencies are identified to within 1 to 2% repeatability; damping ratios to within 5 to 15% repeatability, and mode shapes to within 5 to 10% repeatability. In practice, due to testing limitations, the set of Impulse Hammer structural modal parameters identified experimentally is incomplete with respect to the analytical model. Experimental incompleteness is manifested in two forms, (i) a limited number of measured modes of vibration, (ii) a limited number of eigenvector DOF measurements. 2.4 Analytical and Experimental Model Dimensions Correlation One major prerequisite common to most model refinement algorithms is to match the number of degrees of freedom (DOFs) in the experimentally measured eigenvector and in the discrete analytical model. Two different approaches have been commonly used to resolve this matching problem when an incomplete set of eigenvector DOFs have been measured. The first approach consists of reducing the dimension of the discrete analytical model to the number of the measured DOFs. The other approach is to expand the measured eigenvector to the size of the analytical model. A good evaluative survey of a number of analytical model reduction techniques has been compiled by McGowan (1991). The subject of eigenvector expansion is discussed in fair detail in the papers by Gysin (1990), and Zimmerman and Kaouk (1992a). In the next two sections, some commonly used model reduction and eigenvector expansion algorithms are reported and described. 2.4.1 Model Reduction Methods In this section the general framework of the model reduction concept is first presented. Then formulations of three commonly used model reduction techniques are summarized. Mostly, the concept of model reduction has only been studied for undamped models. In this presentation no attempt has been made to generalize these concepts to damped models. For the purpose of reporting the general concept of model reduction, consider the eigenvalue problem associated with an undamped model, MVA2 + KV = [0] (2.4.1) where, as defined earlier, M and K are, respectively, the mass and stiffness matrix; V is the eigenvector matrix and A is the diagonal eigenvalue matrix. Assume that only a subset of the eigenvector DOFs has been experimentally measured. Equation (2.4.1) can be reordered such that the DOFs associated with the measured DOFs are in the upper rows of the equation, MoVoA2 + KoVo = [0] (2.4.2) Vm Kmm Kmu Mmm Mmu where Vo = Vu Ko = Kum Kuu Mo Mum Mu The matrices Mo, Ko, and Vo are, respectively, the reordered mass, stiffness and eigenvector matrices. In the above equation, the subscripts "m" and "u" denote, respectively, the components associated with the measured and unmeasured DOFs. A transformation matrix P, that relates matrices Vu and Vm can be defined as Vu = PVm (2.4.3) A substitution of this relationship in the reordered eigenvector matrix Vo results in Vo = p Vm = TVm (2.4.4) Substituting Eq. (2.4.4) into Eq. (2.4.2) and premultiplying by TT yields the eigenvalue problem of the reduced model, MrVmA2 + KrVm = [0] (2.4.5) where Mr = TTMoT Kr = TTKoT where Mr and Kr are the reduced mass and stiffness matrices, respectively. In terms of the partitioned matrices, the reduced matrices are defined as Mr = Mmm + pTMum + MmuP + pTMuuP (2.4.6) Kr = Kmm + PTKum + KmuP + PTKuuP (2.4.7) At this point, the only condition placed on P is the relationship of Eq. (2.4.3). Naturally, matrix P can be computed directly from Eq. (2.4.3) if the eigenvectors of the system are available. This approach is known as the exact reduction method and has been discussed in the papers by Kammer (1987) and O'Callahan et al. (1989). The exact reduction method requires solving for a large number of eigenvectors, which can be computationally expensive. The three reduction methods that are presented in the forthcoming discussions propose alternative techniques to compute matrix P. The first two do not require the computation of the system eigenvalue problem. The last one requires the knowledge of one eigenvalue which is computationally admissible. 2.4.1.1 Static Reduction This reduction method is often referred to as Guyan (1965) reduction. In the static reduction, the mass properties associated to the unmeasured DOFs are assumed negligible. With that assumption, Eq. (2.4.2) can be written as Mmm 0 Vm 2 Kmm Kmu Vm 0 [Mm O][VMA2 + [mm :] : = (2.4.8) 0 0 Vu Kum Kuu Vu 0 The second row of this matrix equation can then be manipulated as Vu = KuKumVm (2.4.9) From comparing Eq. (2.4.9) to Eq. (2.4.3), it can deduced that the transformation matrix, P, computed using the Guyan approach is given by Pg = KualKum (2.4.10) The reduced mass and stiffness matrices can then be computed by substituting matrix Pg for matrix P in Eqs. (2.4.6) and (2.4.7). Naturally, the Guyan assumption (Eq. (2.4.8)) suggests that if the mass properties of the omitted DOFs are not small, the accuracy of the Guyan reduced model could be lacking. 2.4.1.2 IRS Reduction The improved reduction method (IRS) was formulated by O'Callahan (1989). It is an improvement over the Guyan reduction in that it accounts for the mass properties of the unmeasured DOFs. In the formulation of the IRS method, the Guyan reduced model is corrected to include the mass influence of the unmeasured DOFs. This formulation is somewhat lengthy and the interested reader is referred back to the paper of O'Callahan (1989) or the thesis of McGowan (1991). The transformation matrix P computed using the IRS reduction is PIRS = Pg + KuU Mum + MuuPg MrKr, (2.4.11) The reduced IRS model is then computed by substituting matrix PIRS in Eqs. (2.4.67). 2.4.1.3 Dynamic Reduction The dynamic reduction was proposed as another improvement to the Guyan reduction (Kidder, 1973; Miller, 1980; and Paz, 1984). This reduction utilizes the dynamic equation associated with a single mode of vibration to compute the transformation matrix P. In this technique, the transformation matrix P is arrived at by considering the reordered dynamic equation associated with the ith mode, [2Mmm + Kmm Mmu + Kmu m] (2.4.12) I (2.4.12) ?2Mum + Kum XMuu + Kuu JYu 0 where Xj is the ith eigenvalue; vmi and vYu are, respectively, the measured and unmeasured eigenvector DOFs associated with the ith mode of vibration. Based on this partition, two equations can be generated. By using the rows that correspond to the unmeasured DOFs (lower rows), the following relationship is obtained, Yu = [Muuux + Kuu] Mum + Kum] I, (2.4.13) Hence, the transformation matrix associated with the ith mode is defined by Pd, = [MuuX? + Kuu] [Mumk32 + Kum] (2.4.14) The reduced mass and stiffness matrices are then computed by using Eqs. (2.4.6) and (2.4.7). Note that different modes will result in different transformation matrices Pd and, hence, different reduced mass and stiffness matrices. 2.4.2 Eigenvector Expansion Methods Alternatively, the dimension of the measured eigenvectors can be correlated to the dimension of the analytical model by using eigenvector expansion algorithms. The common basis of these algorithms is the interpolation of the unmeasured eigenvector components. In the forthcoming sections, two eigenvector expansion algorithms are reviewed. 2.4.2.1 Dynamic Expansion The dynamic expansion technique (Berman and Nagy, 1983) is one of the most commonly used eigenvector expansion algorithms. A slight modification of the Berman and Nagy formulation is presented here to accommodate damped systems (Fuh et al., 1984). In the formulation of the dynamic expansion, it is assumed that the measured modes satisfy the eigenvalue problem involving the property matrices of the original model (M, D, K). For the ith measured mode, this assumption takes the form (k2iM + XeD + K) ve, = 0 (2.4.15) where Xe. and ve. are, respectively, the ith experimental eigenvalue and eigenvector. The matrices M, D and K have the same definitions as in the earlier sections. Assume that only a subset of the DOFs of eigenvector ve. has been measured. Equation (2.4.15) can be reordered, as in Section 2.4.1, such that the measured eigenvector DOFs reside in the upper half of the equation, Mmm Mmu] Dmm Dmu Kmm Kmu 1 e[ 0ei n211 Mum MuuJ e Dum Duu Kum Kuu LVXeuJ, where Yemand Vu, are, respectively, the measured and unmeasured DOFs of ve.. The subscripts "m" and "u" denote measured and unmeasured components. A rearrangement of Eq. (2.4.16) yields [Mmmn + keDmm + Kmm Mmu + ke,Dmu + Kmu em,i I = (2.4.17) L Mum + XeDum + Kum 2Muu + ke.Duu + Kuu jYeu, From Eq. (2.4.17), two matrix equations in function of Vem, and veu, can be generated. By using the equation associated with the second row of the partition, the unmeasured components of the ith experimental eigenvector are computed to be eu = [Muu + DXe + Ku] [MumX2J + Dumke, + Kum vemi (2.4.18) Note that this expansion works on a single mode at a time. Also, notice that it involves the original analytical model (M, D, K). This implies that the accuracy of the expansion depends on the validity of the original analytical model. 2.4.2.2 Orthogonal Procrustes Expansion Another expansion process that has shown great promise is the orthogonal Procrustes (OP) expansion method presented in the papers of Smith and Beattie (1990) and Zimmerman and Kaouk (1992a). The technique uses the general mathematical framework of the orthogonal Procrustes problem (Golub and Van Loan, 1983). Let Vem be the experimentally measured eigenvector component matrix and Vain be the corresponding analytical eigenvector component matrix. The essence of the OP expansion is to find a unitary transformation matrix Pop that closely rotates Vam into Vem. This is attempted by solving the following problem, Minimize 11 Vem VamPop IF (2.4.19) subject to pTpPop = The solution to this general problem is discussed in the book of Golub and Van Loan and is given by Pop = YZT (2.4.20) where Y and Z are, respectively the left and right singular matrices of matrix D defined by (D = VTVem (2.4.21) Let Vau be the eigenvector matrix associated with the unmeasured DOFs. In the orthogonal Procrustes expansion, it is assumed that the transformation matrix computed above also rotates Vau into the unmeasured "experimental" eigenvector component matrix, Veu, as Veu = VauPop (2.4.22) Two different approaches have been defined to generate the expanded experimental eigenvector matrix. The first is suggested in the paper by Smith and Beattie (1990), Vam Ve = Vau Pop (2.4.23) In this definition, the expanded "experimental" eigenvector matrix is the rotated analytical eigenvector matrix. The actual experimental measured eigenvector components Vem are replaced by matrix VamPop. The advantage of the approach is that the resulting "experimental" eigenvector satisfies the crossorthogonality conditions (see Section 2.4). The second viewpoint was proposed by Zimmerman and Kaouk (1992a), [Vemn Ve = [VauPopj (2.4.24) Here, the unaltered eigenvector components measured experimentally are inserted in Ve. In this viewpoint, if crossorthogonalization of the expanded experimental eigenvectors is required, a separated orthogonality algorithm can be used (Section 2.5). In the paper by Smith at al. (1993), it is shown that for actual model refinement problems, both viewpoints give equivalent eigenvector expansion results. However, for damage detection problems, a preliminary study indicates that eigenvectors expanded by using the viewpoint defined in Eq. (2.4.24) give better assessment of the damage. 2.5 Eigenvector Orthogonalization Most matrix update algorithms require the measured eigenvectors to satisfy a crossorthogonality condition. This is especially true in matrix update algorithms in which (i) the model of the structure is assumed undamped and the modelling errors are assumed to be in only one of the two property matrices (M or K is assumed correct) or (ii) the system is modeled by a proportionally damped model with errors in only two of the three property matrices (M, D or K). In these situations, in order to insure symmetry of the updated property matrices, it is required that the expanded experimental eigenvectors be orthogonal with respect to the property matrix assumed accurate. This situation is encountered in the optimal update algorithms developed by Baruch and Bar Itzhack (1978), Kabe (1985), Kammer (1985), and Smith and Beattie (1991), Zimmerman and Kaouk (1992b), Kaouk and Zimmerman (1993b) among others. In most of these algorithms, it is assumed that the mass matrix is correct. This assumption is used in a number of the model refinement algorithms since the inertial properties of structures are known to a good extent. In these cases, one would expect the expanded experimental eigenvectors to be mass orthogonal. However, because of measurements errors, this condition rarely occurs. For this reason, a great deal of effort was focused on the development of mass orthogonalization techniques. Some of the most notable work in that area was performed by Targoff (1976), Baruch and Bar Itzhack (1978), and Baruch (1979). In the next two sections, two orthogonalization techniques (Baruch and Bar Itzhack, 1978; Baruch, 1979) are discussed. Both techniques are mass orthogonalization techniques; however, with obvious modifications, these techniques can be adopted to solve the orthogonalization problem of the eigenvectors with respect to the stiffness or the damping matrices. 2.5.1 Optimal Weighted Orthogonalization The essence of the standard mass orthogonalization technique is to modify the measured eigenvectors such that the mass crossorthogonality condition is satisfied. Baruch and Bar Itzhack (1978) proposed an elegant solution to that problem. An overview of their problem statement and solution is given below. Assuming that Ve is a matrix of expanded experimental eigenvectors that need to be mass orthogonalized. The present formulation searches for the optimal mass weighted change of matrix Ve such that the mass crossorthogonality condition is satisfied. This problem is cast as Minimize N (Veo Ve) 1F (2.5.1) subject to VT M Veo = I (2.5.2) where N=M112 and M is the mass matrix. By means of a Lagrange multiplier, Eq. (2.5.2) can be incorporated into Eq. (2.5.1); then the application of the optimality conditions yields the following expression for Veo, Veo= Ve(V M Ve)1/2 (2.5.3) Before being incorporated into the orthogonalization process, the measured eigenvectors have to be unit mass normalized, i.e., Vej = ve,(vT M ve 12 (2.5.4) where ve. is the it expanded experimental eigenvector (ith column of Ve). 2.5.2 Selective Optimal Orthogonalization Some structures exhibit rigid body modes (modes with zero eigenvalues). It is desirable to preserve these rigid body modes in the refinement process. However, some matrix update algorithms require the rigid body modes and the experimental eigenvectors to be mass orthogonal to insure symmetry of the updated property matrices (see Chapter 6). Naturally, the rigid body modes will be corrupted if they are incorporated along with the expanded experimental eigenvectors in the above orthogonalization process. Thus, Baruch (1979) presented a modification of the procedure in Section 2.5.1 to deal with such a problem. The resulting problem is a selective orthogonalization and is formulated as follows, minimize 11 N (Veo Ve) 1F (2.5.5) Subject to Veo M Veo = I (2.5.6) and Veo M Vr = [0] (2.5.7) In the above equations, Ve and Vr are, respectively, the expanded experimental eigenvector matrix and the rigid body mode matrix. Again, the Lagrange multiplier is used, and the orthogonalized experimental eigenvector matrix that satisfies the conditions in Eqs. (2.5.6) and (2.5.7) is found to be Veo= Q(QT M Q)/2 (2.5.8) where Q = Ve VrVTMVe Note that, as in previous process, the expanded experimental eigenvectors have to be unit mass normalized. 2.6 Load Path Preservation Many matrix update algorithms introduce additional load paths in their updated models, i.e., elements of the mass, damping or stiffness matrices that were originally zero may become nonzero. Whether or not preserving the original load path is a practical problem is still a matter of current debate. It seems that for damage assessment of truss structures it is desired to maintain load paths. In the paper of Zimmerman and Kaouk (1992a), an iterative approach to preserve the load path of the original property matrices was developed. The approach was presented in the context of the symmetric eigenstructure assignment model refinement algorithm (discussed in Chapter 4); however, its application can also be extended to other model refinement algorithms. In Figure 2.3, a flow chart of the iterative load preservation algorithm is presented. The procedure is illustrated for a general model refinement scenario in which all three property matrices (M, D, K) are being updated. However, it can be easily modified to accommodate other refinement problems. Figure 2.3 Flow Chart of the Iterative Load Path Preservation Algorithm In the flow chart, the matrices Ma,m, Da,m, and Ka,m are respectively the adjusted masked mass, damping and stiffness matrices defined by Ma,m = Ma 0 Mm Da,m = Da 0 Dm (2.6.2) Ka,m = Ka 0 Km where Ma, Da, and Ka are the adjusted mass damping and stiffness matrices. The matrices Mm, Dm, and Km are the masking matrices associated with the original mass, damping and stiffness matrix. By definition, the masking matrix, Am, associated with matrix A is given by Am(i,j) = 1 if A(i,j) 0 (2.6.1) Am(i,j) = 0 if A(i,j) = 0 In Eqs. (2.6.2), the operator 0 is the elementbyelement (scalar) matrix multiplication. Let B and C be two nxn matrices, then the elementbyelement multiplication of B and C is given by S = B 0 C => S(i,j) = B(i,j) C(i,j) i,j = l,..., n (2.6.3) At every iteration, the norms of the matrix differences between corresponding adjusted and adjusted masked property matrices are computed. At a given iteration, if the three computed norms are equal to zero or within user set limits, then the load paths of the original three property matrices have been exactly achieved or achieved within user state guidelines. Thus, the procedure is halted, and the refined model consists of the adjusted property matrices computed at that particular iteration. It should be noted that there is no formal guarantee of convergence in using this iterative procedure. Experience gained in using the present algorithm indicates that if the experimental modal data are consistent with the sparsity pattern, the procedure will converge. Consistent data means that there exist mass, damping and stiffness matrices that have the same sparsity pattern as the original matrices and also exhibit the measured test data. Otherwise, if the data are inconsistent, the original sparsity patterns will not be exactly preserved. In this case, the added load path terms of Ma, Da and Ka which should be zero will be closer to zero after application of the algorithm. CHAPTER 3 INVERSE / HYBRID APPROACH FOR FINITE ELEMENT MODEL REFINEMENT 3.1 Introduction The inverse eigenvalue problem is concerned with the construction of the property matrices (mass, damping or stiffness) of a dynamic model using experimentally measured modal data. These techniques require complete modal properties. Thus, for an nDOF model, n natural frequencies, damping ratios and mode shapes (eigenvectors) must be measured, and the identified mode shapes must be of dimension n. Due to practical testing limitations, this is rarely accomplished for typical large structures. In this chapter, the application of the inverse problem is extended to model updating by combining experimental measurements and original analytical FEM modal information. Again, refinement implies correlating the measured and analytical modal properties. 3.2 Theoretical Formulation The dynamic structure under consideration is assumed to be successfully modelled by an nDOF proportionally damped nongyroscopic and noncirculatory (symmetric property matrices) FEM. The free vibration motion of such a dynamic structure can be analytically represented by a differential equation of the form Mw(t) + Dw(t) + Kw(t) = 0 (3.2.1) where the variables M, D, and K are nxn real symmetric matrix models of the mass, damping and stiffness properties of the structure. The nxl time varying vector w(t) represents the n displacements of the nDOF model of the system. The overdots represent differentiation with respect to time. The eigenvalue problem associated with the differential equation shown in Eq. (3.2.1) is of the form Mvy. X + Dv.i X + Kvy = 0 (3.2.2) where Xi and yi are, respectively, the eigenvalue and eigenvector of the ith mode of vibration. In this problem, it is assumed that the accuracy of the original FEM is lacking and, hence, needs improvement. Furthermore, it is assumed that the inaccuracy of the original FEM is solely due to modeling errors in the stiffness and damping properties. The model refinement, proposed herein, exploits the crossorthogonality relations that arise from the symmetric nature of the property matrices and the proportional damping assumption. As discussed in Chapter 2, by proper normalization of the eigenvectors these crossorthogonality relations have the form VT M V = Inxn (3.2.3a) VT D V = diag(2tlow, ,2tn n) = 5: (3.2.3b) VTKV = diag(01 2, ,on2) = Q (3.2.3c) V = [ ... V] where wi and i are the natural frequency and damping ratio, respectively, of the ith mode of the structure. The matrix Inxn is the nxn identity matrix. It is important to recognize that Eq. (3.2.3a) represents necessary and sufficient conditions for conserving symmetry and damping proportionality when updating the stiffness and damping properties of proportionally damped systems. Suppose that p (p << n) modes of an existing structure have been experimentally identified (mode shapes or eigenvectors, frequencies and damping ratios). Assume that the dimension of measured eigenvectors is equal to the dimension of the FEM, i.e., all n components of the measured eigenvectors are available. It is widely accepted that in the absence of specific experimental measurements a good approximation to the unmeasured modes is their corresponding analytical modal information. With that in mind, a complete hybrid set of modal data is generated by combining experimental and analytical information Vea = [Ve Va] [Qe 0 0 Qa (3.2.4) le 0 lea = 0 a where V is the eigenvector matrix; 2 and I are diagonal matrices of frequencies squared and damping ratios, respectively. The subscripts e and a denote, respectively, experimental and analytical sets. At this point, the complete "hybrid" set of eigenvectors, Vea, does not satisfy the crossorthogonality conditions defined in Eq. (3.2.3a); thus the conditions in Eqs. (3.2.3b) and (3.2.3c) are not met. One possible solution to this problem is to modify all of Vea in an optimal way to comply with the orthogonality requirement. This approach treats all parts of Vea equally, and thus overlooks the fact that the experimental modes are known with a higher confidence. Naturally, a technique that assigns a higher credibility to the experimental eigenvectors, Ve, is preferable. This can be achieved by incorporating Vea into the orthogonalization process group by group in the order of their descending credibility (experimental then analytical). If the problem is set such that the experimental modes, Ve, are corrected first, it is clear that the analytical modes, Va, will be subject to larger correction when incorporated into the orthogonalization process since they will be subject to more constraints. The experimental modes, Ve, are orthogonalized by using the orthogonalization technique formulated by Baruch and Bar Itzhack (1978). The general formulation of this orthogonalization technique is discussed in Chapter 2. For this particular application, the problem consists of finding the matrix ,Veo, that satisfies the crossorthogonality condition, Vo M Veo = Ipxp (3.2.5) and that minimizes the weighted Euclidean norm, (D = N (Veo Ve) I (3.2.6) where N = M1/2 The solution to this problem, as reported in Chapter 2, is Veo= Ve(V M Ve)1/2 (3.2.7) The next step is to invoke the orthogonality requirement on the analytical eigenvector matrix, Va, by searching for a matrix Vao that satisfies the following two conditions, Vao M Vao = I(np)x(np) (3.2.8) and Vao M Veo = [0] (3.2.9) while minimizing the objective function, F =1 N (Vao Va) IIF (3.2.10) where N = M1/2 A similar problem was also treated by Baruch (1980) in a different context. A brief discussion of the solution approach can be found in Chapter 2. The set of eigenvectors, Vao that satisfies this problem is given by Vao = Q(QT M Q)/2 (3.2.11) Q = Va VeoV oMVa Clearly, the resultant matrix, Veao = [ Veo Vao ], satisfies Eq. (3.2.3a). The corrected stiffness and damping matrices are then computed using Eqs. (3.2.2b) and (3.2.2c). Ka = MVeao Q ea VTaoM (3.2.12a) Da = MVeao E eaVeLoM (3.2.12a) where Oea = 2Ileagea Note that the matrices Ka and Da computed from Eqs. (3.2.12a,b) will be symmetric. The above formulation suggests that the system modelled by the original mass matrix (M) and the stiffness (Ka) and damping (Da) matrices computed from Eqs. (3.2.12a,b) will have eigenvectors Veao, frequencies 4ea, and damping ratios Hea. Some structures exhibit rigid body modes of vibration. Commonly, it is desirable to preserve these rigid body modes in the updated model. The above formulation also suggest that the updated model will preserve the original rigid body modes. The procedure developed above can be easily contracted to address the case when the system model does not account for the effects of damping (undamped model). The contraction can be obtained by setting to zeros matrices D, Oea, and FIea in Eqs. (3.2.24). The computational burden limits the size of the FEM which can be updated (order of 200). Essentially, the limiting factor is that all mode shapes of the structure that are not available from experimental measurements must be calculated analytically by solving the eigenvalue / eigenvector problem. 3.3 Numerical Illustration The system addressed in this investigation is the commonly used eight degrees of freedom model shown in Figure 3.1. This model was developed by Kabe (1985) to give a common testbed for the evaluation of the performance of model refinement algorithms. An original undamped analytical model of the system was generated by using the mass and stiffness properties shown in Figure 3.1. The elements of the original analytical stiffness matrix are displayed in the second column of Table 3.1. In this problem, it is assumed that the original stiffness matrix of the model is incorrect. These inaccuracies were simulated by mi =0.001 m8= 0.002 mj=1.0 j=2,..., 7 k1 = 1000 k2 = 10 k3 = 900 k4 = 100 k5 = 1.5 k6 = 2.0 Figure 3.1 Kabe's Problem: Analytical Test Structure. using incorrect stiffness constants for most of the springs. The elements of the actual correct stiffness matrix are given in the last column of Table 3.1. Note that the present model refinement problem is very challenging and because of the large difference between the stiffness matrices of the original and exact model. In this study, two cases will be considered. In the first case, it is assumed that only the modal parameters (eigenvalue and eigenvector) of the first mode were measured. In the other, the modal parameters of first three modes are assumed to be available. In both cases, the measured eigenvectors are supposed to be full (all degrees of freedom of the eigenvector(s) are measured). The main objective of this investigation is to compare the performance of the inverse/hybrid method to the algorithm proposed by Baruch and Bar Itzhack (1978). The Baruch and Bar Itzhack model update technique is one of the most commonly used model refinement algorithm. One of the main reasons for its common use is because the algorithm is computationally efficient. The updated stiffness matrices generated by using the Baruch and Bar Itzhack approach for the one mode and three mode cases are shown in the third and fourth column of Table 3.1, respectively. The fifth and sixth columns of Table 3.1 display the results of using the inverse/hybrid algorithm for the one mode and three modes cases. For both cases, it is clear that the performances of both algorithms is lacking in predicting the exact stiffness matrix. As expected, It can also be seen that both algorithms generate better results when using three measured modes. A comparison of the results generated using the Baruch and Bar Itzhack approach and the inverse/hybrid method shows that both algorithms give the same type of results. This similarity of results was also encountered in other example problems. 3.4 Summary A model refinement approach that uses a hybrid set of experimental and analytical modal properties was formulated. The developed approach, termed the inverse/hybrid algorithm, was illustrated by using a computer simulated example. Part of the evaluation of the proposed algorithm was the comparison of its performance with the performance of the Baruch and Bar Itzhack algorithm. It was found that both algorithms give the same type of results. However, the computations involved in the inverse/hybrid approach exceed those involved in the Baruch and Bar Itzhack algorithm. Essentially, the inverse/hybrid approach requires the computation of all eigenvalues and eigenvectors of the structure that are not available from experimental measurements. As will be illustrated in the forthcoming chapter, the Baruch type approaches are not suited for damage assessment applications. For these reasons, further development of the Inverse/Hybrid algorithm was not investigated and new formulations (Chapters 4 & 6) were considered. Table 3.1 Kabe's Problem: Elemental Stiffness Components. Baruch Inverse/Hybrid Element # Original Mode 1 Modes 13 Mode 1 Modes 13 Exact (1,1) 2.0 2.0 2.0 2.0 2.0 1.5 (1,2) 2.0 2.0 3.0 2.0 3.0 1.5 (1,3) 0.0 0.0 0.1 0.0 0.1 0.0 (1,4) 0.0 0.0 0.1 0.0 0.1 0.0 (1,5) 0.0 0.0 0.1 0.0 0.1 0.0 (1,6) 0.0 0.0 0.1 0.0 0.1 0.0 (1,7) 0.0 0.0 0.1 0.0 0.0 0.0 (1,8) 0.0 0.0 0.0 0.0 0.0 0.0 (2,2) 1512.0 1508.6 1024.2 1510.6 1024.3 1011.5 (2,3) 10.0 31.5 68.5 21.5 71.4 10.0 (2,4) 0.0 8.9 9.0 15.0 8.4 0.0 (2,5) 0.0 8.9 20.9 15.0 23.8 0.0 (2,6) 0.0 21.6 38.5 11.5 35.2 0.0 (2,7) 0.0 3.9 9.1 1.6 8.7 0.0 (2,8) 0.0 0.1 0.1 0.1 0.1 0.0 (3,3) 1710.0 1574.6 1560.8 1624.1 1612.2 1110.0 (3,4) 0.0 44.9 49.8 75.6 76.2 0.0 (3,5) 200.0 244.8 244.1 275.6 276.9 100.0 (3,6) 0.0 136.5 123.4 86.2 74.5 0.0 (3,7) 0.0 24.6 50.8 13.1 46.8 0.0 (3,8) 0.0 0.4 0.1 0.4 0.0 0.0 (4,4) 850.0 1083.1 1087.9 1102.1 1099.9 1100.0 (4,5) 200.0 3.27 25.1 51.8 48.0 100.0 (4,6) 200.0 254.3 242.2 276.5 274.9 100.0 (4,7) 0.0 10.1 27.0 17.2 30.8 0.0 (4,8) 0.0 0.2 0.1 0.2 0.2 0.0 (5,5) 850.0 1082.2 1089.5 1101.4 1101.1 1100.0 (5,6) 0.0 45.2 49.5 76.5 75.1 0.0 (5,7) 0.0 10.1 12.4 17.2 11.7 0.0 (5,8) 0.0 0.2 0.1 0.2 0.2 0.0 (6,6) 1714.0 1576.4 1565.0 1617.5 1610.8 1112.0 (6,7) 10.0 34.8 82.6 23.1 76.2 10.0 (6,8) 4.0 4.4 4.7 4.4 4.5 2.0 (7,7) 1512.0 1507.5 1027.9 1510.1 1028.0 1011.5 (7,8) 2.0 2.1 4.1 2.1 4.2 1.5 (8,8) 6.0 6.0 6.0 6.0 6.0 3.5 CHAPTER 4 SYMMETRIC EIGENSTRUCTURE ASSIGNMENT MODEL REFINEMENT ALGORITHM 4.1 Introduction Eigenstructure assignment is a control concept used to alter the transient response of linear systems. This is done by forcing the system to have some predetermined eigenvalues and eigenvectors. A detailed overview of eigenstructure assignment theories can be found in the paper by Andry et al. (1983). Inman and Minas (1990), Zimmerman and Widengren (1989, 1990), and Widengren (1989) have developed model refinement algorithms based on the mathematical framework of eigenstructure assignment. The basic idea of these model refinement techniques is to design the pseudocontroller which is required to produce the measured modal properties (natural frequencies, damping ratios and mode shapes) with the original finite element model (FEM) of the structure. The pseudocontroller is then translated into matrix adjustments applied to the initial FEM. In this work, the eigenstructure assignment based model refinement algorithm proposed by Zimmerman and Widengren (1989, 1990) is extended to better approach the damage assessment problem. A subspace rotation algorithm is developed to enhance eigenvector assignability. Finally, the enhanced algorithm is tested and compared to other techniques on both "simulated" and actual experimental data. 4.2 Problem Formulation In this section, a review of the Zimmerman and Widengren (1989, 1990) refinement technique, which is termed the symmetric eigenstructure assignment model refinement algorithm (SEAMRA), is presented. This review is essential in order to properly introduce and discuss the extension and improvement proposed in this work. 4.2.1 Standard Eigenstructure Assignment Formulation Consider the standard differential equation of motion of an n degrees of freedom damped, nongyroscopic and noncirculatory structure with control feedback, Mw(t) + Dw(t) + Kw(t) = Bou(t) (4.2.1) Again, M, D, and K are nbyn real symmetric matrix models of the mass, damping and stiffness properties of the structure. Assume that these matrices were generated using the finite element method. The nxl time varying vector, w(t), represents the n displacements of the nDOF FEM of the system. The overdots represent differentiation with respect to time. In control terminology, BO is the nxm (m << n) control influence matrix describing the actuator force distributions and u(t) is the mx 1 vector of output feedback control forces defined by u(t) = Fy(t) (4.2.2) In Eq. (4.2.2), F is the mxr feedback gain matrix and y(t) is the rxl output of sensor measurements defined by y(t) = Cow + C1w (4.2.3) in which Co and C1 are the rxn output influence matrices corresponding to position and velocity, respectively. A substitution of Eqs. (4.2.2) and (4.2.3) into Eq. (4.2.1) yields Mw(t) + (D BoFC1)w(t) + (K BoFCo)w(t) = 0 (4.2.4) It is clear, from Eq. (4.2.4), that the feedback controller results in residual changes, BoFCo and BoFC1, to the stiffness and damping matrices, respectively. These changes can be viewed as perturbations to the initial finite element model (FEM) such that the adjusted FEM matches closely the experimentally measured modal properties. The adjusted FEM consists of the original mass matrix and the adjusted stiffness and damping matrices given by Ka K Da = D  BoFCo  BoFC1 (4.2.5) Assume that modal analysis of the structure under consideration has been performed and that p modes (p eigenvalues Xei, and p eigenvectors Ve) have been identified. As discussed earlier in Chapter 2, in practice p is typically much less than n. The feedback gain matrix F, such that the adjusted FEM eigendata matches the experimental modal parameters, is computed using standard eigenstructure assignment theories (Andry et al. 1983): F = [Z A f[C C 1 where (4.2.6) A0 Inxn A M1K MiD B = M1K T = [B P] A = T'AT = A] V = T1 W W WA WA STB = Imxm 0 A = diag(Xe, ,e2, ., ep) w = [eaivea,, .,Veap] Z = S1[ W A [ A WA WA The overbar in the above equations indicates the complex conjugate operator. The vectors Yea, in matrix W are the expanded "best achievable" eigenvectors associated with the experimentally measured eigenvectors ve,. An explanation of the concept of "best T S T =1 S2 achievable" eigenvectors is discussed in Section 4.2.3. The submatrix P of matrix T is arbitrary as long as T is invertible. At this point, the variables BO, Co and C1 are still arbitrary. A random selection of these variables will usually result in nonsymmetric perturbation matrices and, consequently nonsymmetric adjusted stiffness and damping matrices. This clearly conflicts with the fundamental symmetry requirement of most structures' FEM. In the formulation of Inman and Minas (1990), the resulting perturbation matrices from the pseudocontroller are forced to be symmetric through a nonlinear unconstrained optimization problem. Zimmerman and Widengren (1989, 1990) proposed a noniterative and computationally more efficient approach to satisfy the symmetry requirement. This approach in discussed in the following section. 4.2.2 Symmetric Eigenstructure Assignment Formulation The perturbation matrices are symmetric if the following conditions are met, BoFCo = CF TBo (a) (4.2.7) BoFC1 = CF TB (b) At this point, two additional assumptions are made. As a prerequisite to the existence of the inverse of some matrices used in the computations, it is assumed that the number of pseudo sensors and actuators is equal to twice the number of measured modes (m=r=2p). The other assumption consists of restricting the matrices Co and C1 by the conditions CO = GoBo (a) 0 (4.2.8) C1 = GIBT (b) where Go and G1 are mxm invertible matrices. A substitution of Eq. (4.2.8) into Eq. (4.2.7) simplifies the symmetry conditions to the following relationships, FGo = GTFT (a) (4.2.9) FGi = G TFT (b) By using the conditions in Eq. (4.2.9), along with the expression for the feedback gain matrix (Eq. (4.2.6)), a necessary but not sufficient condition on Go and GI, for symmetric perturbation matrices, is expressed in the form of a generalized algebraic Riccati equation, A1X + XA2 + XA3X + A4 = [0] (4.2.10) where XGG'G0 A, = ) T] A2 =* 1 1 * A3 = *11(a ** a )~* A4 = o*a10la* Imxm [w*BO' "AW*BO] T= a = a= Z AIV W Bo AW Bo The matrices Al, W, Z, and V are defined in Eqs. (4.2.6). The superscript ()* indicates the inverse of the complex conjugate transpose matrix. Equation (4.2.10) can be solved for X by using the techniques described in the papers of Potter (1966) or Martensson (1971). In general, there exist multiple solutions (X's) to this generalized algebraic Riccati equation. With all solutions computed, the next step is to decompose these solutions into Go's and GI's. It is shown in the paper by Zimmerman and Widengren (1989) that for a given solution X, any selection of Go and GI satisfying X = G 1Go results in the same adjusted damping (Da) and stiffness (Ka) matrices. Hence, either GI (or Go) can be chosen arbitrarily, as long as its inverse exists. Then, Go (or GI) is calculated from the relationship X = GI 'G0. For each set (Go, G1), a feedback gain matrix F is calculated from Eq. (4.2.6), and the corresponding adjusted damping (Da) and stiffness (Ka) matrices are computed using Eqs. (4.2.5). At this point, a rationale is proposed to choose the most meaningful adjusted damping and stiffness matrices. Among all computed sets (Da, Ka), it is apparent that only the ones that are real and symmetric are acceptable. When dealing with a model refinement problem, among all acceptable solutions, the final selection could be made by choosing the set (Da, Ka) that minimizes the cost function, J = q I K Ka IIF + I D Da IIF (4.2.11) where q K IF Clearly, this process selects the set (Da, Ka) that results in a minimum change from the original set (D, K). The scale factor q in Eq. (4.2.11) is used to give equal weight to the changes in D and K. For the damage detection problem, there is no unique rationale to choose the "best" set (Da, Ka). A physically intuitive approach is to use engineering judgement in selecting the "best" updated model. Thus, all acceptable "adjusted" sets of solutions should be inspected to determine which best provides information concerning the state of damage. 4.2.3 Best Achievable Eigenvectors From standard eigenstructure assignment theory (Andry, et al. 1983), it is shown that the measured eigenvectors are not always exactly assignable to the adjusted finite element model. In fact, it can be shown that the measured eigenvectors are assigned exactly if and only if they lie in their respective achievable subspace. The achievable subspace associated to the ith mode is defined by L, = (MX? + DXi + K)B0 (4.2.12) where 1i is the measured eigenvalue of the ith mode. When all n components of the experimental eigenvectors are available, the ith best achievable eigenvectors is defined as the least square projection of the ith experimental eigenvector Ve, on the ith achievable subspace Li. This projection is schematically illustrated in Figure 4.1. Achievable Subspace Figure 4.1 Best Achievable Eigenvector Projection. This best achievable eigenvector is given by yea = LL L] L ve (4.2.13) When only a subset s of the eigenvector components are measured, s < n, the least square projection discussed above can be used to simultaneously expand and project the measured eigenvectors. In this case, the ith expanded best achievable experimental eigenvector is given by Yea, = Li i Li LiVe (4.2.14) where Li are the rows of Li which correspond to the measured eigenvector components. Notice that the calculation of the p achievable subspace using Eq. (4.2.13) requires p inversions of an nxn matrix. Although the matrix to be inverted is typically banded, this may present a practical computational burden when dealing with large FEMs. The next section discusses an approach that does not require the actual computation of the achievable subspaces and hence avoids this computational burden. 4.2.4 Selection of Bu : The Subspace Rotation Method So far, the control influence matrix has not yet been completely defined. The preceding formulation suggest that different Bo may possibly result in different adjusted FEM. Hence, it is essential to develop a physically meaningful rationale to select B0. Zimmerman and Widengren (1989, 1990) proposed an approach, termed the mode selection method, that consists in selecting Bo such that the unmeasured modes of the structure are nearly unchanged. In other word, BO is selected such that only the measured modes of the structure are corrected. This selection technique fixes the achievable subspaces in which the eigenvectors must lie, and hence places a limitation on the assignment process. In most studied cases, the experimental eigenvectors were not assigned exactly since their assignment "success" depends on the locations visavis the achievable subspaces set by the selection of B0. In this work, a new method of selecting BO, termed the subspace rotation method, is proposed. The subspace rotation method is based on selecting BO such that the measured eigenvectors lie exactly in the achievable eigenvectors subspaces. This procedure is illustrated in Figure 4.2 and is accomplished by setting BO as B0 = [br, br2, ... ,brp I bi, bi, ... bi] (4.2.15) where br = real [(MX2 + DXk + K)ve] bi = imaginary [(MX2 + DX, + K)v where ve. is the eigenvector associated with the jth experimentally measured mode, and it is assumed that all n components of the experimental eigenvectors are available. This could be accomplished by any of the procedures discussed in Chapter 2. Clearly, when Bo is selected as shown in Eq. (4.2.15), the measured expanded eigenvectors lie exactly in the achievable subspaces defined in Eq. (4.2.12). Hence, there is no need for the projection operations defined in Eq. (4.2.13). This eliminates the required p inverses of nxn matrices involved in computing the achievable subspaces. As will be seen in Chapters 5 and 6, the elements of BO, as defined by Eq. (4.2.15), give an indication to the pseudocontroller about the extent of modification of each DOF in order for the structure to exhibit the jth measured eigenvalue and eigenvector. Rotated Subspace  Nominal Subspace Figure 4.2 Rotation of the Achievable Subspace. 4.3 Numerical Illustrations In this section, the characteristics of the proposed enhancement to the symmetric eigenstructure assignment model refinement algorithm (SEAMRA) are evaluated and compared to other refinement techniques for two example problems. The first problem is a widelyused springmass computer simulated example (Kabe, 1985). It is used here for the purposes of illustrating model refinement for a large local discrepancy, analogous to a damage detection situation. The phenomena of global/local mode switching and load path preservation are examined in this problem. The second problem is used to illustrate the characteristics of the enhanced SEAMRA in updating the finite element model (FEM) of a laboratory cantilever beam using actual measured modal parameters. 4.3.1 Damage Detection: Kabe's Problem Kabe's eight degree of freedom springmass system is shown in Figure 4.3. The mass and stiffness properties of the system are included in the figure. This problem presents a challenging situation for damage detection in that stiffness values of various magnitudes are included. The model exhibits closelyspaced frequencies and both local and global modes of vibration. m =0.001 m = 0.002 m = 1.0 ki = 1000 k2 = 10 k3 = 900 k4 = 100 j =12,.. 7 Figure 4.3 Kabe's Problem. A variation of Kabe's original problem is used here. Rather than the standard initial model commonly used, which has incorrect values for all of the connecting springs, only a single spring constant is changed. This is reflective of the fact that damage may occur as a large local change in the stiffness of a structural member. 4.3.1.1 Local to Global Mode Change In the first problem, Kabe's initial model is only incorrect for the spring between masses 3 and 5. A value of 500, five times that of the exact spring, is assumed in this problem. Changing the spring value from 500 to 100 also causes a local mode of vibration to be replaced by a global mode, thus presenting a difficult challenge for damage detection. Figure 4.4 presents elementbyelement stiffness matrix results for applying the Baruch and Bar Itzhack update (1978) and the symmetric eigenstructure assignment model refinement algorithm. Baruch Damage indicates that the update was made using Baruch and Bar Itzhack's algorithm. SEAM Damage indicates that the update was made using the SEAMRA with BO selected by using the modal (M) selection method. SEASR Damage indicates that the update was made by using the SEAMRA with BO selected using Subspace Rotation (SR) method. The xcoordinate on all plots are the indices of a column vector constructed by storing the upper triangular portion of the stiffness matrix in a column vector. The ycoordinate on each plots consists of the difference between the updated stiffness matrix elements and the original stiffness matrix. In the first case, as shown in Figure 4.4, it is assumed that only the fundamental mode of vibration is measured, but all eigenvector components have been measured. Thus, no expansion of eigenvectors is required. It is evident from Figure 4.4 that the Baruch update is unable to discern the damage, but that both the SEAM and SEASR are able to clearly locate the damage. In fact, the SEASR was able to exactly reproduce the correct stiffness matrix. This was true independent of which mode was used in the update. Also, it should be noted that the Baruch update tends to focus elemental changes in the third and fifth row of the stiffness matrix, indicating the possibility of damage between these degrees of freedom, but certainly giving no clear indication to the extent of damage. As is evident from the plot, the Baruch update has spread errors over several elements. Using the algorithm of Lin (1990), the damage vector is given as a = [1.0 0.93 0.72 0.83 0.70 0.90 0.97 1.0]T, where the element number corresponds to the structural DOFs and a number less than 1 indicates the 500 Indices Baruch Damage JUULI I I i 0 10 20 Indices 0 10 20 30 Indices 0 10 20 30 Indices Figure 4.4 Results for Kabe's Problem using the 1st Mode, Full Eigenvector. possibility of damage affecting that DOF It is obvious that DOFs 3 and 5 are affected by damage, but the results also indicate strong damage of DOF 4. In the second case, as shown in Figure 4.5, it is assumed that the first three modes of vibration have been measured, but only the first three components of the eigenvectors have been measured. The eigenvectors components were expanded for the Baruch update using dynamic expansion (Berman and Nagy, 1983) with subsequent orthogonalization (Baruch and Bar Itzhack, 1978). The least squares expansion was used for the SEAM update. The SEASR update utilized the orthogonal Procrustes expansion (Chapter 2). In comparing Figure 4.5 to Figure 4.4, it is clear that the damage detection capabilities of all three algorithms have been degraded when using expanded mode shapes, even though more modes have been measured. However, both the SEAM and SEASR updates give a clear enr\ 500W ' 0 10 20 Indices 500 __ SEAM Dan 5001 1 500 0 Baruch Damage 3UUr j I " 0 I I I 10 20 30 Indices 0 10 20 Indices SEASR Dar 0 10 20 30 Indices Figure 4.5 Results for Kabe's Problem using Modes 1, 2, 3, and Eigenvectors Components 1, 2, 3. indication to both the location and extent of damage. Using Lin's algorithm, the damage vector is given as a = [1.0 0.81 0.75 0.83 0.82 0.79 0.85 1.0]1. It is difficult from inspection of a to determine the location of damage. The effect of applying the iterative load path algorithm described in Chapter 2 in the update procedure is shown in Figure 4.6. For the Baruch update, 100 iterations were performed. For the SEAM and SEASR updates 2 and 3 iterations respectively, were performed. The iterations were halted early for both SEA updates because the discrepancy between the eigenstructure before and after masking was within the numerical precision of the symmetric eigenstructure assignment software. It is seen that the load path enforcement further enhances the damage detection capability of both SEA updates. r f 0n  _500 Actual Damage 0 , 500 0 10 20 30 Indices 0d 0 500 L 0 0 10 20 30 Indices Baruch Damage 10 20 Indices 20 Indices Figure 4.6 Results for Kabe's Problem using Load Path Preservation, Modes 1, 2, 3, and Eigenvectors Components 1, 2, 3. 4.3.1.2 Consistent Modes In the second problem, the initial model is only incorrect for the spring between masses 4 and 6. A value of 200, two times that of the exact spring, is assumed in this problem. In this problem, all global and local modes remain global and local modes respectively after damage. It should be noted that finding a problem with this feature was difficult. In the first case, as shown in Figure 4.7, it is assumed that only the fundamental mode of vibration is measured, but all eigenvector components have been measured. It is evident from Figure 4.7 that the Baruch and SEAM update are unable to discern the damage, but that the SEASR is able to clearly locate damage. In fact, the SEASR was able to exactly reproduce the correct stiffness matrix. Again, this was true independent of which mode was Actual Damage Baruch Damage t100 n 100 0 0^ 0 0 jU 100 100 0 10 20 30 0 10 20 30 Indices Indices SEAM Damage SEASR Damage 100 100 100 100 I II 0 10 20 30 0 10 20 30 Indices Indices Figure 4.7 Results for Kabe's Problem using the 1st Mode, Full Eigenvectors. used in the update. It should be noted that the Baruch and SEAM update tends to focus elemental changes in the fourth and sixth rows of the stiffness matrix, indicating the possibility of damage between these degrees of freedom, but certainly giving no clear indication to the extent of damage. Using Lin's algorithm, the damage vector is given as a = [1.0 0.98 0.92 0.85 0.87 0.84 0.95 1.0]T. This algorithm does not clearly identify the damage location. In the second case, as shown in Figure 4.8, it is assumed that the first three modes of vibration have been measured, but only DOFs 1,3, and 6 of the eigenvectors have been measured. In comparing Figure 4.8 to Figure 4.7, it is clear that the damage detection capability of all three algorithms has again been degraded when using expanded mode shapes. Only the SEASR update gives a clear indication to the location of damage, but is unable to predict the exact extent. Using Lin's algorithm, the damage vector is given as a= [0.99 0.18 0.55 0.34 0.52 0.33 0.41 1.0]T. Again, it is difficult from inspection of a to determine the location of damage. In fact, inspection of a indicates that DOF 2 is the most likely damaged DOF. 100 0 100 0 10 20 30 Indices SEAM Damage 10 20 30 Indices 01 Baruch Damage 100l D 10 20 30 Indices SEASR Damage 100K 0 100 0 10 20 30 Indices Figure 4.8 Results for Kabe's Problem using Modes 1, 2, 3, and Eigenvectors Components 1, 2, 6. It should be noted that in this problem, it was critical to have the proper DOFs measured. When the second test case was run with the first three DOFs measured, no algorithm was able to locate damage. In this case, the eigenvectors components were relatively unaffected by damage, thus causing substantial error in the eigenvector expansion process. 4.3.2 Model Refinement of a Cantilever Beam: Experimental Study 4.3.2.1 Modal Test Description The structure used in this investigation is the aluminum cantilevered beam shown in Figure 4.9. The dimensions and material properties of the beam are given in Table 4.1. Experimental modal analysis of the beam was performed to measure its modal properties. Six equally spaced translational degrees of freedom shown in Figure 4.9 were selected as measurement locations. The modal properties of the first three modes of vibration were determined using frequency domain techniques and single degree of freedom curve fitting algorithms. The excitation source used in the testing was an impact hammer and the driving point measurement was an accelerometer mounted at the free end of the beam. Impact and exponential windows were utilized to improve frequency response calculations. At each measured degree of freedom, five frequency response measurements were made and averaged to reduce the effects of measurement noise. Natural frequencies, damping ratios and mode shapes of the beam's first three modes of vibration were identified and are reported in Tables 4.2 and 4.3. 6 5 4 3 2 1 Figure 4.9 Experimental Cantilever Beam. Table 4.1 Structural Properties of the Cantilever Beam. Table 4.2 Measured Natural Frequencies and Damping Ratios of the Cantilever Beam. Mode # Natural Frequency (Hz) Damping Ratio (%) 1 7.25 4.41 2 45.55 0.68 3 127.01 0.33 Table 4.3 Measured Mode Shapes of the Cantilever Beam. Mode 1 2 3 DOF 1 1.00 1.00 1.00 2 0.95 0.16 0.30 3 0.65 0.53 0.61 4 0.36 0.72 0.20 5 0.15 0.52 0.75 6 0.03 0.12 0.28 4.3.2.2 Finite Element Model Description A twelve DOF undamped finite element model (FEM) of the beam was generated using six equal length beam elements as shown in Figure 4.9. The beam element has two degrees of freedom (DOFs) at each node; bending and rotation. This model was then reduced using Length 0.84 m Mass/Length 2.364 kg/m Moment of Inertia 3.02e9 m4 Youngs Modulus 70 GPa Guyan reduction (1965) eliminating the rotational degrees of freedom. There are several possible errors affecting the accuracy of this FEM. The most obvious is the fact that a perfect cantilever condition is assumed. In addition, an artificial error was purposely introduced by selecting the Young's Modulus higher than that often assumed for aluminum. 4.3.2.3 Application of the SEAMRA. Because the "true" finite element model is unknown, a comparison between the "true" and updated structural matrices is not possible. Besides such comparison, a fair and useful judgement on the quality of SEAMRA updating capability can be obtained by comparing actual experimental frequency response functions with those predicted by the initial and updated FEMs. Figure 4.10 shows a comparison of frequency response functions measured between degrees of freedom 1 and 3 (i.e. sensor measurement at DOF 1 and impact excitation at DOF 3). The solid curve corresponds to the experimental data, while the dotted line corresponds to that predicted by the original analytical FEM. It is apparent that the discrepancies between the frequency response function increases as the frequency of excitation increases. This is in part due to the fact that the assumption of a perfect cantilever condition affects the higher modes of vibration to a greater extent. The dashed lines in this figure corresponds to the SEASR updated finite element model. It is clear from this comparison that the SEASR provided a great deal of improvement to the original analytical FEM. Inspection of the updated stiffness matrix indicates that changes occur throughout the matrix, indicating that the discrepancy between the original and refined FEMs was due to degradation of some global structural property (Youngs Modulus), as opposed to some form of local damage, as seen in the previous problem. 4.4 Discussion of the SEAMRA's Formulation In some problems SEAMRA in conjunction with either the subspace rotation or the modal selection method, failed to find a symmetric updated FEM (symmetric Da and Ka). This shortcoming was especially encountered in practical situations when the experimental SEAMRA 102 , Solid Experimental Measurement Dash Modified Analytical Model 103 Dotted Original Analytical Model 104 10.5 , 106 ., 107 108  0 20 40 60 80 100 120 140 160 180 200 FREQUENCY (Hz) Figure 4.10 Experimental and Analytical Frequency Response Functions of the Cantilever Beam. modal properties were corrupted by measurement errors. This can be attributed to the fact that there is no symmetric updated FEM that is consistent with the present SEAMRA's formulation. Recall that SEAMRA's modifications to the initial stiffness and damping matrices in its updating process are given by AK = BoHoBT with Ho = FGo (4.5.1) AD = BoHIBo with H, = FG1 Clearly, from Eq. (4.5.1), the perturbations (i.e. modifications) to the initial stiffness and damping matrices are constrained by the relationship range(AK) = range(AKT) = range(AD) = range(ADT) (4.5.2) This relationship can always be satisfied for the cases when the experimental modal properties are consistent with an update in which (i) either AK or AD is zero or (ii) AK is proportional to AD (AK = P AD, P3 is a scalar). For all other cases, SEAMRA might fail to produce symmetric AK and AD. A more flexible and general formulation that accounts for such shortcoming is proposed in Section 6.5 of Chapter 6. The formulation as presented in Chapter 6 is more elegant, efficient and guarantees, for all situations, a symmetric updated FEM. 4.5 Summary A previously developed model refinement algorithm based on the general mathematical framework of eigenstructure assignment theory (Zimmerman and Widengren, 1989, 1990) has been extended and improved. A technique to enhance eigenvector assignability of the algorithm has been developed. The method consists of rotating the achievable eigenvector subspaces into the experimentally measured eigenvectors. The subspace rotation method, used in conjunction with one of eigenvector expansion techniques discussed in Chapter 2, results in both a decrease in the computational burden as well as an increase in the accuracy of the assigned eigenvectors. Finally, the improved algorithm (SEASR) was tested for its suitability for model refinement and structural damage assessment. The performances of SEASR in a damage assessment problem on a challenging simulated structure was presented and compared to other algorithms. The results acquired using the SEASR were superior. CHAPTER 5 DAMAGE LOCATION: THE SUBSPACE ROTATION ALGORITHM 5.1 Introduction In this chapter, a computationally attractive algorithm is proposed to provide an insight to the location of structural damage. The proposed algorithm is similar to the Modal Force Error Criteria proposed by several researchers ( Ojalvo and Pilon 1988; Gysin, 1990). However, a greater insight of the Modal Force Error criteria is provided. Further, a new viewpoint which allows for the reduction of the effects of measurement errors in the experimental modal parameters for a certain class of structures is also discussed. As will be shown in the next sections, the proposed damage location algorithm requires only matrixscalar and matrixvector multiplication. 5.2 The Subspace Rotation Algorithm: The Direct Method Assume that an nDOF finite element model of the "healthy" (undamaged) structure exists. As seen in the earlier chapters, the standard differential equation governing the dynamic motion of such structures is given by Mw + Dw + Kw = 0 (5.2.1) where M, D, and K are the n x n analytical mass, damping, and stiffness matrices, w is a n x 1 vector of positions and the overdots represent differentiation with respect to time. The eigenvalue problem associated with Eq. (5.2.1) in second order (lambda) form is given as (2M + XhiD + K)vhi = 0 (5.2.2) where kh and vh denote the ith eigenvalue and eigenvector, respectively, of the predamaged "healthy" structure. It is assumed that Eq. (5.2.2) is satisfied for all measured "healthy" eigenvalues/eigenvectors. This can be enforced by correlating the original FEM (M,D, and K), possibly through the use of a model refinement procedure. Next, consider that the p eigenvalues and eigenvectors, 4i and vvi, of a postdamage modal survey of the structure are available, in which Xd e X i, Vdi VYhi. In the present formulation, it is assumed that the dimension of the measured eigenvector is the same as the analytical eigenvector. As discussed in Chapter 2, this is true (i) when all FEM DOFs are measured (ii) after the application of an eigenvector expansion algorithm, or (iii) after the application of a finite element model reduction algorithm. The ideal situation would be to measure all FEM DOFs since the eigenvector expansion process would introduce additional errors in the "expanded" eigenvectors and the model reduction process would introduce errors in the FEM. It should be noted that in both cases the additional errors may become significant as the ratio of measured to unmeasured DOFs become smaller. Let AMd, ADd, and AKd be the exact perturbation matrices that reflect the nature of the structural damage. For localized damage, the exact perturbation matrices are sparse matrices with the nonzero elements reflecting the state of damage. The lambda equation for the damaged structure is, defined by (12(M AMd) + d(D ADd) + (K AKd))Vd = 0 (5.2.3) Although only p of the n eigenvalues/eigenvectors are assumed measured, p << n, Eq. (5.2.3) holds for any particular eigenvalue and eigenvector of the damaged structure because the perturbation matrices are assumed to be exact. Grouping all perturbation matrices on the righthand side defines a damage vector di, d. m Zdd (5.2.4a) = (X2AMd + XdADd + AKd)Vd (5.2.4b) where Zd = M + d dD + K Although Eqs. (5.2.4a, b) yield the same damage vector, it should be noted that Zdj and the coefficient matrix of Eq. (5.2.4b) are not equal. When the measured eigendata are not corrupted by noise, an inspection ofdi in terms of the Eq. (5.2.4b) reveals that the jth element ofdi will be zero when thejth rows of the perturbation matrices are zero, i.e. the finite element model for thejj* degree of freedom is not directly affected by damage. Conversely, a degree of freedom whose finite element model has been affected by damage will result in a nonzero entry in di. Thus, the degrees of freedom which have been affected by damage can be determined by inspecting the elements of di. Vector di as defined in Eq. (5.2.4b) also reveals that only a single mode of vibration needs to be measured exactly to determine the damage locations exactly. This is true in even multiple member damage situations. More importantly, the vector di can be determined from the original finite element model (M,D,K) and the measured eigenvalues and eigenvectors, Xd and vdi, using Eq. (5.2.4a). Thus, there is no need to use a model refinement algorithm to attempt to estimate the exact perturbation matrices in order to locate the damage. If the damping term is ignored, Eq. (5.2.4a) is essentially the Modal Force Error criteria as proposed by Ojalvo (1988) for use as a diagnostic "tool" to locate modelling errors in FEMs. A physical interpretation of Eq. (5.2.4a) provided by Ojalvo was "... di is the applied harmonic force error distribution, applied at frequency Xdi, which is necessary to cause the analytical model to vibrate with mode shape vd ...". However, Eq. (5.2.4b) provides a much clearer interpretation of the damage vector di for the damage location problem in which the perturbation matrices are sparse. In practice, the perfect zero/nonzero pattern of the damage vector di rarely occurs due to errors present in the experimentally measured eigenvalues and eigenvectors. This scenario was studied and discussed for the undamped case by Gysin (1990) in the context of eigenvector expansion techniques. Gysin (1990) observed that in certain specific cases of eigenvector expansion errors, the damage vector defined by Eq. (5.2.1a) may lead to incorrect conclusions concerning the location of damage. The next section proposes a new viewpoint which allows for the reduction of the effects of measurement errors for certain classes of structures. 5.3 The Subspace Rotation Algorithm: The Angle Perturbation Method In order to provide an alternative view of the state of damage, Eq. (5.2.4a) is rewritten as d 4 = 1 IId II dcos(W) (5.3.1) where d! is the jth component (or j* DOF) of the ith damage vector, zJ is the j* row of the matrix Zd and 01 is the angle between the vectors zJ and vd* In the case when the measurements are free of error, a zero dJ corresponds to a 06 of I I ninety degrees, whereas a nonzero d. corresponds to a OQ different from ninety degrees. 1 I Errors in the experimental measurements of modal parameters will cause slight perturbations in the angles 06 that destroy the zero/nonzero pattern of the damage vector. I One would initially expect that the components of di corresponding to the damaged DOFs would be substantially larger than the other elements. However, by inspecting Eq. (5.3.1), a large di component could be due to a zJ row norm substantially larger than other rows of Zd., di , coupled with a slight deviation of O9 from ninety degrees due to measurement noise. Hence, when dealing with a structure whose FEM results in zJ row norms of different order of di magnitude, it is more reasonable to use the deviation of the angles, W1, from ninety degrees for damage location, ai = 18 900 (5.3.2) d! where Oi = cos 1 I d The angle 01 is determined from Eq. (5.3.2) and a1 is the jth component of ai. I I 5.4 Practical Issues 5.4.1 Cumulative Damage Location Vectors The discussion in the previous section suggests that for a given mode, the damage is locatable if the perturbation to the angle, OQ due to the measurement error is less than the I angle perturbation due to the damage. Hence, modes that are highly affected by the damage are expected to provide better assessment to the location of the damage when errors are present in the measured eigendata. As will be seen in example problems (Chapter 7), certain modes are more susceptible to a given state of damage than others. This is mainly due to the fact that different elements of the structure have different levels of contribution to the total strain energy of a given mode (Kashangaki 1992). Furthermore, a particular part of the structure usually has different orders of strain energy contribution for different modes. Usually, if the damage occurs in a region of high strain energy for a given mode, that mode would be highly susceptible to the damage and, hence would reflect the state of damage. To accommodate this type of problem, when the number of measured modes p is greater than one, two different composite damage vectors may be defined as d i= 1 di a 1 i (5.4.2) p i=l In Eq. (5.4.1), the damage vectors, di, are normalized with respect to their corresponding eigenvectors, Vdi. The reason for this normalization is such that the composite vector ignores the inherent "weighting" of I yd 1i, which is usually of different orders of magnitudes for different measured modes. It should be noted that in the multimode measurement case, Eq. (5.4.2) is preferable when the values of 1 zJ  are of different orders of magnitude for different measured modes. Again, in practice, different measured modes. Again, in practice, the DOFs affected by the damage are expected to have substantially larger d or _q. Finally, the damaged areas of the structure can then be located using the knowledge of the "damaged" DOFs and the connectivity of the FEM. It is interesting to note that Eqs. (5.2.4a, b) reveal an interesting relationship between various model refinement algorithms. Model refinement techniques attempt to approximate the exact perturbation matrices by using limited modal data, but do so in different manners. However, Eq. (5.2.4a, b) indicates that if the model refinement technique has satisfaction of the eigenproblem as an equality constraint, the calculated perturbation matrices AM, AD, and AK are constrained to be related to the original finite element model M, D and K and the measured eigendata by Eqs. (5.2.4a, b). 5.4.2 Eigenvector Filtering Algorithm In a modal survey, the errors associated with the measured eigenvectors are typically greater than the error associated with the measured eigenvalues. In addition, in the case of incomplete eigenvector component measurements, these measurement errors are often compounded with eigenvector expansion induced errors. A simple eigenvector noise filtering algorithm is proposed assuming the measured eigenvalues to be correct. From the cumulative damage vector defined in Eqs. (5.2.4a) or (5.2.4b), and the original FEM connectivity, the engineer can deduce which DOFs have been damaged. It is reasonable to assume that nonzero elements in each damage vector di associated with "undamaged DOFs" are due to eigenvector errors. These elements can then be set to zero. In addition, the magnitude of the elements ofdi at the "damaged" DOFs can be adjusted by using knowledge of DOF connectivity and the properties of the element property matrices connecting the "damaged DOFs." The element property matrices provide constraints relating the effect of damage on each element DOF. The noise filtering algorithm consists simply of replacing the di vectors by df, where df is obtained from di as described above. The ith filtered eigenvector, vdf, can then be obtained from solving (dXM + ,dD + K)vdf = df (5.4.3) using Gaussian elimination. In this calculation, the bandedness of typical FEM matrices should be exploited. Essentially, the filtered eigenvector is just the eigenvector that if measured would have produced the damage vector df. Experience gained in using the eigenvector filtering algorithm indicates that it is best to use structural matrix properties (M, D, K) that (i) are finite element consistent, and (ii) have not been "corrupted" by measurement noise. By finite element consistent, it is meant that the property matrices can be achieved by a finite element program. Note that measurement noise can be introduced in the property matrix through a FEM refinement algorithm. Hence, the property matrices that should be used are the original property matrices (unrefined). The present eigenvector filtering algorithm can be useful in improving the damage extent assessment. This concept is discussed in Chapters 6 and 7. 5.5 Summary A computationally attractive algorithm to determine the location of damage in structures was developed. The algorithm completely bypasses the general framework of the model refinement problem and involves only matrixscalar and matrixvector multiplications. The effect of measurement error in the eigendata was discussed and techniques to reduce these effect were presented. Furthermore, a simple eigenvector filtering algorithm was developed. Practical example problems to illustrate and evaluate the performance of the developed algorithm will be presented in Chapter 7. CHAPTER 6 THE MINIMUM RANK PERTURBATION THEORY 6.1 Background The theory developed in Chapter 5 is limited to determining the location of structural damage. In a practical situation, it is essential to determine the extent of the damage to get a good estimate about the overall integrity of the structure. In general, the extent problem, as discussed in Chapter 1, has been approached by several researchers using existing model refinement algorithms. The formulations of these algorithms were obviously based on the "model refinement philosophy": minimum change made to the original FEM. The minimum change constraint has a clear tendency to smear the changes throughout the entire FEM. However, in most cases, this philosophy is not consistent with the effect of structural damage on FEMs. In fact, the effects of structural damage on FEMs are usually "nonminimal" localized perturbations. Structural damage often occurs at discrete locations. The effect of damage on the analytical model is often restricted to just a few elements of the finite element model. The rank of each element mass, damping or stiffness matrix is dependent on the number of degrees of freedom defined by the element and the shape functions utilized. However, it should be noted that in general the element matrices are not of full rank. For example, the rank of the 6x6 element stiffness matrix of a three dimensional truss element is just one. Thus, instead of using the matrix Frobenius norm minimization formulation to arrive at unique perturbation matrices, minimum rank perturbation constraints are enforced. In this Chapter, a computationally attractive damage extent algorithm is proposed. The proposed damage extent algorithm is a minimum rank perturbation, which is consistent with the effects of many classes of structural damage on a FEM. Assume that "p damaged" eigenvalues and eigenvectors have been measured and that the original FEM has been corrected such that its modal properties match the measured modal properties of the healthy model. The eigenvalue problem of a damaged structure shown in Eq. (5.2.3), for all p measured modes, can be written in matrix form, as MVdAd + DVdAd + KVd = AMdVdAd + ADdVdAd + AKdVd B (6.1.1) where Ad = diag(Xd,. d2' ... dp) Vd= [d,,Y.d2, ..., Yd B = [dl,d2 ... dp] where all variables have the same definitions as in the previous chapter. Note that matrix B can be determined from the FEM (M, D, K) and the "p" measured eigenvalues and eigenvectors. As discussed earlier, the damage extent problem consist of finding the perturbation matrices, AMd, ADd, and AKd, such that Eq. (6.1.1) is satisfied. As already discussed in Chapter 2, structures can be modelled using either undamped, proportionally damped or nonproportionally damped finite element models. The proposed extent algorithm is formulated to accommodate all three types of structural models. For each type of model, several scenarios of damage effects are considered. Practical issues that can be used to improve the damage extent estimate are also presented. 6.2 The Minimum Rank Perturbation Theory: Theoretical Background In this section, the theoretical foundation of the Minimum Rank Perturbation Theory (MRPT) is derived. This theory will be extensively used throughout the remainder of this chapter. PROPOSITION 6.1 Suppose that X, Y E RnxP are given where p < n and rank(X)=rank(Y)=p. Define % to be the set of matrices A in R"' that satisfy, AX = Y with AT = A Then, (1.a) If the set % is nonempty, the minimum rank of any matrix, A, in % is p. Next, define %P to be a subset of % comprised of all A such that rank(A) =p. Then (1.b) If the matrix YTX is symmetric, then one member of %P is given by AP = YHYT (6.2.2) with H = (yTX) and (1.c) The matrix defined by Eq. (6.2.2) is the unique member of %9P. Proof: To prove Proposition (6.1.a), note that Eq. (6.2.1) is exactly satisfied if and only if range(Y) is included in range(A), which is also the range(AT) by symmetry. This implies that rank(Y) = p : rank(A). /// To investigate Proposition (6.1.b), assume that the expanded singular value decomposition of one member, AN, of 9%P to be of the form (6.2.3) where Uj = [u{, u ... ..] XJ = diag(oJ{, oJ, ... i) (6.2.1) APJ = Uj:jUjw where the superscript j indicates the jth family member of 9%P, the uj are the left and right singular vectors and the ao are the nonzero singular values of Ap. In the expanded singular value decomposition, the (p+l) to n singular vectors are not shown in the factorization because of their corresponding zero singular values. Note that the left and right singular vectors are the same because AN is restricted to be symmetric. For Eq. (6.2.1) to be satisfied, the range of Y, AP'N, ApJT and Ui must be equal. Thus, any column of Y can be written as a linear combination of the u.'s. The matrices Y and Ui are then related by a unique pxp invertible matrix QJ, Y = UJ Q (6.2.4) Substituting Eq. (6.2.4) into Eq. (6.2.3) gives AP = Y(Qj12jQJT)YT = yHJyT (6.2.5) Thus, each family member is uniquely defined by the factorization of Eq. (6.2.3). From Eq. (6.2.5), it is evident that HJ is of full rank because its inverse exists (HI' = QjTFji~Q). Inspection of Eq. (6.2.5) reveals that the only unknown term in the factorization is Hi. By using the factorization of AN as defined by Eq. (6.2.5), Eq. (6.2.5) can be rewritten as Y = APX = (YHjiYX = Y(HjYTX) (6.2.6) Equation (6.2.6) is satisfied if and only if HJYTX = Ipxp, where Ipxp is the pxp identity matrix. This is true because Y and X are of full column rank. Thus, Hi is uniquely calculated to be Hi = (yTX)1 (6.2.7) /// Proposition (6.1.c) follows immediately by inspecting the right hand side of Eq. (6.2.7). Inspection reveals that HJ is the same for all members of %P. This fact, in conjunction with Eq. (6.2.5) leads to the conclusion that AN'j is the unique member of the set %P. This member is given by Eq. (6.2.2). At this point, the MRPT as defined in Proposition 6.1 assumes that the matrices X and Y are of full rank. In practical uses, as will be seen later, matrix X is usually of full rank. The rank requirement on matrix Y can be of some concern since it is directly related to the rank of matrix A. The next proposition addresses the case in which matrix Y is rank deficient. PROPOSITION 6.2 Suppose that X, Y E Rnxm are given and rank(X)=m and rank(Y)=p, where p < m < n. Further, suppose that the matrix YTX is symmetric. Define cl, to be the set of matrices A in Rnxn that satisfies the problem, APXP = YP with (AP)T = AP (6.2.8) where the superscript p indicates a rank p matrix. In Eq. (6.2.8), XP, YP E Rnxp are corresponding full rank submatrices of X and Y. Then The set U contains a single member, AP, that can be calculated from Eq. (6.2.2) using any corresponding XP and YP. Proof: The jth member of the set cM is given by AP = YPHjYPj (6.2.9) with HJ = (YPJTXPj) where the additional superscript ( )J indicates the jth member of c. Note that Hi is symmetric since YTX is symmetric. The range of any YP is equal to the range of Y, thus the YP'1 and YP'J are related by YP = YPj Qi' (6.2.10) where Qid C Rpxp and rank(QiJ) = p. By utilizing Eq. (6.2.10), Eq. (6.2.9) can be written for the ith member of cU as Ap, = YPJ(Qi'HiQiT)YPJT (6.2.11) with H' = (QijTYPTp,i) 1 Again, Hi is symmetric because YTX is symmetric. In comparing Eqs. (6.2.9) and (6.2.11), it is seen that AN = Ap' if HJ = Q'J H' QjiT (6.2.12) or equivalently, Hij = Qi'THiTQ'ji' (6.2.13) where Eq. (6.2.13) makes use of the symmetry of Hi. By using the definitions of Hi and H9, Eq. (6.2.13) can be written as YPjTXP' = Qi,TXPiYPJ (6.2.14) Premultiplying Eq. (6.2.14) by Qi'j and utilizing the relation in Eq. (6.2.10) gives the condition such that AN = AP", namely YPirXPj = XpirYPj (6.2.15) This condition is clearly satisfied since YTX is symmetric. The actual uses of Proposition 6.1 and Proposition 6.2 will be clearly seen in the next sections. The practical implication of Proposition 6.2 is discussed in detail in Section 6.6. 6.3 Damage Extent: Undamped Structures In some cases, the damping of the system under consideration is assumed to be negligible. For this type of system, MRPT based algorithms will be developed assuming that the structural damage affects (i) only the mass properties, or (ii) only the stiffness properties, or (iii) simultaneously the mass and stiffness properties. 6.3.1 Damage Extent: Mass Properties In this case, it is assumed that the effect of damage on the stiffness properties of the structure is negligible. With this assumption, Eq. (6.1.1) can be rewritten as MVdA2 + KVd = AMVdA2 = B (6.3.1) Note that the eigenvectors are real and the eigenvalues are purely imaginary. Further, the eigenvectors are linearly independent, which implies that the matrix product VdA2 is of full column rank if rigid body modes are not included. Assume, for the moment, that B is of full rank (rank(B) = p). Then, Proposition 6.1 can be applied to determine the perturbation matrix, AMd, as AMd = B(BTVdA) BT (6.3.2) by letting Y=B and X=VdA Note that the required inversion is that of a pxp matrix, where "p" is the number of measured modes. As discussed in Proposition 6.1, this inversion is feasible if matrix B is of full rank and the rigid body modes of the system are omitted in the computations. When matrix B is rank deficient, Proposition 6.2 should be used to render the computation possible. The properties associated with AMd as computed in Eq. (6.3.2) are as follows: PROPERTY 6.3.1 The perturbation matrix, AMd, defined in Eq. (6.3.2) will be symmetric if the eigenvectors, Vd, are stiffness orthogonal, i.e., the eigenvectors are orthogonal with respect to the original stiffness matrix, K. Proof: Proposition (6.1.c) in conjunction with Proposition (6.1.b) implies that the existence of the unique symmetric rank p AMd requires the symmetry of the matrix product BTVdA 2 The symmetric equivalence associated with this matrix product is BTVdA = A VB (6.3.3) Substituting the expression for B from Eq. (6.3.1) into Eq. (6.3.3) gives AdV MVdAd + V KVdA2 AdVdMVdAd + A2VTKVd (6.3.4) where the symmetry of M, K and A2 has been used in writing Eq. (6.3.4). From Eq. (6.3.4), it is clear that the equivalence is true if (VKVd)A == Ad(V KVd) (6.3.5) Equation (6.3.5) will obviously be satisfied if the measured eigenvectors are stiffness orthogonal. Baruch (1978) treated one approach to mass orthogonalize the measured eigenvectors. A similar approach can be used to orthogonalize the measured eigenvectors with respect to the stiffness matrix. PROPERTY 6.3.2 The updated finite element model (FEM) defined by the original mass and stiffness matrix along with the perturbation mass matrix computed using Eq. (6.3.2) preserve the rigid body characteristics of the original FEM. Proof: This is apparent in that the original stiffness matrix is unchanged and that the rigid body modes are defined as modes whose eigenvectors lie in the null space of the stiffness matrix. 6.3.2 Damage Extent: Stiffness Properties Here, it is assumed that the effect of damage on the mass properties of the structure is negligible. With this assumption, Eq. (6.1.1) can be rewritten as MVdA2 + KVd = AKdVd = B (6.3.6) For this problem, the eigenvectors are real and the eigenvalues are purely imaginary. The eigenvectors are also linearly independent, which implies that matrix Vd is of full rank. If matrix B is assumed to be of full rank (rank(B)=p), Proposition 6.1 can be used to determine the perturbation to the original stiffness matrix, AKd = B(BTVd) IB (6.3.7) This expression for AKd is determined by setting Y=B and X=Vd in Eq. (6.1.3). The properties associated with AKd as computed by Eq. (6.3.7) are as follow. PROPERTY 6.3.3 The matrix AKd will be symmetric if the eigenvectors are mass orthogonal, i.e., the eigenvectors are orthogonal with respect to the original mass matrix. The proof of Property 6.3.3 follows very much the same pattern as the one presented for AMd (Property 6.3.1). PROPERTY 6.3.4 The updated FEM defined by the original mass and stiffness matrices and the perturbation stiffness matrix, AKd, preserves the rigid body mode characteristics if the measured eigenvectors and the rigid body modes are mass orthogonal. The original rigid body modes of an undamped system are defined by the eigenvalue problem, KVr = XrMVr = 0 (6.3.8) where the subscript r denotes the rigid body mode(s) and Xr is equal to zero. Thus, the rigid body modes lie in the null space of the original stiffness matrix. The rigid body modes of the system will be preserved in the updated model if the original rigid body modes lie in the null space of the updated stiffness matrix, e = (K AKd)Vr (6.3.9a) = AKdYr (6.3.9b) where vector e is zero if the the rigid body modes are preserved. Equation (6.3.8) has been used to arrive at the expression shown in Eq. (6.3.9b). Substituting Eq. (6.3.7) into (6.3.9b) gives e = B(BTVd) BTr (6.3.10) By utilizing the symmetry of the original mass and stiffness matrices, along with Eq. (6.3.6), Eq. (6.3.10) can be expanded as e = B(BTVd) [VTKKv + AdVdMv,] (6.3.11) The first term in the parenthesis is zero because the matrixvector product Kvr is zero by definition. The second term will be zero if the rigid body modes and the measured mode shapes are mass orthogonal. 6.3.3 Damage Extent: Mass and Stiffness Properties In this case, it is assumed that the structural damage affects simultaneously the mass and stiffness properties of the structure. With this assumption, Eq. (6.1.1) can be rewritten as MVdAd + KVd = AMdVdA2 + AKdVd = B (6.3.12) 6.3.3.1 Application of The MRPT Assume that Eq. (6.1.12) can be decoupled as follows, AMdVd = Bm (6.3.13a) AKdVd = Bk (6.3.13b) Then, the MRPT, as formulated in Proposition 6.1, can be applied to determine the perturbation matrices AMd and AKd, as AMd=Bm (BLVd) 1 B (6.3.14a) AKd = Bk (B Vk) Bk (6.3.14b) Note that the matrices BLVd and B Vd are invertible if Bm and Bk are of full rank. When these rank requirements are not met, Proposition 6.2 can be used to make the computations possible. 6.3.3.2 Decomposition of Matrix B The decomposition problem as illustrated in the previous section is equivalent to the problem of solving for the matrices Bm and Bk. So far, the only constraint that these unknown matrices must satisfy is given by the expression, B = Bm A2 + Bk (6.3.15) which results from Eqs. (6.3.12) and (6.3.13). Naturally, there is an infinite set of solutions (Bm, Bk) that satisfy Eq. (6.3.15). To arrive at a unique solution, additional physically meaningful constraints can be enforced. The decomposition proposed herein exploits the crossorthogonality relations that arise from the symmetric nature of the property matrices and the undamped assumption. By measuring mass normalized "damaged" eigenvectors (which is possible if a driving point measurement is made), the crossorthogonality relations associated with the damaged structure can be written as V(M AMd)Vd = Ipxp (6.3.16a) VTK AKd)Vd = diag( d,2, 2) = 2 d (6.3.16b) in which Odi is the natural frequency of the ith mode of the "damaged" structure. Matrix Ipxp is the pxp identity matrix. A rearrangement of Eq. (6.3.16) yields VT AMd Vd = VT M Vd Ipxp VT Bm (6.3.17a) VdT AKd Vd = V K Vd d = V Bk (6.3.17b) Clearly, the matrices Bm and Bk can be computed from Eqs. (6.3.17). In the rare situation that the number of measured modes is equal to the number of DOFs in the FEM (p = n), these can be computed by simply inverting matrix Vd. Unfortunately, as discussed earlier, the number of measured modes is usually much less than the number of FEM DOFs (p << n). In this case, the solution that naturally comes to mind is to use the pseudoinverse of matrix VT. The inconvenience of this approach is that the sparsity pattern of matrix B will not be reflected in the computed matrices Bm and Bk. Remember that the sparsity pattern of B, as discussed in Chapter 5, indicates the location of the damage affecting the structure. A more physically intuitive approach is to constrain Bm and Bk to exhibit the same sparsity pattern as matrix B. This is done by casting B in an equation similar to the expressions of Eqs. (6.3.17). The problem in question is then to find an nxp matrix P that satisfies P(VTB) = B (6.3.18) Matrix P can be computed as P = B (VTB) (6.3.19) The inverse involved in this computation is that of a pxp matrix which is invertible if matrix B is of full rank. Now that P is computed, Bm and Bk can be computed using Eq. (6.3.19) as B = P(V M Vd Ipxp) (6.3.20a) Bk = P K Vd Qd) (6.3.20b) It is clear from Eq. (6.3.19) that P will have the same sparsity pattern as matrix B. Hence Bm and Bk will also reflect the important sparsity pattern of B. The computed matrices Bm and Bk can also be used to determine the effect of the damage, respectively, on the mass, and stiffness properties. As in Chapter 5, cumulative vectors associated to Bm and Bk can also be defined when more than one measured mode is available. d 1 IdrI Am = I i (6.3.21a) dk Idl (6.3.21b) i=111 di where dm. and dki are, respectively, the ith column of matrix Bm and Bk. PROPERTY 6.3.5 The perturbation matrices (AMd, AKd) computed from the MRPT using the Bm and Bk resulting from the decomposition discussed above will be symmetric. Proof: The perturbation matrices AMd and AKd will be symmetric because they satisfy the relationships in Eqs. (6.3.17) and the right hand sides of these equations are symmetric. 6.4 Damage Extent: Proportionally Damped Structures Since many structures have nonnegligible damping, it is of practical interest to extend the MRPT to address damped structures. In this analysis, the structure under consideration is assumed to exhibit proportional damping. 6.4.1 Damage Extent: Stiffness and Damping Properties It is assumed that the effect of the structural damage on the mass properties is negligible. In this context, Eq. (6.1.1) is rewritten as MVdA2 + DVdAd + KVd = ADdVdAd + AKdVd B (6.4.1) The complex conjugate of Eq. (6.4.1) is ADdVdAd + AKdVd = B (6.4.2) where the overbar indicates the complex conjugate operator, and the fact that ADd, AKd and Vd are real has been used in writing Eq. (6.4.2). Subtracting Eq. (6.4.2) from Eq. (6.4.1) gives ADdVd(Ad Ad) = (B B) (6.4.3) If (B B) is assumed to be of full rank, Proposition 6.1 can be applied to determine the perturbation matrix, ADd, as ADd = (B B)Hd(B BT with Hd = [(B TVd(Ad d)1 (6.4.4) Note that ADd as defined by Eq. (6.4.4) is real. Postmultiplying Eq. (6.4.1) by Ad and Eq. (6.4.2) by Ad and subtracting the two equations leads to AKdVd(Ad Ad) = (BAd BAd) (6.4.5) where the fact that Ad and Ad are diagonal matrices has been used in writing Eq. (6.4.5). If (BAd BAd) is assumed to be of full rank, Proposition 6.1 can also be applied to determine the perturbation matrix, AKd, as AKd = (BAd BAd)Hk(BAd BAd) 1 (6.4.6) with Hk = [(BAd BAd)TVd(Ad Ad)] (6.4.6) Note that AKd as defined by Eq. (33) is also real. PROPERTY 6.4.1 The perturbation matrices ADd and AKd, as computed above, will be symmetric if the measured eigenvectors, Vd, are mass orthogonal; i.e., the eigenvectors are orthogonal with respect to the original unperturbed mass matrix. Proof: Matrix ADd is symmetric if Hd is symmetric or, equivalently, if Hd 1 is symmetric. Hence, to get a symmetric ADd, the following equivalence must be satisfied. (B B)TVd(Ad Ad) = (Ad AVT(B B) (6.4.7) Substituting the expressions for B and B, from Eqs. (6.4.1) and (6.4.2) respectively, into Eq. (6.4.7) yields (A2VTM + AdVTD AdV2TM AdVD Vd Ad) d d d (6.4.8) (Ad Ad)V (MVdA2 + DVdAd MVdd + DVdAd Note that in Eq. (6.4.8) the terms involving matrix AKd canceled out. A further expansion and simplification of Eq. (6.4.8) yields (Ad Ad) VMMV (Ad d) (Ad d) VMVd (A Ad) (6.4.9) which is clearly satisfied if the measured "damaged" eigenvectors, Vd, are mass orthogonal. Likewise, the perturbation matrix AKd as computed in Eq. (6.4.6) is symmetric if Hk is symmetric or, equivalently, if Hk 1 is symmetric. This symmetry requirement yields the following equivalence. (BAd BAd)TVd(Ad Ad) (Ad Ad)VT(BAd BAd) (6.4.10) Substitution of the expressions for B and B into Eq. (6.4.10) yields (AdA2VM + AdVK AdAdV M AdVK) Vd(Ad Ad) T 2 2 (6.4.11) S(Ad Ad)Vd (MVdA dAd + KVdAd dAd KVdAd) in which the terms involving matrix ADd cancel. Manipulating and simplifying Eq. (6.4.11) yields (?dAdA A dAd)VMVd(Ad Ad) = (Ad Ad)V dMVd(XdAd AdAd) (6.4.12) This equivalence is obviously satisfied if the eigenvectors are mass orthogonal. /// PROPERTY 6.4.2 The updated FEM, defined by the original FEM and the perturbation matrices ADd and AKd computed from Eqs. (6.4.4) and (6.4.6), preserves the original rigid body modes if the measured eigenvectors and the rigid body modes are mass orthogonal. Proof: As discussed earlier, a rigid body mode is defined as a mode whose eigenvalue is equal to zero and whose eigenvector lies in the null space of the FEM stiffness matrix. Hence, the rigid body modes of the original system are preserved in the updated FEM if they lie in the null space of the perturbed stiffness matrix. Consider the relationship e = (K AKd)Vr (6.4.13) where Yr is a rigid body mode eigenvector. Clearly, the rigid body mode associated to eigenvector Yr is preserved if e = Q. By definition, _r is a rigid body eigenvector of the original system, hence Eq. (6.4.13) can be simplified as e = AKdYr (6.4.14) Substituting the expression for AKd as defined in Eq. (6.4.6), into Eq. (6.4.14) gives e = (BAd BAd)Hk(BAd BAd) r (6.4.15) Substitution of the expressions for B and B into this equation yields e = (BAd Ad)Hk[ AdAVM + A K AdAdVdM AdVK ]vr (6.4.16) By using the fact that vr is a rigid body eigenvector of the original system (i.e. Kvr = 0), Eq. (6.3.16) can be simplified as e = (BAd BAd)Hk( AdA2 Ad )VMvr (6.4.17) It is clear from Eq. (6.4.17) that e = 0 if the rigid body mode Yr and the measured eigenvectors Vd are mass orthogonal (i.e. VTMvr = 0). 6.4.2 Damage Extent: Mass and Damping Properties In this case it is assumed that the effect of the structural damage on the stiffness properties is negligible. In this context, Eq. (6.1.1) is rewritten as MVdAd + DVdAd + KVd = AMdVdA2 + ADdVdAd = B (6.4.18) By using an approach similar to one used in the preceding section, Eq. (6.4.18) and its complex conjugate can be manipulated to yield the following decomposition AMdVd(Ad Ad) (BAd BA) (6.4.19) ADdVd(AdAd Ad) = (BAd dBA) (6.4.20) Again by applying the MRPT to the preceding equations, AMd and ADd are determined to be AMd = (BAd Ad)Hm(BAd BAd)T SI (6.4.21) with Hm = (BAd 2BA) V d Ad ADd = (BAd RAd)Hd(BAd TA2 with Hd ( BA) Vd(Add dA(6.4.22) Clearly, the perturbation matrices AMd and ADd as defined by Eqs. (6.4.21) and (6.4.22) are real. PROPERTY 6.4.3 The perturbation matrices AMd and ADd, as computed above, will be symmetric if the measured eigenvectors, Vd, are stiffness orthogonal; i.e., the eigenvectors are orthogonal with respect to the original unperturbed stiffness matrix. PROPERTY 6.4.4 The updated FEM, defined by the original FEM and the perturbation matrices, AMd and ADd, preserves the original rigid body modes. The proof of Property 6.4.4 is straightforward since the original stiffness matrix is unchanged (see Property 6.3.2). The proof of Property 6.4.4 follows very much the same pattern as the proof of Property 6.4.2. 6.4.3 Damage Extent: Mass and Stiffness Properties In this problem, it is assumed that the effect of the structural damage on the damping properties is negligible. For this situation, the general eigenvalue problem defined in Eq. (6.1.1) associated to this case can be simplified as MVdA2 + DVdAd + KVd = AMdVdA2 + AKdVd B (6.4.23) Algebraic manipulations of Eq. (6.4.23) and its complex conjugate yield the following decomposition AMdVd(A Ad) = (B B) (6.4.24) AKdVd(A A) = (BAd dBA) (6.4.25) The perturbation matrices AMd and AKd can then be computed using the MRPT. AMd = (B )Hm(B B)T with Hm = (B BTVdA2 1 (6.4.26) T AKd = (BAd BA)HA BAS d T (6.4.27) with Hk (BA A) Vd( (6.4.27) Note that AMd and AKd as defined by Eqs. (6.4.26) and (6.4.27) are real. PROPERTY 6.4.5 The perturbation matrices AMd and AKd, as computed above, will be symmetric if the measured eigenvectors, Vd, are damping orthogonal; i.e., the eigenvectors are orthogonal with respect to the original unperturbed damping matrix. PROPERTY 6.4.6 The updated FEM, defined by the original FEM and the perturbation matrices, AMd and AKd, preserves the original rigid body modes if the measured eigenvectors and the rigid body modes are damping orthogonal. These proofs of the above two properties are not reported here. They follow very much the same pattern as the proofs in Section 6.4.2. 6.4.4 Damage Extent: Mass. Damping and Stiffness Properties The eigenvalue problem of a proportionally damped system with all property matrices simultaneously affected by damage can be rearranged into the form MVdAd + DVdAd + KVd = AMdVdA + ADdVdAd + AKdVd = B (6.4.28) The theory developed in Section 6.3.3 can be expanded to address this particular problem. The crossorthogonality relationships associated with this type of structures are Vj(M AMd)Vd = Ipxp (6.4.29a) VJ(D ADd)Vd = diag(2Odd1, ,2E ddp) = (6.4.29b) VT(K AKd)Vd = diag(wd2, .. ap2) = d (6.4.29c) Notice that the crossorthogonality relationships in Eqs. (6.4.29a) and (6.4.29c) are exactly the same as the ones associated with undamped systems reported in Eqs. (6.3.16a) and (6.3.16b). As before, these crossorthogonality conditions can also be rearranged as V AMd Vd = V M Vd pxp = V Bm (6.4.30a) VT ADd Vd = V D Vd d VT Bd (6.4.30b) VT AKd Vd = V K Vd d = V Bk (6.4.30c) Following the exact same argument discussed for undamped systems in Section 6.3.3, an nxp matrix P that satisfies the relation, P(V dB) = B (6.4.31) is sought, where B is computed using Eq. (6.4.18) and (vdTB) is a pxp matrix. Although B is a complex matrix, the nxp matrix P is real, since Vd is real. Hence, for computational efficiency, matrix P can computed from P = Br(V Br) (6.4.32) where Br is the real part of B. In Eq. (6.4.32), it is assumed that matrix (VTB) is invertible. With P computed, the next step is to determine the decomposed damage vectors that indicate the effects of the damage on the mass, damping and stiffness matrices, Bm = AMVd = P (V M Vd pxp) (6.4.33a) Bd = ADVd = P (V D Vd Yd) (6.4.33b) Bk = AKVd = P (V' K Vd d) (6.4.33c) The minimum rank perturbation theory (MRPT), as formulated in Proposition 6.1, can again be applied to determine the perturbation matrices, AMd, ADd and AKd, as AMd = Bm (BTVd)1 Bm (6.4.34a) ADd =Bd (B Vd)1 BT (6.4.34b) AKd = k (BVk) 1 Bk (6.4.34c) Note that the matrices BTVd, BJVd and B Vd are pxp matrices that are invertible if Bm, Bd and Bk are of full rank. As in all other cases already studied, Proposition 6.2 can be used to deal with the situation when any one of these matrices are rank deficient. The cumulative damage location vector associated to Bm and Bk, defined in Eqs. 6.3.21, are also applicable to this problem. An additional cumulative damage vector associated to the perturbations in the damping properties can be similarly defined as dd = I (6.4.35) pi=l 1 d where dd is the ith column of matrix Bd. 