Substructural identification with incomplete measurement for structural damage assessment

SUBSTRUCTURAL IDENTIFICATION WITH INCOMPLETE MEASUREMENT FOR STRUCTURAL DAMAGE ASSESSMENT TEE KONG FAH NATIONAL UNIVERSITY OF SINGAPORE 2004 SUBSTRUCTURAL IDENTIFICATION WITH INCOMPLETE MEASUREMENT FOR STRUCTURAL DAMAGE ASSESSMENT TEE KONG FAH B.Eng. (Hons.) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF CIVIL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2004 ACKNOWLEDGEMENT Firstly, I sincerely thank my supervisors, Prof. Koh Chan Ghee and Assoc. Prof Quek Ser Tong, for their useful advice and continuous guidance throughout my graduate study. Their invaluable comments and suggestions were helpful in completion of this thesis. In addition, I would like to thank Prof. Zhang Lingmi for his advice and interest. I have learnt much valuable knowledge as well as serious research attitude from them in the past three years and a half. The financial support by means of research scholarship provided by the National University of Singapore is greatly appreciated. I would also like to thank the technologists in Structural Engineering Laboratory for their assistance in the experimental work. I am thankful to my family, the ones I love, for their care and encouragement, and for the happiness they give me. Finally, I thank all my friends within and outside the Department of Civil Engineering, with whom I spent a lot of good time during my graduate study. Many of them helped me, in one way or another, through difficult times. ii TABLE OF CONTENTS Title Page i Acknowledgement ii Table of Contents iii Summary ix Nomenclature xii List of Figures xv List of Tables xxi CHAPTER 1: Introduction 1.1 Background 1.1.1 System Identification in Structural Damage Assessment 1.2 Literature Review 1.2.1 Time Domain Identification 1.2.2 Observer Kalman Filter Identification (OKID) 1.2.3 System Realization Theory 1.2.4 Model Updating 1.2.5 Second-Order Model Identification 1.2.6 Model Condensation Methods 10 1.2.7 Substructural Identification 11 1.2.8 Genetic Algorithm 13 1.3 Objectives and Scope of Study 14 1.4 Organization of Thesis 16 iii CHAPTER 2: First-Order and Second-Order Model Structural Identification 21 2.1 General Remarks 21 2.2 Basic Formulation 22 2.3 Observer Kalman Filter Identification (OKID) 24 2.4 Eigensystem Realization Algorithm (ERA) 27 2.5 Identification of Mass, Stiffness and Damping Matrices 30 2.5.1 Method 1: Identification with Full Set of Sensors or Actuators 31 2.5.2 Method 2: Identification with Mixed Sensors and Actuators 34 2.6 Damage Detection 36 2.7 Numerical Results 36 2.7.1 Significance of OKID 37 2.7.2 Significance of ERA 38 2.7.3 Comparison of ERA and ERA/DC 39 2.7.4 Comparison of Method and Method 40 2.7.5 Effects of Noise on Method 42 2.7.6 Identification for Damaged Case 44 2.7.7 Damage Detection 46 2.8 Concluding Remarks 47 CHAPTER 3: Condensed Model Identification and Recovery Method for Incomplete Measurement 65 3.1 General Remarks 65 3.2 Condensed Model Identification and Recovery (CMIR) Method 66 iv 3.3 CMIR-Static Condensation Method (CMIR-SC) 68 3.4 CMIR-Dynamic Condensation Method (CMIR-DC) 70 3.5 CMIR-System Equivalent Reduction Expansion Process (CMIR-SEREP) 74 3.6 Fixed and Non-fixed Sensor Approaches 77 3.6.1 Approach 1: Incomplete Measurement with Fixed Sensors 78 3.6.2 Approach 2: Incomplete Measurement with Non-fixed Sensors 79 3.7 Numerical Results and Discussion 3.7.1 Four-DOF Lump-mass System 80 81 3.7.1.1 Stiffness Identification from Condensed Stiffness Matrix 81 3.7.1.2 Damage Detection 84 3.7.2 Twelve-storey Shear Building 85 3.7.2.1 Determination of the First-order State Space Model 85 3.7.2.2 Stiffness Identification from Reduced Stiffness Matrix 86 3.7.2.3 Damage Detection 89 3.8 Concluding Remarks 90 CHAPTER 4: Substructural First and Second Order Model Identification 105 4.1 General Remarks 105 4.2 Substructural First-Order Model Identification (sub-FOMI) 107 4.2.1 Two-Substructure Case 107 4.2.2 Multiple-Substructure Case 109 4.3 Substructural Second-Order Model Identification (sub-SOMI) 112 4.3.1 sub-SOMI with Absolute Response (sub-SOMI-AR) 113 v 4.3.2 sub-SOMI with Relative Response (sub-SOMI-RR) 115 4.4 Start-up Least-Squares Method 118 4.5 Numerical Results 120 4.5.1 Identification of 12-DOF System 121 4.5.1.1 Effects of I/O noise 123 4.5.1.2 Comparison of Different Approaches 124 4.5.1.3 Effect of number of substructures 127 4.5.1.4 Damage Detection 128 4.5.2 Identification of 50-DOF System 129 4.5.2.1 Identification of Undamped System 130 4.5.2.2 Identification of Damped System 132 4.5.2.3 Damage Detection 133 4.6 Concluding Remarks 134 CHAPTER 5: Substructural Identification of Large Structures with Incomplete Measurement 149 5.1 General Remarks 149 5.2 Combination with CMIR and Substructural Approach 150 5.3 Numerical Results 152 5.3.1 Identification of 12-DOF System 152 5.3.1.1 Complete Measurement 154 5.3.1.2 Incomplete Measurement 156 5.3.2 Identification of 50-DOF System 158 5.3.2.1 Identification with Fixed Sensor Approach 159 vi 5.3.2.2 Identification with Non-fixed Sensor Approach 160 5.3.2.3 Damage Detection 162 5.4 Concluding Remarks 163 CHAPTER 6: Experimental Verification 178 6.1 General Remarks 178 6.2 Description of Laboratory Model 179 6.3 Excitation Force and Dynamic Response 180 6.4 Data Acquisition System 182 6.5 Data Processing 183 6.6 Integration 184 6.7 Static Experiments to Determine Storey Stiffness Values 186 6.8 Dynamic Tests and Identification of Undamaged Frame 187 6.8.1 Stiffness Identification with Complete Measurement 187 6.8.2 Stiffness Identification by CMIR-SEREP Method based on Incomplete Measurement 188 6.8.3 Stiffness Identification by sub-SOMI Method based on Complete Measurement 189 6.8.4 Stiffness Identification by Combined CMIR-SEREP and sub-SOMI-RR 190 Method based on Incomplete Measurement 6.9 Simulated Column Damage of Laboratory Model 191 6.10 Dynamic Tests and Identification of Damaged Frame 193 6.10.1 Damage Assessment with the CMIR-SEREP Method 193 6.10.2 Damage Assessment with the sub-SOMI-RR and CMIR-SEREP Method 197 6.11 Concluding Remarks 200 vii CHAPTER 7: Conclusions and Recommendations 227 7.1 Conclusions 227 7.2 Recommendations for Further Study 232 References 233 Appendix A: ERA with Data Correlations (ERA/DC) 242 Appendix B: Method 1: Identification with Full Set of Sensors and Actuators 246 B.1 Identification with Full Set of Sensors and at least One Actuator 246 B.2 Identification with Velocity and Acceleration Measurements 248 B.3 Identification with Full Set of Actuators and at least One Sensor 249 Appendix C: McMillan Normal Form from First-Order State Space Model 251 Appendix D: Method 2: Identification with Mixed Sensors or Actuators 253 D.1 Identification with Displacement Measurements 253 D.2 Velocity and Acceleration Measurements 255 viii SUMMARY This study aims to develop a system identification methodology for determining structural parameters of linear dynamic system, taking into consideration of practical constraints such as large number of unknowns and insufficient sensors. Based on numerical analysis of measured responses (output) due to known excitations (input), structural parameters such as stiffness values are identified. If the values at the damaged state are compared with the identified values at the undamaged state, damage detection and quantification can be carried out. The main identification tools employed are the Observer/Kalman filter Identification (OKID) using input-output data via Markov parameters and Eigensystem Realization Algorithm (ERA). Furthermore, this study also constitutes an attempt at providing a common framework used in obtaining physical parameters of structural systems from identified state space models. The framework established is used to develop several structural identification methods in this thesis. For structural health monitoring, it is unrealistic to use complete measurement to identify all of the parameters included in the structures. To retrieve second-order parameters from the identified state space model, various methodologies developed thus far impose different restrictions on the number of sensors and actuators employed, assuming that all the modes of the structure have been successfully identified. The restrictions are relaxed in this study by a proposed method called the Condensed Model Identification and Recovery (CMIR) Method. The focus is on estimation of all stiffness values from the condensed stiffness matrices by model condensation making use of static condensation, dynamic condensation or the method of System Equivalent Reduction ix References Signal Processing Toolbox. For Use with Matlab. User’s Guide. Version (2000). The Math Works Inc. Srinivasan, M. G. and Kot, C. A. (1992). “Effects of Damage on the Modal Parameters of a Cylindrical Shell.” Proc. of the 10th International Modal Analysis Conference, 529-535. Su, T. -J and Juang, J. -N (1994) “Substructure System Identification and Synthesis.” J. 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Tseng, D.-H., Longman, R.W., and Juang, J.-N. (1994b). “Identification of the Structure of the Damping Matrix in Second Order Mechanical Systems.” Advances in Astronautical Sciences, 87, 166-190. Udwadia, F.E. and Proskurowski, W. (1998). “A Memory-matrix-based Identification Methodology for Structural and Mechanical System.” Earthquake Engineering and Structural Dynamics, 27(12), 1465-1481. Worden, K. (1990). “Data Processing and Experiment Design for the Restoring Force Surface Method, Part I: Integration and Differentiation of Measured Time Data.” Mechanical Systems and Signal Processing, 4(4), 295-319. 240 References Yang, C.D., and Yeh, F.B. (1990). “Identification, Reduction, and Refinement of Model Parameters by the Eigensystem Realization Algorithm.” J. Guidance, Control, and Dynamics, 13(6), 1051-1059. Young, P.C. (1970). “An Instrumental Variable Method for Real-time Identification of A Noisy Process.” Automatica 6, 271-287. Yun, C.B. and Lee, H. -J. (1997) “Substructural Identification for Damage Estimation of Structures.” J. Structural Safety, 19(1), 121-140. Yun, C.B. and Bahng, E.Y. (2000), “Substructural Identification Using Neural Networks.” Computer and Structures, 77(1), 41-52. Zhang, Q., and Lallement, G. (1987). “Comparison of Normal Eigenmodes Calculation Methods Based on Identified Complex Eigenmodes.” J. Spacecraft and Rockets, 24, 6973. Zhao, Q., Sawada, T., Hirao, K. and Nariyuki, Y. (1995) “Localized Identification of MDOF Structures in the Frequency Domain.” Earthquake Engineering and Structural Dynamics, 24, 325-338. 241 Appendix A Appendix A ERA with Data Correlations (ERA/DC) In order to reduce the bias due to noise in the data, an alternative formulation of the ERA, called the ERA/DC, can be used. While the standard ERA method proceeds with using the Hankel matrix H (0) shown in Eq. (2.12), the ERA method with data correlations (ERA/DC) requires the following square matrix of order γ = mα , where m is the number of outputs. ℜ hh (k ) = H (k )H T (0)  Yk +1 Y k +2 =   M  Yk +α Yk + Yk + M Yk + α +1  β T  ∑ Yk + i Yi  βi =1 T  =  ∑ Yk + i +1 Yi i =1  M β ∑ Yk +α + i −1 YiT  i =1 Yk + β  L Yk +β +1  O M   L Yk + α +β −1  L β ∑ Yk +i YiT+1 i =1 β ∑Y k + i +1 i =1 β ∑Y i =1 YiT+1 M k + α + i −1 YiT+1  Y1 Y   M  Yα Yβ  L Yβ +1  O M   L Yα +β −1  Y2 L Y3 M Yα +1  YαT+ i −1  i =1  β T L ∑ Yk + i +1 Yα + i −1   i =1  O M  β T L ∑ Yk +α + i −1 Yα + i −1   i =1 T β L ∑Y k +i (A.1) Here Yk is an m x r matrix whose columns are the Markov parameters (pulse response samples) corresponding to the m outputs. The size of H (k ) and H (0) is α m × β r , whereas the size of ℜ hh (k ) is smaller in size than the Hankel matrix H (k ) in 242 Appendix A particular when the number of columns of the Hankel matrix is sufficiently large. For the case when k = , the correlation matrix ℜ hh (0) becomes ℜ hh (0) = H (0)H T (0)  Y1 Y =   M  Yα Y2 Y3 M Yα +1  β T  ∑ Yi Yi  βi =1 T  =  ∑ Yi +1 Yi i =1  M β ∑ Yα + i −1 YiT  i =1 Yβ   Y1 Yβ +1   Y2 O M   M   L Yα +β −1  Yα L L β ∑ Yi YiT+1 i =1 β ∑Y i +1 i =1 β ∑Y i =1 YiT+1 M α + i −1 YiT+1 Y2 Yβ  L Yβ +1  O M   L Yα +β −1  L Y3 M Yα +1 T   i =1  β T L ∑ Yi +1 Yα + i −1   i =1  O M  β T L ∑ Yα + i −1 Yα + i −1   i =1 β L ∑Y Y i T α + i −1 (A.2) The matrix ℜ hh (0) consists of auto-correlations of Markov parameter such as β ∑ Yi YiT and cross-correlations between outputs such as i =1 β ∑Y Y i =1 i T i +1 at lag time values in the range ± α , summed over the r inputs. If noises in the Markov parameters are not correlated, the correlation matrix ℜ hh (0) will contain less noise than the Hankel matrix H (0) . Indeed, let a block correlation Hankel matrix ( ξ × ζ ) be formed as ℜ hh (k + τ )  ℜ hh (k )  ℜ (k + τ ) ℜ hh (k + 2τ ) h (k ) =  hh  M M  ℜ hh (k + ξτ ) ℜ hh (k + (ξ + 1)τ ) ℜ hh (k + ζτ )  L ℜ hh (k + (ζ + 1)τ )   O M  L ℜ hh (k + (ξ + ζ )τ ) L (A.3) or, for k = , 243 Appendix A ℜ hh (τ )  ℜ hh (0)  ℜ (τ ) ℜ hh (2τ ) h (0) =  hh  M M  ℜ hh (ξτ ) ℜ hh ((ξ + 1)τ ) ℜ hh (ζτ )  L ℜ hh ((ζ + 1)τ )   O M  L ℜ hh ((ξ + ζ )τ ) L (A.4) where k is an integer chosen to avoid correlation terms which give rise to bias when noise is present, and τ is an integer chosen to prevent significant overlap of adjacent ℜ blocks. The integers ξ and ζ define how many correlation lags are included in the analysis. Similar to the ERA, the ERA/DC process continues with the factorization of the block correlation matrix h(0) (as opposed to H(0) in the ERA) using singular value decomposition so that h(0) = R Σ S T (A.5) Next, let R n and S n be the matrices formed by the first n columns of R and S , respectively. The columns of R n and S n are orthonormal and Σ is a diagonal matrix containing the n singular values that are considered significant, based on some truncation procedure. Note that the above factorisation is approximate if noise is present because the discarded singular values are nonzero. Hence, the matrix h(0) becomes h(0) = R n Σ n S nT where R nT R n = I n = S nT S n (A.6) ˆ , Bˆ , C ˆ ] can be shown to be Hence, a realization for [ A ˆ = Σ −1/2 R T h (1)S Σ −1/2 , A n n n n + Bˆ = [E Tγ R n Σ 1/2 n ] H (0)E r , ˆ = E T R Σ 1/2 C m n n (A.7) In order to avoid overlap of adjacent correlation terms in the block correlation matrix, it is required that τ ≥ α (see Eq. (A.3)). The structure of the ℜ hh (k ) matrix, and hence the block correlation Hankel matrix h(k ) , is significantly affected by the choice of 244 Appendix A the parameter ξ . When ξ = , the structure is simplest, but does not necessarily yield the best answer. However, it is the easiest way of implementing the ERA/DC method. The ˆ , Bˆ , C ˆ and D ˆ can be realized discrete-time model represented by the matrices A transformed to the continuous-time model. The system frequencies and damping ratio may then be computed from the eigenvalues of the estimated continuous-time state matrix. The eigenvectors allow the determination of the complex (or damped) mode shapes. 245 Appendix B Appendix B Method 1: Identification with Full Set of Sensors or Actuators B.1 Identification with Full Set of Sensors and at least One Actuator The realization (Eq. 2.21), which is obtained from data, and the desired realization (Eq. 2.18) are representations of the same system. Therefore, there must exist a similarity transformation N . N A N −1 = A ; NB = B ; C N −1 = C (B.1) The transformation matrix N is written as a product of two matrices in partitioned form as follows N N = NR =  11 N 21 N 12  N 22   R 11 R  21 R 12  R 22  (B.2) In the case of full set of sensors and at least one actuator, Eq (B.1) is solved by choosing matrix R to make the product C R T create a new matrix whose right partition is zero, and whose left partition is a full rank n by n matrix, as required. Assume for the moment that there are exactly n independent sensors, and that they are displacement sensors. Since C is full rank, its singular value decomposition is of the form below, and R is selected as follows [C1M C 2M ] = U c [S c 0] VcT = [U c S c 0] VcT = [U c S c 0] R (B.3) 246 Appendix B R = VcT Equation (B.1) shows that N 11 is also nonsingular, and N 12 = 0 = Cp N 12 ; −1 Cp = U c S c N 11 (B.4) The matrix R in Eq. (B.3) also produces the product RB whose lower partition gives −1 B c = N 11 (N 11 N 22 )(R 21 B 1M + R 22 B 2M ) (B.5) after making use of the fact that N 12 = . Define the matrix  W11 W  21 W12   I = −1 W22  N 11 N 21  R −1 N 11 N 22   − Ω B  I  T R − Σ   I N −1 N  11 21  −1 N 11 N 22  −1 (B.6) and note that this matrix is known since all terms on the right have been determined. In terms of this matrix, Eq. (B.1) becomes N 11     W11 N 11   W21 W12   = W22  − Ω I  N 11 − E   N 11  (B.7) The bottom left partition produces an eigenvalue problem for determining N 11 N 11 W21 = − Ω N 11 (B.8) The elements of the diagonal matrix Ω are the negatives of the eigenvalues of the known matrix W21 , and the rows of N 11 are its left eigenvectors. The first step of this method produces the modal model (Eq. 2.17), given the state space realization. The second step starts with the resulting modal model, and produces a mass-stiffness-damping model. In order to convert modal model (Eq. 2.17) to physical model (Eq. 2.1) one uses the standard method of diagonalizing two symmetric matrices simultaneously. Select a transformation of variables as follows 247 Appendix B q = Sa −1 η ; a −1 = diag (a1−1 , a 2−1 , L, a n−1 ) (B.9) where a scaling factor a −1 is introduced. The undamped mode shapes S is chosen to satisfy S T KS = Ω 2A = diag (Ω12 Ω 22 L Ω 2n ) S T MS = I (B.10) (B.11) Then B c = aS T B f ; Cp = C p Sa −1 ; aS T LSa −1 = E (B.12) The information available to determine S and a , is in the input and output Eqs (B.12), where the four matrices B c , B f , Cp and C p are known. S = (C p+ Cp ) a ; B c = a (C p+ Cp ) T B f (B.13) Once S and a are determined, the mass matrix M , the damping matrix L , and the stiffness matrix K are all uniquely determined: M = (SS T ) −1 B.2 ; L = S −T a −1EaS −1 ; K = S − T Ω S −1 (B.14) Identification with Velocity and Acceleration Measurements Instead of displacement measurements, the case where velocity or acceleration measurements are used is investigated. Since the state vector in Eq. (2.18) contains the [ ] T modal displacements and velocities (x (t ) = ηT (t ) η& T (t ) ) , it can be easily shown (simply by substitution) that 248 Appendix B  η(t )  η(t ) = Cv A  Cv     η& (t )  η& (t )  η(t )  η(t ) = Ca A 12  Ca &η&(t ) = Ca    + Du(t )  η& (t )  η& (t ) [0 [ ] [ ] [ ] ] (B.15) so that the singular value decomposition is performed on modified matrices, i.e.: C A 2−1 = [U c S c 0] VcT C A 2−2 = [U c S c 0] VcT for acceleration measurements for velocity measurements (B.16) (B.17) From this point on, the solution proceeds in exactly the same way as for the case of displacement measurements. B.3 Identification with Full Set of Actuators and at least One Sensor The matrix R is chose in order to make the product RB create a new input matrix whose top partition is zero, and whose bottom partition is a full rank n by n matrix, as required. Since B is full rank, its singular value decomposition is of the form below, and we can insert a matrix to reverse the upper and lower partitions S b VbT    B 1M  S b  T 0 I    = = U V U = RT   = Ub  b b b  B    T  T  0  I 0 S b Vb   2M  S b Vb    (B.18) Here we have chosen R to be equal to 0 I  T R=  Ub  I 0 (B.19) Note that this matrix is unitary so that R T = R −1 . Equation (B.1) results in −1 N 12 S b VbT = ; B c = N 22 S b VbT = N 11 (N 11 N 22 )S b VbT (B.20) 249 Appendix B This implies that N 22 is nonsingular, N 12 = , and therefore N 11 is also nonsingular, since the transformation matrix N must be invertible. In Eq. (B.1), multiply by the transpose of R on the right of both sides of the equation and solve for the unknown Cp from the first partition to obtain T −1 Cp = (C1M R 11 + C 2M R T12 )N 11 (B.21) In the case of full set of actuators and at least one sensor, we reverse which equation is solved for S and for a and obtain S = ( B c B f+ ) T a −1 ; Cp = C p ( B c B f+ ) T a −2 (B.22) 250 Appendix C Appendix C McMillan Normal Form from First-Order State Space Model Let ψ be a matrix that diagonalizes system (Eq. 2.2) x = ψθ ; where Γ = diag (γ , γ , L, γ n ) . θ& (t ) = Γθ(t ) + ψ −1B f u(t ) (C.1) Let Γ possess l eigenvalues that are in complex conjugate pairs, and n − l real eigenvalues. Order the entries in Γ so that the complex eigenvalues appear first, and so that the eigenvalues of a complex conjugate pair appear in succession. Then γ j −1 = γ j , j = j = 1,2, L, l / and the eigenvector matrix ψ = [ψ ψ L ψ n ] has the property that ψ j −1 = ψ j , indicate complex conjugate. j = 1,2,L, l / . Overbars Consider one such pair and make the following transformation to real valued form γ j = σ j + iω j 1 M cj =  γ j 1 γ j ; M cj   γ j 0 ; γ j = σ j − iω j  −1   M cj =    =θj 2 γ j − (σ j + ω j ) 2σ j  (C.2) (C.3) Doing this for each eigenvalue pair simultaneously, the McMillan form from state space representation is obtained z& (t ) = Ξz (t ) + B ' u(t ) (C.4a) 251 Appendix C y (t ) = C ' z (t ) (C.4b) where Ξ = diag (θ1 θ L θ n ) ; B ' = M c ψ −1B f ; C ' = Cψ M c−1 The system matrix Ξ is block diagonal with all real entries. Then reordering the sequence of state variables appearing in the McMillan normal form (Eq. C.4) produces the following realization (A2, B2, C2), which is in terms of real valued quantities only. z& r (t ) = A z r (t ) + B u(t ) (C.5a) y (t ) = C z r (t ) (C.5b) with the following system matrices  A2 =  − Ω B I  ; − Σ  B  B =  1M  B 2M  ; C = [C 1M C 2M ] (C.6) where Σ is a diagonal matrix of the damping factors − 2σ j , and Ω B2 is a diagonal matrix of the (σ 2j + ω 2j ) which reduces to the undamped natural frequencies only when there is no damping present. 252 Appendix D Appendix D Method 2: Identification with Mixed Sensors or Actuators D.1 Identification with Displacement Measurements If the first order system of Eq. (2.20) was identified using data that actually came ˆ , from the second order model of Eq. (2.26), one can look for a transformation matrix, N that relates these two representations, i.e.: ˆ −1 Γ N ˆ =Γ, N ˆ =C P ˆ −1 ψ −1B = P T B , Cψ N N c f p (D.1) ˆ = diag (n , n ,L , n ) It is easy to show that the transformation is diagonal, i.e. N 2N and its values are complex conjugate. The input and output matrices ( B f and C p , respectively) of the finite element model are known. These input and output matrices are assumed to contain binary information. If the co-located sensor-actuator pair is at the i th DOF, the well-known co-location requirement can be written as C p (i, :)P = (P T B f (:, i )) T (D.2) ˆ can be evaluated Using the co-location requirement, the transformation matrix N from Eqs. (D.1) and (D.2) as: 253 Appendix D ˆ = (N ˆ −1 ψ −1B (:, i )) T C(i, :)ψ N c ⇒ ˆ = (ψ −1B (:, i )) T C(i, :)ψ N c (D.3) ˆ is diagonal, each n (i = 1,2,L ,2 N ) can be uniquely Since the matrix N i determined from Eq. (D.3). The information pertaining to a certain DOF is embedded either in the input matrix or in the output matrix. Going back to Eq. (D.1), the output matrices essentially contain information about only m DOFs (with m < N ). If there is a sensor at the k th DOF, then the k th row of the matrix P can be evaluated, i.e. ˆ P(k , :) = C(k , :)ψ N (D.4) However, since a DOF has either a sensor or an actuator, the k th row of the matrix P can be evaluated using Eq. (D.1) as ˆ −1 ψ −1B (:, k )) T P(k , :) = (N c (D.5) In general, these eigenvectors can be arbitrarily scaled; however, if the scaling is chosen such that (see Sestieri and Ibrahim 1994, Balmes 1997) then the real and imaginary parts of the components of these complex eigenvectors are equal in magnitude for a proportionally damped system. T  P   L M  P   P Γ  M   P Γ  = I      (D.6a) T  P   P  K P Γ   − M  P Γ  = − Γ      (D.6b) 254 Appendix D Once the properly scaled eigenvector matrix is evaluated, the mass, damping, and stiffness matrices of the finite element model can be obtained using the orthogonality conditions in Eq (D.6): M = (PΓP T ) −1 , L = −MP Γ P T M, K = −(P Γ −1 P T ) −1 , PP T = D.2 (D.7) Velocity and Acceleration Measurements If instead of displacement measurements one uses velocity or acceleration measurements, the output equation in Eq. (2.26) can be rewritten as:  P  y (t ) = [0 C v ]  ζ(t ) = C v P Γζ(t ) for velocity measurements P Γ (D.8)  P  y (t ) = [0 C a ]  ζ& (t ) for acceleration measurements P Γ  = C a P Γ ζ(t ) + C a P Γ P T Bu(t ) (D.9) Clearly, these changes lead to some alterations in Eq. (D.1), according to the type of measurements used: ˆ = C PΓ CψN v for velocity measurements (D.10) ˆ = C P Γ2 CψN a for acceleration measurements (D.11) From this point on, the solution proceeds in exactly the same way as for the case of displacement measurements. 255 [...]... damping matrix with Method 2 63 Figure 2.10 Damage quantification chart for identified damage (Damage Scenario 1) with and without noise using Method 2 64 xv Figure 2.11 Damage quantification chart for identified damage (Damage Scenario 2) with and without noise using Method 2 64 Figure 3.1 Flowchart for identification of storey stiffness values using CMIR-SC 99 Figure 3.2 Flowchart for identification. .. integrity indices with the sub-SOMI-RR method under 0% and 5% noise 148 xvii Figure 5.1 Flowchart for identification of storey stiffness values with the combined CMIR and substructural approach 171 Figure 5.2 A 12-DOF lumped-mass shear building with 2 substructures (Substructural identification with overlap) 172 Figure 5.3 Identified stiffness integrity indices for Damage Scenario 1 with complete measurement. .. efficient for damage estimation of large structures The proposed CMIR method and substructural method address different aspects of large-scale structural identification The former allows the use of incomplete measurement and the latter represents a divide-and-conquer approach to reduce the size of system identification These two methods are thus combined for the identification of stiffness values at substructural. .. noise 173 Figure 5.4 Identified stiffness integrity indices for Damage Scenario 2 with complete measurement and 10% I/O noise 173 Figure 5.5 Identified stiffness integrity indices for Damage Scenario 1 with incomplete measurement and 10% I/O noise 174 Figure 5.6 Identified stiffness integrity indices for Damage Scenario 2 with incomplete measurement and 10% I/O noise 174 Figure 5.7 Ratio of identified... method In the sub-FOMI method, identification at the substructure level is performed by means of the OKID/ERA whereas identification at the global level is performed to obtain second-order model In the subSOMI method, identification is performed at the substructural level throughout the identification process Furthermore, two variations of substructural identification with the sub-SOMI method are presented... stiffness integrity indices with the sub-SOMI-RR method 139 xxiv Table 5.1 Identified storey stiffness values for undamaged case with complete measurement and 10% I/O noise 165 Table 5.2 Identified stiffness integrity indices for Damage Scenario 1 with complete measurement and 10% I/O noise 165 Table 5.3 Identified stiffness integrity indices for Damage Scenario 2 with complete measurement and 10% I/O... identification of storey stiffness values with the sub-FOMI method 141 xvi Figure 4.3 Flowchart for identification of storey stiffness values with the sub-SOMI-AR method 142 Figure 4.4 Flowchart for identification of storey stiffness values with the sub-SOMI-RR method 143 Figure 4.5 A 12-DOF lumped-mass shear building with 3 substructures (Substructural identification with overlap) 144 Figure 4.6 Comparison... integrity indices with the CMIR-SEREP 221 method for Damage Scenario 2 Figure 6.16 Identified storey stiffness integrity indices with the CMIR-SEREP 222 method for Damage Scenario 3 Figure 6.17 Identified storey stiffness integrity indices with the CMIR-SEREP 222 method for Damage Scenario 4 Figure 6.18 Identified storey stiffness integrity indices with the CMIR-SEREP 223 method for Damage Scenario 5... integrity indices with the CMIR-SEREP 223 method for Damage Scenario 6 Figure 6.20 Identified storey stiffness integrity indices with the sub-SOMI-RR 224 method for Damage Scenario 1 Figure 6.21 Identified storey stiffness integrity indices with the sub-SOMI-RR 224 method for Damage Scenario 2 Figure 6.22 Identified storey stiffness integrity indices with the sub-SOMI-RR 225 method for Damage Scenario... sensors 50 for the 4-storey shear building, with and without noise using OKID/ERA Table 2.6 Identified natural frequencies (rad/sec) with ERA and ERA/DC 50 for the 4-storey shear building, with and without noise (2 sensors) Table 2.7 Identified modal damping ratios with ERA and ERA/DC 51 for the 4-storey shear building, with and without noise (2 sensors) Table 2.8 Identified natural frequencies with various . SUBSTRUCTURAL IDENTIFICATION WITH INCOMPLETE MEASUREMENT FOR STRUCTURAL DAMAGE ASSESSMENT TEE KONG FAH B.Eng. (Hons.) A THESIS SUBMITTED FOR THE. SUBSTRUCTURAL IDENTIFICATION WITH INCOMPLETE MEASUREMENT FOR STRUCTURAL DAMAGE ASSESSMENT TEE KONG FAH . 134 CHAPTER 5: Substructural Identification of Large Structures with Incomplete Measurement 149 5.1 General Remarks 149 5.2 Combination with CMIR and Substructural Approach 150