Safak, E., and Celebi, M. (1991). Seismic Response of Transamerica Building. II: System

Transcription

1 tural Systems. II: Applications. Journal of the Engineering Mechanics, Vol.115, 11, 4-4. Safak, E., and Celebi, M. (1991). Seismic Response of Transamerica Building. II: System Identification. Journal of the Structural Engineering Division, ASCE, Vol. 117, 8, Safak, E., and Celebi, M. (199). Seismic Response of Pacific Park Plaza. II: System Identification. Journal of the Structural Engineering Division, ASCE, Vol. 118,, S-PLUS, Version.3. (199). Statistical Sciences, Inc., Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, California. Tajimi, H. (19). A Standard Method of Determining the Maximum Response of a Building Structure During an Earthquake. Proceedings, nd World Conference on Earthquake Engineering, Vol., Tokyo, Japan, Udwadia, F. E., and Marmarelis, P. Z. (197). The Identification of Building Structural Systems. I. The Linear Case. Bulletin of the Seismological Society of America. Vol., 1, Welch, P. (197). The Use of Fast Fourier Transform for the Estimation of Power Spectra: A Method Based on Time Averaging Over Short, Modified Periodograms. IEEE Transactions on Audio and Electroacoustics, Vol. AU-15,, Werner, S. D., Beck, J. L., and Levine M. B. (1987). Seismic Response Evaluation of Meloland Road Overpass Using 1979 Imperial Valley Earthquake Records. Earthquake Engineering and Structural Dynamics, Vol. 15, Wolfram, S. (1991). Mathematica. Addison-Wesley Publishing Company, Inc., Redwood City, California. 7

2 Liu, S.-C., and Yao, J. T. P. (1978). Structural Identification Concept. Journal of the Structural Engineering Division, ASCE, 14(ST1), Li, Yi, and Mau, S. T. (1991). A Case Study of MIMO System Identification Applied to Building Seismic Records. Earthquake Engineering and Structural Dynamics, Vol., Ljung, L. (1987). System Identification: Theory for the user. Prentice-Hall Inc., Englewood Cliffs, New Jersey. Marmarelis, P. Z., and Udwadia, F. E. (197). The Identification of Building Structural Systems. II. The Nonlinear Case. Bulletin of the Seismological Society of America. Vol., 1, Marquardt, D. (193). An Algorithm for Least-Squares Estimation of Nonlinear Parameters, SIAM Journal on Applied Mathematics, 11, MATLAB, Version 4.. (1991). The MathWorks, Inc., 1 Eliot Street, South Natick, MA 17, USA. Mau S. T., and Wang S. (1989). Arch Dam System Identification Using Vibration Test Data. Earthquake Engineering and Structural Dynamics, Vol. 18, McVerry, G. H. (198). Structural Identification in the Frequency Domain from Earthquake Records. Earthquake Engineering and Structural Dynamics, Vol. 8, Nigam, N. C. (1983). Introduction to Random Vibrations. The MIT Press, Cambridge. Papageorgiou, A. S., and Lin, B. C. (1989). Influence of Lateral-Load-Resisting System on the Earthquake Response of Structures - A System Identification Study. Earthquake Engineering and Structural Dynamics, Vol. 18, Paz, M. (1985). Structural Dynamics. Van Nostrand Reinhold Company, New York. Raggett, J. D. (1974). Time Domain Analysis of Structural Motions. Presented at ASCE National Structural Engineering Meeting, Cincinnati, April -. Safak, E. (1989). Adaptive Modeling, Identification, and Control of Dynamic Structural Systems. I: Theory. Journal of the Engineering Mechanics, Vol.115, 11, Safak, E. (1989). Adaptive Modeling, Identification, and Control of Dynamic Struc- 9

3 Conte, J. P. (1991). Lecture Notes - Probabilistic Structural Dynamics. Civil Engineering Department, Rice University, Houston, Texas. Conte, J. P., and Marshall P. W. (1994). Low Frequency Forces on Tubular Spaceframe Towers: Analysis of Cognac Data. Proceedings, OMAE 94. Houston, Texas. Dennis, J. E., Jr., and Schnabel, R. B. (1983). Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice Series in Computational Mathematics. DiStefano, N., and Pena-Pardo, B. (197). System Identification of Frames Under Seismic Loads. Journal of the Engineering Mechanics Division, Proceedings of the American Society of Civil Engineers, 1(EM). DiStefano, N., and Rath, A. (1974). Modeling and Identification in Nonlinear Structural Dynamics. Report EERC, University of California, Berkeley, California. Eykhoff, P. (1974). System Identification. Wiley & Sons, New York. Hart, G. C., and Yao, J. T. P. (1977). System Identification in Structural Dynamics. Journal of the Engineering Mechanics Division, Proceedings of the American Society of Civil Engineers, 13(EM). Hong, K. S., and Yun, C. B. (1993). Improved Method for Frequency Domain Identifications of Structures. Engineering Structures, 15(3), Hoshiya, M., and Saito, E. (1984). Structural Identification by Extended Kalman Filter, Journal of the Engineering Mechanics, Vol.11, 1, Ibrahim, S. R., and Mikulcik, E. C. (1973). A Time Domain Modal Vibration Test Technique. The Shock and Vibration Bulletin, 43, Part 4, Kanai, K. (197). Semi-Empirical Formula for Seismic Characterization of the Ground. Bulletin of Earthquake Research Institute, 35, University of Tokyo, Tokyo, Japan. Levenberg, K. (1944). A Method for the Solution of Certain Problems in Least- Squares, Quarterly of Applied Mathematics,, Lin, Y. K. (197). Probabilistic Theory of Structural Dynamics. McGraw-Hill, New York. 8

4 References Armen Der Kiureghian (198). Structural Response to Stationary Excitation. Journal of the Engineering Mechanics Division, ASCE, 1(EM), Beck, J. L. (1979). Determining Model of Structures from Earthquake Records. Ph.D. Thesis at California Institute of Technology, Pasadena, California. Beck, J. L., and Jennings, P. C. (198). Structural Identification Using Linear Models and Earthquake Records. Earthquake Engineering and Structural Dynamics, Vol. 8, Bekey, G. A. (197). System Identification - An Introduction and a Survey. Simulation, 15, Bendat, J. S. (199). Nonlinear System Analysis and Identification from Random Data. John Wiley, New York. Bendat, J. S., and Piersol, G. (1991). Random Data: Analysis and Measurement Procedures. Wiley-Interscience, New York. Bowles, R. L. and Straeter, T. A. (197). System Identification Computational Considerations in System Identification of Vibrating Structures, W. D. Pilkey and R. Cohen (eds). ASME, New York. Celebi, M. (1993). Seismic Responses of Two Adjacent Buildings. I: Data and Analyses. Journal of the Structural Engineering Division, ASCE, Vol. 119, 8, Celebi, M. (1993). Seismic Responses of Two Adjacent Buildings. II: Interaction. Journal of Structural Engineering Division, ASCE, Vol. 119, 8, Celebi, M., and Safak, E. (1991). Seismic Response of Transamerica Building. I: Data and Preliminary Analysis. Journal of the Structural Engineering Division, ASCE, Vol. 117, 8, Celebi, M., and Safak, E. (199). Seismic Response of Pacific Park Plaza. I: Data and Preliminary Analysis. Journal of the Structural Engineering Division, ASCE, Vol. 118,, Clough, R. W., and Penzien, J. (198). Dynamics of Structures. McGraw-Hill. 7

5 7 m m * Gm Dividing through by : R m ( t) = z m w m Ṡ m ( t) w msm ( t) R ( t) = AṠ ( t) BS ( t) where A = [ z m w m ] diag and B = w m diag., m = 1,..., N (A.3) (A.4) Finally, we can get the absolute acceleration response as, U t ( t) = FG R ( t) = Y R ( t) (A.5) From Eqs. (A.) and (A.3), R m ( t) = z m w m Ṡ m ( t) w msm ( t) = Ṡ m ( t) + Ẋ g ( t) acceleration response. : standardized modal absolute Note: If Ẋ g ( t) is white-noise (not a second-order process - infinite variance), then Ṡ m ( t) is not a second-order process either. However, their sum is a second-order process (finite variance Kanai-Tajimi process). To verify Eqs. (A.3), (A.4) and (A.5), assuming S m (t) satisfies Eq. (A.), we have to check that the following equations, satisfy the equation of motion, Upon substitution, the left-hand-side (LHS) of the equation of motion becomes = I. Hence, Eqs. (A.3), (A.4) and (A.5) are verified. Here, the assumed modeshape normalization is: U ( t) = FG S ( t) = Y S ( t) U ( t) = FG Ṡ ( t) = Y Ṡ ( t) U t ( t) = FG R ( t) = Y R ( t) = FG ( AṠ ( t) BS ( t) ) MU t ( t) + CU ( t) + KU ( t) = Ł M * G C * + C * Gł Ṡ ( t) + Ł M * G K * + K * Gł S ( t) M * and this is equal to since F T MF = I

6 Appendix A : From Relative Displacement and Velocity to Absolute Acceleration Response Equation of motion: 71 MU t ( t) + CU ( t) + KU ( t) = (A.1) U ( t) = FG S ( t) = Y S ( t) U ( t) = FG Ṡ ( t) = Y Ṡ ( t) U t ( t) = U ( t) + ( 1) Ẋ g ( t) : absolute acceleration response vector F = [f (1),f (),...,f (N) ]: Eigen matrix T G N N = f i M 1 ( ) diag S ( t) = S 1 ( t) S ( t)... S N ( t) T : vector of standardized modal relative displacement responses. S m (t) is the solution of: Let us express Ṡ m ( t) + x m w m Ṡ m ( t) + w msm ( t) = Ẋ g ( t) U t ( t) = FG R ( t) = Y R ( t) (A.) R ( t) = R 1 ( t) R ( t)... R N ( t) T : vector of standardized modal absolute acceleration responses to be found. Substituting this in Eq. (A.1) and premultiplying by : T T T f mmfg R ( t) + f mcfg Ṡ ( t) + f mkfg S ( t) = Using orthogonality properties: f i T m m * * * * m m Gm R m ( t) + c m Gm Ṡ m ( t) + K m Gm S m ( t) =, m = 1,..., N,, : generalized mass, damping coefficient, and stiffness coefficient of c m * mode m. K m *

7 Safak, E., and Celebi, M. (1991). Seismic Response of Transamerica Building. II: System Identification. Journal if Structural Engineering, Vol. 117, 8, Safak, E., and Celebi, M. (199). Seismic Response of Pacific Park Plaza. II: System Identification. Journal if Structural Engineering, Vol. 118,, S-PLUS, Version.3. (199). Statistical Sciences, Inc., Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, California. Tajimi, H. (19). A Standard Method of Determining the Maximum Response of a Building Structure During an Earthquake. Proceedings, nd World Conference on Earthquake Engineering, Vol., Tokyo, Japan, Udwadia, F. E., and Marmarelis, P. Z. (197). The Identification of Building Structural Systems. I. The Linear Case. Bulletin of the Seismological Society of America. Vol., 1, Welch, P. (197). The Use of Fast Fourier Transform for the Estimation of Power Spectra: A Method Based on Time Averaging Over Short, Modified Periodograms. IEEE Transactions on Audio and Electroacoustics, Vol. AU-15,, Werner, S. D., Beck, J. L., and Levine M. B. (1987). Seismic Response Evaluation of Meloland Road Overpass Using 1979 Imperial Valley Earthquake Records. Earthquake Engineering and Structural Dynamics, Vol. 15, Wolfram, S. (1991). Mathematica. Addison-Wesley Publishing Company, Inc., Redwood City, California. 7

8 Li, Yi, and Mau, S. T. (1991). A Case Study of MIMO System Identification Applied to Building Seismic Records. Earthquake Engineering and Structural Dynamics, Vol., Ljung, L. (1987). System Identification: Theory for the user. Prentice-Hall Inc., Englewood Cliffs, New Jersey. Marmarelis, P. Z., and Udwadia, F. E. (197). The Identification of Building Structural Systems. II. The Nonlinear Case. Bulletin of the Seismological Society of America. Vol., 1, Marquardt, D. (193). An Algorithm for Least-Squares Estimation of Nonlinear Parameters, SIAM Journal on Applied Mathematics, 11, MATLAB, Version 4.. (1991). The MathWorks, Inc., 1 Eliot Street, South Natick, MA 17, USA. Mau S. T., and Wang S. (1989). Arch Dam System Identification Using Vibration Test Data. Earthquake Engineering and Structural Dynamics, Vol. 18, McVerry, G. H. (198). Structural Identification in the Frequency Domain from Earthquake Records. Earthquake Engineering and Structural Dynamics, Vol. 8, Nigam, N. C. (1983). Introduction to Random Vibrations. The MIT Press, Cambridge. Papageorgiou, A. S., and Lin, B. C. (1989). Influence of Lateral-Load-Resisting System on the Earthquake Response of Structures - A System Identification Study. Earthquake Engineering and Structural Dynamics, Vol. 18, Paz, M. (1985). Structural Dynamics. Van Nostrand Reinhold Company, New York. Raggett, J. D. (1974). Time Domain Analysis of Structural Motions. Presented at ASCE National Structural Engineering Meeting, Cincinnati, April -. Safak, E. (1989). Adaptive Modeling, Identification, and Control of Dynamic Structural Systems. I: Theory. Journal of the Engineering Mechanics, Vol.115, 11, Safak, E. (1989). Adaptive Modeling, Identification, and Control of Dynamic Structural Systems. II: Applications. Journal of the Engineering Mechanics, Vol.115, 11,

9 Conte, J. P., and Marshall P. W. (1994). Low Frequency Forces on Tubular Spaceframe Towers: Analysis of Cognac Data. Proceedings, OMAE 94. Houston, Texas. Dennis, J. E., Jr., and Schnabel, R. B. (1983). Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice Series in Computational Mathematics. DiStefano, N., and Pena-Pardo, B. (197). System Identification of Frames Under Seismic Loads. Journal of the Engineering Mechanics Division, Proceedings of the American Society of Civil Engineers, 1(EM). DiStefano, N., and Rath, A. (1974). Modeling and Identification in Nonlinear Structural Dynamics. Report EERC, University of California, Berkeley, California. Eykhoff, P. (1974). System Identification. Wiley & Sons, New York. Hart, G. C., and Yao, J. T. P. (1977). System Identification in Structural Dynamics. Journal of the Engineering Mechanics Division, Proceedings of the American Society of Civil Engineers, 13(EM). Hong, K. S., and Yun, C. B. (1993). Improved Method for Frequency Domain Identifications of Structures. Engineering Structures, 15(3), Hoshiya, M., and Saito, E. (1984). Structural Identification by Extended Kalman Filter, Journal of the Engineering Mechanics, Vol.11, 1, Ibrahim, S. R., and Mikulcik, E. C. (1973). A Time Domain Modal Vibration Test Technique. The Shock and Vibration Bulletin, 43, Part 4, Kanai, K. (197). Semi-Empirical Formula for Seismic Characterization of the Ground. Bulletin of Earthquake Research Institute, 35, University of Tokyo, Tokyo, Japan. Levenberg, K. (1944). A Method for the Solution of Certain Problems in Least- Squares, Quarterly of Applied Mathematics,, Lin, Y. K. (197). Probabilistic Theory of Structural Dynamics. McGraw-Hill, New York. Liu, S.-C., and Yao, J. T. P. (1978). Structural Identification Concept. Journal of the Structural Engineering Division, ASCE, 14(ST1),

10 References Armen Der Kiureghian (198). Structural Response to Stationary Excitation. Journal of the Engineering Mechanics Division, ASCE, 1(EM), Beck, J. L. (1979). Determining Model of Structures from Earthquake Records. Ph.D. Thesis at California Institute of Technology, Pasadena, California. Beck, J. L., and Jennings, P. C. (198). Structural Identification Using Linear Models and Earthquake Records. Earthquake Engineering and Structural Dynamics, Vol. 8, Bekey, G. A. (197). System Identification - An Introduction and a Survey. Simulation, 15, Bendat, J. S. (199). Nonlinear System Analysis and Identification from Random Data. John Wiley, New York. Bendat, J. S., and Piersol, G. (1971). Random Data: Analysis and Measurement Procedures. Wiley-Interscience, New York. Bowles, R. L. and Straeter, T. A. (197). System Identification Computational Considerations in System Identification of Vibrating Structures, W. D. Pilkey and R. Cohen (eds). ASME, New York. Celebi, M. (1993). Seismic Responses of Two Adjacent Buildings. I: Data and Analyses. Journal if Structural Engineering, Vol. 119, 8, Celebi, M. (1993). Seismic Responses of Two Adjacent Buildings. II: Interaction. Journal if Structural Engineering, Vol. 119, 8, Celebi, M., and Safak, E. (1991). Seismic Response of Transamerica Building. I: Data and Preliminary Analysis. Journal if Structural Engineering, Vol. 117, 8, Celebi, M., and Safak, E. (199). Seismic Response of Pacific Park Plaza. I: Data and Preliminary Analysis. Journal if Structural Engineering, Vol. 118,, Clough, R. W., and Penzien, J. (198). Dynamics of Structures. McGraw-Hill. Conte, J. P. (1991). Lecture Notes - Probabilistic Structural Dynamics. Civil Engineering Department, Rice University, Houston, Texas. 7

11 systems, the accuracy of the estimated loading and structural parameters is strongly dependent on the accuracy of the estimated auto/cross-correlation functions (or target ACF/CCFs) of the measured response quantities. (5) In both the deterministic and stochastic cases, the modal minimization and the global minimization techniques appear to be efficient tools for structural identification. 4.3 Future Research The identification techniques selected, adjusted, developed, investigated and tested during this study are currently being applied to the case of a simplified structural model of the Cognac platform, a fixed offshore platform in the Gulf of Mexico. Cognac is subjected to wave loading which is simulated using linear wave theory (Airy theory) and Morison equation with the drag and inertia terms. Then, the identification techniques will be applied to the recorded ambient vibration data of Cognac.

12 5 converged results for the examples considered. However, the global algorithm is computationally more efficient, but less robust, than the modal sweeps algorithm. () The identification techniques based on continuous-time linear dynamic systems were able to extract both loading and structural modal parameters from the statistics (auto/cross-correlation functions) of the measured response. In general, the loading parameter estimates are not as good as the structural parameter estimates. Moreover, for some of the examples considered, the objective function of the identification algorithm had multiple local minima due to the compensation effects between loading parameters, modal effective participation factors and damping parameters. These compensation effects can be mitigated by incorporating the cross-terms of the correlation matrix in the objective function. The cross-correlation functions act as constraining data sets which reduce the multiplicity of solutions to a unique solution corresponding to the exact solution of the problem. However, when the number of model parameters is excessively large, as in realistic MIMO cases, including the full correlation matrix in the objective function might not be enough for guaranteeing unicity of solution. In such cases, the initial parameter set should be selected extremely carefully (maybe based on some preliminary simplified data analyses) so as to position it in the region of convergence to the global minimum. In the case of multiple solutions, different initial parameter sets lead to different sets of converged parameter sets. Physical arguments and a priori knowledge of the system should be used to discriminate between those solutions. (3) In identification of structures (deterministic and stochastic) responding predominantly in their lower modes, the use of acceleration response data provides a higher degree of identifiability of the system than the use of displacement response data. (4) In the stochastic case of system identification based on continuous-time dynamic

13 4 4. Conclusions 4.1 Summary of Work This study approaches the problem of structural identification from the viewpoint of continuous-time linear dynamic models of real structures. In the second chapter, the deterministic case is considered, namely the case for which both excitation and structural response are measured. Two identification techniques are investigated: (i) an existing structural identification technique based on sequences of minimizations of a modal objective function, and (ii) a global identification scheme based on minimization of a global objective function using the Levenberg-Marquardt algorithm. The two identification techniques are formulated for the general MIMO case. These two methods are applied to shear-beam type of structures. Several applications (SISO, SIMO and MIMO) are performed and the results are presented and discussed. In Chapter 3, using the framework of random vibration theory, the two deterministic identification techniques investigated in Chapter are customized to the statistical case, wherein only the structural response is measured and the statistical features of the loading enter the identification problem as unknown parameters. The developed scheme is then applied to several examples, and the results are presented. Analytical expressions are derived for the auto/cross-correlation and power spectral density functions of the relative displacement and absolute acceleration story responses of a shear building subjected to either random ground motion or lateral story loads. The analytical expressions are then used in formulating the objective function of the identification algorithm. 4. Summary of Findings The most significant findings of the present study are summarized below. (1) In the deterministic case of system identification based on continuous-time dynamic systems, both the global and modal sweeps algorithms led to the same

14 3 Story Level 4 First Mode Story Level 4 Second Mode -4-4 Story Level 4 Third Mode Story Level 4 Fourth Mode Story Level 4 Fifth Mode Story Level 4 Sixth Mode Figure 3.59 Comparison of and Mode Shapes - Identification Using Full Displacement Correlation Matrix

15 Story Level 4 First Mode Story Level 4 Second Mode -4 - Story Level 4 Third Mode Story Level 4 Fourth Mode Story Level 4 Fifth Mode Story Level 4 Sixth Mode Figure 3.58 Comparison of and Mode Shapes - Identification Using Full Displacement Correlation Matrix

16 Table 3.1 Identification Results Using Full Displacement Auto/Cross-Correlation Matrix - MIMO Case (Continued) 1 Parameter Values Initial ID Using AC Matrix ID Using Auto/Cross- Correlation matrix b b b b b b b b b b b b b b f, f, f, J ACF estimation based on: (1) Dt =.1 sec. () Seed = 831; N = 15-5 = 145; n d =.

17 Table 3.1 Identification Results Using Full Displacement Auto/Cross-Correlation Matrix - MIMO Case (Continued) Parameter Values Initial ID Using AC Matrix ID Using Auto/Cross- Correlation matrix b b b b b b b b b b b b b b b b b b b b b b b b b b b

18 Table 3.1 Identification Results Using Full Displacement Auto/Cross-Correlation Matrix - MIMO Case (Continued) 59 Parameter Values Initial ID Using AC Matrix ID Using Auto/Cross- Correlation matrix b b b b b b b b b b b b b b b b b b b b b b b b b b b

19 Table 3.1 Identification Results Using Full Displacement Auto/Cross-Correlation Matrix - MIMO Case (Continued) 58 Parameter Values Initial ID Using AC Matrix ID Using Auto/Cross- Correlation matrix b b b b b b b b b b b b b b b b b b b b b b b b b b b

20 57 Table 3.1 Identification Results Using Full Displacement Auto/Cross-Correlation Matrix - MIMO Case Parameter Values Initial ID Using AC Matrix ID Using Auto/Cross- Correlation matrix T T T T T T z z z z z z b b b b b b b b b b b b b

21 3.9.9 MDOF System Subjected to White-Noise Story Loads (MIMO Case) - Identification Using Story Displacement Responses The last application example is a MIMO case corresponding to the -DOF shear building of Figure. subjected to multiple white-noise story loads. The multiple input consists of three independent white-noise loads applied at the fourth, fifth, and sixth story levels, whereas the multiple output is given by the full displacement response vector. The identification is performed using the full correlation matrix of the displacement response. Stationary white-noise loads were applied at the fourth, fifth and sixth stories. The converged results are reported in Table 3.1. Although the correlation matrix of the displacement response is primarily contributed by the first two modes, four natural periods and three to four damping ratios are identified well. The compensation effect discussed earlier comes into play again. In this example, even though the dimension of the response vector used in the identification process is the same as in the SIMO case, the number of model parameters is almost three times of that in the previous SIMO case, the total number of model parameters has almost tripled. This effectively leads to multiplicity of solution. Hence, the estimated modal effective participation factors and the load intensity parameters are not as accurate as the structural modal parameters. In spite of that, the first and second mode shapes are estimated well as shown in Figure 3.58 and Figure

22 55 Story Level 4 First Mode Story Level 4 Second Mode -4 - Story Level 4 Third Mode Story Level 4 Fourth Mode Story Level 4 Fifth Mode Story Level 4 Sixth Mode - 4 Figure 3.57 Comparison of and Mode Shapes (Identification using estimated full correlation matrix, Seed = 74341)

23 Cross PSD - Stories 1 & Cross PSD - Stories 1 & [in /sec 3 ] Freq [Hz] [in /sec 3 ] Freq [Hz] 1 4 Cross PSD - Stories 3 & Cross PSD - Stories 3 & [in /sec 3 ] Freq [Hz] [in /sec 3 ] Freq [Hz] 1 4 Cross PSD - Stories 4 & 1 4 Cross PSD - Stories 4 & [in /sec 3 ] Freq [Hz] [in /sec 3 ] Freq [Hz] Auto PSD - Story Auto PSD - Story Freq [Hz] Freq [Hz] Figure 3.5 (Contd.) Identification Using Full Auto/Cross-Correlation Matrix (Seed = 74341, Real parts of cross-psds or Co-spectra are plotted)

24 Auto PSD - Story Freq [Hz] [in /sec 3 ] 1 Auto PSD - Story Freq [Hz] [in /sec 3 ] 1 4 Cross PSD - Stories 1 & Cross PSD - Stories 1 & [in /sec 3 ] Freq [Hz] [in /sec 3 ] Freq [Hz] 1 4 Cross PSD - Stories 1 & Cross PSD - Stories 1 & [in /sec 3 ] Freq [Hz] [in /sec 3 ] Freq [Hz] Auto PSD - Story Auto PSD - Story Freq [Hz] Freq [Hz] Figure 3.5 (Contd.) Identification Using Full Auto/Cross-Correlation Matrix (Seed = 74341, Real part of cross-psds or Co-spectra are plotted)

25 5 8 4 CCF - Stories & CCF - Stories & [in /sec 4 ] [in /sec 4 ] 8 4 CCF - Stories & CCF - Stories & [in /sec 4 ] [in /sec 4 ] 8 4 CCF - Stories & CCF - Stories & [in /sec 4 ] [in /sec 4 ] 8 4 ACF - Story 8 4 ACF - Story Figure 3.5 (Contd.) Identification Using Full Auto/Cross-Correlation Matrix (Seed = 74341)

26 ACF - Story ACF - Story [in /sec 4 ] [in /sec 4 ] 8 4 CCF - Stories 5 & 8 4 CCF - Stories 5 & [in /sec 4 ] [in /sec 4 ] 8 4 CCF - Stories & CCF - Stories & [in /sec 4 ] [in /sec 4 ] 8 4 CCF - Stories & 8 4 CCF - Stories & Figure 3.5 (Contd.) Identification Using Full Auto/Cross-Correlation Matrix (Seed = 74341)

27 5 8 4 CCF - Stories 5 & CCF - Stories 5 & [in /sec 4 ] [in /sec 4 ] 8 4 CCF - Stories 5 & 8 4 CCF - Stories 5 & [in /sec 4 ] [in /sec 4 ] 8 4 CCF - Stories 5 & CCF - Stories 5 & [in /sec 4 ] [in /sec 4 ] 8 4 CCF - Stories 5 & CCF - Stories 5 & Figure 3.5 (Contd.) Identification Using Full Auto/Cross-Correlation Matrix (Seed = 74341)

28 CCF - Stories 4 & CCF - Stories 4 & [in /sec 4 ] [in /sec 4 ] 8 4 ACF - Story ACF - Story [in /sec 4 ] [in /sec 4 ] 8 4 CCF - Stories 4 & CCF - Stories 4 & [in /sec 4 ] [in /sec 4 ] 8 4 CCF - Stories 4 & 8 4 CCF - Stories 4 & Figure 3.5 (Contd.) Identification Using Full Auto/Cross-Correlation Matrix (Seed = 74341)

29 CCF - Stories 3 & CCF - Stories 3 & [in /sec 4 ] [in /sec 4 ] 8 4 CCF - Stories 3 & 8 4 CCF - Stories 3 & [in /sec 4 ] [in /sec 4 ] 8 4 CCF - Stories 4 & CCF - Stories 4 & [in /sec 4 ] [in /sec 4 ] 8 4 CCF - Stories 4 & 8 4 CCF - Stories 4 & Figure 3.5 (Contd.) Identification Using Full Auto/Cross-Correlation Matrix (Seed = 74341)

30 CCF - Stories 3 & CCF - Stories 3 & [in /sec 4 ] [in /sec 4 ] 8 4 CCF - Stories 3 & 8 4 CCF - Stories 3 & [in /sec 4 ] [in /sec 4 ] 8 4 ACF - Story ACF - Story [in /sec 4 ] [in /sec 4 ] 8 4 CCF - Stories 3 & CCF - Stories 3 & Figure 3.5 (Contd.) Identification Using Full Auto/Cross-Correlation Matrix (Seed = 74341)

31 4 8 4 CCF - Stories & CCF - Stories & [in /sec 4 ] [in /sec 4 ] 8 4 CCF - Stories & CCF - Stories & [in /sec 4 ] [in /sec 4 ] 8 4 CCF - Stories & CCF - Stories & [in /sec 4 ] [in /sec 4 ] 8 4 CCF - Stories & 8 4 CCF - Stories & Figure 3.5 (Contd.) Identification Using Full Auto/Cross-Correlation Matrix (Seed = 74341)

32 CCF - Stories 1 & CCF - Stories 1 & [in /sec 4 ] [in /sec 4 ] 8 4 CCF - Stories 1 & 8 4 CCF - Stories 1 & [in /sec 4 ] [in /sec 4 ] 8 4 CCF - Stories & CCF - Stories & [in /sec 4 ] [in /sec 4 ] 8 4 ACF - Story 8 4 ACF - Story Figure 3.5 (Contd.) Identification Using Full Auto/Cross-Correlation Matrix (Seed = 74341)

33 ACF - Story ACF - Story [in /sec 4 ] [in /sec 4 ] 8 4 CCF - Stories 1 & 8 4 CCF - Stories 1 & [in /sec 4 ] [in /sec 4 ] 8 4 CCF - Stories 1 & CCF - Stories 1 & [in /sec 4 ] [in /sec 4 ] 8 4 CCF - Stories 1 & CCF - Stories 1 & Figure 3.5 Identification Using Full Auto/Cross-Correlation Matrix - (Seed = 74341)

34 43 Story Level 4 First Mode Story Level 4 Second Mode Story Level 4 Third Mode Story Level 4 Fourth Mode Story Level 4 Fifth Mode Story Level 4 Sixth Mode - 4 Figure 3.55 Comparison of and Mode Shapes (Identification using exact full correlation matrix)

35 Auto PSD - Story 1 1 Cross PSD - Stories 1 & Freq [Hz] Freq [Hz] [in /sec 3 ] [in /sec 3 ] Cross PSD - Stories 1 & Auto PSD - Story Freq [Hz] [in /sec 3 ] Freq [Hz] [in /sec 3 ] Cross PSD - Stories & Cross PSD - Stories 3 & [in /sec 3 ] Freq [Hz] [in /sec 3 ] Freq [Hz] Cross PSD - Stories 4 & Auto PSD - Story Freq [Hz] Freq [Hz] Figure 3.54 (Contd.) Identification using Full Auto/Cross Correlation Matrix (Real parts of cross-psds or Co-spectra are plotted)

36 CCF - Stories & CCF - Stories & [in /sec 4 ] [in /sec 4 ] CCF - Stories & ACF - Story Figure 3.54 (Contd.) Identification using Full Auto/Cross-Correlation Matrix

37 4 8 4 CCF - Stories 5 & CCF - Stories 5 & [in /sec 4 ] [in /sec 4 ] 8 4 CCF - Stories 5 & CCF - Stories 5 & [in /sec 4 ] [in /sec 4 ] ACF - Story CCF - Stories 5 & [in /sec 4 ] [in /sec 4 ] CCF - Stories & CCF - Stories & Figure 3.54 (Contd.) Identification using Full Auto/Cross-Correlation Matrix

38 CCF - Stories 3 & CCF - Stories 3 & [in /sec 4 ] [in /sec 4 ] 8 4 CCF - Stories 4 & CCF - Stories 4 & [in /sec 4 ] [in /sec 4 ] CCF - Stories 4 & ACF - Story [in /sec 4 ] [in /sec 4 ] CCF - Stories 4 & CCF - Stories 4 & Figure 3.54 (Contd.) Identification using Full Auto/Cross-Correlation Matrix

39 CCF - Stories & CCF - Stories & [in /sec 4 ] [in /sec 4 ] 8 4 CCF - Stories & CCF - Stories & [in /sec 4 ] [in /sec 4 ] CCF - Stories 3 & CCF - Stories 3 & [in /sec 4 ] [in /sec 4 ] ACF - Story CCF - Stories 3 & Figure 3.54 (Contd.) Identification using Full Auto/Cross-Correlation Matrix

40 ACF - Story CCF - Stories 1 & [in /sec 4 ] [in /sec 4 ] 8 4 CCF - Stories 1 & CCF - Stories 1 & [in /sec 4 ] [in /sec 4 ] CCF - Stories 1 & CCF - Stories 1 & [in /sec 4 ] [in /sec 4 ] CCF - Stories & ACF - Story Figure 3.54 Identification using Full Auto/Cross-Correlation Matrix

41 Auto PSD - Story 1 Cross PSD - Stories 1 & [in /sec 3 ] Freq [Hz] [in /sec 3 ] Freq [Hz] 1 5 Cross PSD - Stories 1 & Auto PSD - Story Freq [Hz] [in /sec 3 ] Freq [Hz] [in /sec 3 ] 1 5 Cross PSD - Stories & Cross PSD - Stories 3 & [in /sec 3 ] Freq [Hz] [in /sec 3 ] Freq [Hz] Cross PSD - Stories 4 & Auto PSD - Story Freq [Hz] Freq [Hz] Figure 3.53 Initial Auto/Cross Power Spectral Densities Used for Identification (Real parts of cross-psds or Co-spectra are plotted)

42 CCF - Stories & CCF - Stories & [in /sec 4 ] [in /sec 4 ] CCF - Stories & 5 8 ACF - Story Figure 3.5 (Contd.) Initial ACFs and CCFs Used for Identification

43 CCF - Stories 5 & CCF - Stories 5 & [in /sec 4 ] [in /sec 4 ] 8 4 CCF - Stories 5 & CCF - Stories 5 & [in /sec 4 ] [in /sec 4 ] ACF - Story 5 8 CCF - Stories 5 & [in /sec 4 ] [in /sec 4 ] CCF - Stories & 1 8 CCF - Stories & Figure 3.5 (Contd.) Initial ACFs and CCFs Used for Identification

44 CCF - Stories 3 & CCF - Stories 3 & [in /sec 4 ] [in /sec 4 ] 8 4 CCF - Stories 4 & CCF - Stories 4 & [in /sec 4 ] [in /sec 4 ] CCF - Stories 4 & 3 8 ACF - Story [in /sec 4 ] [in /sec 4 ] CCF - Stories 4 & 5 8 CCF - Stories 4 & Figure 3.5 (Contd.) Initial ACFs and CCFs Used for Identification

45 3 8 4 CCF - Stories & CCF - Stories & [in /sec 4 ] [in /sec 4 ] 8 4 CCF - Stories & CCF - Stories & [in /sec 4 ] [in /sec 4 ] CCF - Stories 3 & 1 8 CCF - Stories 3 & [in /sec 4 ] [in /sec 4 ] ACF - Story 3 8 CCF - Stories 3 & Figure 3.5 (Contd.) Initial ACFs and CCFs Used for Identification

46 ACF - Story CCF - Stories 1 & [in /sec 4 ] [in /sec 4 ] 8 4 CCF - Stories 1 & CCF - Stories 1 & [in /sec 4 ] [in /sec 4 ] CCF - Stories 1 & 5 8 CCF - Stories 1 & [in /sec 4 ] [in /sec 4 ] CCF - Stories & 1 ACF - Story Figure 3.5 Initial ACFs and CCFs Used for Identification

47 3 Story Level 4 First Mode Story Level 4 Second Mode -4 - Story Level 4 Third Mode Story Level 4 Fourth Mode Story Level 4 Fifth Mode Story Level 4 Sixth Mode Figure 3.51 Comparison of and Mode Shapes (Identification using diagonal terms of estimated correlation matrix, Seed = 85737)

48 9 8 4 ACF - Story ACF - Story [in /sec 4 ] [in /sec 4 ] ACF - Story 8 4 ACF - Story Figure 3.5 (Contd.) Identification using Auto/Cross-Correlation Matrix - (Seed = 85737, only diagonal terms are used)

49 8 8 4 ACF - Story ACF - Story [in /sec 4 ] [in /sec 4 ] 8 4 ACF - Story 8 4 ACF - Story [in /sec 4 ] [in /sec 4 ] ACF - Story ACF - Story [in /sec 4 ] [in /sec 4 ] ACF - Story 4 ACF - Story Figure 3.5 Identification using Auto/Cross-Correlation Matrix - (Seed = 85737, only diagonal terms are used)

50 7 Story Level 4 First Mode Story Level 4 Second Mode -4 - Story Level 4 Third Mode Story Level 4 Fourth Mode Story Level 4 Fifth Fifth Mode Mode Story Level 4 Sixth Sixth Mode Mode -4-4 Figure 3.49 Comparison of and Mode Shapes (Identification using diagonal terms of estimated correlation matrix, Seed = 74341)

51 8 4 ACF - Story ACF - Story [in /sec 4 ] [in /sec 4 ] ACF - Story 8 4 ACF - Story Figure 3.48 (Contd.) Identification using Auto/Cross-Correlation Matrix - (Seed = 74341, only diagonal terms are used)

52 5 8 4 ACF - Story ACF - Story [in /sec 4 ] [in /sec 4 ] 8 4 ACF - Story 8 4 ACF - Story [in /sec 4 ] [in /sec 4 ] ACF - Story ACF - Story [in /sec 4 ] [in /sec 4 ] ACF - Story 4 ACF - Story Figure 3.48 Identification using Correlation Matrix - (Seed = 74341, only diagonal terms are used)

53 4 Story Level 4 First Mode Story Level 4 Second Mode -4 - Story Level 4 Third Mode Story Level 4 Fourth Mode -1 1 Story Level 4 Fifth Mode Story Level 4 Sixth Mode -4-4 Figure 3.47 Comparison of and Mode Shapes (Identification using diagonal terms of exact correlation matrix)

54 3 8 4 ACF - Story ACF - Story [in /sec 4 ] [in /sec 4 ] 8 4 ACF - Story ACF - Story [in /sec 4 ] [in /sec 4 ] ACF - Story ACF - Story Figure 3.4 Identification Using Auto/Cross-Correlation Matrix (only diagonal terms are used)

55 8 4 ACF - Story ACF - Story [in /sec 4 ] [in /sec 4 ] 8 4 ACF - Story ACF - Story [in /sec 4 ] [in /sec 4 ] ACF - Story 5 8 ACF - Story Figure 3.45 Initial ACFs Used for Identification

56 Table 3.9 Identification Results Using Full Auto/Cross-Correlation Matrix - SIMO Case (Continued) 1 Parameter Values Initial ID Using AC Matrix ID Using AC matrix Seed 1 Seed b b b b b b b f J ACF estimation based on: (1) Dt =.1second. () Seed1 = 85737; N = 15-5 = 145; n d = ; Seed = 74341; N = 15-5 = 145; n d = ;

57 Table 3.9 Identification Results Using Full Auto/Cross-Correlation Matrix - SIMO Case (Continued) Parameter Values Initial ID Using AC Matrix ID Using AC matrix Seed 1 Seed b b b b b b b b b b b b b b b b b b b b b b b b b b b

58 Table 3.8 Identification Results Using Diagonal Terms of Auto/Cross-Correlation Matrix - SIMO Case (Continued) 19 Parameter Values Initial ID Using AC Matrix ID Using AC matrix Seed 1 Seed b b f J Case : Identification using both autocorrelations and crosscorrelations between story absolute acceleration responses. Table 3.9 Identification Results Using Full Auto/Cross-Correlation Matrix - SIMO Case Parameter Values Initial ID Using AC Matrix ID Using AC matrix Seed 1 Seed T T T T T T z z z z z z b b

59 Table 3.8 Identification Results Using Diagonal Terms of Auto/Cross-Correlation Matrix - SIMO Case (Continued) 18 Parameter Values Initial ID Using AC Matrix ID Using AC matrix Seed 1 Seed b b b b b b b b b b b b b b b b b b b b b b b b b b b

60 17 the presence of all six modes. However, only three of them have a significant contribution overall (the first, second, and fifth modes). Hence, even though four natural frequencies were identified accurately, only three mode shapes were estimated well. Case 1: Identification using only the autocorrelations of story absolute acceleration responses. Table 3.8 Identification Results Using Diagonal Terms of Auto/Cross-Correlation Matrix - SIMO Case Parameter Values Initial ID Using AC Matrix ID Using AC matrix Seed 1 Seed T T T T T T z z z z z z b b b b b b b

61 3.9.8 MDOF System Subjected to White-Noise Ground Acceleration (SIMO Case) - Identification Using Absolute Acceleration Response The application example of concern in this section deals with a SIMO case related to the -DOF shear building defined in Figure. and subjected to white-noise ground acceleration. The correlation matrix of the absolute story acceleration response vector represents the target to be matched as closely as possible by the identification procedure. Two cases are considered: (1) Identification performed using only the diagonal terms of the correlation matrix, i.e., matching only the auto-correlation functions of the story absolute accelerations and leaving out the cross-correlation functions between stories; () Identification performed using the full correlation matrix. The identification results for both cases are summarized in Table 3.8 and Table 3.9. It is noted that four to five modal frequencies and damping ratios are identified well in both cases. A compensation effect exists between the bs and f parameters in case 1, when the crosscorrelation functions are ignored, leading to bad identification results for these parameters. Consequently, only two mode shapes are identified well (Figure 3.47, Figure 3.49 and Figure 3.51). The bs and f parameters are estimated well in case which enables the accurate estimation of three mode shapes (Figure 3.55 and Figure 3.57). For this example, by considering the full correlation matrix in the identification process, the compensation effect between the bs and f parameters has been cancelled. The agreement between the exact, estimated and identified auto/cross-correlation functions is very good in general as shown in Figure 3.4 to Figure 3.51 for case 1 and in Figure 3.54 to Figure 3.57 for case and this is reflected in the accurate estimation of all the dominant parameters. However, when using the full correlation matrix in the identification process, the extra information provided by the cross-terms enables the accurate estimation of larger number of parameters, including the load intensity parameter, f. The identification results obtained in the time domain are also examined in the frequency domain. A few selected auto/cross-psd functions plotted on semilog scale are shown in Figure 3.53, Figure 3.54 and Figure 3.5. Only the real parts of the cross-psds, called Co-spectra, are plotted. The exact PSD functions reveal 1

62 15 f X X ( f ) [in -sec] PSD Freq [Hz] f X X ( f ) [in -sec] PSD Freq [Hz] Figure 3.44 (Contd.) Identification Using ACF - (Seed = )

63 14 R X X ( t ) [in ] ACF f f X X ( f ) [in X X ( f ) [in -sec] -sec] R X X ( t ) [in ] Freq [Hz] PSD PSD ACF Freq [Hz] Figure 3.44 Identification Using ACF - (Seed = )

64 13 f X X ( f ) [in -sec] PSD Freq [Hz] f X X ( f ) [in -sec] PSD Freq [Hz] Figure 3.43 (Contd.) Identification Using ACF - (Seed = 854)

65 1 R X X ( t ) [in ] ACF f f X X ( f ) [in X X ( f ) [in -sec] -sec] R X X ( t ) [in ] Freq [Hz] PSD PSD ACF Freq [Hz] Figure 3.43 Identification Using ACF - (Seed = 854)

66 11 f X X ( f ) [in -sec] PSD Freq [Hz] f X X ( f ) [in -sec] PSD Freq [Hz] Figure 3.4 (Contd.) Identification Using ACF - (Seed = )

67 1 R X X ( t ) [in ] ACF f f X X ( f ) [in X -sec] X ( f ) [in -sec] R X X ( t ) [in ] Freq [Hz] PSD PSD ACF Freq [Hz] Figure 3.4 Identification Using ACF - (Seed = )