DOI QR코드

DOI QR Code

A novel Metropolis-within-Gibbs sampler for Bayesian model updating using modal data based on dynamic reduction

  • Ayan Das (Department of Civil Engineering, Indian Institute of Technology Delhi) ;
  • Raj Purohit Kiran (Department of Civil Engineering, Indian Institute of Technology Delhi) ;
  • Sahil Bansal (Department of Civil Engineering, Indian Institute of Technology Delhi)
  • 투고 : 2022.09.30
  • 심사 : 2023.05.15
  • 발행 : 2023.07.10

초록

The paper presents a Bayesian Finite element (FE) model updating methodology by utilizing modal data. The dynamic condensation technique is adopted in this work to reduce the full system model to a smaller model version such that the degrees of freedom (DOFs) in the reduced model correspond to the observed DOFs, which facilitates the model updating procedure without any mode-matching. The present work considers both the MPV and the covariance matrix of the modal parameters as the modal data. Besides, the modal data identified from multiple setups is considered for the model updating procedure, keeping in view of the realistic scenario of inability of limited number of sensors to measure the response of all the interested DOFs of a large structure. A relationship is established between the modal data and structural parameters based on the eigensystem equation through the introduction of additional uncertain parameters in the form of modal frequencies and partial mode shapes. A novel sampling strategy known as the Metropolis-within-Gibbs (MWG) sampler is proposed to sample from the posterior Probability Density Function (PDF). The effectiveness of the proposed approach is demonstrated by considering both simulated and experimental examples.

키워드

과제정보

The authors would like to gratefully acknowledge the Grant-in-aid scheme received from Aeronautics R&D Board, Government of India.

참고문헌

  1. Au, S.K. (2017), "Operational modal analysis: Modeling, bayesian inference, uncertainty laws", Oper. Modal Anal. Model. Bayesian Inference, Uncertain. Laws, Springer Singapore. https://doi.org/10.1007/978-981-10-4118-1/COVER.
  2. Au, S.K. and Zhang, F.L. (2012), "Fast Bayesian ambient modal identification incorporating multiple setups", J. Eng. Mech., 138(7), 800-815. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000385.
  3. Au, S.K. and Zhang, F.L. (2016), "Fundamental two-stage formulation for Bayesian system identification, Part I: General theory", Mech. Syst. Signal Pr., 66-67, 31-42. https://doi.org/10.1016/j.ymssp.2015.04.025.
  4. Baisthakur, S. and Chakraborty, A. (2020), "Modified Hamiltonian Monte Carlo-based Bayesian finite element model updating of steel truss bridge", Struct. Control Heal. Monit., 27(8), e2556. https://doi.org/10.1002/STC.2556.
  5. Bansal, S. (2015), "A new Gibbs sampling based Bayesian model updating approach using modal data from multiple setups", Int. J. Uncertain. Quantif., 5(4), 361-374. https://doi.org/10.1615/int.j.uncertaintyquantification.2015013581.
  6. Bansal, S. (2020), "Bayesian model updating using modal data based on dynamic condensation", J. Eng. Mech., 146(2), 04019123. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001714.
  7. Bansal, S. and Cheung, S.H. (2016), "Stochastic sampling based Bayesian model updating with incomplete modal data", Int. J. Uncertain. Quantif., 6(3), 229-244. https://doi.org/10.1615/int.j.uncertaintyquantification.2016017194.
  8. Beck, J.L. (1996), "System identification methods applied to measured seismic response", Proceedings 11th World Conference on Earthquake Engineering, Elsevier.
  9. Beck, J.L. and Au, S.K. (2002), "Bayesian updating of structural models and reliability using Markov Chain Monte Carlo simulation", J. Eng. Mech., 128(4), 380-391. https://doi.org/10.1061/(ASCE)0733-9399(2002)128:4(380).
  10. Beck, J.L. and Katafygiotis, L.S. (1998), "Updating models and their uncertainties. I: Bayesian statistical framework", J. Eng. Mech., 124(4), 455-461. https://doi.org/10.1061/(ASCE)0733-9399(1998)124:4(455).
  11. Beck, J.L., Siu-Kui, A. and Vanik, M.W. (2001), "Monitoring structural health using a probabilistic measure", Comput. Civil Infrastr. Eng., 16(1), 1-11. https://doi.org/10.1111/0885-9507.00209.
  12. Beck, J.L. and Yuen, K.V. (2004), "Model selection using response measurements: Bayesian probabilistic approach", J. Eng. Mech., 130(2), 192-203. https://doi.org/10.1061/(asce)0733-9399(2004)130:2(192).
  13. Behmanesh, I., Moaveni, B., Lombaert, G. and Papadimitriou, C. (2015), "Hierarchical Bayesian model updating for structural identification", Mech. Syst. Signal Pr., 64-65, 360-376. https://doi.org/10.1016/j.ymssp.2015.03.026.
  14. Bendat, J.S. and Piersol, A.G. (1993), Engineering Applications of Correlation and Spectral Analysis, Wiley.
  15. Betz, W., Papaioannou, I. and Straub, D. (2016), "Transitional Markov Chain Monte Carlo: Observations and improvements", J. Eng. Mech., 142(5), 04016016. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001066.
  16. Boulkaibet, I., Mthembu, L., Marwala, T., Friswell, M.I. and Adhikari, S. (2015), "Finite element model updating using an evolutionary Markov Chain Monte Carlo algorithm", Dynamics of Civil Structures, Volume 2: Proceedings of the 33rd IMAC, A Conference and Exposition on Structural Dynamics, Springer New York.
  17. Boulkaibet, I., Mthembu, L., Marwala, T., Friswell, M.I. and Adhikari, S. (2017), "Finite element model updating using Hamiltonian Monte Carlo techniques", Invers. Probl. Sci. Eng., 25(7), 1042-1070. https://doi.org/10.1080/17415977.2016.1215446.
  18. Brincker, R., Zhang, L. and Andersen, P. (2001), "Modal identification of output-only systems using frequency domain decomposition", Smart Mater. Struct., 10(3), 441-445. https://doi.org/10.1088/0964-1726/10/3/303.
  19. Cheung, S.H. and Bansal, S. (2017), "A new Gibbs sampling based algorithm for Bayesian model updating with incomplete complex modal data", Mech. Syst. Signal Pr., 92, 156-172. https://doi.org/10.1016/j.ymssp.2017.01.015.
  20. Ching, J. and Chen, Y.C. (2007), "Transitional Markov Chain Monte Carlo method for Bayesian model updating, model class selection, and model averaging", J. Eng. Mech., 133(7), 816-832. https://doi.org/10.1061/(asce)0733-9399(2007)133:7(816).
  21. Ching, J., Muto, M. and Beck, J.L. (2006), "Structural model updating and health monitoring with incomplete modal data using Gibbs sampler", Comput. Aid. Civil Infrastr. Eng., 21(4), 242-257. https://doi.org/10.1111/j.1467-8667.2006.00432.x.
  22. Christodoulou, K., Ntotsios, E., Papadimitriou, C. and Panetsos, P. (2008), "Structural model updating and prediction variability using Pareto optimal models", Comput. Meth. Appl. Mech. Eng., 198(1), 138-149. https://doi.org/10.1016/j.cma.2008.04.010.
  23. Christodoulou, K. and Papadimitriou, C. (2007), "Structural identification based on optimally weighted modal residuals", Mech. Syst. Signal Pr., 21(1), 4-23. https://doi.org/10.1016/j.ymssp.2006.05.011.
  24. Das, A. and Debnath, N. (2018), "A Bayesian finite element model updating with combined normal and lognormal probability distributions using modal measurements", Appl. Math. Model., 61, 457-483. https://doi.org/10.1016/j.apm.2018.05.004.
  25. Das, A. and Debnath, N. (2020), "A Bayesian model updating with incomplete complex modal data", Mech. Syst. Signal Pr., 136, 106524. https://doi.org/10.1016/j.ymssp.2019.106524.
  26. Das, A. and Debnath, N. (2021a), "Gibbs sampling for damage detection using complex modal data from multiple setups", ASCE-ASME J. Risk Uncertain. Eng. Syst. Part A Civil Eng., 7(2), 04021018. https://doi.org/10.1061/ajrua6.0001135.
  27. Das, A. and Debnath, N. (2021b), "Limited sensor-based probabilistic damage detection using combined normal-lognormal distributions", Arab. J. Sci. Eng., 46(5), 4639-4663. https://doi.org/10.1007/S13369-020-05056-7/FIGURES/11.
  28. Das, A. and Debnath, N. (2022), "Gibbs sampler-based probabilistic damage detection of structures using reduced order model", Int. J. Struct. Stab. Dyn., 2350075. https://doi.org/10.1142/S021945542350075X.
  29. DiazDelaO, F.A., Garbuno-Inigo, A., Au, S.K. and Yoshida, I. (2017), "Bayesian updating and model class selection with Subset Simulation", Comput. Meth. Appl. Mech. Eng., 317, 1102-1121. https://doi.org/10.1016/j.cma.2017.01.006.
  30. Ding, Y.J., Wang, Z.C., Chen, G., Ren, W.X. and Xin, Y. (2022), "Markov Chain Monte Carlo-based Bayesian method for nonlinear stochastic model updating", J. Sound Vib., 520, 116595. https://doi.org/10.1016/j.jsv.2021.116595.
  31. Eltouny, K.A. and Liang, X. (2021), "Bayesian-optimized unsupervised learning approach for structural damage detection", Comput. Civil Infrastr. Eng., 36(10), 1249-1269. https://doi.org/10.1111/MICE.12680.
  32. Ereiz, S., Duvnjak, I. and Fernando Jimenez-Alonso, J. (2022), "Review of finite element model updating methods for structural applications", Struct., 41, 684-723. https://doi.org/10.1016/j.istruc.2022.05.041.
  33. Esfandiari, A. (2019), "Structural parameter estimation and damage detection using experimental transfer function data", Invers. Prob. Sci. Eng., 28(1), 2-20. https://doi.org/10.1080/17415977.2019.1568426.
  34. Ewins, D.J. (2000), "Adjustment or updating of models", Sadhana-Acad. Proc. Eng. Sci., 25(3), 235-245. https://doi.org/10.1007/BF02703542.
  35. Fang, C., Liu, H.J., Lam, H.F., Adeagbo, M.O. and Peng, H.Y. (2022), "Practical model updating of the Ting Kau Bridge through the MCMC-based Bayesian algorithm utilizing measured modal parameters", Eng. Struct., 254, 113839. https://doi.org/10.1016/j.engstruct.2022.113839.
  36. Friswell, M.I., Garvey, S.D. and Penny, J.E.T. (1995), "Model reduction using dynamic and iterated IRS techniques", J. Sound Vib., 186(2), 311-323. https://doi.org/10.1006/JSVI.1995.0451.
  37. Friswell, M. and Mottershead, J.E. (1995), Finite Element Model Updating in Structural Dynamics, Vol. 38, Springer Science & Business Media.
  38. Gelman, A. and Rubin, D.B. (1992), "Inference from Iterative Simulation Using Multiple Sequences", Stat. Sci., 457-472.
  39. Geman, S. and Geman, D. (1984), "Stochastic relaxation, gibbs distributions, and the bayesian restoration of images", IEEE Trans. Pattern Anal. Mach. Intell., (6), 721-741. https://doi.org/10.1109/TPAMI.1984.4767596.
  40. Girardi, M., Padovani, C., Pellegrini, D. and Robol, L. (2021), "A finite element model updating method based on global optimization", Mech. Syst. Signal Pr., 152, 107372. https://doi.org/10.1016/J.YMSSP.2020.107372.
  41. Goller, B., Beck, J.L. and Schueller, G.I. (2012), "Evidence-based identification of weighting factors in bayesian model updating using modal data", J. Eng. Mech., 138(5), 430-440. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000351.
  42. Goller, B. and Schueller, G.I. (2011), "Investigation of model uncertainties in Bayesian structural model updating", J. Sound Vib., 330(25), 6122-6136. https://doi.org/10.1016/j.jsv.2011.07.036.
  43. Green, P.L. (2015), "Bayesian system identification of dynamical systems using large sets of training data: A MCMC solution", Probab. Eng. Mech., 42, 54-63. https://doi.org/10.1016/j.probengmech.2015.09.010.
  44. Hastings, W.K. (1970), "Monte Carlo sampling methods using Markov chains and their applications", Biometrika, 57(1), 97. https://doi.org/10.2307/2334940.
  45. Hizal, C. and Aktas, E. (2021), "Structural health monitoring-integrated reliability assessment of engineering structures", Reliab. Anal. Des. Struct. Infrastr., 117-128.
  46. Hizal, C. and Turan, G. (2020), "A two-stage Bayesian algorithm for finite element model updating by using ambient response data from multiple measurement setups", J. Sound Vib., 469, 115139. https://doi.org/10.1016/j.jsv.2019.115139.
  47. Hizal, C., Turan, G., Aktas, E. and Ceylan, H. (2019), "A mode shape assembly algorithm by using two stage Bayesian fast fourier transform approach", Mech. Syst. Signal Pr., 134, 106328. https://doi.org/10.1016/j.ymssp.2019.106328.
  48. Huang, Y. and Beck, J.L. (2018), "Full gibbs sampling procedure for bayesian system identification incorporating sparse bayesian learning with automatic relevance determination", Comput. Civil Infrastr. Eng., 33(9), 712-730. https://doi.org/10.1111/mice.12358.
  49. Jaishi, B. and Ren, W.X. (2007), "Finite element model updating based on eigenvalue and strain energy residuals using multiobjective optimisation technique", Mech. Syst. Signal Pr., 21(5), 2295-2317. https://doi.org/10.1016/j.ymssp.2006.09.008.
  50. Juang, J.N. (1994), Applied System Identification, Prentice Hall, Englewood Cliffs, N.J..
  51. Jung, D.S. and Kim, C.Y. (2009), "FE model updating based on hybrid genetic algorithm and its verification on numerical bridge model", Struct. Eng. Mech., 32(5), 667-683. https://doi.org/10.12989/sem.2009.32.5.667.
  52. Kamariotis, A., Chatzi, E. and Straub, D. (2022), "Value of information from vibration-based structural health monitoring extracted via Bayesian model updating", Mech. Syst. Signal Pr., 166, 1-21. https://doi.org/10.1016/j.ymssp.2021.108465.
  53. Karimpour, A. and Rahmatalla, S. (2021), "Identification of structural parameters and boundary conditions using a minimum number of measurement points", Front. Struct. Civil Eng., 14(6), 1331-1348. https://doi.org/10.1007/S11709-020-0686-4.
  54. Lam, H.F., Alabi, S.A. and Yang, J.H. (2017a), "Identification of rail-sleeper-ballast system through time-domain Markov chain Monte Carlo-based Bayesian approach", Eng. Struct., 140, 421-436. https://doi.org/10.1016/j.engstruct.2017.03.001.
  55. Lam, H.F., Hu, J. and Yang, J.H. (2017b), "Bayesian operational modal analysis and Markov chain Monte Carlo-based model updating of a factory building", Eng. Struct., 132, 314-336. https://doi.org/10.1016/j.engstruct.2016.11.048.
  56. Lam, H.F., Yang, J.H. and Au, S.K. (2018), "Markov chain Monte Carlo-based Bayesian method for structural model updating and damage detection", Struct. Control Hlth. Monit., 25(4), e2140. https://doi.org/10.1002/STC.2140.
  57. Li, J., Huang, Y. and Asadollahi, P. (2021), "Sparse Bayesian learning with model reduction for probabilistic structural damage detection with limited measurements", Eng. Struct., 247, 113183. https://doi.org/10.1016/j.engstruct.2021.113183.
  58. Liu, Y., Li, L. and Chang, Z. (2023), "Efficient Bayesian model updating for dynamic systems", Reliab. Eng. Syst. Saf., 236, 109294. https://doi.org/10.1016/j.ress.2023.109294.
  59. Lye, A., Cicirello, A. and Patelli, E. (2021), "Sampling methods for solving Bayesian model updating problems: A tutorial", Mech. Syst. Signal Pr., 159, 107760. https://doi.org/10.1016/j.ymssp.2021.107760.
  60. Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H. and Teller, E. (1953), "Equation of state calculations by fast computing machines", J. Chem. Phys., 21(6), 1087. https://doi.org/10.1063/1.1699114.
  61. Mottershead, J.E., Link, M. and Friswell, M.I. (2011), "The sensitivity method in finite element model updating: A tutorial", Mech. Syst. Signal Pr., 25(7), 2275-2296. https://doi.org/10.1016/j.ymssp.2010.10.012.
  62. Mustafa, S., Debnath, N. and Dutta, A. (2015), "Bayesian probabilistic approach for model updating and damage detection for a large truss bridge", Int. J. Steel Struct., 15(2), 473-485. https://doi.org/10.1007/S13296-015-6016-3.
  63. Nagel, J.B. and Sudret, B. (2016), "Hamiltonian Monte Carlo and borrowing strength in hierarchical inverse problems", ASCE-ASME J. Risk Uncertain. Eng. Syst. Part A Civil Eng., 2(3), B4015008. https://doi.org/doi:10.1061/AJRUA6.0000847.
  64. Ni, P., Li, J., Hao, H., Han, Q. and Du, X. (2021), "Probabilistic model updating via variational Bayesian inference and adaptive Gaussian process modeling", Comput. Meth. Appl. Mech. Eng., 383, 113915. https://doi.org/10.1016/j.cma.2021.113915.
  65. Ni, Y.C., Lam, H.F. and Zhang, F.L. (2023), "Assessing uncertainty in fast Bayesian modal identification based on seismic structural responses", Mech. Syst. Signal Pr., 185, 109686. https://doi.org/10.1016/j.ymssp.2022.109686.
  66. Ni, Y.C. and Zhang, F.L. (2018), "Fast Bayesian approach for modal identification using forced vibration data considering the ambient effect", Mech. Syst. Signal Pr., 105, 113-128. https://doi.org/10.1016/j.ymssp.2017.11.007.
  67. Ni, Y.C. and Zhang, F.L. (2021), "Uncertainty quantification in fast Bayesian modal identification using forced vibration data considering the ambient effect", Mech. Syst. Signal Pr., 148, 107078. https://doi.org/10.1016/j.ymssp.2020.107078.
  68. Niu, Z. (2020), "Frequency response-based structural damage detection using Gibbs sampler", J. Sound Vib., 470, 115160. https://doi.org/10.1016/J.JSV.2019.115160.
  69. Paz, M. and Kim, Y.H. (2019), Structural Dynamics, Springer International Publishing, Cham.
  70. Peeters, B. and De Roeck, G. (2001), "Stochastic system identification for operational modal analysis: A review", J. Dyn. Syst. Meas. Control, 123(4), 659-667. https://doi.org/10.1115/1.1410370.
  71. Prajapat, K. and Ray-Chaudhuri, S. (2016), "Prediction error variances in Bayesian model updating employing data sensitivity", J. Eng. Mech., ASCE, 142(12), 1-9. https://doi.org/10.1061/(asce)em.1943-7889.0001158.
  72. Qin, S., Hu, J., Zhou, Y.L., Zhang, Y. and Kang, J. (2019), "Feasibility study of improved particle swarm optimization in kriging metamodel based structural model updating", Struct. Eng. Mech., 70(5), 513-524. https://doi.org/10.12989/sem.2019.70.5.513.
  73. Rather, S. and Bansal, S. (2023), "Bayesian modal identification of non-classically damped systems using time-domain data", J. Sound Vib., 197, 117712. https://doi.org/10.1016/J.JSV.2023.117712.
  74. Rosenbluth, M.N. and Rosenbluth, A.W. (2004), "Monte Carlo calculation of the average extension of molecular chains", J. Chem. Phys., 23(2), 356. https://doi.org/10.1063/1.1741967.
  75. Rosenkrantz, R.D. (1989), "Where do we stand on maximum entropy? (1978)", E. T. Jaynes Pap. Probab. Stat. Stat. Phys., 210-314. https://doi.org/10.1007/978-94-009-6581-2_10.
  76. Schneider, F., Papaioannou, I., Straub, D., Winter, C. and Muller, G. (2022), "Bayesian parameter updating in linear structural dynamics with frequency transformed data using rational surrogate models", Mech. Syst. Signal Pr., 166, 108407. https://doi.org/10.1016/j.ymssp.2021.108407.
  77. Sehgal, S. and Kumar, H. (2015), "Structural dynamic model updating techniques: A state of the art review", Arch. Comput. Meth. Eng.., 23, 515-533. https://doi.org/10.1007/S11831-015-9150-3.
  78. Sengupta, P. and Chakraborty, S. (2022), "Markov Chain Monte Carlo simulation based Bayesian updating of model parameters and their uncertainties", Struct. Eng. Mech., 81(1), 103. https://doi.org/10.12989/sem.2022.81.1.103.
  79. Sohn, H. and Law, K.H. (1997), "A Bayesian probabilistic approach for structure damage detection", Earthq. Eng. Struct. Dyn., 26(12), 1259-1281. https://doi.org/10.1002/(SICI)1096-9845(199712)26:12<1259::AID-EQE709>3.0.CO,2-3.
  80. Sun, H. and Buyukozturk, O. (2016), "Probabilistic updating of building models using incomplete modal data", Mech. Syst. Signal Pr., 75, 27-40. https://doi.org/10.1016/j.ymssp.2015.12.024.
  81. Teixeira, J.S., Stutz, L.T., Knupp, D.C. and Neto, A.J.S. (2017), "Structural damage identification via time domain response and Markov Chain Monte Carlo method", Invers. Probl. Sci. Eng., 25(6), 909-935. https://doi.org/10.1080/17415977.2016.1209749.
  82. Vanik, M.W. (1997), A Bayesian Probabilistic Approach to Structural Health Monitoring.
  83. Vanik, M.W., Beck, J.L. and Au, S.K. (2000), "Bayesian probabilistic approach to structural health monitoring", J. Eng. Mech., 126(7), 738-745. https://doi.org/10.1061/(ASCE)0733-9399(2000)126:7(738).
  84. Xu, M., Guo, J., Wang, S., Li, J. and Hao, H. (2021), "Structural damage identification with limited modal measurements and ultra-sparse Bayesian regression", Struct. Control Hlth. Monit., 28(6), e2729. https://doi.org/10.1002/stc.2729.
  85. Yan, W.J. and Katafygiotis, L.S. (2015a), "A two-stage fast Bayesian spectral density approach for ambient modal analysis. Part I: Posterior most probable value and uncertainty", Mech. Syst. Signal Pr., 54, 139-155. https://doi.org/10.1016/j.ymssp.2014.07.027.
  86. Yan, W.J. and Katafygiotis, L.S. (2015b), "A novel Bayesian approach for structural model updating utilizing statistical modal information from multiple setups", Struct. Saf., 52, 260-271. https://doi.org/10.1016/j.strusafe.2014.06.004.
  87. Yan, W.J. and Katafygiotis, L.S. (2015c), "A two-stage fast Bayesian spectral density approach for ambient modal analysis. Part II: Mode shape assembly and case studies", Mech. Syst. Signal Pr., 54, 156-171. https://doi.org/10.1016/j.ymssp.2014.08.016.
  88. Yin, T. (2022), "A practical bayesian framework for structural model updating and prediction", ASCE-ASME J. Risk Uncertain. Eng. Syst. Part A Civil Eng., 8(1), 1-15. https://doi.org/10.1061/ajrua6.0001196.
  89. Yin, T., Jiang, Q.H. and Yuen, K.V. (2017), "Vibration-based damage detection for structural connections using incomplete modal data by Bayesian approach and model reduction technique", Eng. Struct., 132, 260-277. https://doi.org/10.1016/j.engstruct.2016.11.035.
  90. Yuen, K.V. (2010a), Bayesian Methods for Structural Dynamics and Civil Engineering, John Wiley Sons Pte Ltd., John Wiley and Sons.
  91. Yuen, K.V. (2010b), "Recent developments of Bayesian model class selection and applications in civil engineering", Struct. Saf., 32(5), 338-346. https://doi.org/10.1016/j.strusafe.2010.03.011.
  92. Yuen, K.V. and Katafygiotis, L.S. (2001), "Bayesian time-domain approach for modal updating using ambient data", Prob. Eng. Mech., 16(3), 219-231. https://doi.org/10.1016/S0266-8920(01)00004-2.
  93. Yuen, K.V. and Katafygiotis, L.S. (2003), "Bayesian fast Fourier transform approach for modal updating using ambient data", Adv. Struct. Eng., 6(2), 81-95. https://doi.org/10.1260/136943303769013183.
  94. Yuen, K.V. and Kuok, S.C. (2011), "Bayesian methods for updating dynamic models", Appl. Mech. Rev., 64(1), 010802. https://doi.org/10.1115/1.4004479.
  95. Zhang, F.L. and Au, S.K. (2016), "Fundamental two-stage formulation for Bayesian system identification, Part II: Application to ambient vibration data", Mech. Syst. Signal Pr., 66-67, 43-61. https://doi.org/10.1016/j.ymssp.2015.04.024.
  96. Zhang, F.L., Au, S.K. and Lam, H.F. (2015), "Assessing uncertainty in operational modal analysis incorporating multiple setups using a Bayesian approach", Struct. Control Hlth. Monit., 22(3), 395-416. https://doi.org/10.1002/STC.1679.
  97. Zhou, K., Zhi, L.H., Wang, J.F., Hong, X., Xu, K. and Shu, Z.R. (2023), "An improved stochastic subspace modal identification method considering uncertainty quantification", Struct., 51, 1083-1094. https://doi.org/10.1016/j.istruc.2023.03.101.
  98. Zhou, X., Kim, C.W., Zhang, F.L. and Chang, K.C. (2022), "Vibration-based Bayesian model updating of an actual steel truss bridge subjected to incremental damage", Eng. Struct., 260, 114226. https://doi.org/10.1016/j.engstruct.2022.114226.
  99. Zhu, Z., Au, S.K., Li, B. and Xie, Y.L. (2021), "Bayesian operational modal analysis with multiple setups and multiple (possibly close) modes", Mech. Syst. Signal Pr., 150, 107261. https://doi.org/10.1016/j.ymssp.2020.107261.