Structural Engineering and Mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

The structured multiparameter eigenvalue problems in finite element model updating problems

  • Zhijun Wang (School of Mathematical Science, Dalian University of Technology) ;
  • Bo Dong (School of Mathematical Science, Dalian University of Technology) ;
  • Yan Yu (Business School, Dalian University of Foreign Languages) ;
  • Xinzhu Zhao (School of Mathematics, Liaoning University) ;
  • Yizhou Fang (School of Mathematical Science, Dalian University of Technology)
  • Received : 2023.02.04
  • Accepted : 2023.11.17
  • Published : 2023.12.10
⟨ Previous Next ⟩

Abstract

The multiparameter eigenvalue method can be used to solve the damped finite element model updating problems. This method transforms the original problems into multiparameter eigenvalue problems. Comparing with the numerical methods based on various optimization methods, a big advantage of this method is that it can provide all possible choices of physical parameters. However, when solving the transformed singular multiparameter eigenvalue problem, the proposed method based on the generalised inverse of a singular matrix has some computational challenges and may fail. In this paper, more details on the transformation from the dynamic model updating problem to the multiparameter eigenvalue problem are presented and the structure of the transformed problem is also exposed. Based on this structure, the rigorous mathematical deduction gives the upper bound of the number of possible choices of the physical parameters, which confirms the singularity of the transformed multiparameter eigenvalue problem. More importantly, we present a row and column compression method to overcome the defect of the proposed numerical method based on the generalised inverse of a singular matrix. Also, two numerical experiments are presented to validate the feasibility and effectiveness of our method.

Keywords

Acknowledgement

This work was supported by the National Natural Science Foundation of China under Grant 12371376, 11801382 and Liaoning Provincial Natural Science Foundation of China under Grant 2020-MS-278 and Scientific Research Fund of Liaoning Provincial Education Department under Grant 2020JYT04.

References

  1. Atkinson, F.V. (1972), Multiparameter Eigenvalue Problems, 1, Academic Press, New York.
  2. Atkinson, F.V. and Mingarelli, A.B. (2010), Multiparameter Eigenvalue Problems, CRC Press.
  3. Bai, Z.J., Chu, D. and Sun, D. (2007), "A dual optimization approach to inverse quadratic eigenvalue problems with partial eigenstructure", SIAM J. Scientif. Comput., 29(6), 2531-2561. https://doi.org/10.1137/060656346.
  4. Bai, Z.J., Datta, B.N. and Wang, J. (2010), "Robust and minimum norm partial quadratic eigenvalue assignment in vibrating systems: A new optimization approach", Mech. Syst. Signal Pr., 24(3), 766-783. https://doi.org/10.1016/j.ymssp.2009.09.014.
  5. Baybordi, S. and Esfandiari, A. (2022), "A novel sensitivity-based finite element model updating and damage detection using time domain response", J. Sound Vib., 537, 117187. https://doi.org/10.1016/j.jsv.2022.117187.
  6. Chan, T.H., Guo, L. and Li, Z. (2003), "Finite element modelling for fatigue stress analysis of large suspension bridges", J. Sound Vib., 261(3), 443-464. https://doi.org/10.1016/S0022-460X(02)010866.
  7. Chen, M.X. (2014), "An augmented Lagrangian dual optimization approach to the H-weighted model updating problem", J. Comput. Appl. Math., 29(6), 111-120. https://doi.org/10.1016/j.cam.2014.02.027.
  8. Chiewanichakorn, M., Aref, A.J. and Alampalli, S. (2007), "Dynamic and fatigue response of a truss bridge with fiber reinforced polymer deck", Int. J. Fatigue, 29(8), 1475-1489. https://doi.org/10.1016/j.ijfatigue.2006.10.031.
  9. Chu, M.T. and Golub, G.H. (2005), Inverse Eigenvalue Problems: Theory, Algorithms, and Applications, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, USA.
  10. Cottin, N. (2001), "Dynamic model updating-a multiparameter eigenvalue problem", Mech. Syst. Signal Pr., 15(4), 649-665. https://doi.org/10.1006/mssp.2000.1362.
  11. Cottin, N. and Reetz, J. (2006), "Accuracy of multiparameter eigenvalues used for dynamic model updating with measured natural frequencies only", Mech. Syst. Signal Pr., 20(1), 65-77. https://doi.org/10.1016/j.ymssp.2004.10.005.
  12. Dong, B., Lin, M.M. and Chu, M.T. (2009), "Parameter reconstruction of vibration systems from partial eigeninformation", J. Sound Vib., 327(3-5), 391-401. https://doi.org/10.1016/j.jsv.2009.06.026.
  13. Friswell, M.I. and Mottershead, J.E. (1995), Solid Mechanics and its Applications, 38, Kluwer Academic Publishers, Dordrecht, Holland.
  14. Gantmakher, F.R. (1959), The Theory of Matrices, 131, American Mathematical Society.
  15. Gladwell, G.M.L. (2004), Solid Mechanics and its Applications, 119, Kluwer Academic Publishers, Dordrecht, Holland.
  16. Gohberg, I., Lancaster, P. and Rodman, L. (1982), Matrix Polynomials, Academic Press Inc., New York, USA.
  17. Hemez, F.M. and Doebling, S.W. (2001), "Review and assessment of model updating for non-linear, transient dynamics", Mech. Syst. Signal Pr., 15(1), 45-74. http://doi.org/10.1006/mssp.2000.1351.
  18. Kuchak, A.J.T., Marinkovic, D. and Zehn, M. (2020), "Finite element model updating-Case study of a rail damper", Struct. Eng. Mech., 73(1), 27-35. https://doi.org/10.12989/sem.2020.73.1.027.
  19. Moreno, J., Datta, B. and Raydan, M. (2009), "A symmetry preserving alternating projection method for matrix model updating", Mech. Syst. Signal Pr., 23(6), 1784-1791. https://doi.org/10.1016/j.ymssp.2008.06.011.
  20. Mottershead, J. and Friswell, M. (1993), "Model updating in structural dynamics: A survey", J. Sound Vib., 167(2), 347-375. https://doi.org/10.1006/jsvi.1993.1340.
  21. 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.
  22. Shadan, F., Khoshnoudian, F. and Esfandiari, A. (2016), "A frequency response-based structural damage identification using model updating method", Struct. Control Hlth. Monit., 23(2), 286-302. https://doi.org/10.1002/stc.1768.
  23. Tian, W., Weng, S., Xia, Q. and Xia, Y. (2021), "Dynamic condensation approach for response-based finite element model updating of large-scale structures", J. Sound Vib., 506, 116176. https://doi.org/10.1016/j.jsv.2021.116176.
  24. Tisseur, F. and Meerbergen, K. (2001), "The quadratic eigenvalue problem", SIAM Rev., 43(2), 235-286. https://doi.org/10.1137/S0036144500381988.
  25. Wang, B., Yan, G.X. and Allahyari, S. (2021), "Optimization and mathematical modelling of multilayer beam based on sinusoidal theory", Struct. Eng. Mech., 79(1), 109-116. https://doi.org/10.12989/sem.2021.79.1.109.
  26. Wang, W., Mottershead, J.E., Ihle, A., Siebert, T. and Schubach, H.R. (2007), "Finite element model updating from full-field vibration measurement using digital image correlation", J. Sound Vib., 330(8), 1599-1620. http://doi.org/10.1016/j.jsv.2010.10.036.
  27. Yu, Y., Dong, B. and Yu, B. (2017), "Reconstruction of structured models using incomplete measured data", Struct. Eng. Mech., 62(3), 303-310. https://doi.org/10.12989/sem.2017.62.3.303.
  28. Zhang, J., Au, F.T.K. and Yang, D. (2020), "Finite element model updating of long-span cablestayed bridge by Kriging surrogate model", Struct. Eng. Mech., 74(2), 157-173. https://doi.org/10.12989/sem.2020.74.2.157.
  29. Zhao, K. (2022), "Robust partial quadratic eigenvalue assignment for the damped vibroacoustic system", Mech. Syst. Signal Pr., 162, 108001. https://doi.org/10.1016/j.ymssp.2021.108001.