Finite Linear Representation of Nonlinear Structural Dynamics Using Phase Space Embedding Coordinate
Publication: Journal of Engineering Mechanics
Volume 150, Issue 5
Abstract
Modeling of structural nonlinear dynamic behavior is a central challenge in civil and mechanical engineering communities. The phase space embedding of response time series has been demonstrated to be an efficient coordinate basis for data-driven approximation of the modern Koopman operator, which can fully capture the global evolution of nonlinear dynamics by a linear representation. This study demonstrates that linear and nonlinear structural dynamic vibrations can be represented by a universal forced linear model in a finite dimension space projected by time-delay coordinates. Compared with the existing methods, the proposed approach improves the performance of finite linear representation of nonlinear structural dynamics on two essential issues including the robustness to measurement noise and applicability to multidegree-of-freedom (MDOF) systems. For linear structures, the dynamic mode shapes and the corresponding natural frequencies can be accurately identified by using the time-delay dynamic mode decomposition (DMD) algorithm with acceleration response data experimentally measured from an 8-story shear-type linear steel frame. Modal parameters extracted from the time-delay DMD matched well with those identified from traditional modal identification methods, such as frequency domain decomposition (FDD) and complex mode indicator function (CMIF). In addition, numerical and experimental studies on nonlinear structures are conducted to demonstrate that the finite-dimensional DMD based on the discrete Hankel singular value decomposition (SVD) coordinate is highly symmetrically structured, and is able to accurately obtain a linear representation of structural nonlinear vibration. The resulting linearized data-driven equation-free model can be used to accurately predict the responses of nonlinear systems with limited training data sets.
Get full access to this article
View all available purchase options and get full access to this article.
Data Availability Statement
All data, models, or codes that support the findings of this study are available from the corresponding author upon reasonable request.
Acknowledgments
The support from Australian Research Council Future Fellowship FT190100801, “Innovative Data Driven Techniques for Structural Condition Monitoring,” is acknowledged.
References
Banyaga, A. 1997. “The structure of classical diffeomorphisms.” In Mathematics and its applications, 400. Berlin: Springer Science & Business Media.
Bhowmick, S., and S. Nagarajaiah. 2020. “Identification of full-field dynamic modes using continuous displacement response estimated from vibrating edge video.” J. Sound Vib. 489 (Dec): 115657. https://doi.org/10.1016/j.jsv.2020.115657.
Brincker, R., L. Zhang, and P. Andersen. 2001. “Modal identification of output-only systems using frequency domain decomposition.” Smart Mater. Struct. 10 (3): 441. https://doi.org/10.1088/0964-1726/10/3/303.
Brunton, S. L., B. W. Brunton, J. L. Proctor, E. Kaiser, and J. N. Kutz. 2017. “Chaos as an intermittently forced linear system.” Nat. Commun. 8 (1): 1–9. https://doi.org/10.1038/s41467-017-00030-8.
Brunton, S. L., J. L. Proctor, and J. N. Kutz. 2016. “Discovering governing equations from data by sparse identification of nonlinear dynamical systems.” Proc. Natl. Acad. Sci. U.S.A. 113 (15): 3932–3937. https://doi.org/10.1073/pnas.1517384113.
Buzug, T., and G. Pfister. 1992. “Optimal delay time and embedding dimension for delay-time coordinates by analysis of the global static and local dynamical behavior of strange attractors.” Phys. Rev. A 45 (10): 7073. https://doi.org/10.1103/PhysRevA.45.7073.
Cerrada, M., R.-V. Sánchez, C. Li, F. Pacheco, D. Cabrera, J. V. de Oliveira, and R. E. Vásquez. 2018. “A review on data-driven fault severity assessment in rolling bearings.” Mech. Syst. Signal Process. 99 (Jan): 169–196. https://doi.org/10.1016/j.ymssp.2017.06.012.
Challa, V. R., M. Prasad, Y. Shi, and F. T. Fisher. 2008. “A vibration energy harvesting device with bidirectional resonance frequency tenability.” Smart Mater. Struct. 17 (1): 015035. https://doi.org/10.1088/0964-1726/17/01/015035.
Champion, K. P., S. L. Brunton, and J. N. Kutz. 2019. “Discovery of nonlinear multiscale systems: Sampling strategies and embeddings.” SIAM J. Appl. Dyn. Syst. 18 (1): 312–333. https://doi.org/10.1137/18M1188227.
Deyle, E. R., and G. Sugihara. 2011. “Generalized theorems for nonlinear state space reconstruction.” PLoS One 6 (3): e18295. https://doi.org/10.1371/journal.pone.0018295.
Dylewsky, D., D. Barajas-Solano, T. Ma, A. M. Tartakovsky, and J. N. Kutz. 2020a. “Dynamic mode decomposition for forecasting and analysis of power grid load data.” Preprint, submitted October 8, 2020. http://arxiv.org/abs/2010.04248.
Dylewsky, D., E. Kaiser, S. L. Brunton, and J. N. Kutz. 2020b. “Principal component trajectories (PCT): Nonlinear dynamics as a superposition of time-delayed periodic orbits.” Preprint, submitted May 28, 2020. http://arxiv.org/abs/2005.14321.
Dylewsky, D., E. Kaiser, S. L. Brunton, and J. N. Kutz. 2022. “Principal component trajectories for modeling spectrally continuous dynamics as forced linear systems.” Phys. Rev. E 105 (1): 015312. https://doi.org/10.1103/PhysRevE.105.015312.
Eftekhari, A., H. L. Yap, M. B. Wakin, and C. J. Rozell. 2018. “Stabilizing embedology: Geometry-preserving delay-coordinate maps.” Phys. Rev. E 97 (2): 022222. https://doi.org/10.1103/PhysRevE.97.022222.
Gavish, M., and D. L. Donoho. 2014. “The optimal hard threshold for singular values is 4/3–√.” IEEE Trans. Inf. Theory 60 (8): 5040–5053. https://doi.org/10.1109/TIT.2014.2323359.
Giagopoulos, D., and A. Arailopoulos. 2017. “Computational framework for model updating of large scale linear and nonlinear finite element models using state of the art evolution strategy.” Comput. Struct. 192 (Nov): 210–232. https://doi.org/10.1016/j.compstruc.2017.07.004.
Héas, P., C. Herzet, and B. Combes. 2020. “Generalized kernel-based dynamic mode decomposition.” In Proc., ICASSP 2020-2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 3877–3881. New York: IEEE.
Hong, M., Q. Wang, Z. Su, and L. Cheng. 2014. “In situ health monitoring for bogie systems of CRH380 train on Beijing–Shanghai high-speed railway.” Mech. Syst. Signal Process. 45 (2): 378–395. https://doi.org/10.1016/j.ymssp.2013.11.017.
Jiang, A.-H., X.-C. Huang, Z.-H. Zhang, J. Li, Z.-Y. Zhang, and H.-X. Hua. 2010. “Mutual information algorithms.” Mech. Syst. Signal Process. 24 (8): 2947–2960. https://doi.org/10.1016/j.ymssp.2010.05.015.
Kamb, M., E. Kaiser, S. L. Brunton, and J. N. Kutz. 2020. “Time-delay observables for Koopman: Theory and applications.” SIAM J. Appl. Dyn. Syst. 19 (2): 886–917. https://doi.org/10.1137/18M1216572.
Kerschen, G., K. Worden, A. F. Vakakis, and J.-C. Golinval. 2006. “Past, present and future of nonlinear system identification in structural dynamics.” Mech. Syst. Signal Process. 20 (3): 505–592. https://doi.org/10.1016/j.ymssp.2005.04.008.
Korda, M., and I. Mezić. 2018. “On convergence of extended dynamic mode decomposition to the Koopman operator.” J. Nonlinear Sci. 28 (2): 687–710. https://doi.org/10.1007/s00332-017-9423-0.
Lai, Z., and S. Nagarajaiah. 2019. “Sparse structural system identification method for nonlinear dynamic systems with hysteresis/inelastic behavior.” Mech. Syst. Signal Process. 117 (Feb): 813–842. https://doi.org/10.1016/j.ymssp.2018.08.033.
Lai, Z., T. Sun, and S. Nagarajaiah. 2019. “Adjustable template stiffness device and SDOF nonlinear frequency response.” Nonlinear Dyn. 96 (2): 1559–1573. https://doi.org/10.1007/s11071-019-04871-4.
Lee, H. S., Y. H. Hong, and H. W. Park. 2010. “Design of an FIR filter for the displacement reconstruction using measured acceleration in low-frequency dominant structures.” Int. J. Numer. Methods Eng. 82 (4): 403–434. https://doi.org/10.1002/nme.2769.
Leng, Y., D. Tan, J. Liu, Y. Zhang, and S. Fan. 2017. “Magnetic force analysis and performance of a tri-stable piezoelectric energy harvester under random excitation.” J. Sound Vib. 406 (Apr): 146–160. https://doi.org/10.1016/j.jsv.2017.06.020.
Liao, Z., L. Song, P. Chen, Z. Guan, Z. Fang, and K. Li. 2018. “An effective singular value selection and bearing fault signal filtering diagnosis method based on false nearest neighbors and statistical information criteria.” Sensors 18 (7): 2235. https://doi.org/10.3390/s18072235.
Lusch, B., J. N. Kutz, and S. L. Brunton. 2018. “Deep learning for universal linear embeddings of nonlinear dynamics.” Nat. Commun. 9 (1): 1–10. https://doi.org/10.1038/s41467-018-07210-0.
Mezić, I. 2013. “Analysis of fluid flows via spectral properties of the Koopman operator.” Annu. Rev. Fluid Mech. 45 (Jan): 357–378. https://doi.org/10.1146/annurev-fluid-011212-140652.
Ni, P., Y. Xia, J. Li, and H. Hao. 2018. “Improved decentralized structural identification with output-only measurements.” Measurement 122 (Jul): 597–610. https://doi.org/10.1016/j.measurement.2017.09.029.
Noël, J.-P., and G. Kerschen. 2017. “Nonlinear system identification in structural dynamics: 10 more years of progress.” Mech. Syst. Signal Process. 83 (Jan): 2–35. https://doi.org/10.1016/j.ymssp.2016.07.020.
Pan, S., and K. Duraisamy. 2020. “Physics-informed probabilistic learning of linear embeddings of nonlinear dynamics with guaranteed stability.” SIAM J. Appl. Dyn. Syst. 19 (1): 480–509. https://doi.org/10.1137/19M1267246.
Peng, Z., and J. Li. 2022. “Phase space reconstruction and Koopman operator based linearization of nonlinear model for damage detection of nonlinear structures.” Adv. Struct. Eng. 25 (7): 1652–1669. https://doi.org/10.1177/13694332221082729.
Peng, Z., J. Li, and H. Hao. 2022. “Structural damage detection via phase space based manifold learning under changing environmental and operational conditions.” Eng. Struct. 263 (Jul): 114420. https://doi.org/10.1016/j.engstruct.2022.114420.
Peng, Z., J. Li, and H. Hao. 2023a. “Development and experimental verification of an IoT sensing system for drive-by bridge health monitoring.” Eng. Struct. 293 (Oct): 116705. https://doi.org/10.1016/j.engstruct.2023.116705.
Peng, Z., J. Li, H. Hao, and C. Li. 2021a. “Nonlinear structural damage detection using output-only Volterra series model.” Struct. Control Health Monit. 28 (9): e2802. https://doi.org/10.1002/stc.2802.
Peng, Z., J. Li, H. Hao, and Z. Nie. 2021b. “Improving identifiability of structural damage using higher order responses and phase space technique.” Struct. Control Health Monit. 28 (10): e2808. https://doi.org/10.1002/stc.2808.
Peng, Z., J. Li, H. Hao, and N. Yang. 2023b. “Mobile crowdsensing framework for drive-by-based dense spatial-resolution bridge mode shape identification.” Eng. Struct. 292 (Oct): 116515. https://doi.org/10.1016/j.engstruct.2023.116515.
Plaza, E. G., and P. N. López. 2017. “Surface roughness monitoring by singular spectrum analysis of vibration signals.” Mech. Syst. Signal Process. 84 (Feb): 516–530. https://doi.org/10.1016/j.ymssp.2016.06.039.
Proctor, J. L., S. L. Brunton, and J. N. Kutz. 2016. “Dynamic mode decomposition with control.” SIAM J. Appl. Dyn. Syst. 15 (1): 142–161. https://doi.org/10.1137/15M1013857.
Qu, C. X., T. H. Yi, and H. N. Li. 2019. “Mode identification by eigensystem realization algorithm through virtual frequency response function.” Struct. Control Health Monit. 26 (10): e2429. https://doi.org/10.1002/stc.2429.
Reynders, E. 2012. “System identification methods for (operational) modal analysis: Review and comparison.” Arch. Comput. Methods Eng. 19 (1): 51–124. https://doi.org/10.1007/s11831-012-9069-x.
Sabeerali, C., R. Ajayamohan, D. Giannakis, and A. J. Majda. 2017. “Extraction and prediction of indices for monsoon intraseasonal oscillations: An approach based on nonlinear Laplacian spectral analysis.” Clim. Dyn. 49 (9–10): 3031–3050. https://doi.org/10.1007/s00382-016-3491-y.
Saito, A., and T. Kuno. 2020. “Data-driven experimental modal analysis by dynamic mode decomposition.” J. Sound Vib. 481 (Sep): 115434. https://doi.org/10.1016/j.jsv.2020.115434.
Salova, A., J. Emenheiser, A. Rupe, J. P. Crutchfield, and R. M. D’Souza. 2019. “Koopman operator and its approximations for systems with symmetries.” Chaos: Interdiscip. J. Nonlinear Sci. 29 (9): 093128. https://doi.org/10.1063/1.5099091.
Shih, C., Y. Tsuei, R. Allemang, and D. Brown. 1988. “Complex mode indication function and its applications to spatial domain parameter estimation.” Mech. Syst. Signal Process. 2 (4): 367–377. https://doi.org/10.1016/0888-3270(88)90060-X.
Takeishi, N., Y. Kawahara, and T. Yairi. 2017. “Learning Koopman invariant subspaces for dynamic mode decomposition.” In NIPS’17: Proc., 31st Int. Conf. on Neural Information Processing Systems, 1130–1140. Red Hook, NY: Curran Associates. https://doi.org/10.5555/3294771.3294879.
Todd, M., J. Nichols, L. Pecora, and L. Virgin. 2001. “Vibration-based damage assessment utilizing state space geometry changes: Local attractor variance ratio.” Smart Mater. Struct. 10 (5): 1000. https://doi.org/10.1088/0964-1726/10/5/316.
Williams, M. O., I. G. Kevrekidis, and C. W. Rowley. 2015. “A data–driven approximation of the Koopman operator: Extending dynamic mode decomposition.” J. Nonlinear Sci. 25 (6): 1307–1346. https://doi.org/10.1007/s00332-015-9258-5.
Xu, B., J. He, R. Rovekamp, and S. J. Dyke. 2012. “Structural parameters and dynamic loading identification from incomplete measurements: Approach and validation.” Mech. Syst. Signal Process. 28 (Apr): 244–257. https://doi.org/10.1016/j.ymssp.2011.07.008.
Zeng, Q., G. Fan, D. Wang, W. Tao, and A. Liu. 2024. “A systematic approach to pixel-level crack detection and localization with a feature fusion attention network and 3D reconstruction.” Eng. Struct. 300 (Feb): 117219. https://doi.org/10.1016/j.engstruct.2023.117219.
Zhang, W., J. Li, H. Hao, and H. Ma. 2017. “Damage detection in bridge structures under moving loads with phase trajectory change of multi-type vibration measurements.” Mech. Syst. Signal Process. 87 (Mar): 410–425. https://doi.org/10.1016/j.ymssp.2016.10.035.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 150 • Issue 5 • May 2024
Copyright
© 2024 American Society of Civil Engineers.
History
Received: May 2, 2023
Accepted: Dec 1, 2023
Published online: Feb 22, 2024
Published in print: May 1, 2024
Discussion open until: Jul 22, 2024
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.