Skip to main content
SpringerLink
Log in

Data-driven identification of nonlinear normal modes via physics-integrated deep learning

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

Identifying the characteristic coordinates or modes of nonlinear dynamical systems is critical for understanding, analysis, and reduced-order modeling of the underlying complex dynamics. While normal modal transformation exactly characterizes any linear systems, there exists no such a general mathematical framework for nonlinear dynamical systems. Nonlinear normal modes (NNMs) are natural generalization of the normal modal transformation for nonlinear systems; however, existing research for identifying NNMs has relied on theoretical derivation or numerical computation from the closed-form equation of the system, which is usually unknown. In this work, we present a new data-driven framework based on physics-integrated deep learning for nonlinear modal identification of unknown nonlinear dynamical systems from the system response data only. Leveraging the universal modeling capacity and learning flexibility of deep neural networks, we first represent the forward and inverse nonlinear modal transformations through the physically interpretable deep encoder–decoder architecture, generalizing the modal superposition to nonlinear dynamics. Furthermore, to guarantee correct nonlinear modal identification, the proposed deep learning architecture integrates prior physics knowledge of the defined NNMs by embedding a unique dynamics-coder with physics-based constraints, including generalized modal properties, dynamics evolution, and future-state prediction. We test the proposed method by a series of study on the conservative and non-conservative Duffing systems with cubic nonlinearity and observe that the proposed data-driven framework is able to identify NNMs with invariant manifolds, energy-dependent nonlinear modal spectrum, and future-state prediction for unknown nonlinear dynamical systems from response data only; these identification results are found consistent with those from theoretically derived or numerically computed from closed-form equations. We also discuss its implementations and limitations for nonlinear modal identification of dynamical systems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Strogatz, Steven, Mark Friedman, A., Mallinckrodt, John, McKay, Susan: Nonlinear dynamics and chaos: with applications to physics biology chemistry and engineering. Comput. Phys. 8(5), 532 (1994)

  2. Heylen, Ward: Stefan Lammens, and Paul Sas. Modal Analysis Theory and Testing, Technical report (1997)

    Google Scholar 

  3. Haller, George: Ponsioen, Sten: Nonlinear normal modes and spectral submanifolds: existence, uniqueness and use in model reduction. Nonlin. Dyn. 86(3), 1493–1534 (2016)

    Article  Google Scholar 

  4. Komatsu, Keiji: Sano, Masaaki, Kai, Takashi, Tsujihata, Akio, Mitsuma, Hidehiko: Experimental modal analysis for dynamic models of spacecraft. J. Guid. Control Dyn. 14(3), 686–688 (1991)

    Article  Google Scholar 

  5. Likins, Peter W.: Modal method for analysis of free rotations of spacecraft. AIAA J. 5(7), 1304–1308 (1967)

    Article  Google Scholar 

  6. Touzé, C., Amabili, M., Thomas, O.: Reduced-order models for large-amplitude vibrations of shells including in-plane inertia. Comput. Methods Appl. Mech. Eng. 197(21–24), 2030–2045 (2008)

    Article  MathSciNet  Google Scholar 

  7. Bladh, R., Pierre, C., Castanier, M.P., Kruse, M.J.: Dynamic response predictions for a mistuned industrial turbomachinery rotor using reduced-order modeling. J. Eng. Gas Turbines Power 124(2), 311–324 (2002)

    Article  Google Scholar 

  8. Castanier, M.P., Bladh, R.: Component-mode-based reduced order modeling techniques for mistuned bladed disks-Part 1: theoretical models. J. Eng. Gas Turbines Power 123(1), 89–99 (2001)

    Article  Google Scholar 

  9. Peeters, Bart: Hendricx, Wim, Debille, Jan, Climent, Hector: Modern solutions for ground vibration testing of large aircraft. Sound Vibr. 43(1), 8–15 (2009)

    Google Scholar 

  10. Vakakis, A.F.: Non-linear normal modes (NNMs) and their applications in vibration theory: an overview. Mech. Syst. Signal Process. 11(1), 3–22 (1997)

    Article  Google Scholar 

  11. Kerschen, G., Peeters, M., Golinval, J.C., Vakakis, A.F.: Nonlinear normal modes, Part I: a useful framework for the structural dynamicist. Mech. Systems Signal Process. 23(1), 170–194 (2009)

    Article  Google Scholar 

  12. Mezić, Igor: Spectral properties of dynamical systems, model reduction and decompositions. Nonlin. Dyn. 41(1–3), 309–325 (2005)

    Article  MathSciNet  Google Scholar 

  13. Mezić, Igor: Analysis of fluid flows via spectral properties of the koopman operator. Annual Rev. Fluid Mech. 45(1), 357–378 (2013)

    Article  MathSciNet  Google Scholar 

  14. Koopman, B.O.: Hamiltonian Systems and Transformation in Hilbert Space. Proceedings of the National Academy of Sciences, 17(5):315–318, (may 1931)

  15. Rowley, Clarence W., Mezi, Igor, Bagheri, Shervin, Schlatter, Philipp, Henningson, Dan S.: Spectral analysis of nonlinear flows. J. Fluid Mech. 641, 115–127 (2009)

    Article  MathSciNet  Google Scholar 

  16. Nathan Kutz, J., Brunton, Steven L., Brunton, Bingni W., Proctor, Joshua L.: Dynamic Mode Decomposition: Data-Driven Modeling of. Complex Systems. (2016)

  17. Takeishi, Naoya: Kawahara, Yoshinobu, Yairi. Learning Koopman invariant subspaces for dynamic mode decomposition. Technical report, Takehisa (2017)

    Google Scholar 

  18. Lusch, Bethany, Kutz, J. Nathan., Brunton, Steven L.: Deep learning for universal linear embeddings of nonlinear dynamics. Nat. Commun. 9(1), 4950 (2018)

  19. Yeung, Enoch, Kundu, Soumya, Hodas, Nathan: Learning deep neural network representations for koopman operators of nonlinear dynamical systems. In Proceedings of the American Control Conference, volume 2019-July, pages 4832–4839. Institute of Electrical and Electronics Engineers Inc., (jul 2019)

  20. Shaw, S.W., Pierre, C.: Normal modes for non-linear vibratory systems. J. Sound Vibr. 164(1), 85–124 (1993)

    Article  Google Scholar 

  21. Cirillo, G.I., Mauroy, A., Renson, L., Kerschen, G., Sepulchre, R.: A spectral characterization of nonlinear normal modes. J. Sound Vibr. 377, 284–301 (2016)

    Article  Google Scholar 

  22. Kuether, Robert J., Allen, Matthew S.: A numerical approach to directly compute nonlinear normal modes of geometrically nonlinear finite element models. Mech. Syst. Signal Process. 46(1), 1–15 (2014)

    Article  Google Scholar 

  23. Peeters, M., Viguié, R., Sérandour, G., Kerschen, G., Golinval, J.C.: Nonlinear normal modes, Part II: toward a practical computation using numerical continuation techniques. Mech. Syst. Signal Process. 23(1), 195–216 (2009)

    Article  Google Scholar 

  24. Ponsioen, Sten: Pedergnana, Tiemo, Haller, George: Automated computation of autonomous spectral submanifolds for nonlinear modal analysis. J. Sound Vibr. 420, 269–295 (2018)

    Article  Google Scholar 

  25. Noël, Jean Philippe, Renson, L., Grappasonni, C., Kerschen, G.: Identification of nonlinear normal modes of engineering structures under broadband forcing. Mech. Syst. Signal Process. 74, 95–110 (2016)

  26. Poon, Chun-Wing, Chang, Chih-Chen: Identification of nonlinear normal modes of structures using the empirical mode decomposition method. In Masayoshi Tomizuka, editor, Smart Structures and Materials 2005: Sensors and Smart Structures Technologies for Civil, Mechanical, and Aerospace Systems, volume 5765, page 881. SPIE, (may 2005)

  27. Frank Pai, P.: Time-frequency characterization of nonlinear normal modes and challenges in nonlinearity identification of dynamical systems. Mech. Syst. Signal Process. 25(7), 2358–2374 (2011)

    Article  Google Scholar 

  28. Eriten, Melih, Kurt, Mehmet, Guanyang Luo, D., McFarland, Michael, Bergman, Lawrence A., Vakakis, Alexander F.: Nonlinear system identification of frictional effects in a beam with a bolted joint connection. Mech. Syst. Signal Process. 39(1–2), 245–264 (2013)

  29. Worden, K., Green, P.L.: A machine learning approach to nonlinear modal analysis. Mech. Syst. Signal Process. 84, 34–53 (2017)

    Article  Google Scholar 

  30. Dervilis, Nikolaos, Simpson, Thomas E., Wagg, David J., Worden, Keith: Nonlinear modal analysis via non-parametric machine learning tools. Strain 55(1), e12297 (2019)

    Article  Google Scholar 

  31. Hornik, Kurt: Stinchcombe, Maxwell, White, Halbert: Universal approximation of an unknown mapping and its derivatives using multilayer feedforward networks. Neural Netw. 3(5), 551–560 (1990)

    Article  Google Scholar 

  32. Akhtar, Naveed, Mian, Ajmal: Threat of Adversarial Attacks on Deep Learning in Computer Vision: A Survey, (feb 2018)

  33. Lagaris, Isaac Elias, Likas, Aristidis, Fotiadis, Dimitrios I.: Artificial neural networks for solving ordinary and partial differential equations. IEEE Trans. Neural Netw. 9(5), 987–1000 (1998)

    Article  Google Scholar 

  34. Jin, Xiaowei: Cheng, Peng, Chen, Wen Li, Li, Hui: Prediction model of velocity field around circular cylinder over various Reynolds numbers by fusion convolutional neural networks based on pressure on the cylinder. Phys. Fluids 30(4), 047105 (2018)

    Article  Google Scholar 

  35. Sirignano, Justin: Spiliopoulos, Konstantinos: DGM: a deep learning algorithm for solving partial differential equations. J. Comput. Phys. 375, 1339–1364 (2018)

    Article  MathSciNet  Google Scholar 

  36. Wei, Shiyin: Jin, Xiaowei, Li, Hui: General solutions for nonlinear differential equations: a rule-based self-learning approach using deep reinforcement learning. Comput. Mech. 64(5), 1361–1374 (2019)

    Article  Google Scholar 

  37. Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys. 378, 686–707 (2019)

    Article  MathSciNet  Google Scholar 

  38. Raissi, Maziar, Wang, Zhicheng, Triantafyllou, Michael S., Karniadakis, George Em: Deep learning of vortex-induced vibrations. J. Fluid Mech. 861, 119–137 (2019)

    Article  MathSciNet  Google Scholar 

  39. Raissi, Maziar: Babaee, Hessam, Givi, Peyman: Deep learning of turbulent scalar mixing. Phys. Rev. Fluids 4(12), 124501 (2019)

    Article  Google Scholar 

  40. Bao, Yuequan, Chen, Zhicheng, Wei, Shiyin, Yang, Xu., Tang, Zhiyi, Li, Hui: The state of the art of data science and engineering in structural health monitoring. Engineering 5(2), 234–242 (2019)

  41. Iten, Raban, Metger, Tony, Wilming, Henrik, Rio, Del, Renato, L.R.: Discovering physical concepts with neural networks. Phys. Rev. Lett. 124(1), 010508 (2020)

  42. Geneva, Nicholas: Zabaras, Nicholas: Modeling the dynamics of PDE systems with physics-constrained deep auto-regressive networks. J. Comput. Phys. 403, 109056 (2020)

    Article  MathSciNet  Google Scholar 

  43. Jin, Xiaowei: Laima, Shujin, Chen, Wen Li, Li, Hui: Time-resolved reconstruction of flow field around a circular cylinder by recurrent neural networks based on non-time-resolved particle image velocimetry measurements. Exp. Fluids 61(4), 1–23 (2020)

    Article  Google Scholar 

  44. Li, Wenjie: Laima, Shujin, Jin, Xiaowei, Yuan, Wenyong, Li, Hui: A novel long short-term memory neural-network-based self-excited force model of limit cycle oscillations of nonlinear flutter for various aerodynamic configurations. Nonlin. Dyn. 100(3), 2071–2087 (2020)

    Article  Google Scholar 

  45. Brincker, Rune: Zhang, Lingmi, Andersen, Palle: Modal identification of output-only systems using frequency domain decomposition. Smart Mater. Struct. 10(3), 441–445 (2001)

    Article  Google Scholar 

  46. Brunton, Steven L., Proctor, Joshua L., Nathan Kutz, J., Bialek, William: Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc. Natl. Acad. Sci. USA 113(15), 3932–3937 (2016)

  47. Schmidt, Michael: Lipson, Hod: Distilling free-form natural laws from experimental data. Science 324(5923), 81–85 (2009)

    Article  Google Scholar 

  48. Nee, Sean, Colegrave, Nick, West, Stuart A., Grafen, Alan: Evolution: the illusion of invariant quantities in life histories. Science 309(5738), 1236–1239 (2005)

    Article  MathSciNet  Google Scholar 

  49. Nathan Kutz, J.: Deep learning in fluid dynamics. J. Fluid Mech. 814, 1–4 (2017)

    Article  Google Scholar 

  50. Reichstein, Markus: Camps-Valls, Gustau, Stevens, Bjorn, Jung, Martin, Denzler, Joachim, Carvalhais, Nuno: Deep learning and process understanding for data-driven Earth system science. Nature 566(7743), 195–204 (2019)

    Article  Google Scholar 

  51. Rosenberg, R.M.: The normal modes of nonlinear n-degree-of-freedom systems. J. Appl. Mech., Trans. ASME 29(1), 7–14 (1960)

  52. Rosenberg, R.M.: On nonlinear vibrations of systems with many degrees of freedom. Adv. Appl. Mech. 9(C):155–242, Jan (1966)

  53. Abadi, Martín, Agarwal, Ashish, Barham, Paul, Brevdo, Eugene, Chen, Zhifeng, Citro, Craig, Corrado, Greg S., Davis, Andy, Dean, Jeffrey, Devin, Matthieu, Ghemawat, Sanjay, Goodfellow, Ian, Harp, Andrew, Irving, Geoffrey, Isard, Michael, Jia, Yangqing, Jozefowicz, Rafal, Kaiser, Lukasz, Kudlur, Manjunath, Levenberg, Josh, Mane, Dan, Monga, Rajat, Moore, Sherry, Murray, Derek, Olah, Chris, Schuster, Mike, Shlens, Jonathon, Steiner, Benoit, Sutskever, Ilya, Talwar, Kunal, Tucker, Paul, Vanhoucke, Vincent, Vasudevan, Vijay, Viegas, Fernanda, Vinyals, Oriol, Warden, Pete, Wattenberg, Martin, Wicke, Martin, Yuan, Yu.: and Xiaoqiang Zheng. Large-Scale Machine Learning on Heterogeneous Distributed Systems. Technical report, TensorFlow (2016)

  54. Kingma, Diederik P., Ba, Jimmy Lei: Adam: A method for stochastic optimization. In 3rd International Conference on Learning Representations, ICLR 2015 - Conference Track Proceedings. International Conference on Learning Representations, ICLR, (2015)

  55. Peeters, Maxime: Theoretical and Experimental Modal Analysis of Nonlinear Vibrating Structures using Nonlinear Normal Modes. University of Liege, Liege (2010). (PhD thesis)

  56. Callier, Frank M., Desoer, Charles A.: Linear system theory. Springer, Berlin (2012)

    MATH  Google Scholar 

Download references

Acknowledgements

This research was partially funded by the Physics of Artificial Intelligence Program of US Defense Advanced Research Projects Agency (DARPA) and the Michigan Technological University faculty startup fund.

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering-Engineering Mechanics, Michigan Technological University, Houghton, Michigan, 49931, USA

    Shanwu Li & Yongchao Yang

Authors
  1. Shanwu Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yongchao Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yongchao Yang.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Data availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, S., Yang, Y. Data-driven identification of nonlinear normal modes via physics-integrated deep learning. Nonlinear Dyn 106, 3231–3246 (2021). https://doi.org/10.1007/s11071-021-06931-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-021-06931-0

Keywords

Search

Navigation