Skip to main content
SpringerLink
Log in

Parameter identification of systems with multiple disproportional local nonlinearities

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

Abstract

Identification of nonlinear systems, especially with multiple local nonlinearities exhibiting disproportional ratios of the degree of nonlinearity and present at a single or multiple spatial locations, is a highly challenging inverse problem. Identification of such complex nonlinear systems cannot be handled easily by the existing conventional restoring force or describing function methods. Further, noise-corrupted measured time history responses make the parameter identification process much more difficult. Keeping this in view, we propose a new meta support vector machine (meta-SVM) model to precisely identify the type, spatial location(s) and also the nonlinear parameters present in disproportionate levels using the noisy measurements. Apart from the conventional SVM model, we also explore the effectiveness of the non-batch processing models like incremental learning for lesser computational cost and increased efficiency. Both incremental and conventional support vector regression models are explored to precisely identify the nonlinear parameters. A numerically simulated multi-degree of freedom spring-mass system with limited multiple local nonlinearities at a few selected spatial locations is considered to illustrate the proposed meta-SVM model for nonlinear parametric identification. However, the extension of the proposed meta-SVM model is rather straightforward to include all types of nonlinearities and cases with the simultaneous existence of multiple numbers of same or different nonlinearities (i.e. combined nonlinearities) at single or multiple locations. It is also clearly established from the numerical simulation studies that the proposed incremental meta-SVM model paves way for online real-time identification of nonlinear parameters which is not yet been addressed in the existing literature.

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
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

References

  1. Kerschen, G., Worden, K., Vakakis, A.F., Golinval, J.C.: Past, present and future of nonlinear system identification in structural dynamics. Mech. Syst. Signal Process. 20(3), 505–592 (2006)

    Article  Google Scholar 

  2. Worden, K., Tomlinson, G.R.: Nonlinearity in Structural Dynamics: Detection, Identification and Modelling. CRC Press, Boca Raton (2019)

    Book  MATH  Google Scholar 

  3. Noël, J.P., Kerschen, G.: Nonlinear system identification in structural dynamics: 10 more years of progress. Mech. Syst. Signal Process. 83, 2–35 (2017)

    Article  Google Scholar 

  4. Prawin, J., Rao, A.R.M.: Detection of nonlinear structural behavior using time-frequency and multivariate analysis. Smart Struct. Syst. 22(6), 711–725 (2018)

    Google Scholar 

  5. Brewick, P.T., Masri, S.F.: An evaluation of data-driven identification strategies for complex nonlinear dynamic systems. Nonlinear Dyn. 85(2), 1297–1318 (2016)

    Article  MathSciNet  Google Scholar 

  6. Nie, Z.Y., Liu, R.J., Wang, Q.G., Guo, D.S., Ma, Y.J., Lan, Y.H.: Novel identification approach for nonlinear systems with hysteresis. Nonlinear Dyn. 95, 1053–1066 (2018)

    Article  MATH  Google Scholar 

  7. Prawin, J., Rao, A.R.M.: Nonlinear identification of MDOF systems using Volterra series approximation. Mech. Syst. Signal Process. 84, 58–77 (2017)

    Article  Google Scholar 

  8. Gondhalekar, A.C.: Strategies for non-linear system identification (Doctoral dissertation, Imperial College London) (2009)

  9. Herrera, C.A., McFarland, D.M., Bergman, L.A., Vakakis, A.F.: Methodology for nonlinear quantification of a flexible beam with a local, strong nonlinearity. J. Sound Vib. 388, 298–314 (2017)

    Article  Google Scholar 

  10. Özer, M.B., Özgüven, H.N., Royston, T.J.: Identification of structural non-linearities using describing functions and the Sherman–Morrison method. Mech. Syst. Signal Process. 23(1), 30–44 (2009)

    Article  Google Scholar 

  11. Prawin, J., Rao, A.R.M.: Damage detection in nonlinear systems using an improved describing function approach with limited instrumentation. Nonlinear Dyn. 96, 1447–1470 (2019)

    Article  Google Scholar 

  12. Burges, C.J.: A tutorial on support vector machines for pattern recognition. Data Min. Knowl. Discov. 2(2), 121–167 (1998)

    Article  Google Scholar 

  13. Bornn, L., Farrar, C.R., Park, G.: Damage detection in initially nonlinear systems. Int. J. Eng. Sci. 48(10), 909–920 (2010)

    Article  Google Scholar 

  14. Cortes, C., Vapnik, V.: Support-vector networks. Mach. Learn. 20(3), 273–297 (1995)

    MATH  Google Scholar 

  15. Sharma, S.: Application of support vector machines for damage detection in structures. Msc Thesis, Worcester Polytechnic Institute (2009)

  16. Widodo, A., Yang, B.S.: Support vector machine in machine condition monitoring and fault diagnosis. Mech. Syst. Signal Process. 21(6), 2560–2574 (2007)

    Article  Google Scholar 

  17. Santos, J.D.A., Barreto, G.A.: An outlier-robust kernel RLS algorithm for nonlinear system identification. Nonlinear Dyn. 90(3), 1707–1726 (2017)

    Article  MathSciNet  Google Scholar 

  18. Tian, J., Gu, H.: Anomaly detection combining one-class SVMs and particle swarm optimization algorithms. Nonlinear Dyn. 61(1–2), 303–310 (2010)

    Article  MATH  Google Scholar 

  19. Hou, S., Li, Y.: Detecting nonlinearity from a continuous dynamic system based on the delay vector variance method and its application to gear fault identification. Nonlinear Dyn. 60(1–2), 141–148 (2010)

    Article  MATH  Google Scholar 

  20. Jin, X., Shao, J., Zhang, X., An, W., Malekian, R.: Modeling of nonlinear system based on deep learning framework. Nonlinear Dyn. 84(3), 1327–1340 (2016)

    Article  MathSciNet  Google Scholar 

  21. Soeffker, D., Wei, C., Wolff, S., Saadawia, M.S.: Detection of rotor cracks: comparison of an old model-based approach with a new signal-based approach. Nonlinear Dyn. 83(3), 1153–1170 (2016)

    Article  Google Scholar 

  22. Muhamad, P., Sims, N.D., Worden, K.: On the orthogonalised reverse path method for nonlinear system identification. J. Sound Vib. 331(20), 4488–4503 (2012)

    Article  Google Scholar 

  23. Liu, J., Li, B., Miao, H., Zhang, X., Li, M.: A modified time domain subspace method for nonlinear identification based on nonlinear separation strategy. Nonlinear Dyn. 94(4), 2491–2509 (2018)

    Article  Google Scholar 

  24. Wei, S., Peng, Z.K., Dong, X.J., Zhang, W.M.: A nonlinear subspace-prediction error method for identification of nonlinear vibrating structures. Nonlinear Dyn. 91(3), 1605–1617 (2018)

    Article  MATH  Google Scholar 

  25. Ganyun, L.V., Haozhong, C., Haibao, Z., Lixin, D.: Fault diagnosis of power transformer based on multi-layer SVM classifier. Electr. Power Syst. Res. 74(1), 1–7 (2005)

    Article  Google Scholar 

  26. Zhang, J., Sato, T., Iai, S.: Support vector regression for online health monitoring of large-scale structures. Struct. Saf. 28(4), 392–406 (2006)

    Article  Google Scholar 

  27. Cauwenberghs, G., Poggio, T.: Incremental and decremental support vector machine learning. In: Advances in Neural Information Processing Systems, pp. 409–415 (2001)

  28. Ma, J., Theiler, J., Perkins, S.: Accurate on-line support vector regression. Neural Comput. 15(11), 2683–2703 (2003)

    Article  MATH  Google Scholar 

  29. Yang, X.S.: Firefly algorithms for multimodal optimization. In: International Symposium on stochastic algorithms, pp. 169–178. Springer, Berlin, Heidelberg (2009)

  30. Yang, X.S.: Firefly algorithm, stochastic test functions and design optimisation. Int. J. Bio-inspired Comput. 2(2), 78–84 (2010)

    Article  Google Scholar 

  31. Yang, X.S.: Review of metaheuristics and generalized evolutionary walk algorithm. Int. J. Bio-inspired Comput. 3(2), 77–84 (2011)

    Article  Google Scholar 

  32. Yang, X.S.: Multiobjective firefly algorithm for continuous optimization. Eng. Comput. 29(2), 175–184 (2013)

    Article  Google Scholar 

  33. Yang, X.S., Deb, S.: Eagle strategy using Lévy walk and firefly algorithms for stochastic optimization. In: Nature Inspired Cooperative Strategies for Optimization (NICSO 2010), pp. 101–111. Springer, Berlin (2010)

  34. Prawin, J., Rao, A.R.M., Lakshmi, K.: Nonlinear parametric identification strategy combining reverse path and hybrid dynamic quantum particle swarm optimization. Nonlinear Dyn. 84(2), 797–815 (2016)

    Article  MathSciNet  Google Scholar 

  35. Kelley, C.T.: Detection and remediation of stagnation in the Nelder–Mead algorithm using a sufficient decrease condition. SIAM J. Optim. 10(1), 43–55 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  36. Rao, A.R.M., Lakshmi, K.: Optimal design of stiffened laminate composite cylinder using a hybrid SFL algorithm. J. Compos. Mater. 46(24), 3031–3055 (2012)

    Article  Google Scholar 

  37. Rao, A.R.M., Lakshmi, K., Ganesan, K.: Structural system identification using quantum-behaved particle swarm optimisation algorithm. Struct. Durab. Health Monit. 9(2), 99–128 (2013)

    Article  Google Scholar 

  38. Rao, A.R.M.: Parallel mesh-partitioning algorithms for generating shape optimised partitions using evolutionary computing. Adv. Eng. Softw. 40(2), 141–157 (2009)

    Article  MATH  Google Scholar 

  39. Rao, A.R.M., Rao, T.V.S.R.A., Dattaguru, B.: Generating optimised partitions for parallel finite element computations employing float-encoded genetic algorithms. Comput. Model. Eng. Sci. 5(3), 213–234 (2004)

    MathSciNet  MATH  Google Scholar 

Download references

Funding

The author(s) received no financial support for the research, authorship and/or publication of this article.

Author information

Authors and Affiliations

  1. CSIR-Structural Engineering Research Centre, CSIR Campus, Chennai, TN, India

    J. Prawin & A. Rama Mohan Rao

  2. BITS Pilani, Pilani, India

    Abhinav Sethi

Authors
  1. J. Prawin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Rama Mohan Rao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Abhinav Sethi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. Prawin.

Ethics declarations

Conflict of interest

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

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

Prawin, J., Rao, A.R.M. & Sethi, A. Parameter identification of systems with multiple disproportional local nonlinearities. Nonlinear Dyn 100, 289–314 (2020). https://doi.org/10.1007/s11071-020-05538-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-020-05538-1

Keywords

Search

Navigation