Skip to main content
SpringerLink
Log in

The harmonic probing method for output-only nonlinear mechanical systems

  • Technical Paper
  • Published:
Journal of the Brazilian Society of Mechanical Sciences and Engineering Aims and scope Submit manuscript

Abstract

Most engineering applications involving vibrating structures are nonlinear in nature and many techniques have been recently investigated to provide a better understanding of such problems. Among the large variety of methods, the harmonic probing has presented useful properties for identification and dynamic analysis of nonlinear systems. The method is conventionally described by the multi-dimensional Fourier transform of the Volterra kernels and it depends on the knowledge of the equations of motion and the respective input and output data. However, this white-box methodology is limited to applications where the input signal is either unknown or even unmeasured. Thus, the present paper is concerned with the development of an extended version of the harmonic probing method to deal with applications where only the outputs are available. The algebraic expressions of the extended Volterra kernels transform and their theoretical properties are provided. The main advantages, novelties and drawbacks of this new approach are discussed and compared with the conventional approach. It is verified that the new kernels can be expressed as a combination of the conventional ones. Numerical tests based on a classical 2DOF Duffing oscillator are carried out and the results reveal the effectiveness and potential of the extended harmonic probing method based on a nonparametric model using new kernels to describe a prediction of vibrating systems in nonlinear regime of motion.

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

Similar content being viewed by others

References

  1. Abed-Merain K (1997) Blind system identification. Proc IEEE 85(8):1310–1322. doi:10.1109/5.622507

    Google Scholar 

  2. Billings S (2013) Nonlinear system identification: NARMAX methods in the time, frequency, and spatio-temporal domain. Wiley, Chichester, UK

    Book  MATH  Google Scholar 

  3. Billings S, Tsang K (1989) Spectral analysis for non-linear systems, part I: parametric non-linear spectral analysis. Mech Syst Signal Process 3(4):319–339. doi:10.1016/0888-3270(89)90041-1

    Article  MATH  Google Scholar 

  4. Billings S, Tsang K (1989) Spectral analysis for non-linear systems, part II: interpretation of non-linear frequency response functions. Mech Syst Signal Process 3(4):341–359. doi:10.1016/0888-3270(89)90042-3

    Article  MATH  Google Scholar 

  5. Cafferty S, Tomlinson G (1997) Characterization of automotive dampers using higher order frequency response function. J Automob Eng 211(3):181–203. http://pid.sagepub.com/content/211/3/181.refs

  6. Carassale L, Kareem A (2010) Modeling nonlinear systems by Volterra series. J Eng Mech 136(6):801818. doi:10.1061/(ASCE)EM.1943-7889.0000113

    Article  Google Scholar 

  7. Chatterjee A (2010) Structural damage assessment in a cantilever beam with a breathing crack using higher order frequency response functions. J Sound Vib 329(16):3325–3334. doi:10.1016/j.jsv.2010.02.026

    Article  Google Scholar 

  8. Chatterjee A, Vyas N (2000) Convergence analysis of Volterra series response of nonlinear systems subjected to harmonic excitation. J Sound Vib 236(2):339–358. doi:10.1006/jsvi.2000.2967

    Article  MathSciNet  MATH  Google Scholar 

  9. Chatterjee A, Vyas NS (2003) Nonlinear parameter estimation in rotor-bearing system using Volterra series and method of harmonic probing. J Vib Acoust 125(3):299–306. doi:10.1115/1.1547486

    Article  Google Scholar 

  10. Chatterjee A, Vyas NS (2004) Non-linear parameter estimation in multi-degree-of-freedom systems using multi-input Volterra series. Mech Syst Signal Process 18(3):457–489. doi:10.1016/S0888-3270(03)00016-5

    Article  Google Scholar 

  11. Cheng C, Peng Z, Zhang W, Meng G (2017) Volterra-series-based nonlinear system modeling and its engineering applications: a state-of-the-art review. Mech Syst Signal Process 87:340–364. doi:10.1016/j.ymssp.2016.10.029

    Article  Google Scholar 

  12. Cherif I, Abid S, Fnaiech F (2012) Nonlinear blind identification with three-dimensional tensor analysis. Math Probl Eng 2012:22, Article ID 284,815. doi:10.1155/2012/284815

  13. Chesné S, Deraemaeker A (2013) Damage localization using transmissibility functions: a critical review. Mech Syst Signal Process 38(2):569–584. doi:10.1016/j.ymssp.2013.01.020

    Article  Google Scholar 

  14. Grosel J, Sawicki W, Pakos W (2014) Application of classical and operational modal analysis for examination of engineering structures. Proc Eng 91:136–141. doi:10.1016/j.proeng.2014.12.035. http://www.sciencedirect.com/science/article/pii/S1877705814030537

  15. Kalouptsidis N, Koukoulas P (2003) Blind identification of Volterra-Hammerstein systems. In: Statistical signal processing, 2003 IEEE Workshop, pp 202–205. doi:10.1109/SSP.2003.1289379

  16. Kerschen G, Worden K, Vakakis AF, Golinval J (2006) Past, present and future of nonlinear system identification in structural dynamics. Mech Syst Signal Process 20(3):505–592. doi:10.1016/j.ymssp.2005.04.008

    Article  Google Scholar 

  17. Lang Z, Park G, Farrar C, Todd M, Mao Z, Zhao L, Worden K (2011) Transmissibility of non-linear output frequency response functions with application in detection and location of damage in MDOF structural systems. Int J Non Linear Mech 46(6):841–853. doi:10.1016/j.ijnonlinmec.2011.03.009. http://www.sciencedirect.com/science/article/pii/S0020746211000333

  18. Lee G (1997) Estimation of non-linear system parameters using higher-order frequency response functions. Mech Syst Signal Process 11(2):219–228. doi:10.1006/mssp.1996.0080

    Article  Google Scholar 

  19. Li D, Ren WX, Hu YD, Yang D (2016) Operational modal analysis of structures by stochastic subspace identification with a delay index. Struct Eng Mech 59(1):187–207. doi:10.12989/sem.2016.59.1.187

    Article  Google Scholar 

  20. Maia NMM, Silva JMM, Ribeiro AMR (2001) The transmissibility concept in multi-degree-of-freedom systems. Mech Syst Signal Process 15(1):129–137. doi:10.1006/mssp.2000.1356

    Article  Google Scholar 

  21. Peng Z, Lang Z (2007) On the convergence of the Volterra-series representation of the Duffing’s oscillators subjected to harmonic excitations. J Sound Vib 305(12):322–332. doi:10.1016/j.jsv.2007.03.062

    Article  MathSciNet  MATH  Google Scholar 

  22. Rugh WJ (1981) Nonlinear system theory–the Volterra/Wiener approach. University Press, The Johns Hopkings, Baltimore

    MATH  Google Scholar 

  23. Schetzen M (1980) The Volterra and Wiener theories of nonlinear systems. Wiley, New York

    MATH  Google Scholar 

  24. Scussel O, da Silva S (2016) Output-only identification of nonlinear systems via Volterra series. ASME J Vib Acoust 138(4):041,012 (13 pages). doi:10.1115/1.4033458

  25. Shiki SB, Lopes V Jr, da Silva S (2014) Identification of nonlinear structures using discrete-time Volterra series. J Braz Soc Mech Sci Eng 36(3):523–532. doi:10.1007/s40430-013-0088-9

    Google Scholar 

  26. da Silva S (2011) Non-linear model updating of a three-dimensional portal frame based on Wiener series. Int J Non Linear Mech 46(1):312–320. doi:10.1016/j.ijnonlinmec.2010.09.014

    Article  Google Scholar 

  27. Storer D, Tomlinson G (1993) Recent developments in the measurement and interpretation of higher order transfer functions from non-linear structures. Mech Syst Signal Process 7(2):173–189. doi:10.1006/mssp.1993.1006

    Article  Google Scholar 

  28. Tan HZ, Huang Y, Fu J (2008) Blind identification of sparse Volterra systems. Int J Adapt Control Signal Process 22(7):652–662. doi:10.1002/acs.1011

    Article  MathSciNet  MATH  Google Scholar 

  29. Tawfiq I, Vinh T (2003) Contribution to the extension of modal analysis to non-linear structure using Volterra funcitonal series. Mech Syst Signal Process 17(2):379–407. doi:10.1006/mssp.2002.1499

    Article  Google Scholar 

  30. Thouverez F (1998) A new convergence criteria of Volterra series for harmonic inputs. In: IMAC XVI—16th international modal analysis conference, 2–5 February, Santa Barbara, vol 758, pp 723 – 727

  31. Tomlinson G, Manson G, Lee G (1996) A simple criterion for establishing an upper limit to the harmonic excitation level of the Duffing oscillator using the Volterra series. J Sound Vib 190(5):751–762. doi:10.1006/jsvi.1996.0091. http://www.sciencedirect.com/science/article/pii/S0022460X96900917

  32. Weijtjens W, Sitter GD, Devriendt C, Guillaume P (2014) Operational modal parameter estimation of MIMO systems using transmissibility functions. Automatica 50(2):559–564. doi:10.1016/j.automatica.2013.11.021

    Article  MathSciNet  MATH  Google Scholar 

  33. Worden K, Tomlinson GR (2001) Nonlinearity in structural dynamics. Institute of Physics, London

    Book  MATH  Google Scholar 

  34. Worden K, Manson G, Tomlinson G (1997) A harmonic probing algorithm for the multi-input Volterra series. J Sound Vib 201(1):67–84. doi:10.1006/jsvi.1996.0746

    Article  MathSciNet  MATH  Google Scholar 

  35. Xia X, Zhou J, Xiao J, Xiao H (2016) A novel identification method of Volterra series in rotor-bearing system for fault diagnosis. Mech Syst Signal Process 66–67:557–567. doi:10.1016/j.ymssp.2015.05.006

    Article  Google Scholar 

  36. Yang Y, Nagarajaiah S (2013) Output-only modal identification with limited sensors using sparse component analysis. J Sound Vib 332(19):4741–4765. doi:10.1016/j.jsv.2013.04.004

    Article  Google Scholar 

  37. Zhao XY, Lang ZQ, Park G, Farrar CR, Todd MD, Mao Z, Worden K (2015) A new transmissibility analysis method for detection and location of damage via nonlinear features in mdof structural systems. IEEE/ASME Trans Mechatron 20(4):1933–1947. doi:10.1109/TMECH.2014.2359419

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to acknowledge the financial support provided by the São Paulo Research Foundation (FAPESP) by Grant Number 12/09135-3 and National Council for Scientific and Technological Development (CNPq) under Grants Numbers 47058/2012-0 and 203610/2014-8. The first author acknowledges his scholarship from the Coordination for the Improvement of Higher Education Personnel (CAPES). Additionally, the authors would like to thank the anonymous reviewers and professor Michael J. Brennan for their helpful review, relevant comments and useful suggestions.

Author information

Authors and Affiliations

  1. Departamento de Engenharia Mecânica, Faculdade de Engenharia de Ilha Solteira, Unesp-Universidade Estadual Paulista, Av. Brasil 56, Ilha Solteira, SP, 15385-000, Brazil

    Oscar Scussel & Samuel da Silva

Authors
  1. Oscar Scussel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Samuel da Silva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Oscar Scussel.

Additional information

Technical Editor: Kátia Lucchesi Cavalca Dedini.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Scussel, O., da Silva, S. The harmonic probing method for output-only nonlinear mechanical systems. J Braz. Soc. Mech. Sci. Eng. 39, 3329–3341 (2017). https://doi.org/10.1007/s40430-017-0723-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40430-017-0723-y

Keywords

Search

Navigation