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Quantification of nonlinear output frequency responses for a general input based on volterra series and conditioned spectral analysis

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Abstract

This paper focuses on the issue of quantifying nonlinear output frequency response components of different orders of nonlinear dynamical systems for a general excitation. An alternative approach is presented based on Volterra series and conditioned spectral analysis (CSA) theories. Firstly, a multiple-input/single-output (MISO) linear system with a series of power characterized inputs is obtained by decomposing the nonlinear system under analyzed based on Volterra series. Secondly, the correlations among the inputs of different orders are removed by utilizing CSA approach, obtaining an algorithm of identifying the nonlinear output frequency response functions (NOFRFs) and evaluating the contributions of different order nonlinearities to the output of the system. Two kinds of nonlinear systems were simulated numerically to verify the accuracy of the method. The results reached by the proposed method are very close to the numerical results obtained by the fourth order Runge–Kutta method. Finally, an experiment analysis was carried out, in which the vibration transmission properties of a bolt connection were tested when the bolt was fastened and loose respectively. The results of experiments reflected further the effectiveness of the method on distinguishing quantitatively the contributions of each order nonlinearities to the output of a nonlinear system.

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References

  1. Liu, L.F., Xiang, H.Y., Li, X.J.: A novel perturbation method to reduce the dynamical degradation of digital chaotic maps. Nonlinear Dyn. 103(1), 1099–1115 (2021)

    Article  Google Scholar 

  2. Roy, T., Maiti, D.K.: An optimal and modified homotopy perturbation method for strongly nonlinear differential equations. Nonlinear Dyn. 111(16), 15215–15231 (2023)

    Article  Google Scholar 

  3. Rayguru, M.M., Kar, I.N.: A singular perturbation approach to saturated controller design with application to bounded stabilization of wing rock phenomenon. Nonlinear Dyn. 93(4), 2263–2272 (2018)

    Article  Google Scholar 

  4. Shou, D.H.: The homotopy perturbation method for nonlinear oscillators. Comput. Math. Appl. 58, 2456–2459 (2009)

    Article  MathSciNet  Google Scholar 

  5. Hu, H.Y., Wang, Z.H.: Singular perturbation methods for nonlinear dynamic systems with time delays. Chaos Solitons Fractals 40, 13–27 (2009)

    Article  MathSciNet  Google Scholar 

  6. Rafei, M., Van Horssen, W.T.: Solving systems of nonlinear difference equations by the multiple scales perturbation method. Nonlinear Dyn. 69(4), 1509–1516 (2012)

    Article  MathSciNet  Google Scholar 

  7. Razzak, M.A., Alam, M.Z., Sharif, M.N.: Modified multiple time scale method for solving strongly nonlinear damped forced vibration systems. Results Phys. 8, 231–238 (2018)

    Article  Google Scholar 

  8. Jain, S., Tiso, P.: Model order reduction for temperature-dependent nonlinear mechanical systems: a multiple scales approach. J. Sound Vib. 465, 115022 (2020)

    Article  Google Scholar 

  9. Ihsan, A.F., van Horssen, W.T., Tuwankotta, J.M.: On a multiple timescales perturbation approach for a stefan problem with a time-dependent heat flux at the boundary. Nonlinear Dyn. 110(3), 2673–2683 (2022)

    Article  Google Scholar 

  10. Ali Akbar, M., Shamsul, A.M., Sattar, M.A.: KBM unified method for solving an nth order non-linear differential equation under some special conditions including the case of internal resonance. Int. J. Nonlin. Mech. 41(1), 26–42 (2006)

    Article  Google Scholar 

  11. Cai, J.P., Wu, X.F., Li, Y.P.: Comparison of multiple scales and KBM methods for strongly nonlinear oscillators with slowly varying parameters. Mech. Res. Commun. 31(5), 519–524 (2004)

    Article  MathSciNet  Google Scholar 

  12. Peyton-Jones, J.C., Yaser, K.A.: Recent advances and comparisons between harmonic balance and Volterra-based nonlinear frequency response analysis methods. Nonlinear Dyn 91(1), 131–145 (2018)

    Article  MathSciNet  Google Scholar 

  13. Radecki, R., Leamy, M.J., Packo, P., Klepka, A.: Prediction of higher-order harmonics generation due to contact stiffness hysteresis using Harmonic Balance: theory and experimental validation. Nonlinear Dyn. 103(1), 541–556 (2021)

    Article  Google Scholar 

  14. Shaw, S.W., Rosenberg, S., Shoshani, O.: A hybrid averaging and harmonic balance method for weakly nonlinear asymmetric resonators. Nonlinear Dyn. 11(5), 3969–3979 (2023)

    Article  Google Scholar 

  15. Wang, S., Zhang, Y.O., Guo, W.Y., Pi, T., Li, X.F.: Vibration analysis of nonlinear damping systems by the discrete incremental harmonic balance method. Nonlinear Dyn. 111(3), 2009–2028 (2023)

    Article  Google Scholar 

  16. Volterra, V.: Theory of functionals and of integral and integro-differential equations. Dover Publications, New York (2005)

    Google Scholar 

  17. Helie, T., Laroche, B.: Input/output reduced model of a damped nonlinear beam based on Volterra series and modal decomposition with convergence results. Nonlinear Dyn. 105(1), 515–540 (2021)

    Article  Google Scholar 

  18. Annabestani, M., Naghavi, N.: Practical realization of discrete-time Volterra series for high-order nonlinearities. Nonlinear Dyn. 98(3), 2309–2325 (2019)

    Article  Google Scholar 

  19. de Paula, N.C.G., Marques, F.D.: Multi-variable Volterra kernels identification using time-delay neural networks: application to unsteady aerodynamic loading. Nonlinear Dyn. 97(1), 767–780 (2019)

    Article  Google Scholar 

  20. Liu, Q.Q., He, Y.G.: A family of quaternion-valued pipelined second-order Volterra adaptive filters for nonlinear system identification. Nonlinear Dynam. 108(4), 3951–3967 (2022)

    Article  MathSciNet  Google Scholar 

  21. Rahrooh, A., Shepard, S.: Identification of nonlinear systems using NARMAX model. Nonlinear Anal. 71, e1198–e1202 (2009)

    Article  Google Scholar 

  22. Wen, G.X., Liu, Y.J.: Adaptive fuzzy-neural tracking control for uncertain nonlinear discrete-time systems in the NARMAX form. Nonlinear Dyn. 66(4), 745–753 (2011)

    Article  MathSciNet  Google Scholar 

  23. Huang, H.L., Mao, H.Y., Mao, H.L., Zheng, W.X., Huang, Z.F., Li, X.X., Wang, X.H.: Study of cumulative fatigue damage detection for used parts with nonlinear output frequency response functions based on NARMAX modelling. J. Sound Vib. 411, 75–87 (2017)

    Article  Google Scholar 

  24. Richter, H., Stein, G.: On Taylor series expansion for chaotic nonlinear systems. Chaos Solitons Fractals 13(9), 1783–1789 (2002)

    Article  Google Scholar 

  25. Guillot, L., Cochelin, B., Vergez, C.: A Taylor series-based continuation method for solutions of dynamical systems. Nonlinear Dyn. 98(4), 2827–2845 (2019)

    Article  Google Scholar 

  26. Ni, Z., Fan, Y.C., Hang, Z.Y., Zhu, F., Wang, Y., Feng, C., Yang, J.: Damped vibration analysis of graphene nanoplatelet reinforced dielectric membrane using Taylor series expansion and differential quadrature methods. Thin-Walled Struct. 184, 110493 (2023)

    Article  Google Scholar 

  27. Xiong, W.L., Ma, J.X., Ding, R.F.: An iterative numerical algorithm for modeling a class of Wiener nonlinear systems. Appl. Math. Lett. 26, 487–493 (2013)

    Article  MathSciNet  Google Scholar 

  28. Kazemi, M., Arefi, M.M.: A fast iterative recursive least squares algorithm for Wiener model identification of highly nonlinear systems. ISA Trans. 67, 382–388 (2017)

    Article  Google Scholar 

  29. Sersour, L., Djamah, T., Bettayeb, M.: Nonlinear system identification of fractional Wiener models. Nonlinear Dyn. 92(4), 1493–1505 (2018)

    Article  Google Scholar 

  30. Ozer, S., Zorlu, H.: Mete, S: system identification application using Hammerstein model. Sadhana-Acad. Proc. Eng. 41(6), 597–605 (2016)

    Article  Google Scholar 

  31. Filipovic, V.Z.: Outlier robust stochastic approximation algorithm for identification of MIMO Hammerstein models. Nonlinear Dyn. 90(2), 1427–1441 (2017)

    Article  MathSciNet  Google Scholar 

  32. Cheng, C.M., Peng, Z.K., Zhang, W.M., Meng, G.: A novel approach for identification of cascade of Hammerstein model. Nonlinear Dyn. 86(1), 513–522 (2016)

    Article  Google Scholar 

  33. Haryanto, A., Hong, K.S.: Maximum likelihood identification of Wiener-Hammerstein models. Mech. Syst. Signal Pr. 41(1–2), 54–70 (2013)

    Article  Google Scholar 

  34. Lawrynczuk, M.: Nonlinear predictive control of dynamic systems represented by Wiener-Hammerstein models. Nonlinear Dyn. 86(2), 1193–1214 (2016)

    Article  MathSciNet  Google Scholar 

  35. Hammar, K., Djamah, T., Bettayeb, M.: Nonlinear system identification using fractional Hammerstein-Wiener models. Nonlinear Dyn. 98(3), 2327–2338 (2019)

    Article  Google Scholar 

  36. Cheng, C.M., Peng, Z.K., Zhang, W.M., Meng, G.: Volterra-series-based nonlinear system modeling and its engineering applications: a state-of-the-art review. Mech. Syst. Signal Pr. 87, 340–364 (2017)

    Article  Google Scholar 

  37. Zhang, B., Billings, S.A.: Volterra series truncation and kernel estimation of nonlinear systems in the frequency domain. Mech. Syst. Signal Pr. 84, 39–57 (2017)

    Article  Google Scholar 

  38. Vitaliy, P., Aleksandr, F., Yuriy, G.: Identification accuracy of nonlinear system based on Volterra model in frequency domain. AASRI Procedia 4, 297–305 (2013)

    Article  Google Scholar 

  39. Prawin, J., Rama Mohan Rao, A.: Nonlinear identification of MDOF systems using Volterra series approximation. Mech. Syst. Signal Pr. 84, 58–77 (2017)

    Article  Google Scholar 

  40. Hong, J.Y., Kim, Y.C., Powers, E.J.: On modeling the nonlinear relationship between fluctuations with Nonlinear transfer functions. P. IEEE 68(8), 1026–1027 (1980)

    Article  Google Scholar 

  41. Liu, W.T., Zhang, Y., Feng, Z.J., Zhao, J.S., Wang, D.F.: A study on waviness induced vibration of ball bearings based on signal coherence theory. J. Sound Vib. 333(23), 6107–6120 (2014)

    Article  Google Scholar 

  42. Cho, Y.S., Kim, S.B., Powers, E.J.: A digital technique to estimate second-order distortion using higher order coherence spectra. IEEE Trans. Signal Process. 40(5), 1029–1040 (1992)

    Article  Google Scholar 

  43. An, C.K., Powers, E.J., Ritz, C.P.: A digital method of modeling two-input quadratic systems with general random inputs. IEEE Trans. Signal Process. 39(10), 2320–2323 (1991)

    Article  Google Scholar 

  44. Scussel, O., da Silva, S.: Output-only identification of nonlinear systems via Volterra series. J. Vib. Acoust. 138(4), 041012 (2016)

    Article  Google Scholar 

  45. Lin, R.M., Ng, T.Y.: Higher-order FRFs and their applications to the identifications of continuous structural systems with discrete localized nonlinearities. Mech. Syst. Signal Pr. 108, 326–346 (2018)

    Article  Google Scholar 

  46. Marzocca, P., Nichols, J.M., Milanese, A., Seaver, M., Trickey, S.T.: Second-order spectra for quadratic nonlinear systems by Volterra functional series: analytical description and numerical simulation. Mech. Syst. Signal Pr. 22, 1882–1895 (2008)

    Article  Google Scholar 

  47. Scussel, O., da Silva, S.: The harmonic probing method for output only nonlinear mechanical systems. J. Braz. Soc. Mech. Sci. 39(9), 3329–3341 (2017)

    Article  Google Scholar 

  48. Chatterjee, A.: Identification and parameter estimation of a bilinear oscillator using Volterra series with harmonic probing. Int. J. Nonlin. Mech. 45, 12–20 (2010)

    Article  Google Scholar 

  49. Chatterjee, A.: Structural damage assessment in a cantilever beam with a breathing crack using higher order frequency response functions. J. Sound Vib. 329, 3325–3334 (2010)

    Article  Google Scholar 

  50. Peng, Z.K., Lang, Z.Q.: On the convergence of the Volterra-series representation of the duffing’s oscillators subjected to harmonic excitations. J. Sound Vib. 305, 322–332 (2007)

    Article  MathSciNet  Google Scholar 

  51. Müller, F., Woiwode, L., Gross, J., Scheel, M., Krack, M.: Nonlinear damping quantification from phase-resonant tests under base excitation. Mech. Syst. Signal Pr. 177, 109170 (2022)

    Article  Google Scholar 

  52. Scheel, M., Peter, S., Leine, R.I., Krack, M.: A phase resonance approach for modal testing of structures with nonlinear dissipation. J. Sound Vib. 435, 56–73 (2018)

    Article  Google Scholar 

  53. Lacayo, R.M., Deaner, B.J., Allen, M.S.: A numerical study on the limitations of modal Iwan models for impulsive excitations. J. Sound Vib. 390, 118–140 (2017)

    Article  Google Scholar 

  54. Kwarta, M., Allen, M.S.: Nonlinear normal mode backbone estimation with near-resonant steady state inputs. Mech. Syst. Signal Pr. 162, 108046 (2022)

    Article  Google Scholar 

  55. Sun, Y.K., Vizzaccaro, A., Yuan, J., Salles, L.: An extended energy balance method for resonance prediction in forced response of systems with non-conservative nonlinearities using damped nonlinear normal mode. Nonlinear Dynam. 103(4), 3315–3333 (2021)

    Article  Google Scholar 

  56. Sadeqi, A., Moradi, S., Shirazi, K.H.: Nonlinear subspace system identification based on output-only measurements. J. Frankl. Inst.-Eng. Appl. Math. 357(17), 12904–12937 (2020)

    Article  MathSciNet  Google Scholar 

  57. Sadeqi, A., Moradi, S., Shirazi, K.H.: System identification based on output-only decomposition and subspace appropriation. J. Dyn. Sys., Meas., Control 141(9), 1091012 (2019)

    Article  Google Scholar 

  58. Sadeqi, A., Moradi, S.: Nonlinear system identification based on restoring force transmissibility of vibrating structures. Mech. Syst. Signal Pr. 172, 108978 (2022)

    Article  Google Scholar 

  59. Karaağaçlı, T., Özgüven Nevzat, H.: Experimental modal analysis of nonlinear systems by using response-controlled stepped-sine testing. Mech. Syst. Signal Pr. 146, 107023 (2021)

    Article  Google Scholar 

  60. Karaağaçlı, T., Özgüven Nevzat, H.: Experimental identification of backbone curves of strongly nonlinear systems by using response-controlled stepped-Sine testing (RCT). Vibration 3(3), 266–280 (2020)

    Article  Google Scholar 

  61. Lang, Z.Q., Billings, S.A.: Energy transfer properties of non-linear systems in the frequency domain. Int. J. Control. 78(5), 345–362 (2005)

    Article  MathSciNet  Google Scholar 

  62. Peng, Z.K., Lang, Z.Q., Billings, S.A., Lu, Y.: Analysis of bilinear oscillators under harmonic loading using nonlinear output frequency response functions. Int. J. Mech. Sci. 49(11), 1213–1225 (2007)

    Article  Google Scholar 

  63. Peng, Z.K., Lang, Z.Q., Billings, S.A., Tomlinson, G.R.: Comparisons between harmonic balance and nonlinear output frequency response function in nonlinear system analysis. J. Sound Vib. 311, 56–73 (2008)

    Article  Google Scholar 

  64. Cheng, C.M., Peng, Z.K., Dong, X.J., Zhang, W.M., Meng, G.: Locating non-linear components in two dimensional periodic structures based on NOFRFs. Int. J. Nonlin. Mech. 67, 198–208 (2014)

    Article  Google Scholar 

  65. Lang, Z.Q., Park, G., Farrar, C.R., Todd, M.D., Mao, Z., Zhao, L., Worden, K.: Transmissibility of non-linear output frequency response functions with application in detection and location of damage in MDOF structural systems. Int. J. Nonlin. Mech. 46(6), 841–853 (2011)

    Article  Google Scholar 

  66. Bayma, S.R., Zhu, Y.P., Lang, Z.Q.: The analysis of nonlinear systems in the frequency domain using nonlinear output frequency response functions. Automatica 94, 452–457 (2018)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was supported by Ministry of Industry and Information Technology Manufacturing High Quality Development Project (Grant No. TC200H02J), National Natural Science Foundation of China (Grant No. 51605190) and Shandong Innovation Capability Improvement Project of Scientific and Technological Small and Medium-sized Enterprises (Grant No. 2021TSGC1366).

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Authors and Affiliations

  1. School of Mechanical Engineering, University of Jinan, Jinan, 250022, China

    Wentao Liu, Shengshuan Jiao, Jiafeng Wu & Yu’e Yang

  2. Beijing Key Lab of Precision/Ultra-Precision Manufacturing Equipment and Control, Tsinghua University, Beijing, 100084, China

    Wentao Liu & Yun Zhang

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  1. Wentao Liu
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Liu, W., Zhang, Y., Jiao, S. et al. Quantification of nonlinear output frequency responses for a general input based on volterra series and conditioned spectral analysis. Nonlinear Dyn (2024). https://doi.org/10.1007/s11071-024-09602-y

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