Skip to main content
SpringerLink
Log in

Vibration analysis of nonlinear damping systems by the discrete incremental harmonic balance method

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

Abstract

An improved incremental harmonic balance method (IHBM) is proposed by Wang (J Sound Vib 441:111–125, 2019) to solve the periodic responses of the continuous nonlinear stiffness systems. However, the nonlinear damping systems remain unsolved. This paper aims to investigate the nonlinear damping parts by the proposed IHBM method, which is based on the principle that any continuous curve can be approximated by a piecewise-linear curve with discrete nodes. The piecewise-linear function can be considered a unified benchmark function that can convert the complex IHBM Galerkin process of arbitrary nonlinear damping systems to that of unified piecewise-linear damping systems. The general process of the proposed method for this piecewise-linear system is derived considering the stability of the solutions. Then, a polynomial nonlinear damping system is investigated to validate the accuracy of the method. Furthermore, five typical cases of single-degree-of-freedom (SDOF) nonlinear damping systems are carried out, and this method is also extended to multi-degree-of-freedom (MDOF) systems where each nonlinear force in the systems is expressed by the function of only one independent DOF. The results illustrate that the proposed method shows convenience and accuracy in obtaining the dynamics of nonlinear 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
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17

Similar content being viewed by others

Data availability

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

References

  1. Karlii, D., Caji, M., Paunovi, S., et al.: Nonlinear energy harvester with coupled Duffing oscillators. Commun. Nonlinear Sci. 91, 105394 (2020)

    MathSciNet  Google Scholar 

  2. Ranjbarzadeh, H., Kakavand, F.: Determination of nonlinear vibration of 2DOF system with an asymmetric piecewise-linear compression spring using incremental harmonic balance method. Eur. J. Mech. A Solids. 73, 161–168 (2019)

    MathSciNet  MATH  Google Scholar 

  3. Kong, X., Sun, W., Wang, B., et al.: Dynamic and stability analysis of the linear guide with time-varying, piecewise-nonlinear stiffness by multi-term incremental harmonic balance method. J. Sound Vib. 346, 265–283 (2015)

    Google Scholar 

  4. Sun, X., Jing, X.: A nonlinear vibration isolator achieving high-static-low-dynamic stiffness and tunable anti-resonance frequency band. Mech. Syst. Signal Process. 80, 166–188 (2016)

    Google Scholar 

  5. Gao, X., Teng, H.-D.: Dynamics and nonlinear effects of a compact near-zero frequency vibration isolator with HSLD stiffness and fluid damping enhancement. Int. J. Nonlin Mech. 128, 103632 (2021)

    Google Scholar 

  6. Wang, S., Hua, L., Yang, C., et al.: Nonlinear vibrations of a piecewise-linear quarter-car truck model by incremental harmonic balance method. Nonlinear Dyn. 92, 1719–1732 (2018)

    Google Scholar 

  7. Ghandchi, T.-M., Elliott, S.-J.: Extending the dynamic range of an energy harvester using nonlinear damping. J. Sound Vib. 333, 623–629 (2014)

    Google Scholar 

  8. Xiao, Z., Zhao, J., et al.: The transmissibility of vibration isolators with cubic nonlinear damping under both force and base excitations. J. Sound Vib. 332, 1335–1354 (2013)

    Google Scholar 

  9. Habib, G., Cirillo, G.-I., Kerschen, G.: Isolated resonances and nonlinear damping. Nonlinear Dyn. 93, 979–994 (2018)

    MATH  Google Scholar 

  10. Huang, X., Sun, J., Hua, H., et al.: The isolation performance of vibration systems with general velocity-displacement-dependent nonlinear damping under base excitation: numerical and experimental study. Nonlinear Dyn. 85, 777–796 (2016)

    MathSciNet  Google Scholar 

  11. Pierre, C., Ferri, A.-A., Dowell, E.-H.: Multi-harmonic analysis of dry friction damped systems using an incremental harmonic balance method. J. Appl. Mech. 52, 958–964 (1985)

    MATH  Google Scholar 

  12. Niknam, A., Farhang, K.: Friction-induced vibration in a two-mass damped system. J. Sound Vib. 456, 454–475 (2019)

    Google Scholar 

  13. Pesek, L., Pust, L., Snabl, P., Bula, V., Hajzman, M., Byrtus, M.: Dry-friction damping in vibrating systems, theory and application to the bladed disc assembly. In: Jauregui, J. (ed.) Nonlinear Structural Dynamics and Damping. Mechanisms and Machine Science, vol. 69. Springer, Cham (2019)

    MATH  Google Scholar 

  14. Hui, Y., Law, S.-S., Zhu, W., et al.: Extended IHB method for dynamic analysis of structures with geometrical and material nonlinearities. Eng. Struct. 205, 1–14 (2020)

    Google Scholar 

  15. Vaiana, N., Sessa, S., Marmo, F., et al.: A class of uniaxial phenomenological models for simulating hysteretic phenomena in rate-independent mechanical systems and materials. Nonlinear Dyn. 93, 1647–1669 (2018)

    Google Scholar 

  16. Vaiana, N., Sessa, S., Rosati, L.: A generalized class of uniaxial rate-independent models for simulating asymmetric mechanical hysteresis phenomena. Mech. Syst. Signal Process. 146, 106984 (2021)

    Google Scholar 

  17. Caughey, T.-K.: Sinusoidal excitation of a system with bilinear hysteresis. J. Appl. Mech. 27, 640–643 (1960)

    MathSciNet  Google Scholar 

  18. Kalmárnagy, T., Shekhawat, A.: Nonlinear dynamics of oscillators with bilinear hysteresis and sinusoidal excitation. Phys. D 238, 1768–1786 (2009)

    MathSciNet  MATH  Google Scholar 

  19. Balasubramanian, P., Franchini, G., Ferrari, G., et al.: Nonlinear vibrations of beams with bilinear hysteresis at supports: interpretation of experimental results. J. Sound Vib. 499, 115998 (2021)

    Google Scholar 

  20. Wong, C.-W., Ni, Y.-Q., Lau, S.-L.: Steady-state oscillation of hysteretic differential model. I. J. Eng. Mech. 120, 2271–2298 (1994)

    Google Scholar 

  21. Wong, C.-W., Ni, Y.-Q., Ko, J.-M.: Steady-state oscillation of hysteretic differential model. II. J. Eng. Mech. 120, 2299–2325 (1994)

    Google Scholar 

  22. Jalali, H.: An alternative linearization approach applicable to hysteretic systems. Commun. Nonlinear Sci. 19, 245–257 (2014)

    MathSciNet  MATH  Google Scholar 

  23. Civera, M., Grivet-talocia, S., Surace, C., et al.: A generalised power-law formulation for the modelling of damping and stiffness nonlinearities. Mech. Syst. Signal Process. 153, 107531 (2021)

    Google Scholar 

  24. Miguel, L.-P.: Some practical regards on the application of the harmonic balance method for hysteresis models. Mech. Syst. Signal Process. 143, 106842 (2020)

    Google Scholar 

  25. Zucca, S., Firrone, C.-M.: Nonlinear dynamics of mechanical systems with friction contacts: coupled static and dynamic Multi-Harmonic Balance Method and multiple solutions. J. Sound Vib. 333, 916–926 (2014)

    Google Scholar 

  26. Cheung, Y.-K., Chen, S.-H., Lau, S.-L.: Application of the incremental harmonic balance method to cubic non-linearity systems. J. Sound Vib. 140, 273–286 (1990)

    Google Scholar 

  27. Zhang, Z., Tian, X., Ge, X.: Dynamic characteristics of the Bouc-Wen nonlinear isolation system. Appl. Sci. 11, 6106 (2021)

    Google Scholar 

  28. Fang, L., Wang, J., Tan, X.: An incremental harmonic balance-based approach for harmonic analysis of closed-loop systems with Prandtl-Ishlinskii operator. Automatica 88, 48–56 (2018)

    MathSciNet  MATH  Google Scholar 

  29. Xiong, H., Kong, X., Li, H., et al.: Vibration analysis of nonlinear systems with the bilinear hysteretic oscillator by using incremental harmonic balance method. Commun. Nonlinear Sci. 42, 437–450 (2017)

    MathSciNet  MATH  Google Scholar 

  30. Wang, X.-F., Zhu, W.-D.: A modified incremental harmonic balance method based on the fast Fourier transform and Broyden’s method. Nonlinear Dyn. 81, 981–989 (2015)

    MathSciNet  MATH  Google Scholar 

  31. Sun, M.: Effect of negative stiffness mechanism in a vibration isolator with asymmetric and high-static-low-dynamic stiffness. Mech. Syst. Signal Process. 124, 388–407 (2019)

    Google Scholar 

  32. Hui, Y., Law, S.-S., Zhu, W., et al.: Internal resonance of structure with hysteretic base-isolation and its application for seismic mitigation. Eng. Struct. 229, 111643 (2021)

    Google Scholar 

  33. Wang, S., Hua, L., Yang, C., et al.: Applications of incremental harmonic balance method combined with equivalent piecewise-linearization on vibrations of nonlinear stiffness systems. J. Sound Vib. 441, 111–125 (2019)

    Google Scholar 

  34. Lau, S.-L.: Nonlinear vibrations of piecewise-linear systems by incremental harmonic balance method. J. Appl Mech. 59, 153–160 (1992)

    MathSciNet  MATH  Google Scholar 

  35. Dhooge, A., Govaerts, W., Kouznetsov, Y.A., et al.: New features of the software MatCont for bifurcation analysis of dynamical systems. Math. Comput. Model. Dyn. 14, 1–18 (2008)

    MathSciNet  Google Scholar 

  36. Dankowicz, H., Schilder, F.: Recipes for Continuation. SIAM, New Delhi (2013)

    MATH  Google Scholar 

  37. Teloli, R.D., da Silva, S.: A new way for harmonic probing of hysteretic systems through nonlinear smooth operators. Mech. Syst. Signal Process. 121, 856–875 (2019)

    Google Scholar 

  38. Okuizumi, N., Kimura, K.: Multiple time scale analysis of hysteretic systems subjected to harmonic excitation. J. Sound Vib. 272, 675–701 (2004)

    Google Scholar 

  39. Carpineto, N., Lacarbonara, L., et al.: Hysteretic tuned mass dampers for structural vibration mitigation. J. Sound Vib. 333, 1302–1318 (2014)

    Google Scholar 

  40. Zhang, Y., Kong, X., Yue, C., et al.: Dynamic analysis of 1-dof and 2-dof nonlinear energy sink with geometrically nonlinear damping and combined stiffness. Nonlinear Dyn. 105, 167–190 (2021)

    Google Scholar 

  41. Zhang, Y., Spanos, P.-D.: Efficient response determination of a M-D-O-F gear model subject to combined periodic and stochastic excitations. Int. J. Nonlinear Mech. 120, 103378 (2020)

    Google Scholar 

Download references

Funding

There is no available fundings for this paper.

Author information

Authors and Affiliations

  1. College of Power Engineering, Naval University of Engineering, Wuhan, 430033, China

    Sheng Wang, Wenyong Guo, Ting Pi & Xiaofeng Li

  2. School of Transportation, Wuhan University of Technology, Wuhan, 430070, China

    Yongou Zhang

Authors
  1. Sheng Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yongou Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wenyong Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ting Pi
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Xiaofeng Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Sheng Wang or Yongou Zhang.

Ethics declarations

Conflict of interest

We have no competing interests. All authors gave final approval for publication and agree to be accountable for all aspects of the work presented herein.

Human and animal rights

No human or animal subjects were used in this work.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, S., Zhang, Y., Guo, W. et al. Vibration analysis of nonlinear damping systems by the discrete incremental harmonic balance method. Nonlinear Dyn 111, 2009–2028 (2023). https://doi.org/10.1007/s11071-022-07953-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-022-07953-y

Keywords

Search

Navigation