Skip to main content

Advertisement

SpringerLink
Log in

A nonlinear vibration isolator supported on a flexible plate: analysis and experiment

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

Abstract

To address low-frequency vibration isolation, an issue that engineers often face, this paper first studies the nonlinear energy transfer of a flexible plate, with arbitrary boundary, with the coupling of high-static-low-dynamic-stiffness (HSLDS) isolator. The nonlinear coupled dynamic equation was derived via the Lagrange method, and the improved Fourier series and Rayleigh–Ritz methods provide modal coefficients of the arbitrary boundary flexible plate with nonlinear vibration isolators. The Galerkin and harmonic balance methods approximate the frequency response functions of power flow for the coupled system. The numerical method, via direct integration of the dynamic equation, validates the analytical results of the frequency response functions. In addition, the finite element simulation, used here, validates the analytical results of the mode shapes for flexible plate. The experiment is carried out to validate the isolation performance of the nonlinear vibrator supported on a flexible plate. On these bases, increasing damping and controlling HSLDS can improve the low-frequency isolation efficiency, and nonlinear jumping-phenomena could disappear over a low-frequency range (either frequency overlap or frequency jump). Hence, a properly configured flexible plate could improve the bearing capacity and low-frequency isolation efficiency while avoiding frequency mistune. An explanation for these is offered in the article.

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

Similar content being viewed by others

Data availability

The data used to support the findings of this study are available from the corresponding author upon request.

References

  1. Lacarbonara, W.: Nonlinear Structural Mechanics. Springer, New York (2013). https://doi.org/10.1007/978-1-4419-1276-3

    Book  MATH  Google Scholar 

  2. Zhang, Y., Du, J.T.: Influence of boundary conditions on three-dimensional vibration characteristics of thick rectangular plates. Sci. Progress. 103, 1–16 (2020). https://doi.org/10.1177/0036850420969548

    Article  Google Scholar 

  3. Liu, C., Jing, X.J., Daley, S., Li, F.: Recent advances in micro-vibration isolation. Mech. Syst. Signal Process. 56–57, 55–80 (2015). https://doi.org/10.1016/j.ymssp.2014.10.007

    Article  Google Scholar 

  4. Ibrahim, R.A.: Recent advances in nonlinear passive vibration isolators. J. Sound Vib. 314, 371–452 (2008). https://doi.org/10.1016/j.jsv.2008.01.014

    Article  Google Scholar 

  5. Zhang, F., Xu, M., Shao, S., Xie, S.: A new high-static-low-dynamic stiffness vibration isolator based on magnetic negative stiffness mechanism employing variable reluctance stress. J. Sound Vib. 476, 115322 (2020). https://doi.org/10.1016/j.jsv.2020.115322

    Article  Google Scholar 

  6. Sun, Y., Zhao, J., Wang, M., Sun, Y., Pu, H.Y., Luo, J., Peng, Y., Xie, S., Yang, Y.: High-static–low-dynamic stiffness isolator with tunable electromagnetic mechanism. IEEE ASME Trans. Mechatron. 25, 316–326 (2020). https://doi.org/10.1109/TMECH.2019.2954910

    Article  Google Scholar 

  7. Pu, H.Y., Yuan, S., Peng, Y., Meng, K., Zhao, J., Xie, R., Huang, Y., Sun, Y., Yang, Y., Xie, S.: Multi-layer electromagnetic spring with tunable negative stiffness for semi-active vibration isolation. Mech. Syst. Signal Process. 121, 942–960 (2019). https://doi.org/10.1016/j.ymssp.2018.12.028

    Article  Google Scholar 

  8. Liu, X., Huang, X., Hua, H.: On the characteristics of a quasi-zero stiffness isolator using Euler buckled beam as negative stiffness corrector. J. Sound Vib. 332, 3359–3376 (2013). https://doi.org/10.1016/j.jsv.2012.10.037

    Article  Google Scholar 

  9. Yao, Y., Li, H., Yun, L.: Analytical and experimental investigation of a high-static-low-dynamic stiffness isolator with cam-roller-spring mechanism. Int. J. Mech. Sci. 185, 105888 (2020). https://doi.org/10.1016/j.ijmecsci.2020.105888

    Article  Google Scholar 

  10. Fan, H., Yang, L., Tian, Y., Wang, Z.: Design of metastructures with quasi-zero dynamic stiffness for vibration isolation. Compos. Struct. 243, 112244 (2020). https://doi.org/10.1016/j.compstruct.2020.112244

    Article  Google Scholar 

  11. Wang, X., Liu, H., Chen, Y., Gao, P.: Beneficial stiffness design of a high-static-low-dynamic-stiffness vibration isolator based on static and dynamic analysis. Int. J. Mech. Sci. 142, 235–244 (2018). https://doi.org/10.1016/j.ijmecsci.2018.04.053

    Article  Google Scholar 

  12. Sun, Y., Zhou, J., Thompson, D., Yuan, T., You, T.: Design, analysis and experimental validation of high static and low dynamic stiffness mounts based on target force curves. Int. J. Nonlinear Mech. 126, 103559 (2020). https://doi.org/10.1016/j.ijnonlinmec.2020.103559

    Article  Google Scholar 

  13. Yang, J., Xiong, Y.P., Xing, J.T.: Dynamics and power flow behaviour of a nonlinear vibration isolation system with a negative stiffness mechanism. J. Sound Vib. 332, 167–183 (2013). https://doi.org/10.1016/j.jsv.2012.08.010

    Article  Google Scholar 

  14. Gatti, G., Kovacic, I., Brennan, M.J.: On the response of a harmonically excited two degree-of-freedom system consisting of a linear and a nonlinear quasi-zero stiffness oscillator. J. Sound Vib. 329, 1823–1835 (2010). https://doi.org/10.1016/j.jsv.2009.11.019

    Article  Google Scholar 

  15. Ding, H., Zhu, M.H., Chen, L.Q.: Nonlinear vibration isolation of a viscoelastic beam. Nonlinear Dyn. 92, 325–349 (2018). https://doi.org/10.1007/s11071-018-4058-8

    Article  Google Scholar 

  16. Ding, H., Lu, Z.Q., Chen, L.Q.: Nonlinear isolation of transverse vibration of pre-pressure beams. J. Sound Vib. 442, 738–751 (2019). https://doi.org/10.1016/j.jsv.2018.11.028

    Article  Google Scholar 

  17. Shaw, A.D., Neild, S.A., Wagg, D.J., Weaver, M.P., Carrella, A.: A nonlinear spring mechanism incorporating a bistable composite plate for vibration isolation. J. Sound Vib. 332, 6265–6275 (2013). https://doi.org/10.1016/j.jsv.2013.07.016

    Article  Google Scholar 

  18. Shaw, A.D., Neild, S.A., Wagg, D.J.: Dynamic analysis of high static low dynamic stiffness vibration isolation mounts. J. Sound Vib. 332, 1437–1455 (2013). https://doi.org/10.1016/j.jsv.2012.10.036

    Article  Google Scholar 

  19. Kovacic, I., Brennan, M.J., Waters, T.P.: A study of a nonlinear vibration isolator with a quasi-zero stiffness characteristic. J. Sound Vib. 315, 700–711 (2008). https://doi.org/10.1016/j.jsv.2007.12.019

    Article  Google Scholar 

  20. Dong, G., Zhang, X., Xie, S., Yan, B., Luo, Y.: Simulated and experimental studies on a high-static-low-dynamic stiffness isolator using magnetic negative stiffness spring. Mech. Syst. Signal Process. 86, 188–203 (2017). https://doi.org/10.1016/j.ymssp.2016.09.040

    Article  Google Scholar 

  21. Lu, Z.Q., Yang, T.J., Brennan, M.J., Li, X., Liu, Z.G.: On the performance of a two-stage vibration isolation system which has geometrically nonlinear stiffness. J. Vib. Acoust. 136, 064501 (2014). https://doi.org/10.1115/1.4028379

    Article  Google Scholar 

  22. Yan, B., Wang, Z., Ma, H., Bao, H., Wu, C.Y.: A novel lever-type vibration isolator with eddy current damping. J. Sound Vib. 494, 115862 (2021). https://doi.org/10.1016/j.jsv.2020.115862

    Article  Google Scholar 

  23. Jing, B., Jing, X.J.: Analysis and design of a novel and compact X-structured vibration isolation mount (X-Mount) with wider quasi-zero-stiffness range. Nonlinear Dyn. 101, 2195–2222 (2020). https://doi.org/10.1007/s11071-020-05878-y

    Article  Google Scholar 

  24. Zhang, W., Zhao, J.B.: Analysis on nonlinear stiffness and vibration isolation performance of scissor-like structure with full types. Nonlinear Dyn. 86, 17–36 (2016). https://doi.org/10.1007/s11071-016-2869-z

    Article  Google Scholar 

  25. Lu, Z.Q., Gu, D.H., Ding, H., Lacarbonara, W., Chen, L.Q.: Nonlinear vibration isolation via a circular ring. Mech. Syst. Signal Process. 136, 106490 (2019). https://doi.org/10.1016/j.ymssp.2019.106490

    Article  Google Scholar 

  26. Zhou, J.X., Xiao, Q.Y., Xu, D.L., Ouyang, H.J., Li, Y.L.: A novel quasi-zero-stiffness strut and its applications in six-degree-of-freedom vibration isolation platform. J. Sound Vib. 394, 59–74 (2017). https://doi.org/10.1016/j.jsv.2017.01.021

    Article  Google Scholar 

  27. Zheng, Y.S., Li, Q.P., Yan, B., Luo, Y.J., Zhang, X.N.: A Stewart isolator with high-static-low-dynamic stiffness struts based on negative stiffness magnetic springs. J. Sound Vib. 422, 390–408 (2018). https://doi.org/10.1016/j.jsv.2018.02.046

    Article  Google Scholar 

  28. Wang, M., Hu, Y., Sun, Y., Ding, J., Luo, J.: An adjustable low-frequency vibration isolation Stewart platform based on electromagnetic negative stiffness. Int. J. Mech. Sci. 181, 105714 (2020). https://doi.org/10.1016/j.ijmecsci.2020.105714

    Article  Google Scholar 

  29. Lu, Z.Q., Wu, D., Ding, H., Chen, L.Q.: Vibration isolation and energy harvesting integrated in a Stewart platform with high static and low dynamic stiffness. Appl. Math. Model. 89, 249–267 (2020). https://doi.org/10.1016/j.apm.2020.07.060

    Article  MathSciNet  MATH  Google Scholar 

  30. Du, J.T., Li, W.L., Liu, Z.G., Yang, T.J., Jin, G.Y.: Free vibration of two elastically coupled rectangular plates with uniform elastic boundary restraints. J. Sound Vib. 330, 788–804 (2011). https://doi.org/10.1016/j.jsv.2010.08.044

    Article  Google Scholar 

  31. Li, W.L.: Free vibration of beams with general boundary conditions. J. Sound Vib. 237, 709–725 (2000). https://doi.org/10.1006/jsvi.2000.3150

    Article  Google Scholar 

  32. Li, W.L., Zhang, X., Du, J.T., Liu, Z.G.: An exact series solution for the transverse vibration of rectangular plates with general elastic boundary supports. J. Sound Vib. 321, 254–269 (2009). https://doi.org/10.1016/j.jsv.2008.09.035

    Article  Google Scholar 

  33. Zhao, X., Chen, B., Li, Y.H., Zhu, W.D., Nkiegaing, F.J., Shao, Y.B.: Forced vibration analysis of Timoshenko double-beam system under compressive axial load by means of Green’s functions. J. Sound Vib. 464, 115001 (2019). https://doi.org/10.1016/j.jsv.2019

    Article  Google Scholar 

  34. Wang, Q., Shi, D., Liang, Q., Ahad, F.E.: A unified solution for free in-plane vibration of orthotropic circular, annular and sector plates with general boundary conditions. Appl. Math. Model. 40, 9228–9253 (2016). https://doi.org/10.1016/j.apm.2016.06.005

    Article  MathSciNet  MATH  Google Scholar 

  35. Jin, G.Y., Ye, T., Ma, X., Chen, Y., Su, Z., Xiang, X.: A unified approach for the vibration analysis of moderately thick composite laminated cylindrical shells with arbitrary boundary conditions. Int. J. Mech. Sci. 75, 357–376 (2013). https://doi.org/10.1016/j.ijmecsci.2013.08.003

    Article  Google Scholar 

  36. Zhang, T., Ouyang, H.J., Zhang, Y.O., Lv, B.L.: Nonlinear dynamics of straight fluid-conveying pipes with general boundary conditions and additional springs and masses. Appl. Math. Model. 40, 7880–7900 (2016). https://doi.org/10.1016/j.apm.2016.03.050

    Article  MathSciNet  MATH  Google Scholar 

  37. Mao, X.Y., Shu, S., Fan, X., Ding, H., Chen, L.Q.: An approximate method for pipes conveying fluid with strong boundaries. J. Sound Vib. 505, 116157 (2021). https://doi.org/10.1016/j.jsv.2021.116157

    Article  Google Scholar 

Download references

Funding

This study was funded by the National Natural Science Foundation of China (Grant Nos. 11872037, 11872159, and 11572182) and the Innovation Program of Shanghai Municipal Education Commission (Grant No. 2017-01-07-00-09-E00019).

Author information

Authors and Affiliations

  1. School of Mechanics and Engineering Science, Shanghai University, 99 Shangda Road, Shanghai, 200444, China

    Rong-Biao Hao, Ze-Qi Lu, Hu Ding & Li-Qun Chen

Authors
  1. Rong-Biao Hao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ze-Qi Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hu Ding
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Li-Qun Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ze-Qi Lu.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

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

Appendix

Appendix

Detailed expressions for zi, wi

$$ \left\{ \begin{gathered} z_{1} = x_{{{\text{rig}}}} + \frac{b}{2}\sin \alpha_{{{\text{rig}}}} + \frac{a}{2}\sin \beta_{{{\text{rig}}}} ,\;z_{2} = x_{{{\text{rig}}}} - \frac{b}{2}\sin \alpha_{{{\text{rig}}}} + \frac{a}{2}\sin \beta_{{{\text{rig}}}} \hfill \\ z_{3} = x_{{{\text{rig}}}} - \frac{b}{2}\sin \alpha_{{{\text{rig}}}} - \frac{a}{2}\sin \beta_{{{\text{rig}}}} ,\;z_{4} = x_{{{\text{rig}}}} + \frac{b}{2}\sin \alpha_{{{\text{rig}}}} - \frac{a}{2}\sin \beta_{{{\text{rig}}}} \hfill \\ \end{gathered} \right. $$
(24)

Compared with the vertical displacement x, α, and β are smaller, and the above equation can be rewritten as,

$$ \left\{ \begin{gathered} z_{1} = x_{{{\text{rig}}}} + 0.2\alpha_{{{\text{rig}}}} + 0.2\beta_{{{\text{rig}}}} ,\;z_{2} = x_{{{\text{rig}}}} - 0.2\alpha_{{{\text{rig}}}} + 0.2\beta_{{{\text{rig}}}} \hfill \\ z_{3} = x_{{{\text{rig}}}} - 0.2\alpha_{{{\text{rig}}}} - 0.2\beta_{{{\text{rig}}}} ,\;z_{4} = x_{{{\text{rig}}}} + 0.2\alpha_{{{\text{rig}}}} - 0.2\beta_{{{\text{rig}}}} \hfill \\ \end{gathered} \right. $$
(24)
$$ w_{i} = \varphi_{{{\text{mn}}}} (x_{{{\text{p}}i}} ,y_{{{\text{p}}i}} )q_{mn} (t) $$
(25)

where, xpi and ypi represent the connected position coordinates in op−xpyp.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hao, RB., Lu, ZQ., Ding, H. et al. A nonlinear vibration isolator supported on a flexible plate: analysis and experiment. Nonlinear Dyn 108, 941–958 (2022). https://doi.org/10.1007/s11071-022-07243-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-022-07243-7

Keywords

Search

Navigation