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Acta Mechanica

, Volume 230, Issue 7, pp 2563–2579 | Cite as

On the analysis of nonlinear dynamic behavior of an isolation system with irrational restoring force and fractional damping

  • Y. Y. DongEmail author
  • Y. W. Han
  • Z. J. ZhangEmail author
Original Paper
  • 81 Downloads

Abstract

An archetypal isolation system with rational restoring force and fractional damping is proposed and investigated based on the nonlinear mechanism of geometric kinematics. The equations of motion of this nonlinear isolator subject to nonlinear damping and external excitation are derived based on the Lagrange equation. For the free vibration system, the nonlinear irrational restoring force, nonlinear stiffness behaviors, and fractional damping are investigated to show the complex transitions of multi-stability. For the forced vibration system, the analytical expressions of force transmissibility of the nonlinear isolator with single-well potential under the perturbation of viscous damping and harmonic forcing are formulated by applying the harmonic balance method. The shock response spectra of the perturbed system subject to half-sine input are evaluated by the maximum responses. The Melnikov analysis and empirical method are employed to determine the analytical criteria of chaotic thresholds for the homoclinic orbit of the perturbed system with symmetric double-well characteristics. The numerical simulations are carried out to demonstrate periodic solutions, periodic doubling bifurcation, and chaotic solutions. The maximum displacements have been obtained to show the isolation characteristics in the case of chaotic vibration.

Notes

Acknowledgements

The authors would like to acknowledge the Fundamental Research Funds for the Central Universities, No. NS2018055.

References

  1. 1.
    Lei, Z.X., Zhang, L.W., Liew, K.M.: Modeling large amplitude vibration of matrix cracked hybrid laminated plates containing CNTR-FG layers. Appl. Math. Model. 55, 33–48 (2018)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Harris, C.M., Paez, T.L.: Shock and Vibration Handbook, 6th edn. McGraw-Hill, New York (2009)Google Scholar
  3. 3.
    Ibrahim, R.A.: Recent advances in non-linear passive vibration isolators. J. Sound Vib. 314, 371–452 (2008)CrossRefGoogle Scholar
  4. 4.
    Liu, C.C., Jing, X.J., Daley, S.: Recent advances in micro-vibration isolation. Mech. Syst. Signal Process. 56, 55–80 (2015)CrossRefGoogle Scholar
  5. 5.
    Fulcher, B.A., Shahan, D.W., Haberman, M.R.: Analytical and experimental investigation of buckled beams as negative stiffness elements for passive vibration and shock isolation systems. J. Vib. Acoust. 136, 031009 (2014)CrossRefGoogle Scholar
  6. 6.
    Yang, C., Yuan, X.W., Wu, J.: The research of passive vibration isolation system with broad frequency field. J. Vib. Control 19, 1348–1356 (2013)CrossRefGoogle Scholar
  7. 7.
    Zhang, L.W., Zhang, Y., Liew, K.M.: Modeling of nonlinear vibration of graphene sheets using a meshfree method based on nonlocal elasticity theory. Appl. Math. Model. 49, 691–704 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Xu, D.L., Yu, Q.P., Zhou, J.X.: Theoretical and experimental analyses of a nonlinear magnetic vibration isolator with quasi-zero-stiffness characteristic. J. Sound Vib. 1332, 3377–3389 (2013)CrossRefGoogle Scholar
  9. 9.
    Zhou, N., Liu, K.A.: Tunable high-static-low-dynamic stiffness vibration isolator. J. Sound Vib. 329, 1254–1273 (2010)CrossRefGoogle Scholar
  10. 10.
    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)CrossRefGoogle Scholar
  11. 11.
    Le, T.D., Ahn, K.K.: Experimental investigation of a vibration isolation system using negative stiffness structure. Int. J. Mech. Sci. 70, 99–112 (2013)CrossRefGoogle Scholar
  12. 12.
    Schuster, H.G.: Reviews of Nonlinear Dynamics and Complexity. Wiley-VCH, Weinheim (2008)CrossRefzbMATHGoogle Scholar
  13. 13.
    Ruzicka, J.E., Derby, T.F.: Influence of Damping in Vibration Isolation. The Shock Vib. Inf. Center, Washington (1971)Google Scholar
  14. 14.
    Yang, P., Yang, J.M., Ding, J.N.: Dynamic transmissibility of a complex nonlinear coupling isolator. Tsinghua Sci. Technol. 54, 538–542 (2006)zbMATHGoogle Scholar
  15. 15.
    Paola, M.D., Mendola, L.L., Navarra, G.: Stochastic seismic analysis of structures with nonlinear viscous dampers. Tsinghua Sci. Technol. 133, 1475–1478 (2007)Google Scholar
  16. 16.
    Lang, Z.Q., Jing, X.J., Billings, S.A., Tomlinson, G.R., Peng, Z.K.: Theoretical study of the effects of nonlinear viscous damping on vibration isolation of SDOF systems. J. Sound Vib. 323, 352–365 (2009)CrossRefGoogle Scholar
  17. 17.
    Guo, P.F., Lang, Z.Q., Peng, Z.K.: Analysis and design of the force and displacement transmissibility of nonlinear viscous damper based vibration isolation systems. Nonlinear Dyn. 67, 2671–2687 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Yan, S., Lin, B., Fei, J., Liu, P.: Nonlinear dynamic behavior of a rotor bearing system with non-linear viscous damping suspension. In: ASME International Mechanical Engineering Congress Exposition, Tampa, USA (2017)Google Scholar
  19. 19.
    Guo, P.F., Lang, Z.Q., Peng, Z.K.: Application of a weakly nonlinear absorber to suppress the resonant vibrations of a forced nonlinear oscillator. J. Vib. Acoust. 134, 044502 (2012)CrossRefGoogle Scholar
  20. 20.
    Tang, B., Brennan, M.J.: A comparison of two nonlinear damping mechanisms in a vibration isolator. J. Sound Vib. 332, 510–520 (2013)CrossRefGoogle Scholar
  21. 21.
    Zhang, Y., Zhang, L.W., Liew, K.M., Yu, J.L.: Free vibration analysis of bilayer graphene sheets subjected to in-plane magnetic fields. Compos. Struct. 144, 86–95 (2016)CrossRefGoogle Scholar
  22. 22.
    Wang, Y., Wang, R.C., Meng, H.D.: Analysis and comparison of the dynamic performance of one-stage inerter-based and linear vibration isolators. Int. J. Appl. Mech. 10, 1850005 (2018)CrossRefGoogle Scholar
  23. 23.
    Li, J., Li, S.: Generating ultra wide low-frequency gap for transverse wave isolation via inertial amplification effects. Phys. Lett. A 383, S0375960117311507 (2017)Google Scholar
  24. 24.
    Chen, Y., Wang, L.: Isolation of surface wave-induced vibration using periodically modulated piles. Int. J. Appl. Mech. 6, 1450042 (2014)CrossRefGoogle Scholar
  25. 25.
    Zhou, J.X., Wang, K., Xu, D.L., Ouyang, H.J.: Multi-low-frequency flexural wave attenuation in Euler-Bernoulli beams using local resonators containing negative-stiffness mechanisms. Phys. Lett. A 381, 3141–3148 (2017)CrossRefGoogle Scholar
  26. 26.
    Moon, F.C., Gollub, J.P.: Chaotic Vibrations: An Introduction for Applied Scientists and Engineers. Wiley-Interscience, Hoboken (2004)CrossRefGoogle Scholar
  27. 27.
    Lim, C.W., Lai, S.K., Wu, B.S.: Accurate higher-order analytical approximate solutions to large-amplitude oscillating systems with a general non-rational restoring force. Nonlinear Dyn. 42, 267–281 (2005)CrossRefzbMATHGoogle Scholar
  28. 28.
    Lim, C.W., Wu, B.S., Sun, W.P.: Higher accuracy analytical approximations to the Duffing-harmonic oscillator. J. Sound Vib. 296, 1039–1045 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Lai, S.K., Lim, C.W., Wu, B.S., Wang, C., Zeng, Q.C., He, X.F.: Newton-harmonic balancing approach for accurate solutions to non-linear cubic–quantic Duffing oscillators. Appl. Math. Model. 33, 852–866 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of AstronauticsNanjing University of Aeronautics and AstronauticsNanjingChina
  2. 2.School of Civil EngineeringHenan University of Science and TechnologyLuoyangChina

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