Nonlinear Dynamics

, Volume 93, Issue 4, pp 2339–2356 | Cite as

Enhanced isolation performance of a high-static–low-dynamic stiffness isolator with geometric nonlinear damping

  • Guangxu Dong
  • Yahong Zhang
  • Yajun Luo
  • Shilin Xie
  • Xinong ZhangEmail author
Original Paper


To enhance the low-frequency vibration isolation performance of the high-static–low-dynamic stiffness (HSLDS) isolator, a novel design of the geometric nonlinear damping (GND) comprising semi-active electromagnetic shunt damping is proposed. The GND is dependent on the vibration displacement and velocity, which can make the HSLDS isolator attain different damping characteristics in different frequency bands. Firstly, the configuration of the HSLDS isolator assembled with GND is presented, and then the restoring force, stiffness, and damping are derived. The dynamics of the mount under both base and force excitations are investigated based on the harmonic balance method, which are then verified by numerical simulations. After that, the effects of GND on the displacement and force transmissibility are studied, and the excellent performance caused by GND is analyzed based on the equivalent viscous damping mechanism. Finally, the comparison between the GND and cubic nonlinear damping is performed. The results demonstrate that the HSLDS isolator assembled with GND can realize the requirements of an isolation system under both base and force excitations of broadband vibration isolation performance and a low resonance peak with the high-frequency attenuation unaffected. Moreover, the GND outperforms the linear damping no matter the base excitation or force excitation is applied. For base excitation, the GND exhibits some desirable properties that the cubic nonlinear damping does not have at high frequencies.


Geometric nonlinear damping High-static–low-dynamic stiffness Harmonic balance method Base and force excitations Vibration isolation 



Geometric nonlinear damping


Electromagnetic shunt damping


High-static–low-dynamic stiffness


Quasi-zero stiffness


Negative stiffness mechanisms


Displacement–velocity-dependent damping


Spiral flexure spring


Magnetic negative stiffness spring


Magnetic negative stiffness


Electromagnetic devices


External negative impedance circuits



This work is supported by the National Natural Science Foundation of China Academy of Engineering Physics and jointly set up “NSAF” joint fund (Grant No. U1630120).


  1. 1.
    Rivin, E.I.: Passive vibration isolation. Appl. Mech. Rev. 57(6), B31–B32 (2004)CrossRefGoogle Scholar
  2. 2.
    Thomson, W.: Theory of Vibration with Applications. CRC Press, Boca Raton (1996)Google Scholar
  3. 3.
    Alabuzhev, P.M., Rivin, E.I.: Vibration Protecting and Measuring Systems with Quasi-Zero Stiffness. Hemisphere Pub. Corp, Washington (1989)Google Scholar
  4. 4.
    Carrella, A., Brennan, M.J., Waters, T.P., Lopes, V.: Force and displacement transmissibility of a nonlinear isolator with high-static--low-dynamic-stiffness. Int. J. Mech. Sci. 55(1), 22–29 (2012). CrossRefGoogle Scholar
  5. 5.
    Carrella, A., Brennan, M.J., Waters, T.P.: Static analysis of a passive vibration isolator with quasi-zero-stiffness characteristic. J. Sound Vib. 301(3–5), 678–689 (2007). CrossRefGoogle Scholar
  6. 6.
    Carrella, A., Brennan, M.J., Kovacic, I., Waters, T.P.: On the force transmissibility of a vibration isolator with quasi-zero-stiffness. J. Sound Vib. 322(4–5), 707–717 (2009). CrossRefGoogle Scholar
  7. 7.
    Xu, D.L., Zhang, Y.Y., Zhou, J.X., Lou, J.J.: On the analytical and experimental assessment of the performance of a quasi-zero-stiffness isolator. J. Vib. Control 20(15), 2314–2325 (2014). CrossRefGoogle Scholar
  8. 8.
    Platus, D.L.: Negative-stiffness-mechanism vibration isolation systems. In: Derby, E.A., Gordon, C.G., Vukobratovich, D., Yoder, P.R., Zweben, C. (eds.) Optomechanical Engineering and Vibration Control, Proceedings of the Society of Photo-Optical Instrumentation Engineers (Spie), vol. 3786, pp. 98–105. Spie-International Society for Optical Engineering, Bellingham (1999)Google Scholar
  9. 9.
    Huang, X.C., Liu, X.T., Sun, J.Y., Zhang, Z.Y., Hua, H.X.: Vibration isolation characteristics of a nonlinear isolator using Euler buckled beam as negative stiffness corrector: a theoretical and experimental study. J. Sound Vib. 333(4), 1132–1148 (2014). CrossRefGoogle Scholar
  10. 10.
    Liu, X.T., Huang, X.C., Hua, H.X.: On the characteristics of a quasi-zero stiffness isolator using Euler buckled beam as negative stiffness corrector. J. Sound Vib. 332(14), 3359–3376 (2013). CrossRefGoogle Scholar
  11. 11.
    Huang, X.C., Liu, X.T., Hua, H.X.: Effects of stiffness and load imperfection on the isolation performance of a high-static--low-dynamic-stiffness non-linear isolator under base displacement excitation. Int. J. Non Linear Mech. 65, 32–43 (2014). CrossRefGoogle Scholar
  12. 12.
    Huang, X.C., Liu, X.T., Sun, J.Y., Zhang, Z.Y., Hua, H.X.: Effect of the system imperfections on the dynamic response of a high-static--low-dynamic stiffness vibration isolator. Nonlinear Dyn. 76(2), 1157–1167 (2014). MathSciNetCrossRefGoogle Scholar
  13. 13.
    Carrella, A., Brennan, M.J., Waters, T.P., Shin, K.: On the design of a high-static--low-dynamic stiffness isolator using linear mechanical springs and magnets. J. Sound Vib. 315(3), 712–720 (2008). CrossRefGoogle Scholar
  14. 14.
    Robertson, W.S., Kidner, M.R.F., Cazzolato, B.S., Zander, A.C.: Theoretical design parameters for a quasi-zero stiffness magnetic spring for vibration isolation. J. Sound Vib. 326(1–2), 88–103 (2009). CrossRefGoogle Scholar
  15. 15.
    Xu, D.L., Yu, Q.P., Zhou, J.X., Bishop, S.R.: Theoretical and experimental analyses of a nonlinear magnetic vibration isolator with quasi-zero-stiffness characteristic. J. Sound Vib. 332(14), 3377–3389 (2013). CrossRefGoogle Scholar
  16. 16.
    Wu, W.J., Chen, X.D., Shan, Y.H.: Analysis and experiment of a vibration isolator using a novel magnetic spring with negative stiffness. J. Sound Vib. 333(13), 2958–2970 (2014). CrossRefGoogle Scholar
  17. 17.
    Dong, G.X., Zhang, X.N., Xie, S.L., Yan, B., Luo, Y.J.: 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). CrossRefGoogle Scholar
  18. 18.
    Zhou, N., Liu, K.: A tunable high-static--low-dynamic stiffness vibration isolator. J. Sound Vib. 329(9), 1254–1273 (2010). CrossRefGoogle Scholar
  19. 19.
    Ravindra, B., Mallik, A.K.: Hard Duffing-type vibration isolator with combined Coulomb and viscous damping. Int. J. Non Linear Mech. 28(4), 427–440 (1993). CrossRefzbMATHGoogle Scholar
  20. 20.
    Ho, C., Lang, Z.Q., Billings, S.A.: A frequency domain analysis of the effects of nonlinear damping on the Duffing equation. Mech. Syst. Signal Process. 45(1), 49–67 (2014). CrossRefGoogle Scholar
  21. 21.
    Ho, C., Lang, Z.Q., Billings, S.A.: Design of vibration isolators by exploiting the beneficial effects of stiffness and damping nonlinearities. J. Sound Vib. 333(12), 2489–2504 (2014). CrossRefGoogle Scholar
  22. 22.
    Peng, Z.K., Meng, G., Lang, Z.Q., Zhang, W.M., Chu, F.L.: Study of the effects of cubic nonlinear damping on vibration isolations using harmonic balance Method. Int. J. Non Linear Mech. 47(10), 1073–1080 (2012). CrossRefGoogle Scholar
  23. 23.
    Laalej, H., Lang, Z.Q., Daley, S., Zazas, I., Billings, S.A., Tomlinson, G.R.: Application of non-linear damping to vibration isolation: an experimental study. Nonlinear Dyn. 69(1–2), 409–421 (2012). CrossRefGoogle Scholar
  24. 24.
    Milovanovic, Z., Kovacic, I., Brennan, M.J.: On the displacement transmissibility of a base excited viscously damped nonlinear vibration isolator. J. Vib. Acoust. 131(5), 054502 (2009). CrossRefGoogle Scholar
  25. 25.
    Tang, B., Brennan, M.J.: A comparison of two nonlinear damping mechanisms in a vibration isolator. J. Sound Vib. 332(3), 510–520 (2013). CrossRefGoogle Scholar
  26. 26.
    Freischlag, J.A., Lawrence, P.F., Perler, B.A.: The Society for Vascular Surgery will participate in a new global vascular practice guidelines initiative. J. Sound Vib. 59(2), 510–510 (2014). Google Scholar
  27. 27.
    Sun, X.T., Jing, X.J.: Analysis and design of a nonlinear stiffness and damping system with a scissor-like structure. Mech. Syst. Signal Process. 66–67, 723–742 (2016). CrossRefGoogle Scholar
  28. 28.
    Sun, X.T., Xu, J., Jing, X.J., Cheng, L.: Beneficial performance of a quasi-zero-stiffness vibration isolator with time-delayed active control. Int. J. Mech. Sci. 82, 32–40 (2014). CrossRefGoogle Scholar
  29. 29.
    Xu, J., Sun, X.T.: A multi-directional vibration isolator based on Quasi-Zero-Stiffness structure and time-delayed active control. Int. J. Mech. Sci. 100, 126–135 (2015). CrossRefGoogle Scholar
  30. 30.
    Huang, X.C., Sun, J.Y., Hua, H.X., Zhang, Z.Y.: The isolation performance of vibration systems with general velocity–displacement-dependent nonlinear damping under base excitation: numerical and experimental study. Nonlinear Dyn. 85(2), 777–796 (2016). MathSciNetCrossRefGoogle Scholar
  31. 31.
    Rahim, M.A., Waters, T.P., Rustighi, E.: Active damping control for vibration isolation of high-static--low-dynamic-stiffness isolators. In: Proceedings of International Conference on Noise and Vibration Engineering (ISMA2014) and International Conference on Uncertainty in Structural Dynamics (USD2014), pp. 173–186 (2014)Google Scholar
  32. 32.
    Cheng, C., Li, S.M., Wang, Y., Jiang, X.X.: Force and displacement transmissibility of a quasi-zero stiffness vibration isolator with geometric nonlinear damping. Nonlinear Dyn. 87(4), 2267–2279 (2017). CrossRefGoogle Scholar
  33. 33.
    Yan, B., Luo, Y.J., Zhang, X.N.: Structural multimode vibration absorbing with electromagnetic shunt damping. J. Vib. Control 22(6), 1604–1617 (2016). MathSciNetCrossRefGoogle Scholar
  34. 34.
    Yan, B., Zhang, X.N., Luo, Y.J., Zhang, Z.F., Xie, S.L., Zhang, Y.H.: Negative impedance shunted electromagnetic absorber for broadband absorbing: experimental investigation. Smart Mater. Struct. 23(12), 11 (2014). CrossRefGoogle Scholar
  35. 35.
    Zhang, X.N., Niu, H.P., Yan, B.: A novel multimode negative inductance negative resistance shunted electromagnetic damping and its application on a cantilever plate. J. Sound Vib. 331(10), 2257–2271 (2012). CrossRefGoogle Scholar
  36. 36.
    Niu, H.P., Zhang, X.N., Xie, S.L., Wang, P.P.: A new electromagnetic shunt damping treatment and vibration control of beam structures. Smart Mater. Struct. 18(4), 15 (2009). CrossRefGoogle Scholar
  37. 37.
    Ravaud, R., Lemarquand, G., Electromagnetics, A.: Analytical expressions of the magnetic field created by tile permanent magnets of various magnetization directions. In: Piers 2009 Moscow Vols I and II, Proceedings. Electromagnetics Acad, Cambridge (2009)Google Scholar
  38. 38.
    Kovacic, I., Brennan, M.J.: The Duffing Equation: Nonlinear Oscillators and Their Behaviour. Wiley, New York (2011)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  • Guangxu Dong
    • 1
  • Yahong Zhang
    • 1
  • Yajun Luo
    • 1
  • Shilin Xie
    • 1
  • Xinong Zhang
    • 1
    Email author
  1. 1.State Key Laboratory for Strength and Vibration of Mechanical Structures, School of AerospaceXi’an Jiao Tong UniversityXi’anPeople’s Republic of China

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