# Dynamic analysis and performance evaluation of nonlinear inerter-based vibration isolators

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## Abstract

This paper investigates a nonlinear inertance mechanism (NIM) for vibration mitigation and evaluates the performance of nonlinear vibration isolators employing such mechanism. The NIM comprises a pair of oblique inerters with one common hinged terminal and the other terminals fixed. The addition of the NIM to a linear spring-damper isolator and to nonlinear quasi-zero-stiffness (QZS) isolators is considered. The harmonic balance method is used to derive the steady-state frequency response relationship and force transmissibility of the isolators subjected to harmonic force excitations. Different performance indices associated with the dynamic displacement response and force transmissibility are employed to evaluate the performance of the resulting isolators. It is found that the frequency response curve of the inerter-based nonlinear isolation system with the NIM and a linear stiffness bends towards the low-frequency range, similar to the characteristics of the Duffing oscillator with softening stiffness. It is shown that the addition of NIM to a QZS isolator enhances vibration isolation performance by providing a wider frequency band of low amplitude response and force transmissibility. These findings provide a better understanding of the functionality of the NIM and assist in better designs of nonlinear passive vibration mitigation systems with inerters.

## Keywords

Inerter Nonlinear vibration isolator Force transmissibility Nonlinear inertance mechanism Backbone curve Quasi-zero stiffness## Notes

### Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant number 51605233) and by Ningbo Science and Technology Bureau under Natural Science Programme (Grant number 2019A610155).

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

## References

- 1.Rivin, E.I.: Passive Vibration Isolation. ASME Press, New York (2003)CrossRefGoogle Scholar
- 2.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)CrossRefGoogle Scholar - 3.Ibrahim, R.A.: Recent advances in nonlinear passive vibration isolators. J. Sound Vib.
**314**, 371–452 (2008)CrossRefGoogle Scholar - 4.Smith, M.C.: Synthesis of mechanical networks: the inerter. IEEE T. Autom. Contr.
**47**(10), 1648–1662 (2002)MathSciNetCrossRefGoogle Scholar - 5.Smith, M.C., Wang, F.C.: Performance benefits in passive vehicle suspensions employing inerters. Veh. Syst. Dyn.
**42**(4), 235–257 (2004)CrossRefGoogle Scholar - 6.Wang, F.C., Liao, M.K., Liao, B.H., Sue, W.J., Chan, H.A.: The performance improvements of train suspension systems with mechanical networks employing inerters. Veh. Syst. Dyn.
**47**(7), 805–830 (2009)CrossRefGoogle Scholar - 7.Jiang, J.Z., Matamoros-Sanchez, A.Z., Goodall, R.M., Smith, M.C.: Passive suspensions incorporating inerters for railway vehicles. Veh. Syst. Dyn.
**50**(Suppl. 1), 263–276 (2012)CrossRefGoogle Scholar - 8.Wang, F.C., Hong, M.F., Chen, C.W.: Building suspensions with inerters. P. I. Mech. Eng. C-J. Mec.
**224**(8), 1605–1616 (2010)CrossRefGoogle Scholar - 9.Lazar, I.F., Neild, S.A., Wagg, D.J.: Using an inerter-based device for structural vibration suppression. Earthq. Eng. Struct. D.
**43**, 1129–1147 (2014)CrossRefGoogle Scholar - 10.Zhang, S.Y., Jiang, J.Z., Neild, S.A.: Optimal configurations for a linear vibration suppression device in a multi-storey building. Struc. Control Hlth.
**24**, 1887 (2017). https://doi.org/10.1002/stc.1887 CrossRefGoogle Scholar - 11.Li, Y., Jiang, J.Z., Neild, S.A.: Inerter-based configurations for main-landing-gear shimmy suppression. J. Aircr.
**54**(2), 684–693 (2017)CrossRefGoogle Scholar - 12.Bott, R., Duffin, R.J.: Impedance synthesis without use of transformers. J. Appl. Phys.
**20**, 816 (1949)MathSciNetCrossRefGoogle Scholar - 13.Jiang, J.Z., Smith, M.C.: Regular positive-real functions and five-element network synthesis for electrical and mechanical networks. IEEE T. Automat. Contr.
**56**(6), 1275–1290 (2011)MathSciNetCrossRefGoogle Scholar - 14.Zhang, S.Y., Jiang, J.Z., Wang, H.L., Neild, S.A.: Synthesis of essential-regular bicubic impedances. Int. J. Circ. Theor. App.
**45**(11), 1482–1496 (2017)CrossRefGoogle Scholar - 15.Carrella, A., Brennan, M.J., Waters, T.P.: Static analysis of a passive vibration isolator with quasi-zero-stiffness characteristic. J. Sound Vib.
**301**, 678–689 (2007)CrossRefGoogle Scholar - 16.Wang, Y., Li, S., Neild, S.A., Jiang, J.Z.: Comparison of the dynamic performance of nonlinear one and two degree-of-freedom vibration isolators with quasi-zero stiffness. Nonlinear Dyn.
**88**(1), 635–654 (2017)CrossRefGoogle Scholar - 17.Moraes, F.H., Silveira, M., Paupitz Gonclaves, P.J.: On the dynamics of a vibration isolator with geometrically nonlinear inerter. Nonlinear Dyn.
**93**, 1325–1340 (2018)CrossRefGoogle Scholar - 18.Goyder, H.G.D., White, R.G.: Vibrational power flow from machines into built-up structures. J. Sound Vib.
**68**, 59–117 (1980)CrossRefGoogle Scholar - 19.Xiong, Y.P., Xing, J.T., Price, W.G.: A general linear mathematical model of power flow analysis and control for integrated structure-control systems. J. Sound Vib.
**267**, 301–334 (2003)MathSciNetCrossRefGoogle Scholar - 20.Yang, J.: Force transmissibility and vibration power flow behaviour of inerter-based vibration isolators. J. Phys. Conf. Ser.
**744**, 012234 (2016)CrossRefGoogle Scholar - 21.Yang, J., Jiang, J.Z., Zhu, X., Chen, H.: Performance of a dual-stage inerter-based vibration isolator. Proc. Eng.
**199**, 1822–1827 (2017)CrossRefGoogle Scholar - 22.Yang, J., Xiong, Y.P., Xing, J.T.: Nonlinear power flow analysis of the Duffing oscillator. Mech. Syst. Signal Pr.
**45**(2), 563–578 (2014)CrossRefGoogle Scholar - 23.Yang, J., Shi, B.Y., Rudd, C.: On vibration transmission between interactive oscillators with nonlinear coupling interface. Int. J. Mech. Sci.
**137**, 238–251 (2018)CrossRefGoogle Scholar - 24.Shi, B.Y., Yang, J., Rudd, C.: On vibration transmission in oscillating systems incorporating bilinear stiffness and damping. Int. J. Mech. Sci.
**150**, 458–470 (2019)CrossRefGoogle Scholar - 25.Yang, J., Xiong, Y.P., Xing, J.T.: Vibration power flow and force transmission behaviour of a nonlinear isolator mounted on a nonlinear base. Int J. Mech Sci.
**115–116**, 238–252 (2016)CrossRefGoogle Scholar - 26.Yang, J., Xiong, Y.P., Xing, J.T.: Power flow behaviour and dynamic performance of a nonlinear vibration absorber coupled to a nonlinear oscillator. Nonlinear Dyn.
**80**(3), 1063–1079 (2015)CrossRefGoogle Scholar - 27.Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Willey, New York (1979)zbMATHGoogle Scholar
- 28.Wagg, D.J., Neild, S.A.: Nonlinear Vibration with Control. Springer, New York (2009)zbMATHGoogle Scholar
- 29.Cammarano, A., Hill, T.L., Neild, S.A., Wagg, D.J.: Identification of systems containing nonlinear stiffnesses using backbone curves. Nonlinear Dyn.
**77**, 311–320 (2014)CrossRefGoogle Scholar - 30.Londono, J., Cooper, J.E., Neild, S.A.: Identification of systems containing nonlinear stiffnesses using backbone curves. Mech. Syst. Signal Pr.
**84**, 116–135 (2017)CrossRefGoogle Scholar - 31.Xiong, Y.P., Xing, J.T., Price, W.G.: Interactive power flow characteristics of an integrated equipment-nonlinear isolator-travelling flexible ship excited by sea waves. J. Sound Vib.
**287**, 245–276 (2005)CrossRefGoogle Scholar