A torsion–translational vibration isolator with quasi-zero stiffness

  • Qianlong Zhang
  • Shuyan Xia
  • Daolin XuEmail author
  • Zhike Peng
Original paper


In this paper, a torsion–translational quasi-zero-stiffness (TT QZS) isolator with convex ball–roller mechanisms was proposed to attenuate the torsion and translational vibrations along shaft systems simultaneously. The mathematical model of the isolator was established, based on which the static characteristics of the isolator were analyzed and the quasi-zero-stiffness condition that the design parameters should satisfy was derived as well. Subsequently, the isolator was integrated into a shaft system to study its dynamical performances under excitations of two independent external forces. Unique phenomenon of so-called vibration frequency–interference was observed and elaborated theoretically with harmonic balance method. In addition, the amplitude–frequency characteristics of the two degrees-of-freedom (DOFs) system subjected to two independent harmonic excitations were analyzed by employing numerical simulations. Accordingly, the small-oscillation regions of the system were determined through the jump-down frequencies. Meanwhile, the force transmissibility in the small-amplitude-oscillation region was discussed by taking advantage of semi-analytical solutions and compared with that of the corresponding linear system. The results indicated that the TT QZS isolator has extra low-frequency vibration isolation performance in both torsion and translational DOFs.


Torsion–translational isolator Quasi-zero stiffness Nonlinear isolator Transmissibility Low-frequency vibration isolation 

List of symbols

\(k_{\theta }\)

Torsion stiffness of rubber (N m/rad)


Stiffness of axial spring (N/m)


Number of convex ball–roller mechanisms


Stiffness of horizontal spring (N/m)

\(F_{0}, F\)

Axial payloads (N) and axial force (N)

\(M_0, M\)

Torsional payloads (N m) and torsional force (N m)


Axial coordinate of the isolator (m)

\(\theta \)

Torsional coordinate of the isolator (rad)

\(\delta \)

Pre-compressed deflection of horizontal spring (m)

\(r_1,r_2 ,r_3\)

Radius of shaft, convex balls and rollers (m)


Distance between centers of shaft and rollers (m)

\(C(\bar{z}, \theta )\)

Geometrical contact function

\(K_z (\bar{z}, \theta )\)

Axial stiffness of the isolator (N/m)

\(K_{\theta } (\bar{z}, \theta )\)

Torsional stiffness of the isolator (N m/rad)

\(\varepsilon _{\bar{\mathrm{F}}}, \varepsilon _{\bar{\mathrm{M}}}\)

Relative error of axial and torsional forces

\(A_z, A_{\theta }\)

Amplitudes of axial and torsional excitation forces (m)


Frequencies of axial and torsional excitation forces (rad/s)

\(\varphi _z, \varphi _{\theta }\)

Initial phases of axial and torsional excitation forces (rad)

\(\zeta _z, \zeta _{\theta }\)

Damping ratios of axial and torsional excitation forces

\(\gamma \)

Coefficients of Taylor series of axial FD function

\(\lambda \)

Coefficients of Taylor series of torsional FD function

\(T_z, T_{\theta }\)

Transmissibility of axial and torsional directions

\(\varOmega _z, \varOmega _{\theta }\)

Ratios between excitation forces and natural frequencies



This research work was supported by the National Natural Science Foundation of China (Nos. 11472100, 11632011) and the High-tech Ship Research Projects Sponsored by MIIT.

Compliance with ethical standards

Conflict of interest

All authors declare that they have no conflict of interest.

Human and animal rights

This article does not contain any studies with human participants or animals performed by any of the authors.


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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  • Qianlong Zhang
    • 1
  • Shuyan Xia
    • 2
  • Daolin Xu
    • 2
    Email author
  • Zhike Peng
    • 1
  1. 1.State Key Laboratory of Mechanical System and VibrationShanghai Jiao Tong UniversityShanghaiPeople’s Republic of China
  2. 2.State Key Laboratory of Advanced Design and Manufacturing for Vehicle BodyHunan UniversityChangshaPeople’s Republic of China

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