A Novel of Low-Frequency Vibration Isolation with High-Static Low-Dynamic Stiffness Characteristic


The efficiency of quasi-zero stiffness vibration isolator will decrease when getting overloaded and underloaded. Given the variability of load in general engineering application, this paper aims at presenting a newly designed type of vibration isolation system with positive stiffness in parallel with elements of negative stiffness, which has variable carrying capacity. The vibration isolation performance of system requires overall analysis. Firstly, static analysis is applied to obtain the optimal vibration isolation parameters of the system at the equilibrium position and to verify the high-static low-dynamic stiffness characteristic of the system. Then, the nonlinear dynamic equation of the system is established. Meanwhile, the influence of excitation amplitude on the transmissibility of the system is analyzed under three different conditions by harmonic balance method. Finally, the response curves of the system to sinusoidal excitation and multi-frequency excitation are analyzed by numerical simulation. The result of numerical simulation shows that the vibration isolation system still has good low-frequency vibration isolation characteristics when dealing with different loads. The problem of poor vibration isolation performance of general quasi-zero stiffness isolators under load changes is well solved.

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\( A_{0} \) :

Constant term of the steady-state solution

\( A_{1} \) :

Amplitude of harmonic term of the steady-state solution

\( A_{0}^{f} \) :

Peak amplitude of \( A_{0} \) for the force excitation

\( A_{0}^{\text{z}} \) :

Peak amplitude of \( A_{0} \) for the displacement excitation

\( A_{1}^{\text{f}} \) :

Peak amplitude of \( A_{1} \) for the force excitation

\( A_{1}^{z} \) :

Peak amplitude of \( A_{1} \) for the displacement excitation

\( {\text{a}} \) :

The horizontal distance between the tilted spring end and the equilibrium point

\( {\text{a}}_{0} \) :

The original horizontal distance between the tilted spring end and the equilibrium point

c :

Damping coefficient

F :

The external load

F V :

The restoring force of vertical spring

F s :

The restoring force of inclined spring

F 1 :

The restoring force of the system under harmonic force excitation

F 2 :

The restoring force of the system under harmonic displacement excitation

\( {\text{f}}_{m} \) :

Dynamic force transmitted to the base

\( {\text{f}}_{{\text{md}}} \) :

Damping force

\( {\text{f}}_{{\text{me}}} \) :

Elastic force

\( F_{{\text{me}0}} \) :

Constant term of \( {\text{f}}_{{\text{me}}} \)

\( F_{{\text{me}1}} \) :

Amplitude of harmonic term of \( f_{{\text{me}}} \)

\( F_{{\text{md}}} \) :

Amplitude of harmonic term of \( f_{{\text{md}}} \)

\( {\text{h}}_{0} \) :

The distance between the initial position and the initial equilibrium point


The system stiffness

\( k_{\text{s}} \) :

Stiffness of horizontal spring

\( k_{\text{v}} \) :

Stiffness of vertical spring

L :

The length of the inclined spring

\( L_{0} \) :

The original length of the inclined spring



\( T_{\text{f}} \) :

The force transmissibility of ideal system

\( T_{z} \) :

The absolute displacement transmissibility of the system

\( T_{\text{l}} \) :

The force transmissibility of equivalent linear system

\( {\text{u}}_{0} \) :

The deviation from the desired position

\( v \) :

The displacement with the equilibrium position

x :

The displacement at the equilibrium point

y :

Relative displacement (y = u z)

z :

Absolute displacement response of base

Z :

Amplitude of base absolute displacement

\( \alpha \) :

Stiffness ratio

\( \beta \) :

The included angle between the guide track and the horizontal plane

\( \theta \) :

The included angle between the inclined spring and the horizontal plane

\( \xi \) :

Dimensionless viscous damping coefficient

\( \rho \) :

The amplitude of harmonic excitation

\( \omega \) :

Excitation frequency

\( \omega_{\text{n}} \) :

Natural frequency of the system

\( \varphi \) :

Phase difference

\( \varOmega \) :

Frequency ratio \( \omega / \omega_{\text{n}} \)

\( \varOmega^{\text{f}} \) :

Frequency corresponding to the peak response for force excitation

\( \varOmega^{\text{z}} \) :

Frequency corresponding to the peak response for displacement excitation


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This work is supported by the National Natural Science Foundation of China (Project No. 51305444) and the Project Funded by Design of Robot Variable Stiffness Joint based on Electromagnetic Control.

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Correspondence to Jiayu Zheng.

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Zheng, J., Yang, X., Xu, J. et al. A Novel of Low-Frequency Vibration Isolation with High-Static Low-Dynamic Stiffness Characteristic. Iran J Sci Technol Trans Mech Eng (2020). https://doi.org/10.1007/s40997-020-00370-9

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  • Low-frequency vibration isolation
  • High-static low-dynamic
  • Variable load
  • Harmonic balance method