Abstract
This study develops a modified elliptic harmonic balance method (EHBM) and uses it to solve the force and displacement transmissibility of a two-stage geometrically nonlinear vibration isolation system. Geometric damping and stiffness nonlinearities are incorporated in both the upper and lower stages of the isolator. After using the relative displacement of the nonlinear isolator, we can numerically obtain the steady-state response using the first-order harmonic balance method (HBM1). The steady-state harmonic components of the stiffness and damping force are modified using the Jacobi elliptic functions. The developed EHBM can reduce the truncation error in the HBM1. Compared with the HBM1, the EHBM can improve the accuracy of the resonance regimes of the amplitude-frequency curve and transmissibility. The EHBM is simple and straightforward. It can maintain the same form as the balancing equations of the HBM1 but performs better than it.
Highlights
Nonlinearity has become the focus of recent research studies on improving vibration isolation performance [1, 2]
Comparing three kinds of nonlinear 2-DOF vibration isolation models, which are grounded-grounded, bottomsprings grounded, and top-springs grounded isolators, Wang et al found that the bottom-springs grounded isolator has the best isolation performance when the excitation force amplitude is small [12]. e vibrational power flow method was applied to investigate the performance of a 2-DOF nonlinear isolation system [15]
When the elliptic harmonic balance method is applied to solve two or multiple DOFs systems, the number of equations obtained by harmonic balancing is not equal to that of unknowns [30]
Summary
Nonlinearity has become the focus of recent research studies on improving vibration isolation performance [1, 2]. HBM with prior linearization [23], have been presented In this respect, Zhou et al illustrated that an accurate approximation solution for the resonance response of harmonically forced strongly nonlinear oscillator could be obtained by linearizing the governing equation before harmonic balancing [24]. When the elliptic harmonic balance method is applied to solve two or multiple DOFs systems, the number of equations obtained by harmonic balancing is not equal to that of unknowns [30] For this issue, Chen and Liu analysed a 2-DOF self-excited oscillator with strongly cubic nonlinearity by an additional equation prior to harmonic balancing using Jacobi elliptic functions [30]. Wu and Tang modified the harmonic components of the first-order terms of the damping and stiffness force of a single DOF nonlinear isolation system using Jacobi elliptic functions [33]. After calculating the amplitude-frequency characteristics, we can obtain the force and displacement transmissibility. e accuracy of the results is analysed
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