Resonance analysis of vibration isolation system with quasi-zero stiffness and quadratic damping under base excitation

The transmissibility of the forced resonance for the nonlinear vibration isolation system (VIS) coupled with quasi-zero stiffness (QZS) and quadratic damping under base excitation are investigated. By utilizing the averaging method, the approximate analytical solutions of primary resonance (PR) and 1/3 subharmonic resonance (SR) for the nonlinear vibration isolator with QZS and quadratic damping are acquired. Employing Lyapunov's first method, the stability conditions of steady-state solutions for the nonlinear VIS with QZS and quadratic damping are determined. According to the derived conditions for the existence of subharmonic resonance, it is proved that when the considered nonlinear VIS has subharmonic resonance, it only exists within a certain excitation frequency range. The accuracy of the approximate analytical solutions for the amplitude-frequency response, force transmissibility, and relative displacement transmissibility of the PR and SR of the nonlinear VIS is confirmed by comparing them with the numerical results. The effects of QZS and quadratic damping on transmissibility of both force and relative displacement of nonlinear VIS have been discussed. The analysis results indicate that by choosing the appropriate QZS parameter or quadratic damping coefficient, the subharmonic resonance of the nonlinear VIS under a certain base excitation can be completely eliminated. When the amplitude of the base excitation increases to the extent that the system exhibits significant resonance behavior, for the same coefficient value, the nonlinear VIS coupled with QZS and quadratic damping can achieve smaller initial vibration isolation frequency and better amplitude suppression effect than that with linear damping.