On the Optimal Cubic Feedback Gain and Time Delay of Quasi-Zero Stiffness Vibration Isolators: Analytical and Numerical Study

PurposeFor the inevitable time-delay problem in semi-active vibration control, a novel quasi-zero stiffness vibration isolator with a time-delay cubic displacement feedback control (QSVI-TC) is proposed to study the optimal feedback parameters and delay time.MethodsThe dynamic model of the QSVI-TC is established and the approximate analytical solution of the steady-state response of the system under harmonic excitation is obtained using the multi-scale method. The accuracy of the approximate analytical solution is verified through numerical solutions. The effects of damping, feedback gain, and delay time on the amplitude-frequency characteristics and displacement transmissibility of the system under fixed excitation frequency are analyzed, and the simulation analysis of different models is carried out and compared. Also, the stability of the system is analyzed based on Lyapunov's theory.ResultsThe results indicate that increasing the damping can improve the low-frequency vibration isolation performance of the system. The study results show that as the feedback gain is greater than the threshold, the low-frequency vibration isolation can be effectively improved. And the vibration amplitude of the system in the low-frequency region is reduced by 60%. The influence of delay time on vibration isolation performance changes periodically. The effect of delay time on the vibration amplitude of the system is determined and then the optimal value range can be obtained. Moreover, the unstable closed loops would appear in the amplitude-frequency curves, which are uniformly arranged on a straight line determined by the feedback gain. The stable delay time value of the system can be determined by the excitation frequency. The comparative simulation results show that the vibration isolation performance of QSVI-TC is more perfect.ConclusionThe study has shown that setting the optimal solution with feedback gain and delay time can achieve a better low-frequency vibration isolation performance while improving the vibration isolation effect at mid to high frequencies.