Using the delayed feedback to control the vibration of semi-active suspension system for high-speed train

In order to suppress the vertical vibration of high-speed train caused by track irregularity, the delayed feedback control is applied to suppress the vibration of the suspension system. The suspension system is considered as a nonlinear mass-spring-dash vibrating system. The method of multiple scales is employed to obtain the approximate analytical solutions when the primary resonance occurs. The effect of cubic nonlinear stiffness coefficient and damping coefficient on amplitude-frequency response curves is investigated. The focus is to study the effect of gain and time delay on amplitude-frequency response curves. The stability of the system is investigated by Routh-Hurwitz Criterion. The saddle node bifurcation occurs in the amplitude frequency response curves at some values of parameters. The amplitude of the system may jump from one steady state to another one. The results show that the time delay plays an important role in the suspension system. The amplitude of the system could be suppressed at some values of time delay. Moreover, when the suitable values of time delay is chosen the saddle node bifurcation may disappear. However, the system may lose its stability at other values of time delay. The value of time delay should be modified very carefully. The numerical simulation is agreement with the analytical solutions well.