Fractional‑order time‑delayed feedback control of nonlinear dynamics in a giant magnetostrictive actuator

In this paper， a fractional-order time-delayed feedback controller is designed to control the nonlinear dynamic response of a single-degree-of-freedom giant magnetostrictive actuator （GMA）. Considering the effect of geometric nonlinear factors intro? duced by the preloaded disc spring mechanism， a nonlinear mathematical model of the GMA system is established. The amplitudefrequency response equation of the main resonance of the system under the feedback control strategy with fractional-order time-de? layed is obtained by the averaging method， and the stability condition of the system is obtained according to the Routh-Hurwitz cri? terion. The influence of key structural parameters in the GMA system on the amplitude-frequency response characteristics， as well as the characteristic law of the main resonance peak and system stability with each time-delayed feedback parameter are studied through numerical simulation. The bifurcation diagram and Lyapunov exponent diagram are obtained and the influence of the exter? nal excitation amplitude on the chaotic motion of the system is studied； finally， the time-delayed feedback gain and fractional order are used to suppress the chaotic motion of the system. The results show that the time-delayed feedback gain and fractional order can effectively suppress the main resonance peak and unstable region of the system， and the system response can be adjusted from cha? otic motion to stable periodic movement to improve the stability of the system.