Time-delayed feedback control of nonlinear dynamics in a giant magnetostrictive actuator

A time-delayed displacement and velocity feedback controller is designed to control the nonlinear dynamic characteristics, particularly principal resonance response, chaotic motion, and limit cycle amplitude, of a single-degree-of-freedom giant magnetostrictive actuator (GMA) system, thereby improving the stability of the system. This controller is established using the previously reported mechanical model of GMA system based on its structure and working principle. Further, the multi-scale method is used to solve the amplitude–frequency response equation and obtain the stability conditions for the time-delayed feedback control of the system’s primary resonance. The influence of each time-delayed feedback parameter on the stability of the primary resonance, chaotic motion, and limit cycle amplitude is examined. The results show that the displacement feedback gain coefficient can only shift the resonance curve to the left and right, while the velocity feedback gain coefficient and time delay parameters can effectively improve the stability and suppress the nonlinear vibration of the system. By increasing the negative displacement feedback gain coefficient and the negative velocity feedback gain coefficient, the system response can be tuned from chaotic motion to periodic motion. The feedback gain coefficient can be effectively used to control the amplitude of the limit loop of the system. Overall, by selecting appropriate time-delayed feedback parameters, the multi-valued solution of the primary resonance can be avoided, the stability of the system can be improved, the chaotic motion can be circumvented, and the limit cycle amplitude can be controlled.