Vibration characteristics and chaos analysis of a 2-DOF system with a fractional-order time-delay PD controller

This paper investigates the vibration mechanism and chaotic characteristics of a rotor vibration suppression system with fractional-order time-delay PD control. In the rotor system, based on displacement feedback, a Jeffcott rotor model with fractional-order time-delay PD control is established. The multiscale method is used to analyze the relationship between the steady-state vibration amplitude and frequency and further explores the relationship between stable vibration phase and frequency. The Routh–Hurwitz criterion is applied to obtain the equilibrium conditions of the model, and the effect of system parameters on the amplitude–frequency characteristics is studied. Numerical simulations are performed, and the results show a high degree of consistency between the two computational simulations and theoretical solutions. Furthermore, the Melnikov method is used to calculate the chaotic threshold of the rotor system, and comparisons with bifurcation diagrams, Lyapunov exponent plots, and the topology of the safe basin and domain of attraction validate the effectiveness of the proposed model. Finally, a particle swarm optimization algorithm is used to optimize the system parameters, and the optimal control parameters under different harmonic excitations are analyzed.