Tuning Mechanism and Parameter Optimization of a Dynamic Vibration Absorber with Inerter and Negative Stiffness Under Delayed FOPID

The dynamic vibration absorber (DVA) based on delayed fractional PID (DFOPID) can achieve a more superior vibration suppression effect. However, the strong nonlinear characteristics of the system and the computational burden resulting from its high dimensionality make solving and optimizing more challenging. This paper presents a coupled model of DFOPID and DVA, exploring its parameter tuning mechanism and optimization problem. First, using the averaging method and Lyapunov stability theory, the amplitude-frequency equation and the stability condition of the steady-state solution of the primary system are derived. Numerical simulations validate the accuracy of the analytical result. Next, based on the mechanics of vibration, the approximate expressions of the controller under different differential conditions are calculated, and their equivalent action mechanisms are analyzed. Finally, by minimizing the maximum amplitude of the primary system as the objective function, the Particle Swarm Optimization (PSO) algorithm is applied to optimize the parameters of the passive DVA and the DVA models controlled by PID, FOPID, and DFOPID, successfully addressing the parameter optimization challenges posed by traditional fixed-point theory. The vibration reduction performance is compared across different loading environments. The results demonstrate that the model presented in this paper performs the best, exhibiting excellent vibration suppression and robustness.