Mechanical mechanism and parameter optimization of a tuned inerter damper with delayed fractional-order PID

This paper aims to tackle three challenges in the vibration absorber system with the fractional-order PID, i.e., the rarity of vibration mechanism analysis, limitations in order-taking, and performance enhancement. An active tuned inerter damper with delayed fractional-order PID based on the displacement feedback of the primary system is presented, integrating the advantages of the time-delayed control and tuned inerter damper. The analytical expressions of displacement amplitude and stability condition of the primary system are derived by the averaging method, Lyapunov indirect method, and Routh-Hurwitz criterion. The differential order of the controller in the analytical expressions applies to the fully parametric domain, and the integral order applies between zero and one. The obtained analytical solution unifies the analytical forms for two kinds of fractional-order differential terms from previous work. Definitions of equivalent stiffness, damping coefficient, and mass for different ranges of the differential order are given through analysis of the equivalent mechanical mechanism of the control force. The influence of the differential and integral orders on the amplitude-frequency characteristics of the system is analyzed. The parameter tuning is performed through the improved grey wolf algorithm, and the influences of different mass ratios and weight coefficients on the optimal results are analyzed. In most working cases, the differential order is greater than one, indicating viscous inertia characteristics. The presented model is compared with various passive or active vibration absorbers under harmonic excitation, mixed excitation of harmonic and white noise, pulse excitation, as well as far-field and near-field seismic excitation. The presented model exhibits superior vibration suppression capability in various performance evaluation indexes. Under harmonic excitation, the peak displacement transmissibility of the primary system it controls can be as low as 1.0742, and the damping bandwidth is wider. Under mixed excitation, the RMS attenuation ratio of the primary system displacement can reach 79.22%, the highest among the compared models, reflecting a certain robustness. With pulse excitation, the attenuation curve of the primary system it controls has a lower peak of the maximum waveform and a shorter stabilization time. The maximum and RMS values of displacement and acceleration for the primary system it controls are smaller under far-field and