Application of Perturbation Method to Analytical H2 Optimization for a Damped Structure with Inerter-Based Control

Abstract

Inherent damping exists in all real engineering systems and it affects the dynamic behavior of the structures in vibration reduction. However, it is generally neglected in existing analytical H2 optimization for inerter-based vibration control. Hence, an analytical H2 optimization of a tuned inerter damper (TID) based on the perturbation method is presented to control the vibration of a damped structure subjected to stochastic acceleration ground excitations. Since relative displacement and velocity are crucial to structural damage and absolute acceleration is related to vibration comfort, three sets of closed-form solutions for the optimal stiffness ratio and damping ratio are derived to suppress the corresponding responses. The expressions of these optimal parameters are functions of the inherent damping and inerter-mass ratio. The accuracy of these analytical solutions is demonstrated by comparison with the results provided by the numerical optimization procedure. The relevant parameters affecting the vibration control performance and the robustness of the TID in a damped structure are analyzed in frequency domain. Finally, time–history response and energy dissipation of TID are also explored by numerical examples under real and artificial seismic waves. The results highlight that more than 10% reduction ratio of the mean square value can be further achieved for a structure with a low inherent damping ratio ([Formula: see text]) if the present multi-order perturbation solutions are used for optimization, instead of the existing method which neglects the inherent damping. This superiority of the present perturbation method will become more prominent with the increase of the inherent damping ratio. Therefore, the present perturbation method has a great significance for designing a TID to achieve the best control performance.