Optimization design and stability analysis for a new class of inerter-based dynamic vibration absorbers with a spring of negative stiffness

This article investigates the parameter optimization problem for a novel inerter-based dynamic vibration absorber (IDVA) system with a grounded negative stiffness spring, which can be applied to many vibration control systems, such as building vibrations, bridge vibrations, vehicle suspensions, etc. The passive mechanical network in the IDVA system of this article is the parallel connection of a passive spring and a three-element damper-spring-inerter series network. First, the motion equations of the IDVA system are established based on Newton's second law, and the transfer function in the dimensionless form is derived. Then, the H- infinity optimization problem is formulated for the given values of the mass ratio and negative stiffness ratio. Since it is found that the frequency response curves of this system pass through four fixed points that are independent of the damping ratio, the extended fixed-point method is adopted to optimize the system. Then, the analytical solutions of the optimal parameter values in terms of the mass ratio and negative stiffness ratio are derived, and the necessary and sufficient condition for the IDVA system with optimal parameter values to be stable is obtained by the Hurwitz criterion. Compared with other four optimal DVAs in the existing literature, the optimal IDVA system with a grounded negative stiffness spring can provide better H- infinity performance, and the numerical examples illustrate that the system can provide better time-domain performances under random excitations.