Design of structure with inerter system based on stochastic response mitigation ratio

An easy-to-use method for a practical design of single-degree-of-freedom (SDOF) structures with inerter systems was developed. The method adopts the stochastic vibration response mitigation ratio as the design index, and expressions of the closed-form stochastic response are used to evaluate the structural response and optimize the design parameters. Three basic types of inerter systems with tuning and damping magnification mechanisms were considered in this study. The closed-form stochastic responses of SDOF structures with inerter systems under white-noise excitations were derived on the basis of the equations of motion. In addition, the response variation patterns due to changes in key parameters of the inerter system were investigated, and the parameter optimization approaches for different inerter systems are proposed on the basis of the response variation patterns. The use of the closed-form stochastic response mitigation ratio as the design index leads to a rational and simple design process, with the tuning or response extremum conditions employed as complementary conditions for determining the key design parameters. Thus, the proposed method involves the solution of a set of nonlinear equations. Fitted empirical design formulae are provided in this paper for quick calculation of the key parameters of the SDOF structure with the inerter system for a given response mitigation ratio and inherent damping ratio. Moreover, design examples are provided to verify the proposed method. The design results show that the proposed method is effective and more convenient than existing methods. Furthermore, the method can be easily extended for considering seismic excitations and design of structures with other complex-form inerter systems.