Summary 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.

SummaryIn this study, a demand-based optimal design method is proposed for an oscillator (a single-degree-of-freedom system) with a parallel-layout viscous inerter damper (PVID). The proposed design method overcomes some deficiencies of the existing method, which is based on the fixed-point theory and is mainly suitable for tuned mass dampers. Moreover, for the fixed-point method, the inherent damping of the primary structure is neglected, and the global optimal solution cannot be obtained.The proposed method can obtain a more rational and practical design for the actual design by minimizing both the response and the cost. The design problem of a PVID-equipped oscillator is transformed into a multi-objective optimization problem that can be solved using the ε-constraint approach, which is consistent with the concept of demand-based design. The dynamic response of the oscillator and the force of the PVID (i.e., the cost factor) are evaluated according to theories of random vibration to reduce the number of calculations required. A computer program is developed to perform demand-based parametric design of a PVID-equipped oscillator. Several design cases were examined under different excitation conditions using the computer program, and dynamic time history analyses were then conducted to verify the designs obtained. The results show that the proposed optimal design method identifies satisfactory designs more effectively than the existing method by obtaining PVID design parameter values that better meet the performance demand and simultaneously minimize the cost. | INTRODUCTIONStructural vibrations induced by earthquake or wind loads can be attenuated by various means, including modifying the stiffness, masses, damping, or shape of the structure, and by providing passive or active counter forces.[1] Among numerous available methods, passive mitigation measures are the most widely used in building and bridge engineering. Passive vibration mitigation involves providing a structure with additional damping by installing dampers, and the effectiveness of damping has been confirmed by extensive scientific studies and practical applications. Many types of dampers are used in the passive vibration mitigation of building structures all over the world, such as metalyield dampers, friction dampers, viscoelastic dampers, viscous fluid dampers, tuned mass dampers, and tuned liquid dampers. The concept of tuned mass dampers dates back to the 1900s, when Frahm [2] introduced a tuned mass damper (TMD) to absorb the energy of vibrations, thereby reducing the amplitudes of motions. Den Hartog [3] modified the original TMD concept in the

Summary Interactions among an inerter, spring, and energy dissipation element (EDE) in an inerter system can result in a higher energy dissipation efficiency compared to a single identical EDE, which is referred to as the damping enhancement effect. Previous studies have mainly concentrated on the vibration mitigation effect of the inerter system without an explicit consideration or utilization of the damping enhancement mechanism. In this study, the theoretical essence of the damping enhancement effect is discovered, and a universal design principle is proposed for an inerter system. A fundamental equation is found and demonstrated on the basis of closed‐form stochastic responses, which establishes a bridge between the damping deformation enhancement factor (DDEF) and the response mitigation ratio, thus clarifying the relationship of the damping enhancement effect and the response mitigation effect. Inspired by the equation, a novel damping‐enhancement‐based strategy is proposed to determine the key parameters of an inerter system. Following the performance‐demand‐based design philosophy, the parameters of the inerter system can be determined in the design condition of a target‐damping‐enhancement effect. Through the implementation of the damping enhancement equation, the damping parameter of an inerter system can be directly obtained by the prespecified DDEF and the displacement response mitigation ratio. The influence of parameters on the response mitigation effect and the damping enhancement effect is then investigated to determine ways of obtaining the other two parameters in an inerter system. Finally, design examples are conducted to verify the proposed strategy and the theoretical relationship revealed by the damping enhancement equation. The results show that the proposed design strategy explicitly utilizes the damping enhancement effect for vibration control, where the target of the DDEF is effective in enhancing the efficiency of the EDE for energy dissipation. In the design condition of the target DDEF, the implementation of the proposed damping enhancement equation provides an inerter system with a practical equation to determine the key parameters of an inerter system in a direct manner.

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