# Displacement-Dependent Damping Inerter System for Seismic Response Control
## 1 Introduction
As a classic example, the ball-screw inerter system utilizes fluid viscous damping, with the fluid fully filled into a small gap between the fixed inner cylinder and rotating outer cylinder connected to a ball screw [38]. However, the inertance of such a device cannot be adjusted or replaced once it is manufactured even if the mass enhancement ratio is found to be insufficient for vibration control afterwards. In addition, there is also a concern about the leakage risk of the damping fluid, with high pressure within the rotating cylindrical tube requiring thorough sealing of the device [39]. From the perspective of the constitutive relationship of the mechanical behavior, viscous damping and electromagnetic damping (mentioned earlier) are both VDE mechanisms, which means that the damping force is in-phase with the relative velocity of the VDE. Because the maximum displacement and maximum velocity never occur at the same time, there is an inevitable time difference between the maximum damping force and displacement [40; 41]. Owing to this difference, the VDE would be incapable of mitigating the structural response greatly in the time domain if a peak response occurs in the early stage of excitation, which is generally true in most earthquake events. This sometimes unfavorable velocity dependence of a VDE can be replaced by a displacement-dependent damper that uses a displacement-dependent damping element (DDE) for energy dissipation, with the damping force directly depending on the relative deformation of the device [42]. Such systems are widely used because of their stable hysteretic behavior and relatively low production and maintenance costs. [PERSON] et al. [43] were the first to propose a displacement-dependent damper for vibration energy dissipation. Since then, considerable efforts [44; 45; 46; 47] have been devoted to analyzing the dynamic behavior of structures controlled by displacement-dependent dampers and the design of these systems. There is no doubt that a displacement-dependent damper is effective for a vibration mitigation, theoretically as well as experimentally. In particular, this damper is preferentially chosen at the beginning of the excitation, based on the fact that displacement-dependent dampersreduce substantial displacements more effectively compared to viscous dampers with equivalent damping ratios (meaning that the energy dissipated per cycle for DDE and VDE [42] is the same). However, the energy dissipation effect of the damper is limited when the deformation in a DDE is small, especially in the early stage of shaking, in a seismic event. The main contribution of the tuned inerter in a displacement-dependent damping inerter system (DDIS) is to amplify the deformation in a DDE to ensure efficient energy dissipation during the early stage of excitation. Based on these motivations, a novel DDIS is developed that overcomes the aforementioned drawbacks of the VDIS. The proposed DDIS consists of an inerter, a DDE, and a spring. Stochastic dynamic analysis of a structure equipped with the DDIS system is performed, considering a base acceleration modeled with a Kanai-Tajimi power spectral density function. The stochastic linearization technique is used to handle the non-linear terms. It is demonstrated that DDIS combines the advantageous properties of DDE and inerter in a single layout. The structural displacement, acceleration, and energy dissipation mitigation indices are determined from the stochastic analysis results and used to establish an evaluation method for the dynamic performance of the DDIS-equipped structure. A broad parametric analysis is performed within the evaluation framework, taking into consideration the effects of non-linearity of the DDIS and variability of its parameters. Finally, several design cases in the time domain are presented to illustrate the damping enhancement and quick-control provided by the proposed DDIS.