# Displacement-Dependent Damping Inerter System for Seismic Response Control
## 1 Introduction  
Structural control technology is proven to be effective in suppressing structural responses with the aid of various control devices and methods. Among these devices, inerter systems have been found to be very effective owing to their tuning frequency, mass enhancement, and damping enhancement mechanisms [1; 2; 3]. The performance evaluation and benefits of inerter-based systems for the protection of building structures [4; 5; 6; 7; 8], storage tanks [9; 10; 11], wind turbine towers [12; 13; 14], semi-submersible platform [15], and for vibration suppression of cables [16; 17], machine [18] and suspension systems [19; 20] have been studied in recent literature. An inerter is a mechanical element with two terminals [21; 22; 23; 24; 25] and ideally produces a resistive force proportional to its inner relative acceleration and large apparent mass designated as inertance. An essential property of the inerter is that a large inertance can be produced by devices with negligible physical masses. In the past, [PERSON] [26] developed a vibration control device that used fluid inertance, which is designated as a mass pump, to suppress the seismic responses of a structure; this device has an inertial massenhancement mechanism. [PERSON] et al. [27; 28] utilized the rotation mechanism to amplify the effective damping force of a viscous damper, which is a type of velocity-dependent damping element (VDE). However, these devices did not explicitly use the mass enhancement effect until [PERSON] et al. [1] proposed the tuned viscous mass damper, which belongs to a type of velocity-dependent damping inerter system (VDIS). The performance of the tuned viscous mass damper control system was subsequently investigated via shaking table tests conducted on single-story systems equipped with scaled-down versions of the damper [29]. [PERSON] et al. [30] proposed a rotational inertia double-tuned mass damper by replacing the viscous damping of the tuned mass damper with a tuned viscous mass damper, which achieved significantly greater control than the tuned mass damper with similar additional mass ratio. Through the incorporation of an electromagnetic damper, which is a type of VDE, [PERSON] et al. [31] developed an electromagnetic inerter mass damper with variable damping force. [PERSON] et al. [32; 33] achieved enhanced energy-harvesting performance using a tuned inerter. [PERSON] et al. [10; 34] proposed an isolation inerter system that used an inerter and a VDE to mitigate the vibration of a storage tank. The effect of the mechanical layout of the system was also investigated and considered in the development of a demand-based optimal design method for the system. [PERSON] et al. [1] presented the closed-form optimal design formulae for a VDIS based on the fixed-point method (an optimal design method to minimize the \(H_{\infty}\) norm of the transfer function). On the basis of consideration of both the response mitigation effect and control cost, [PERSON] et al. [35] proposed a demand-performance-based optimal design methodology for a structure equipped with a VDIS to achieve the desired seismic performance level from the primary structure. [PERSON] and [PERSON] [36] subsequently derived a closed-form expression of the root-mean-square (RMS) response of a single-degree-of-freedom (SDOF) structure with three representative VDISs, namely, the series layout inerter system, series-parallel layout I inerter system (SPIS-I), and series-parallel layout II inerter system. [PERSON] et al. [37] analyzed the effect of the soil-structure interaction on the vibration mitigation effect of a VDIS. Based on the determined effect, they proposed an optimal design method that utilized a simulated annealing algorithm. As mentioned above, recent studies in the field have mainly focused on the development and optimal design of inerter systems that utilize VDEs for energy dissipation.