# Theoretical Study on a Cable-Bracing Inerter System for Seismic Mitigation
## 1 Introduction
To establish an efficient and practical design method for structures with inerter systems, some theoretical analyses were carried out in the present study. [PERSON] et al. [9] derived a simple formula for optimal design of TVMD based on the fixed-point theory, which can be used as a design method in practice. Taking the inherent damping ability of a single degree-of-freedom (SDOF) structure into consideration, [PERSON] et al. [32; 33] proposed a demand-based optimal design method for a parallel-layout inerter system to satisfy performance demands with minimum control costs. [PERSON] et al. [34] investigated the impact of the mechanical layout of inerter systems on seismic-response mitigation of liquid-storage tanks, and [PERSON] et al. [35] explored the influence of soil-structure interaction on structures equipped with an inerter system. After these designing measures were undertaken in the engineering applications, some novel inerter-based devices were developed. Researchers had different approaches to equipping structures with these inerter-based devices. [PERSON] et al. [36] presented a ball-screw inerter system connected with a toggle brace to magnify the relative displacement between adjacent floors and showed that their system could be utilized effectively in structures even when the drift was small. [PERSON] et al. [23] presented a rack-pinion-flywheel system supported by an infinitely stiff chevron frame and demonstrated that this system was particularly effective in suppressing the peak displacement of structures over long periods of time. [PERSON] et al. [31] installed the TVMD system in a building in Tohoku, Japan. This building has seismic response control systems to upgrade the seismic safety of structures and facilities in the Tohoku building. It has traditional viscous dampers supporting the lower floors and the TVMD system supporting the upper floors. The TVMD was fixed between the adjacent floors using a support member with a relevant stiffness like steel. These braces can transmit bending moments, shear, and torque, and are sensitive to displacement at boundaries, which may induce the non-negligible moment and deformation at the TVMD terminals. Ball joints were used to release the deformation which could have induced the unwanted internal moment and torque in the brace. Cable bracing is the alternative method to ball joints for connecting inerters, since cables can only bear the axial tension force and release deformation other than the axial direction. Tension-only cables are an important element in the seismic control systems and are mainly used to transmit control forces and to direct deformation from the main structure to energy dissipating devices. [PERSON] (1975) uses cables to prevent progressive collapse of buildings. [PERSON] et al. [PERSON] et al. (2018) proposed a rotational friction damper connected to tension-only braces to enhance the seismic resisting capacity of existing structures. [PERSON] et al. [PERSON] et al. (2019) developed a bracing system consisting of cables and a central energy dissipator. The tension-only cable design can increase the speed of construction by adopting simple connections with rapid installation features. In this paper, we propose to use a pair of tension-only cables to transfer the story drift to the rotating flywheels, i.e., the cable-bracing inerter system (CBIS). Section 2 will introduce the concept of CBIS and establish the motion governing equation for a CBIS-equipped SDOF system excited by the ground motion. The frequency response functions of displacement and force output are derived for characteristic study. In Section 3, a parametric analysis is conducted to study the effects of CBIS parameters on structural seismic mitigation. In addition, a performance-based multi-objective \(H_{2}\) norm optimum design method is proposed to design the CBIS. Design cases are carried out to illustrate the effects of CBIS and the effectiveness of the proposed design method. Section 4 draws the conclusions.