# Theoretical Study on a Cable-Bracing Inerter System for Seismic Mitigation
## 2 Theoretical Analysis of a Cable-Bracing Inerter System
### Motion Governing Equation of SDOF System with CBIS  
When the structure is in a balanced state, the prestressed tension forces in both cables are \(T_{0}\). If the structure starts to leave the balance position by moving to the right as illustrated in Figure 4, the diagonal cable on the right side drives the inerter to rotate clockwise. At this time, the force increment in the right cable is \(\Delta T\), and the force becomes \(T_{2}\) (\(T_{2}=T_{0}+\Delta T\)). The tension in the left cable decreases \(\Delta T\) and becomes \(T_{1}\) (\(T_{1}=T_{0}-\Delta T\)). Figure 3: Layout of an SDOF system with a CBIS. Figure 2: Mechanical model of an inerter element. When the structure has a positive deformation, it moves to the right with a relative displacement \(u(t)\), and the shaft and the conductor plates rotate correspondingly. We assume there is no relative slippage between the cable and the shaft. Considering the axial stiffness of the one-sided cable \(\mathrm{k_{b}}^{0}\), the forces in the right cable \(T_{2}\) and the left cable \(T_{1}\) are as follows:  
\[ \begin{array}{l}T_{2}=T_{0}+\mathrm{k_{b}}^{0}(u(t)\cos\theta-\varphi(t)r_{0} )\\ T_{1}=T_{0}-\mathrm{k_{b}}^{0}(u(t)\cos\theta-\varphi(t)r_{0}),\end{array}  \tag{2}\]  
where the rotational angle of the conductor plate is \(\varphi(t)\). As shown in Figure 5, when the conductor plates rotate, the cable moves in its own axial direction. Thus, the angle difference \(\Delta\theta\) from its balance position is trivial and can be ignored. \(u(t)\cos\theta-\varphi(t)r_{0}\) is the axial elongation of the cable. During the operation of a CBIS, the tension force difference between two cables drives the shaft to rotate and is given as:  
\[T_{2}-T_{1}=2\mathrm{k_{b}}^{0}(u(t)\cos\theta-\varphi(t)r_{0}). \tag{3}\]  
The eddy currents cause a damping force that is proportional to the velocity of the conductive metal, which makes the eddy currents function like a viscous damper. According to the force equilibrium conditions, compatibility condition, and the layout of the system (as shown in Figure 3), the motion equation for this SDOF structure with a CBIS under earthquake excitations can be written as:  
\[m\ddot{u}(t)+ci\dot{u}(t)+ku(t)+k_{\mathrm{b}}(u(t)\cos\theta-\varphi(t)r_{0}) \cos\theta=-ma_{g}(t), \tag{4}\]  
where \(u(t)\) is the relative displacement of the SDOF system, and the dots represent the derivative with respect to time \(t\). \(a_{\mathrm{g}}(t)\) is the acceleration of the ground motion, \(k_{\mathrm{b}}\) is the equivalent stiffness of two cables, and it is used to replace \(2\mathrm{k_{b}}^{0}\) in Equation (3). The motion equation for the CBIS is written as:  
\[J\ddot{\varphi}(t)+c_{\mathrm{d}}\dot{\varphi}(t)r_{0}{}^{2}=k_{\mathrm{b}}(u (t)\cos\theta-\varphi(t)r_{0})r_{0}, \tag{5}\]  
where \(J\) is the moment of inertia for the inerter, and \(r_{0}\) is the radius of the shaft. The conductor plate serves as a flywheel whose moment of inertia can be calculated as:  
\[J=m_{\mathrm{I}}R^{2}/2, \tag{6}\]  
where \(m_{\mathrm{I}}\) is the physical mass of two conductor plates and the shaft, and \(R\) is the radius of gyration. Figure 4: Transmission mechanism of cable-bracing inerter system. The governing motion equation of an SDOF structure can be described as:  
\[\left[ \begin{array}{cc}m&0\\ 0&J\end{array} \right]\left\{ \begin{array}{c}\ddot{u}\\ \ddot{\varphi}\end{array} \right\}+\left[ \begin{array}{cc}c&0\\ 0&c_{\mathrm{d}}r_{0}^{2}\end{array} \right]\left\{ \begin{array}{c}\dot{u}\\ \dot{\varphi}\end{array} \right\}+\left[ \begin{array}{cc}k+k_{\mathrm{b}}\cos^ {2}\theta&-k_{\mathrm{b}}r_{0}\cos\theta\\ -k_{\mathrm{b}}r_{0}\cos\theta&k_{\mathrm{b}}r_{0}^{2}\end{array} \right] \left\{ \begin{array}{c}u\\ \varphi\end{array} \right\}=\left[ \begin{array}{c}-ma_{g}\\ 0\end{array} \right]. \tag{7}\]  
Therefore, Equation (7) can be expressed in matrix form as:  
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