# Theoretical Study on a Cable-Bracing Inerter System for Seismic Mitigation
## 2 Theoretical Analysis of a Cable-Bracing Inerter System
### Frequency Response Function  
In this section, the frequency response function of an SDOF structure with a CBIS under seismic excitation is obtained. To normalize the dynamic equation, the parameters can be defined as:  
\[\omega_{\mathrm{s}}=\sqrt{\frac{k}{m}}, \tag{15}\]  
\[\zeta=c/2m\omega_{\mathrm{s}}, \tag{16}\]  
\[\xi=c_{\mathrm{d}}/2m\omega_{\mathrm{s}}, \tag{17}\]where \(\omega_{\rm s}\) and \(\zeta\) are the circular frequency and the inherent damping ratio of the original SDOF structure, respectively. \(\xi\) is the additional damping ratio provided by the CBIS. Through dimensionless processing, the following parameters can be defined for designing CBIS:  
\[\mu=m_{\rm d}/m, \tag{18}\]  
\[\kappa=k_{\rm b}\cos^{2}\theta/k, \tag{19}\]  
where \(\mu\) is inertance-mass ratio, a ratio of the inertance of CBIS to the mass of primary system. \(\kappa\) is the ratio of supporting spring stiffness in the horizontal direction to the primary stiffness \(k\). Substituting the parameters in Equations (15)-(19) into Equation (9), the Laplace transformation of Equation (7) can be written as:  
\[\left[\begin{array}{c}sU\\ s^{2}U\\ s\Phi\\ s^{2}\Phi\\ \end{array}\right]=\left[\begin{array}{cccc}0&1&0&0\\ -\omega_{\rm s}{}^{2}-\kappa\omega_{\rm s}{}^{2}&-2\zeta\omega_{\rm s}&\kappa \omega_{\rm s}{}^{2}r_{0}/\cos\theta&0\\ 0&0&0&1\\ \kappa\omega_{\rm s}{}^{2}mr_{0}/J\cos\theta&0&-\kappa\omega_{\rm s}{}^{2}mr_{ 0}{}^{2}/J\cos^{2}\theta&-2\zeta\omega_{\rm s}r_{0}{}^{2}m/J\\ \end{array}\right]\left[\begin{array}{c}U\\ sU\\ \Phi\\ s\Phi\\ \end{array}\right]-\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ \end{array}\right]A_{\rm g}(s), \tag{20}\]  
where \(s=i\omega\), and \(A_{\rm g}(s)\) is the Laplace transformation of \(a_{\rm g}(t)\). \(U\), \(U\), \(\Phi\) and \(\dot{\Phi}\) are the Laplace transformations of \(u\), \(\dot{u}\), \(\varphi\) and \(\dot{\varphi}\), respectively, and they can be solved from Equation (8):  
\[\left[\begin{array}{c}U(s)\\ \dot{U}(s)\\ \dot{\Phi}(s)\\ \dot{\Phi}(s)\\ \end{array}\right]=\left[\begin{array}{c}\frac{-(s^{2}\mu+2\xi\omega_{\rm s }\cos^{2}\theta+\zeta\omega_{\rm s}{}^{2})A_{\rm g}(s)}{C(s,\xi,\zeta,\mu,\mu, \kappa,\mu,\kappa)}\\ \frac{-\delta(s^{2}\mu+2\xi\omega_{\rm s}\zeta\omega+\theta\kappa\omega_{\rm s }{}^{2})A_{\rm g}(s)}{C(s,\xi,\zeta,\mu,\kappa,\mu,\kappa)}\\ \frac{-\omega_{\rm s}\omega_{\rm s}\cos\theta A_{\rm g}(s)}{D(s,\xi,\zeta,\mu, \mu,\kappa,\mu,\kappa)}\\ \frac{-\omega_{\rm s}\omega_{\rm s}\cos\theta A_{\rm g}(s)}{D(s,\xi,\zeta,\mu, \kappa,\mu,\kappa,\mu,\kappa)}\\ \end{array}\right], \tag{21}\]  
where  
\[\begin{array}{l}C(s,\xi,\zeta,\mu,\kappa,\omega_{\rm s})=s^{4}\mu+2s^{3}( \xi\cos^{2}\theta+\zeta\mu)\omega_{\rm s}+s^{2}(\kappa+4\xi\zeta\cos^{2}\theta +\mu+\kappa\mu)\omega_{\rm s}{}^{2}\\ +2s(\zeta\kappa+\xi\cos^{2}\theta+\kappa\xi\cos^{2}\theta)\omega_{\rm s}{}^{3} +\kappa\omega_{\rm s}{}^{4}\\ D(s,\xi,\zeta,\mu,\kappa,\omega_{\rm s},r_{0})=s^{4}\mu r_{0}+2s^{3}(r_{0}\xi \cos^{2}\theta+r_{0}\zeta\mu)\omega_{\rm s}+s^{2}(\kappa\mu r_{0}+4\xi\zeta \cos^{2}\theta r_{0}\\ +\mu r_{0}+\kappa r_{0})\omega_{\rm s}{}^{2}+2s(\xi\kappa r_{0}+\zeta\kappa r_{ 0}+\xi\cos^{2}\theta r_{0})\omega_{\rm s}{}^{3}+\kappa r_{0}\omega_{\rm s}{}^{4}.\end{array} \tag{22}\]  
The frequency-domain transfer function between \(u(t)\) and input excitations can be easily obtained as:  
\[H_{U}(s)=\frac{U(s)}{A_{g}(s)}=\frac{-(s^{2}\mu+2\xi\omega_{\rm s}\cos^{2} \theta+\kappa\omega_{\rm s}{}^{2})}{C(s,\xi,\zeta,\mu,\kappa,\omega_{\rm s})}. \tag{23}\]  
The normalized force of the CBIS is defined as \(F(t)=k_{\rm b}(u(t)\cos\theta-\varphi(t)r_{0})/m\), which is provided by the inertial mass element and the eddy current damping element. The frequency-domain transfer function between \(F(t)\) and the input excitation is given as:  
\[H_{F}(s)=\frac{F(s)}{A_{g}(s)}=\frac{-(2\kappa\xi\omega_{\rm s}{}^{3}s\cos^{2} \theta+\kappa\omega_{\rm s}{}^{2}\mu s^{2})}{A(s,\xi,\zeta,\mu,\kappa,\omega_{ \rm s})\cos\theta}. \tag{24}\]