# Seismic Study of An Isolated Cable-Stayed Bridge under Near-Fault Ground Motions
## 2 Longitudinal Seismic-Reduction System
### Viscous Damper between Tower and Girder  
Apart from friction pendulum bearings (FPBs), fluid viscous dampers (FVDs) are another type of damping device applied to long-span bridges. The large-span cable-stayed bridge with twin towers is a multi-degree-of-freedom system. Solving its dynamic equations under seismic actions is challenging. This study focuses on a semi-floating twin-tower cable-stayed bridge and examines the longitudinal displacement at its beam ends. Neglecting the friction effects of the longitudinal displacement supports, the longitudinal seismic response of the bridge is similar to that of a floating cable-stayed bridge. Previous research has shown that when there is a significant disparity between the stiffness of the main tower and the beam, the structural behavior is more like a double-degree-of-freedom model (DDFM) system. Furthermore, using a single-degree-of-freedom model for calculations will result in significant errors. Therefore, in this paper, we adopt a DDFM, as described in reference [24]. A DDFM simplifies the bridge by considering two mass blocks, one for the tower and one for the beam. The tower's mass is concentrated at the top of the tower. The parameters of the simplified viscous dampers are determined with equivalent linearization principles. As shown in Figure 2, \(\text{m}_{\text{b}}\) and \(\text{k}_{\text{b}}\) represent the beam's mass and longitudinal resistance stiffness, respectively. \(\text{m}_{\text{t}}\) and \(\text{k}_{\text{t}}\) represent the main tower's mass and longitudinal resistance stiffness.  
Figure 1: Illustration of the FPB: (**a**) schematic of FPB; (**b**) force–displacement relationship.  
In this paper, the force \(F_{\text{rd}}\) of the viscous dampers depends on the relative velocity between the two ends of the structures and can be expressed as Equation (1):  
\[F_{\text{rd}}=c\upsilon^{\alpha} \tag{1}\]  
where \(c\) is the viscous coefficient and \(\alpha\) is the velocity factor.  
According to [PERSON] principle, the equation of motion for the system can be derived as Equation (2):  
\[\begin{bmatrix}m_{b}&0\\ 0&m_{t}\end{bmatrix}\begin{Bmatrix}\tilde{x}_{b}\\ \tilde{x}_{t}\end{Bmatrix}+\begin{bmatrix}c_{b}+c_{1.0}&-c_{b}-c_{1.0}\\ -c_{b}-c_{1.0}&c_{b}+c_{1.0}+c_{t}\end{bmatrix}\begin{Bmatrix}\tilde{x}_{b}\\ \tilde{x}_{t}\end{Bmatrix}+\begin{bmatrix}k_{b}&-k_{b}\\ -k_{b}&k_{b}+k_{t}\end{bmatrix}\begin{Bmatrix}x_{b}\\ x_{t}\end{Bmatrix}=-\begin{bmatrix}m_{b}&0\\ 0&m_{t}\end{bmatrix}\begin{Bmatrix}1\\ 1\end{Bmatrix}\tilde{x}_{g} \tag{2}\]  
According to D'Alembert's principle, the equation of motion for the system can be derived as in Equation (3):  
\[c_{1.0}=2m_{b}\xi_{d}\omega_{b}=2m_{b}\omega_{b}(\frac{\xi_{T}\sqrt{\alpha_{1} }[(1-\gamma)+(1-\alpha_{1})^{2}\gamma]-\xi_{t}\sqrt{((1-\gamma)/R_{t})}(1- \alpha_{1})^{2}}{\alpha_{1}^{2}(1-\gamma)}-\xi_{b}) \tag{3}\]  
where \(\alpha_{1.0}\) is the equivalent damping coefficient of the nonlinear viscous damper between the tower and the beam.  
In the above equation, \(c_{1.0}\) is independent of dynamic input and is solely related to the inherent properties of the structure. Therefore, the damping coefficient of the equivalent linear viscous damper for the structure can be expressed as Equation (4) [25]:  
\[c_{d}=\frac{2\pi m_{b}(1+R_{t})^{\alpha-1}}{\omega_{b}D^{\alpha-1}\lambda}( \frac{\xi_{T}\sqrt{\alpha_{1}}[(1-\gamma)+(1-\alpha_{1})^{2}\gamma]-\xi_{t} \sqrt{((1-\gamma)/R_{t})}(1-\alpha_{1})^{2}}{\alpha_{1}^{2}(1-\gamma)}-\xi_{b}) \tag{4}\]  
where \(\omega\) represents the excitation frequency, it is equal to the structural natural frequency \(\omega_{b}^{2}=k_{b}/m_{b}\); \(\gamma=m_{t}/(m_{t}+m_{b})\); stiffness ratio \(R_{t}=k_{b}/k_{t}\); the velocity exponent \(\alpha_{1}=\omega_{1}^{2}/\omega_{b}^{2}\); \(D\) is the maximum displacement of the main beam.