# Base Pounding Model and Response Analysis of Base-Isolated Structures under Earthquake Excitation
## 2 Models and Equations of Motion
### Equations of Motion  
Assuming that the stiffness of the floor slabs in-plane is infinite and the masses of the floor slabs are lumped at the floor levels, base bounding models were built with linear spring-gap elements. Figures 3 and 4 present the models of a base-isolated structure with lead-rubber bearing and with friction pendulum isolation bearing, respectively. The equations of motion of the superstructure are expressed in Equation (2). \[\left\{ \begin{array}{l}m_{1}\ddot{x}_{1}+c_{1}\dot{x}_{1}\ -c_{1}\dot{x}_{0}+f_{1}-f_{2}=-m_{1}\ddot{x}_{g}\\ m_{i}\ddot{x}_{i}+c_{i}\dot{x}_{i}+f_{i}-f_{n}=-m_{i}\ddot{x}_{g}\\ m_{n}\ddot{x}_{n}\ +c_{n}\dot{x}_{n}+f_{n}=-m_{n}\ddot{x}_{g}\end{array} \right., \tag{2}\]  
where \(\dot{x}_{i},\ddot{x}_{i}(i=1,2\cdot\cdot\cdot n)\) are the relative velocities and accelerations of floor \(i\), respectively, \(\ddot{x}_{g}\) is the earthquake ground motion acceleration, \(m_{i}(i=1,2\cdot\cdot n)\) and \(c_{i}(i=1,2\cdot\cdot\cdot n)\) are the mass and damping of floor \(i\), respectively. The restoring force of floor \(i\) is expressed by the following equation:  
\[f_{i}=k_{i}(x_{i}-x_{i-1})\,(i=1,2\cdot\cdot\cdot n), \tag{3}\]  
where \(k_{i}\) is the stiffness of floor \(i\), and \(x_{i}\) is the relative displacement of floor \(i\). Figure 3: Model of the base-isolated structure with lead-rubber bearing. Figure 2: Force-displacement curve. Rayleigh Damping is calculated by the following equation:  
\[c_{i}=\alpha m_{i}+\beta\mathrm{k}_{i}\,(i=1,2\cdots n), \tag{4}\]  
where \(\alpha\), \(\beta\) are calculated by Equation (5) if the damping ratios \(\xi_{i}\) and \(\xi_{j}\) associated with specific frequencies \(\omega_{i}\), \(\omega_{j}\) are known. \[\left\{ \begin{array}{c}\alpha=2\omega_{i}\xi_{j}(\omega_{j}\xi_{i}-\omega_{ i}\omega_{j})/(\omega_{j}{}^{2}-\omega_{i}{}^{2})\\ \beta=2(\omega_{j}\xi_{j}-\omega_{i}\xi_{i})/(\omega_{j}{}^{2}-\omega_{i}{}^{2 })\end{array} \right., \tag{5}\]  
Equations of motion for the isolation layer (the base-isolated structure with lead-rubber bearing) are given as,  
\[m_{0}\ddot{x}_{0}+(c_{0}+c_{1})\dot{x}_{0}-c_{1}\dot{x}_{1}+f_{0}-f_{1}+f_{p}= -m_{0}\ddot{x}_{g}, \tag{6}\]  
where \(m_{0}\) is the mass of isolation layer, \(\dot{x}_{0}\), \(\ddot{x}_{0}\) are relative the velocities and accelerations of the isolation layer, respectively, \(c_{0}\) is the damping coefficient of the isolation layer, \(f_{0}\) and \(f_{p}\) are the restoring and the impact force the of isolation layer, respectively. Equations for the restoring force have been built using the Bouc-wen model:  
\[f_{0}=\alpha_{0}k_{0}x_{0}+(1-\alpha_{0})k_{0}x_{y}z_{0}, \tag{7}\]  
where \(k_{0}\) is the isolation layer's initial stiffness, \(\alpha_{0}\) is the ratio of the yield stiffness to the pre-yield stiffness of bearing, \(x_{0}\) is the displacement of the isolation layer, \(z_{0}\) is the hysteretic displacement of the isolation system, and \(x_{y}\) is the yield displacement. The first order differential equation of the hysteretic displacement is given as,  
\[\dot{z}_{0}=(-\gamma_{0}|\dot{x}_{0}|z_{0}|z_{0}|^{n_{0}-1}-\beta_{0}\dot{x}_{ 0}|z_{0}|^{n_{0}}+A_{0}\dot{x}_{0})/x_{y}, \tag{8}\]  
where \(\beta_{0}\), \(A_{0}\), \(\gamma_{0}\), and \(n_{0}\) are related to the amplitude of hysteretic displacement, initial stiffness, and hysteretic shape. Equations of motion of the isolation layer (the base-isolated structure with friction pendulum isolation bearing) is given as,  
\[m_{0}\ddot{x}_{0}+c_{1}\dot{x}_{0}-c_{1}\dot{x}_{1}+f_{0}-f_{1}+f_{p}+f_{f}=-m _{0}\ddot{x}_{g}, \tag{9}\]  
Figure 4: Model of the base-isolated structure with friction pendulum isolation bearing. Restoring force can be calculated by Equation (10). \[f_{0}=k_{0}x_{0}, \tag{10}\]  
where \(k_{0}\) is the stiffness of bearing, \(x_{0}\) is the displacement of isolation layer.