# Advanced Hysteretic Model of a Prototype Seismic Isolation System Made of Polymeric Bearings
## 5 Numerical Analysis
### Single-Storey Structure Model  
In order to evaluate dynamic response of the fixed-based single-storey model in numerical analysis, the structure was idealized as 1-DOF system, for which the differential equation of motion is given in Equation (6). This simple numerical model is characterized by three parameters: lumped mass m\({}_{1}\) lumped at the roof level (i.e., first storey level), lateral stiffness k\({}_{1}\), and viscous damping c\({}_{1}\). The exact values of these parameters, which are also given in Equations (7)-(9), were already determined during the previously conducted shaking table investigation (see [28] for details):  
\[\mathrm{m}_{1}\cdot\ddot{\mathrm{u}}(\mathrm{t})+\mathrm{c}\cdot\ddot{\mathrm{u }}(\mathrm{t})+\mathrm{k}_{1}\cdot\mathrm{u}(\mathrm{t})=-\mathrm{m}_{1}\cdot \ddot{\mathrm{u}}_{\mathrm{g}}(\mathrm{t}) \tag{6}\]  
\[\omega=2\cdot\pi\cdot\mathrm{f} \tag{7}\]  
\[\mathrm{k}_{1}=\omega^{2}\cdot\mathrm{m}_{1} \tag{8}\]  
\[\mathrm{c}_{1}=2\cdot\mathrm{m}_{1}\cdot\omega\cdot\xi, \tag{9}\]  
Employing parameters determined in the previous study ([28]), i.e., dominant frequency of the fixed-base single-storey structure, f, equal to 3.34 Hz (and, consequently, angular frequency, \(\omega\), equal to 20.98 rad\({}^{-1}\) according to Equation (7)), lumped-mass concentrated at the top level, m\({}_{1}\), equal to 46.76 kg, and damping ratio, \(\xi\), equal to 0.5%, into the Equations (8) and (9), stiffness and damping was obtained as k\({}_{1}=\)20,571\(\frac{\mathrm{N}}{\mathrm{m}}\) and c\({}_{1}=\) 10.48\(\frac{\mathrm{kg}}{\mathrm{s}}\), respectively. The single-storey structure model mounted on the foundation base plate supported with PBs (as shown in Figure 2) was considered as 2-DOF system, for which the differential equation of motion can be presented in a general matrix form given in Equation (10). \[\mathbf{M}\cdot\ddot{\mathbf{u}}(\mathrm{t})+\mathbf{C}\cdot\dot{\mathbf{u}} (\mathrm{t})+\mathbf{K}\cdot\mathbf{u}(\mathrm{t})=-\mathbf{M}\mathbf{I} \cdot\ddot{\mathrm{u}}_{\mathrm{g}}(\mathrm{t}) \tag{10}\]  
In this formula, \(\mathbf{M}\), \(\mathbf{C}\), and \(\mathbf{K}\) denote mass, damping, and stiffness matrices, respectively, \(\ddot{\mathbf{u}}(\mathrm{t})\), \(\dot{\mathbf{u}}(\mathrm{t})\), \(\mathbf{u}(\mathrm{t})\) are the acceleration, velocity and displacement vectors, respectively, and \(\ddot{\mathrm{u}}_{\mathrm{g}}(\mathrm{t})\) stands for the acceleration input ground motion. Additional matrix \(\mathbf{I}\) is the influence coefficient matrix, having 1 for elements corresponding to degrees of freedom in the direction of the applied ground motion and 0 for other degrees of freedom. The structure was mounted on a base plate of mass m\({}_{\mathrm{b}}\) which is supported on a base isolation system made of PBs characterized by lateral stiffness k\({}_{\mathrm{b}}(\mathrm{u}_{\mathrm{b}}(\mathrm{t}),\dot{\mathrm{u}}_{\mathrm{b}}( \mathrm{t}))\), and viscous damping c\({}_{\mathrm{b}}\), as defined in Equations (4) and (5). Accordingly, for the base-isolated single-storey model idealized as 2-DOF system, matrices \(\mathbf{M}\), \(\mathbf{C}\), \(\mathbf{K}\), \(\ddot{\mathbf{u}}(\mathrm{t})\), \(\dot{\mathbf{u}}(\mathrm{t})\), \(\mathbf{u}(\mathrm{t})\) and \(\mathbf{I}\) are defined in Equations (11)-(15). \[\mathbf{M}=\left[ \begin{array}{cc}\mathrm{m}_{\mathrm{b}}&0\\ 0&\mathrm{m}_{1}\end{array} \right] \tag{11}\]  
\[\mathbf{C}=\left[ \begin{array}{cc}\mathrm{c}_{\mathrm{b}}+\mathrm{c}_{1}&- \mathrm{c}_{1}\\ -\mathrm{c}_{1}&\mathrm{c}_{1}\end{array} \right] \tag{12}\]\[\mathbf{K}=\left[ \begin{array}{cc}\mathrm{k}_{\mathrm{b}}\big{(}\mathrm{u}_{ \mathrm{b}}(\mathrm{t}),\dot{\mathrm{u}}_{\mathrm{b}}(\mathrm{t})\big{)}+ \mathrm{k}_{1}&-\mathrm{k}_{1}\\ -\mathrm{k}_{1}&\mathrm{k}_{1}\end{array} \right] \tag{13}\]  
\[\mathbf{I}=\left\{ \begin{array}{c}1\\ 1\end{array} \right\} \tag{14}\]  
\[\ddot{\mathbf{u}}(\mathrm{t})=\left\{ \begin{array}{c}\ddot{\mathrm{u}}_{ \mathrm{b}}(\mathrm{t})\\ \ddot{\mathrm{u}}_{\mathrm{1}}(\mathrm{t})\end{array}