# Advanced Hysteretic Model of a Prototype Seismic Isolation System Made of Polymeric Bearings
## 5 Numerical Analysis
### Two-Storey Structure Model  
In the next step of the numerical investigation, 2-DOF and 3-DOF systems were applied to simulate the dynamic behaviour of the fixed-base and base-isolated two-storey structure models, respectively. The general matrix form of differential equation of motion has already been presented in Equation (10). Similarly to the single-storey structure model, for the fixed-based two-storey structure model, matrices **M**, **C**, **K**, \(\ddot{\textbf{u}}(\textbf{t})\), \(\dot{\textbf{u}}(\textbf{t})\), \(\textbf{u}(\textbf{t})\) and **I** are defined as given in Equations (16)-(20), whereas for the base-isolated case, these matrices are presented in Equations (21)-(25):  
\[\textbf{M}=\left[ \begin{array}{cc}\textbf{m}_{1}&0\\ 0&\textbf{m}_{2}\end{array} \right] \tag{16}\]  
\[\textbf{C}=\left[ \begin{array}{cc}\textbf{c}_{1}+\textbf{c}_{2}&-\textbf{c} _{2}\\ -\textbf{c}_{2}&\textbf{c}_{2}\end{array} \right] \tag{17}\]  
\[\textbf{K}=\left[ \begin{array}{cc}\textbf{k}_{1}+\textbf{k}_{2}&-\textbf{k} _{2}\\ -\textbf{k}_{2}&\textbf{k}_{2}\end{array} \right] \tag{18}\]  
\[\textbf{I}=\left\{ \begin{array}{c}1\\ 1\end{array} \right\} \tag{19}\]  
\[\ddot{\textbf{u}}(\textbf{t})=\left\{ \begin{array}{c}\ddot{\textbf{u}}_{1}( \textbf{t})\\ \ddot{\textbf{u}}_{2}(\textbf{t})\end{array} \right\},\dot{\textbf{u}}(\textbf {t})=\left\{ \begin{array}{c}\dot{\textbf{u}}_{1}(\textbf{t})\\ \dot{\textbf{u}}_{2}(\textbf{t})\end{array} \right\}\text{, }\textbf{u}(\textbf{t})=\left\{ \begin{array}{c}\textbf{u}_{1}( \textbf{t})\\ \textbf{u}_{2}(\textbf{t})\end{array} \right\} \tag{20}\]  
\[\textbf{M}=\left[ \begin{array}{ccc}\textbf{m}_{\text{b}}&0&0\\ 0&\textbf{m}_{1}&0\\ 0&0&\textbf{m}_{2}\end{array} \right] \tag{21}\]  
\[\textbf{C}=\left[ \begin{array}{ccc}\textbf{c}_{\text{b}}+\textbf{c}_{1}&- \textbf{c}_{1}&0\\ -\textbf{c}_{1}&\textbf{c}_{1}+\textbf{c}_{2}&-\textbf{c}_{2}\\ 0&-\textbf{c}_{2}&\textbf{c}_{2}\end{array} \right] \tag{22}\]  
\[\textbf{K}=\left[ \begin{array}{ccc}\textbf{k}_{\text{b}}(\textbf{u}_{\text {b}}(\textbf{t}),\dot{\textbf{u}}_{\text{b}}(\textbf{t}))+\textbf{k}_{1}&- \textbf{k}_{1}&0\\ -\textbf{k}_{1}&\textbf{k}_{1}+\textbf{k}_{2}&-\textbf{k}_{2}\\ 0&-\textbf{k}_{2}&\textbf{k}_{2}\end{array} \right] \tag{23}\] \begin{table}
\begin{tabular}{l l l l l} \hline \hline \multirow{3}{*}{**Dynamic Excitation**} & \multicolumn{3}{c}{**Numerically Computed Peak Lateral**} & \multirow{3}{*}{**Reduction Based on**} & \multirow{3}{*}{**Reduction Based on**} \\  & **Acceleration for the Single-Storey** & & & **Reduction Based on** \\  & **Structure Model (m/s\({}^{2}\))** & & & **Shaking Table** \\ \cline{2-5}  & **Fixed-Base** & **Base-Isolated** & & **Investigation (\%)**[28] \\  & **(1-DOF System)** & & & \\ \hline
1989 Loma Prieta earthquake & & & & \\ Corralitos station, NS & & & & \\ component, scaled record with & & & & \\ PGA = 3.158 m/s\({}^{2}\) & & & & \\ \hline
1994 Northridge earthquake & & & & & \\ Santa Monica station, EW & & & & \\ component, scaled record with & & & & \\ PGA = 4.332 m/s\({}^{2}\) & & & & \\ \hline
2002 Polkovoice mining tremor & & & & \\ NS component, original record & & & & \\ with PGA = 1.634 m/s\({}^{2}\) & & & & \\ \hline \hline \end{tabular}  
Where: PGA: Peak Ground Acceleration, NS: North-South, EW: East-West.  
\end{table} 
Table 3: _Cont._\[\mathbf{I}=\left\{ \begin{array}{c}1\\ 1\\ 1\end{array} \right\} \tag{24}\]  
\[\ddot{\mathbf{u}}(\mathrm{t})=\left\{ \begin{array}{c}\ddot{\mathbf{u}}_{ \mathrm{b}}(\mathrm{t})\\ \ddot{\mathbf{u}}_{\mathrm{I}}(\mathrm{t})\\ \ddot{\mathbf{u}}_{\mathrm{2}}(\mathrm{t})\end{array} \right\},\dot{\mathbf{u}}( \mathrm{t})=\left\{ \begin{array}{c}\dot{\mathbf{u}}_{\mathrm{b}}(\mathrm{t}) \\ \dot{\mathbf{u}}_{\mathrm{I}}(\mathrm{t})\\ \dot{\mathbf{u}}_{\mathrm{2}}(\mathrm{t})\end{array} \right\},\ \mathbf{u}( \mathrm{t})=\left\{ \begin{array}{c}\mathbf{u}_{\mathrm{b}}(\mathrm{t})\\ \mathbf{u}_{\mathrm{I}}(\mathrm{t})\\ \mathbf{u}_{\mathrm{2}}(\mathrm{t})\end{array}