## 3 Modeling of Elastomeric Bearings
### Algebraic Model Formulation  
In the Algebraic Model, \(u\) (\(f\)) represents the bearing transverse displacement (restoring force) whereas \(\dot{u}\) is the bearing transverse velocity.  
The typical restoring force-displacement hysteresis loop characterizing such a model may be limited by two parallel straight lines (curves), as shown in Figure 5a,b of Ref. [22]. In particular, the restoring force, during the generic loading case (\(\dot{u}>0\)), is evaluated as:  
\[f\Big{(}u,u_{j}^{+}\Big{)}=\left\{\begin{array}{ll}c^{+}\Big{(}u,u_{j}^{+} \Big{)}&u\in\Big{[}u_{j}^{+}-2u_{0},u_{j}^{+}\Big{]}\end{array}\right. \tag{1a}\] \[f\Big{(}u,u_{j}^{-}\Big{)}=\left\{\begin{array}{ll}c^{-}\Big{(}u,u_{j}^{-} \Big{)}&u\in\Big{[}u_{j}^{-},u_{j}^{-}+2u_{0}\Big{]}\end{array}\right. \tag{1b}\]  
whereas, during the generic unloading case (\(\dot{u}<0\)), it is computed as:  
\[f\Big{(}u,u_{j}^{-}\Big{)}=\left\{\begin{array}{ll}c^{-}\Big{(}u,u_{j}^{-} \Big{)}&u\in\Big{[}u_{j}^{-},u_{j}^{-}+2u_{0}\Big{]}\end{array}\right. \tag{2a}\] \[c_{l}(u)\qquad\qquad u\in\Big{(}-\infty,u_{j}^{-}\Big{]}. \tag{2b}\]  
In Equations (1) and (2), \(c^{+}\) and \(c^{-}\) represent, respectively, the generic loading and unloading curves:  
\[c^{+}\Big{(}u,u_{j}^{+}\Big{)}=\beta_{1}u^{3}+\beta_{2}u^{5}+k_{b}u+(k_{a}-k_ {b})\left[\frac{\Big{(}1+u-u_{j}^{+}+2u_{0}\Big{)}^{(1-a)}}{1-\alpha}-\frac{ \left(1+2u_{0}\right)^{(1-a)}}{1-\alpha}\right]+f_{0}, \tag{3}\]  
\[c^{-}\Big{(}u,u_{j}^{-}\Big{)}=\beta_{1}u^{3}+\beta_{2}u^{5}+k_{b}u+(k_{a}-k_ {b})\left[\frac{\Big{(}1-u+u_{j}^{-}+2u_{0}\Big{)}^{(1-a)}}{\alpha-1}-\frac{ \left(1+2u_{0}\right)^{(1-a)}}{\alpha-1}\right]-f_{0}. \tag{4}\]  
whereas \(c_{u}\) and \(c_{l}\) are, respectively, the upper and lower limiting curves:  
\[c_{u}(u)=\beta_{1}u^{3}+\beta_{2}u^{5}+k_{b}u+f_{0}, \tag{5}\]  
\[c_{l}(u)=\beta_{1}u^{3}+\beta_{2}u^{5}+k_{b}u-f_{0}. \tag{6}\]  
The internal variable \(u_{j}^{+}\) (\(u_{j}^{-}\)), representing the displacement where the generic loading (unloading) curve intersects the upper (lower) limiting curve, is given by:  
\[u_{j}^{+}=+1+u_{P}+2u_{0}-\left\{\frac{1-\alpha}{k_{a}-k_{b}}\Big{[}f_{P}- \beta_{1}u_{P}^{3}-\beta_{2}u_{P}^{5}-k_{b}u_{P}\ \ -f_{0}+(k_{a}-k_{b})\frac{\left(1+2u_{0}\right)^{(1-\alpha)}}{1-\alpha}\Big{]} \right\}^{\left(\frac{1}{1-\alpha}\right)}, \tag{7}\]  
\[u_{j}^{-}=-1+u_{P}-2u_{0}+\left\{\frac{\alpha-1}{k_{a}-k_{b}}\Big{[}f_{P}- \beta_{1}u_{P}^{3}-\beta_{2}u_{P}^{5}-k_{b}u_{P}\ \ +f_{0}+(k_{a}-k_{b})\frac{\left(1+2u_{0}\right)^{(1-\alpha)}}{\alpha-1}\Big{]} \right\}^{\left(\frac{1}{1-\alpha}\right)}, \tag{8}\]in which \(u_{P}\) and \(f_{P}\) are the coordinates of a generic point \(P\) belonging to \(c^{+}\) or \(c^{-}\), as shown in Figure 7 of Ref. [22] for a typical hysteresis loop limited by two parallel curves.  
Finally, the internal parameters \(f_{0}\) and \(u_{0}\), representing, respectively, the half of the force difference between the upper and lower limiting curves, and the half of the displacement difference between the starting and ending points of a generic loading or unloading curve, are evaluated as:  
\[f_{0}=\frac{k_{a}-k_{b}}{2}\Bigg{[}\frac{\left(1+2u_{0}\right)^{\left(1-\kappa \right)}-1}{1-\alpha}\Bigg{]}, \tag{9}\]  
\[u_{0}=\frac{1}{2}\Bigg{[}\left(\frac{k_{a}-k_{b}}{\delta_{k}}\right)^{\frac{1 }{\alpha}}-1\Bigg{]}, \tag{10}\]  
where \(\delta_{k}\) is a numerical parameter set equal to \(10^{-20}\), whereas \(k_{a}\), \(k_{b}\), and \(\alpha\), together with \(\beta_{1}\) and \(\beta_{2}\), represent the five model parameters to be calibrated from experimental or numerical data [24]. Specifically, \(k_{a}>k_{b}\), \(k_{a}>0\), \(\alpha>0\), \(\alpha\
eq 1\), whereas \(\beta_{1}\) and \(\beta_{2}\) are reals.