## 2 Mechanical Model
### Model of the Frame Structure Connected to Vibrating Mass and an Inerter Device
\begin{split}& m_{e}(\dot{u}_{e}+\dot{u}_{g})+M_{R}(\dot{u}_{e}+ \ddot{u}_{M}-\ddot{\theta}_{e}h_{e}+\ddot{u}_{g})+M_{I}(\ddot{u}_{e}+\dot{u}_ {M}-\ddot{\theta}_{e}h_{e})+\\ & c_{e}u_{e}-c_{es}(\dot{u}_{s}-\dot{u}_{e})+k_{e}u_{e}-k_{es}(u_ {s}-u_{e})=0,\\ &(I_{e}+I_{M})\ddot{\theta}_{e}-M_{R}(\ddot{u}_{e}+\ddot{u}_{M}- \ddot{\theta}_{e}h_{e}+\ddot{u}_{g})h_{e}-M_{I}(\ddot{u}_{e}+\ddot{u}_{M}- \ddot{\theta}_{e}h_{e})h_{e}+\\ & c_{\theta}\dot{\theta}_{e}-c_{\theta e_{es}}(\dot{\theta}_{s}- \dot{\theta}_{e})+k_{\theta e}\theta_{e}-c_{\theta e_{es}}(\theta_{s}-\theta_{e })=0,\\ & M_{R}(\dot{u}_{e}+\ddot{u}_{M}-\ddot{\theta}_{e}h_{e}+\ddot{u}_ {g})+M_{I}(\dot{u}_{e}+\ddot{u}_{M}-\ddot{\theta}_{e}h_{e})+c_{M}\dot{u}_{M}+ k_{M}u_{M}=0,\\ & m_{s}(\ddot{u}_{s}+\ddot{u}_{g})+m_{1}(\ddot{u}_{s}+\ddot{u}_{1}- \ddot{\theta}_{s}h_{1}+\ddot{u}_{g})+m_{2}(\ddot{u}_{s}+\ddot{u}_{2}-\ddot{ \theta}_{s}h_{2}+\ddot{u}_{g})+\\ & c_{s}\dot{u}_{s}+c_{es}(\dot{u}_{s}-\dot{u}_{e})+k_{s}u_{s}+k_{es }(u_{s}-u_{e})=0,\\ &(I_{s}+I_{1}+I_{2})\ddot{\theta}_{s}-m_{1}(\ddot{u}_{s}+\ddot{u}_ {1}-\ddot{\theta}_{s}h_{1}+\ddot{u}_{g})h_{1}-m_{2}(\ddot{u}_{s}+\ddot{u}_{2}- \ddot{\theta}_{s}h_{2}+\ddot{u}_{g})h_{2}+\\ & c_{\theta s}\dot{\theta}_{s}+c_{\theta e_{es}}(\dot{\theta}_{s}- \dot{\theta}_{e})+k_{\theta s}\theta_{s}+k_{es}(\theta_{s}-\theta_{e})=0,\\ & m_{1}(\ddot{u}_{s}+\ddot{u}_{1}-\ddot{\theta}_{s}h_{1}+\ddot{u}_ {g})+c_{11}\dot{u}_{1}+c_{12}\dot{u}_{2}+k_{1}u_{1}-k_{2}(u_{2}-u_{1})=0,\\ & m_{2}(\ddot{u}_{s}+\ddot{u}_{2}-\ddot{\theta}_{s}h_{2}+\ddot{u}_ {g})+c_{21}\dot{u}_{1}+c_{22}\dot{u}_{2}-k_{2}u_{1}+k_{2}u_{2}=0.\end{split}  \tag{6}\]It worth observing that, contrarily to the oscillating mass \(M_{r}\), the virtual mass of the inerter device \(M_{I}\) responds only to the relative acceleration between its two terminals, \(\ddot{u}_{e}+\ddot{u}_{M}-\ddot{\theta}_{e}h_{e}\). As a consequence, the external seismic acceleration \(\ddot{x}_{\delta}\) does not influence the virtual mass \(M_{I}\) as it does to the real mass \(M_{r}\). In Equation (6), \(c_{ij}\), where \(\{i,j\}=\{1,2\}\), represents the coefficients of the symmetric Rayleigh damping matrix of the standalone frame structure on rigid soil. The Rayleigh damping matrix is obtained according to Equation (7):  
\[\left[ \begin{array}{cc}c_{11}&c_{12}\\ c_{21}&c_{22}\end{array} \right]=\alpha\left[ \begin{array}{cc}m_{1}&0\\ 0&m_{2}\end{array} \right]+\beta\left[ \begin{array}{cc}k_{1}+k_{2}&-k_{2}\\ -k_{2}&k_{2}\end{array} \right], \tag{7}\]  
where coefficients \(\alpha\) and \(\beta\) are derived under the assumption of identical relative damping coefficients for both modes of the standalone frame structure, denoted by \(\tilde{\zeta}=0.05\). Finally, the damping \(c_{M}\) is determined by modeling the moving mass \(M\) as a 1-DOF system with a relative damping of \(\tilde{\zeta}_{M}=0.05\). Specifically, \(c_{M}=2M_{M}^{2}\omega_{M}\), where \(\omega_{M}=\sqrt{k_{M}/M}\) and \(M=M_{r}+M_{I}\). Figure 2: Archetype low-dimensional mechanical model. (**a**) Geometrical and mechanical characteristics of the system. (**b**) Positive Lagrangian parameters.