# Seismic Benefits from Coupling Frame Structures with a Hysteretic Mass Damper Inerter
## 2 Mechanical Model
### Equations of Motion  
Figure 2 shows the simplified structure, which is the reference for the low-dimensional mechanical model used to capture the behavior of the multi-DOF frame structure linked to the HMDI. Specifically, a 2-DOF shear-type structural model is used to describe the main dynamics of the frame structure, regardless of the number of its stories. The mechanical characteristics of such a 2-DOF model are obtained by using the dynamic equivalence criterion defined in [18], which was developed mainly for low- and medium-rise frame structures. The dynamical equivalence consists in determining both the equivalent stiffness coefficients \(k_{1}\) and \(k_{2}\), and the masses \(m_{1}\) and \(m_{2}\) that correspond to the masses of the first story and the sum of the masses of all the stories above the first one, respectively (Figure 2a). According to the procedure presented in [18], to derive the values for the equivalent characteristics, the DOFs \(u_{1}\) and \(u_{2}\) have to correspond to the displacements of the first and top stories of the structure, respectively. The following additional assumptions are at the base of the equivalence criterion:  
* The circular frequencies \(\omega\) of the leading mode of the M-DOF frame structure and of the first mode of the 2-DOF system (Figure 2) are the same;
* The modal components of the first and top stories of the frame structure are equal to the modal components \(u_{1}\) and \(u_{2}\) of the equivalent low-dimensional system. Figure 1: Frame structure coupled with the HMDI at the first story. Furthermore, the Rayleigh formulation is used to derive the damping coefficients of the equivalent 2-DOF model by considering the damping ratio \(\xi=0.05\) for both modes. A simple 1-DOF system with displacement \(u_{ext}\) (Figure 2b) describes the HMDI. The physical mass \(m\) of the external oscillating system is increased through an inerter device with inertance (or virtual mass) \(m_{I}\) and is linked to the first story of the frame structure through a hysteretic connection described by the classical Bouc-Wen model [35]. The equations of motion of the 3-DOF system composed of the 2-DOF share-type structural model plus the protection device are obtained with a direct approach, such as,  
\[ \begin{split}& m_{1}\dot{u}_{1}(t)+c_{1}\dot{u}_{1}(t)-c_{12}\dot{u }_{2}(t)+(k_{1}+k_{2})u_{1}(t)-k_{2}u_{2}(t)-F_{H}=-m_{1}\dot{x}_{g}(t),\\ & m_{2}\dot{u}_{2}(t)-c_{12}\dot{u}_{1}(t)+c_{2}\dot{u}_{2}(t)-k_ {2}u_{1}(t)+k_{2}u_{2}(t)=-m_{2}\ddot{x}_{g}(t),\\ &(m+m_{I})\ddot{u}_{ext}(t)+F_{H}=-m\dot{u}_{g}(t),\end{split}  \tag{1}\]  
where the external excitation is denoted by \(\ddot{u}_{g}\), and \(F_{H}\) is the coupling hysteretic force. The three equations (Equation (1)) represent the equilibrium of the forces acting on the masses \(m_{1}\), \(m_{2}\), and \(m+m_{I}\), respectively. Moreover, Equation (1) (third line equation) shows that the external excitation \(\ddot{u}_{g}\) acts only on the physical mass \(m\) of the protection device, whereas the virtual mass \(m_{I}\) is only affected by the relative acceleration \(\ddot{u}_{ext}\) acting on its connection points. The application of the Rayleigh method provides a full symmetric damping matrix \(\mathbf{C}=\)\(\alpha\mathbf{M}+\beta\mathbf{K}\), where quantities \(c_{12}=c_{21}\), in Equation (1), are the off-diagonal terms of \(\mathbf{C}\). According to the Bouc-Wen model [35], the restoring hysteretic force \(F_{H}\) can be expressed as  
\[F_{H}=\psi k_{c}(u_{ext}(t)-u_{1}(t))+k_{c}u_{y}(1-\psi)z(t). \tag{2}\]  
In Equation (2), the ratio between post-yielding stiffness \(k_{\text{plast}}\) and pre-yielding stiffness is represented by \(\psi\) (i.e., \(k_{\text{plast}}=\psi k_{c}\)). The case with \(\psi=0\) describes the elastic perfectly plastic behavior of the connection device; \(\psi=1\) represents a fully elastic behavior. The yielding force of the hysteretic connection \(F_{y}\) is provided by the product \(k_{c}u_{y}\), with \(u_{y}\) being the yielding displacement.