## 2 Mechanical System
### Mechanical Model  
To model the coupled system, the multi-degree-of-freedom frame structure was represented by a two-degree-of-freedom (2-DOF) equivalent system, which corresponds with a two-story shear-type frame structure. The mechanical characteristics (mass and stiffness) of the 2-DOF system were determined using the dynamic equivalence procedure proposed in [3; 4; 32]. This dynamic equivalence yielded the mass \(m_{i}\), stiffness \(k_{i}\), and damping \(c_{i}\) (\(i=1,2\)) of the two equivalent stories, thus resulting in a low-dimensional mechanical model that effectively captured the main dynamics of the M-DOF frame structure. In contrast, the external structure was modeled as a 1-DOF mechanical system, where the VI-NES was embedded in its inertial mass (Figure 1b). The protection system is characterized by mass \(m_{e}\), stiffness \(k_{e}\), and damping \(c_{e}\), while \(M\) represents the mass of the vibro-impacting mass (see Figure 1b).  
In total, the full coupled system is described by three Lagrangian parameters. Specifically, \(u_{1}\) and \(u_{2}\) represent the two degrees of freedom of the frame structure, whereas \(u_{M}\) is associated with the motion of the vibro-impacting mass (Figure 1c). It is worth noting that the external structure embedding the vibro-impacting mass did not require any additional Lagrangian parameters as it was rigidly connected to the structure to be protected.  
The Lagrangian parameter \(u_{1}\) corresponds to the displacement of the first floor, which is the floor where the protection system is connected to the structure. Similarly, \(u_{2}\) describes the displacement of the top story of the frame structure. In this manner, the quantity \(\Delta u=u_{2}-u_{1}\) represents the drift of the superstructure. The three equations of motion for the coupled mechanical system were obtained through a direct approach, and they can be expressed as follows:  
Figure 1: Mechanical system: (**a**) Coupled frame structure and the proposed external device; (**b**) geometrical characterization of the low-order mechanical model; and (**c**) the Lagrangian parameters and positive directions.  
\[(m_{1}+m_{e})\dot{u}_{1}+(c_{11}+c_{e})\dot{u}_{1}+c_{12}\dot{u}_{2}+(k_{1}+k_{2}+ k_{e})u_{1}-k_{2}u_{2}=-(m_{1}+m_{e})\dot{u}_{g}, \tag{1}\]  
\[m_{2}\ddot{u}_{2}+c_{21}\dot{u}_{1}+c_{22}\dot{u}_{2}-k_{2}u_{1}+k_{2}u_{2}=-m_{ 2}\ddot{u}_{g}, \tag{2}\]  
\[M\ddot{u}_{M}=0, \tag{3}\]  
where \(\dot{u}_{g}(t)\) is the ground acceleration. In Equation (1), which represents the dynamical equilibrium associated with the Lagrangian parameter \(u_{1}\), it is noteworthy that, due to the rigid connection between the structure and the protection system, the mass \(m_{e}\), stiffness \(k_{e}\), and damping \(c_{e}\) simply add to \(m_{1}\), \(k_{1}\), and \(c_{11}\), respectively. The quantities \(c_{ij}\) with \((i,j)=(1,2)\) are the coefficients of the symmetric Rayleigh damping matrix of the stand-alone frame structure. Such a Rayleigh damping matrix is obtained by  
\[\left[\begin{array}{cc}c_{11}&c_{12}\\ c_{21}&c_{22}\end{array}\right]=\alpha\left[\begin{array}{cc}m_{1}&0\\ 0&m_{2}\end{array}\right]+\beta\left[\begin{array}{cc}k_{1}+k_{2}&-k_{2}\\ -k_{2}&k_{2}\end{array}\right]. \tag{4}\]  
In Equation (4), the coefficients \(\alpha\) and \(\beta\) are obtained assuming that the relative damping coefficients of both modes of the stand-alone frame structure have the same value, which are denoted as \(\xi=0.05\). Finally, the damping \(c_{e}\) is obtained by treating the external structure as a 1-DOF system with a relative damping of \(\xi_{e}=0.05\). Specifically, \(c_{e}=2m_{e}\xi_{e}\omega_{e}\), where \(\omega_{e}=\sqrt{k_{e}/m_{e}}\).