## 2 Mechanical Model
### Substructure Synthesis of a SDOF-Main System with SA-TLGD Attached  
The SDOF-main system with the assigned interaction force \(F_{x}\) from the SA-TLGD dynamics under combined seismic \(\ddot{w}_{g}\) and force excitation \(F(t)\), is considered in the next step of the substructure synthesis method (Figure 2b). The external dynamic force \(F(t)\) is interpreted as a wind load. The main system floor deformation is given by the displacement \(w(t)\), whereby the time variant \(P\)-\(\Delta\) effect is neglected. The moving floor mass \(m_{H}\) includes the dead weight \(m_{D,A}\) of the SA-TLGD and the modal masses of the vertical columns. The field stiffness of the columns is denoted with \(k\), and it includes the geometric correction of prestressing by the dead weight considered for the columns according to [39, 41]. Light structural damping is assumed by the damping coefficient \(b\).  
Conservation of momentum of the floor mass \(m_{S}\) yields the relevant linear equation of motion of the SDOF-main system with the attached SA-TLGD (Figure 2b),  
\[\ddot{w}+2\;\zeta_{S}\;\Omega_{S}\;\dot{w}+\Omega_{S}^{2}\;w=-\ddot{w}_{g}+ \frac{F(t)}{m_{S}}-\frac{F_{x}}{m_{S}},\;\zeta_{S}\;=\;\frac{b}{2\;m_{S}\;\Omega _{S}},\;\Omega_{S}^{2}\;=\;\frac{k_{S}}{m_{S}}, \tag{14}\]  
where \(\Omega_{S}\) and \(\zeta_{S}\) represent the undamped circular natural frequency of the SDOF-main system, and the linear viscous damping ratio. Inserting the coupling force \(F_{x}\) from Equation (13) into Equation (14) leads together with Equation (12) to the coupled system of linearized equations of motion for the seismic and force excited 2-DOF system,  
\[\begin{array}{l}\ddot{w}+2\;\zeta_{S}\;\Omega_{S}\;\dot{w}+\Omega_{S}^{2}\;w= -\ddot{w}_{g}+\frac{F(t)}{m_{S}}-\mu\;\ddot{w}_{t}-\mu\;\ddot{u}\;,\\ \ddot{u}+2\;\zeta_{A}\;\omega_{A}\;\dot{u}+\omega_{A}^{2}\;u\;=\;-\ddot{w}_{t} \;,\\ w_{t}=w_{g}+w\;.\end{array} \tag{15}\]  
The ratio of fluid mass to the moving mass of the SDOF-main system is defined by,  
\[\mu=\frac{m_{f}}{m_{S}}. \tag{16}\]  
To provide highest possible vibration suppression of the main system the mass ratio \(\mu\) should be maximized. However, from a practical point of view, i.e., limited installation space and avoidance of detrimental frequency shifts of the main system, the mass ratio \(\mu\) is typically chosen in range of 0.5-5%.  
To prepare for the equations of motion of a MDOF-main system with multiple, differently tuned SA-TLGDs attached, Equation (15) is rewritten in its linearized matrix form,  
\[\begin{array}{l}\mathbf{M_{S}}\left[\begin{array}{c}\ddot{w}\\ \ddot{u}\end{array}\right]+\mathbf{C_{S}}\left[\begin{array}{c}\dot{w}\\ \dot{u}\end{array}\right]+\mathbf{K_{S}}\left[\begin{array}{c}w\\ u\end{array}\right]\;=\;-\left[\begin{array}{cc}m_{S}+m_{f}\\ 1\end{array}\right]\ddot{w}_{g}+\;\left[\begin{array}{c}F(t)\\ 0\end{array}\right],\\ \mathbf{M_{S}}=\left[\begin{array}{cc}m_{S}+m_{f}&m_{f}\\ 1&1\end{array}\right],\;\;\mathbf{C_{S}}=\left[\begin{array}{cc}2\zeta_{S} \Omega_{S}m_{S}&0\\ 0&2\zeta_{A}\omega_{A}\end{array}\right],\;\;\mathbf{K_{S}}=\left[\begin{array} []{cc}k_{S}&0\\ 0&\omega_{A}^{2}\end{array}\right].\end{array} \tag{17}\]