## 2 Equation of Motion for a Structure Coupled with an Equipment
\begin{vmatrix}\mathbf{m_{s}l_{s}\ddot{U}_{g}}\\ m_{ii}\ddot{U}_{g}-F_{i}\end{vmatrix}  \tag{6}\]  
The equation of motion for the structure without equipment, as shown in Figure 2c, can be expressed as Equation (7), which is transformed into Equation (8) in the frequency domain and simplified as Equation (9). \[ \begin{bmatrix}\begin{matrix}\mathbf{m_{ss}}&0\\ \mathbf{0}&m_{ii}\end{bmatrix}\begin{vmatrix}\ddot{u}_{s}^{*}\\ \ddot{u}_{i}^{*}\end{vmatrix} + \begin{bmatrix}\mathbf{c_{ss}}&-\mathbf{c_{si}}\\ -\mathbf{c_{is}}&c_{ii}\end{bmatrix}\begin{vmatrix}\dot{u}_{s}^{*}\\ \dot{u}_{i}^{*}\end{vmatrix} + \begin{bmatrix}\mathbf{k_{ss}}&-\mathbf{k_{si}}\\ -\mathbf{k_{is}}&k_{ii}\end{bmatrix}\begin{vmatrix}\mathbf{u}_{s}^{*}\\ u_{i}^{*}\end{vmatrix} =- \begin{vmatrix}\mathbf{m_{s}l_{s}}\\ m_{ii}\end{vmatrix} \ddot{u}_{g} \tag{7}\]  
\[ \begin{pmatrix}\begin{bmatrix}\mathbf{k_{ss}}&-\mathbf{k_{si}}\\ -\mathbf{k_{is}}&k_{ii}\end{bmatrix}-\mathbf{\omega}^{2}\begin{bmatrix}\mathbf{m_{ss}}&\mathbf{0} \\ \mathbf{0}^{T}&m_{ii}\end{bmatrix}+j\omega\begin{bmatrix}\mathbf{c_{ss}}&-\mathbf{c_{si}}\\ -\mathbf{c_{is}}&c_{ii}\end{bmatrix}\end{pmatrix}\begin{vmatrix}\mathbf{U_{s}}^{*}\\ \dot{U}_{i}^{*}\end{vmatrix} =- \begin{vmatrix}\mathbf{m_{s}l_{s}}\\ m_{ii}\end{vmatrix} \ddot{U}_{g} \tag{8}\]\[\left( \begin{bmatrix}\mathbf{K_{ss}}&\mathbf{K_{si}}\\ \mathbf{K_{is}}&K_{ii}\end{bmatrix} \right)\begin{\{}\mathbf{U}_{s}^{*}\\ \mathbf{U}_{i}^{*}\end{subarray}\}=- \begin{cases}\mathbf{m_{s}I_{s}}\\ \mathbf{m_{ii}}\end{cases} \ddot{U}_{g} \tag{9}\]  
where \(\mathbf{u}_{s}^{*}\) is the displacement vector of the structure and \(u_{i}^{*}\) is the displacement at the node to which the equipment is connected. By subtracting Equation (8) from Equation (6), the relation between the equipment-structure interaction force and the corresponding displacement change at the equipment-structure interface is derived as Equation (14) as follows:  
\[\left( \begin{bmatrix}\mathbf{K_{ss}}&\mathbf{K_{si}}\\ \mathbf{K_{is}}&K_{ii}\end{bmatrix} \right)\begin{\{}\mathbf{U}_{s}-\mathbf{U}_{s}^{*}\\ \mathbf{U}_{i}-\mathbf{U}_{i}^{*}\end{\}}= \begin{cases}\mathbf{0}\\ F_{i}\end{cases}  \tag{10}\]  
\[\mathbf{K_{ss}}(\mathbf{U}_{s}-\mathbf{U}_{s}^{*})+\mathbf{K_{si}}(U_{i}-U_{i}^{*})=\mathbf{0} \tag{11}\]  
\[\mathbf{K_{is}}(\mathbf{U}_{s}-\mathbf{U}_{s}^{*})+K_{ii}(U_{i}-U_{i}^{*})=F_{i} \tag{12}\]  
\[(\mathbf{U}_{s}-\mathbf{U}_{s}^{*})=-\mathbf{K_{ss}^{-1}}\mathbf{K_{si}}(U_{i}-U_{i}^{*}) \tag{13}\]  
\[X_{i}(U_{i}-U_{i}^{*})=F_{i} \tag{14}\]  
where \(X_{i}\) is the impedance function of the structure expressed as  
\[X_{i}=\mathbf{K_{ii}}-\mathbf{K_{is}}\mathbf{K_{ss}}^{-1}\mathbf{K_{si}} \tag{15}\]  
Equation (4), the equation of motion considering the equipment-structure interaction, can be rewritten as Equation (16) by substituting \(F_{i}\) with Equation (14). Equation (16) can be rewritten as a two-DOF equation of motion in the frequency domain, as shown in Equation (17), and the corresponding solution can be obtained as Equation (18). \[ \begin{bmatrix}K_{ii}&K_{ie}\\ \mathbf{K_{ei}}&K_{ee}\end{bmatrix} \begin{\{}U_{i}\\ \mathbf{U}_{e}\end{bmatrix}=- \begin{cases}F_{i}\\ m_{e}\ddot{U}_{g}\end{cases} =- \begin{cases}X_{i}(U_{i}-U_{i}^{*})\\ m_{e}\ddot{U}_{g}\end{cases}  \tag{16}\]  
\[ \begin{bmatrix}X_{i}+K_{ii}&K_{ie}\\ K_{ei}&K_{ee}\end{bmatrix} \begin{\{}U_{i}\\ \mathbf{U}_{e}\end{bmatrix}= \begin{cases}F_{i}^{*}\\ -m_{e}\ddot{U}_{g}\end{cases}  \tag{17}\]  
\[\begin{\{}U_{i}\\ U_{e}\end{subarray}\}= \begin{bmatrix}X_{i}+K_{ii}&K_{ie}\\ K_{ei}&K_{ee}\end{bmatrix} ^{-1} \begin{cases}F_{i}^{*}\\ -m_{e}\ddot{U}_{g}\end{cases}  \tag{18}\]  
where  
\[F_{i}^{*}=X_{i}U_{i}^{*} \tag{19}\]  
is the effective force that can reproduce the structural response at the interface only caused by the ground motion.