## 2 Equation of Motion for a Structure Coupled with an Equipment  
The equations of motion for a structure coupled with equipment are formulated assuming that the primary system is an MDOF structure, and the equipment is represented by an SDOF oscillator, as shown in Figure 2a. The equations of motion for the equipment and structure are expressed by Equations (1) and (2), respectively, as follows:  
\[ \begin{bmatrix}0&0\\ 0&m_{e}\end{bmatrix}\begin{bmatrix}\ddot{u}_{i}\\ \ddot{u}_{e}\end{bmatrix} + \begin{bmatrix}c_{e}&-c_{e}\\ -c_{e}&c_{e}\end{bmatrix} \begin{vmatrix}\dot{u}_{i}\\ \ddot{u}_{e}\end{bmatrix}+ \begin{bmatrix}k_{e}&-k_{e}\\ -k_{e}&k_{e}\end{bmatrix}\begin{vmatrix}u_{i}\\ u_{e}\end{vmatrix} =- \begin{vmatrix}0\\ m_{e}\end{vmatrix} \ddot{u}_{g}- \begin{vmatrix}f_{i}\\ 0\end{vmatrix}  \tag{1}\]  
\[ \begin{bmatrix}\mathbf{m_{ss}}&\mathbf{0}\\ \mathbf{0}^{T}&m_{ii}\end{bmatrix}\begin{vmatrix}\ddot{u}_{s}\\ \ddot{u}_{i}\end{vmatrix} + \begin{bmatrix}\mathbf{c_{ss}}&-\mathbf{c_{si}}\\ -\mathbf{c_{is}}&c_{ii}\end{bmatrix}\begin{vmatrix}\mathbf{u}_{s}\\ \dot{u}_{i}\end{vmatrix} + \begin{bmatrix}\mathbf{k_{ss}}&-\mathbf{k_{si}}\\ -\mathbf{k_{is}}&k_{ii}\end{bmatrix}\begin{vmatrix}\mathbf{u}_{s}\\ u_{i}\end{vmatrix} =- \begin{vmatrix}\mathbf{m_{s}l_{s}}\\ m_{ii}\end{vmatrix} \ddot{u}_{g}+ \begin{vmatrix}0\\ f_{i}\end{vmatrix}  \tag{2}\]  
where \(u_{i}\) is the displacement at the interface between the equipment and structure; \(u_{e}\), \(\mathbf{u}_{s}\), \(f_{i}\) and \(\ddot{u}_{g}\) are the displacements of the equipment and structure, interaction force between the equipment and structure, and ground acceleration, respectively. In addition, \(\mathbf{m}_{e}\), \(\mathbf{c_{e}}\) and \(\mathbf{k_{e}}\) are the mass, damping coefficient, and stiffness of the equipment, respectively; \(\mathbf{m_{ss}}\) and \(m_{ii}\) are the mass of the structure; \(\mathbf{c_{ss}}\), \(\mathbf{c_{si}}\), \(\mathbf{c_{is}}\) and \(c_{ii}\) are the damping of the structure; \(\mathbf{k_{ss}}\), \(\mathbf{k_{si}}\), \(\mathbf{k_{is}}\) and \(k_{ii}\) are the stiffness of the structure; and the subscripts \(i\) and \(s\) indicate the DOFs at the interface of the equipment and the structure, and the other DOFs in the structure, respectively. Equations (1) and (2) can be transformed into Equations (3) and (4), which are defined in the frequency domain using Fourier transform. \[ \begin{pmatrix}\begin{bmatrix}k_{e}&-k_{e}\\ -k_{e}&k_{e}\end{bmatrix}-\omega^{2}\begin{bmatrix}0&0\\ 0&m_{e}\end{bmatrix}+j\omega\begin{bmatrix}c_{e}&-c_{e}\\ -c_{e}&c_{e}\end{bmatrix}\end{pmatrix} \begin{vmatrix}\dot{U}_{i}\\ \dot{U}_{e}\end{pmatrix}=- \begin{vmatrix}0\\ m_{e}\end{vmatrix} \ddot{U}_{g}- \begin{vmatrix}F_{i}\\ 0\end{vmatrix}  \tag{3}\]  
\[ \begin{pmatrix}\begin{bmatrix}\mathbf{k_{ss}}&-\mathbf{k_{si}}\\ -\mathbf{k_{is}}&k_{ii}\end{bmatrix}-\omega^{2}\begin{bmatrix}\mathbf{m_{ss}}&\mathbf{0}\\ \mathbf{0}^{T}&m_{ii}\end{bmatrix}+j\omega\begin{bmatrix}\mathbf{c_{ss}}&-\mathbf{c_{si}}\\ -\mathbf{c_{is}}&c_{ii}\end{bmatrix}\end{pmatrix}\begin{vmatrix}\mathbf{U_{s}}\\ \dot{U}_{i}\end{vmatrix} =- \begin{vmatrix}\mathbf{m_{s}l_{s}}\\ m_{ii}\end{vmatrix} \ddot{U}_{g}+ \begin{vmatrix}0\\ F_{i}\end{vmatrix}  \tag{4}\]  
where \(\omega\) is the circular frequency; \(j\) is the imaginary number; and \(U_{e}\), \(U_{i}\), \(\mathbf{U}_{s}\), \(\ddot{U}_{g}\) and \(F_{i}\) are the Fourier transforms of \(u_{i}\), \(u_{e}\), \(\mathbf{u}_{s}\), \(\ddot{u}_{g}\) and \(f_{i}\), respectively. The frequency-dependent dynamic characteristics of the equipment and structure are combined with the dynamic stiffnesses corresponding to each DOFs so that Equations (3) and (4) are rewritten as follows:  
\[ \begin{bmatrix}K_{ii}&K_{ie}\\ K_{ei}&K_{ee}\end{bmatrix} \begin{vmatrix}\dot{U}_{i}\\ \dot{U}_{e}\end{bmatrix}=- \begin{vmatrix}F_{i}\\ m_{e}\ddot{U}_{g}\end{vmatrix}  \tag{5}\]  
\[ \begin{bmatrix}\mathbf{K_{ss}}&\mathbf{K_{si}}\\ \mathbf{K_{is}}&K_{ii}\end{bmatrix}\begin{vmatrix}\mathbf{U_{s}}\\ \dot{U}_{i}\end{vmatrix} =-