# Error Caused by Damping Formulating in Multiple Support Excitation Problems
## 2 Review of Modeling Methods  
The free vibration EOM of a structure containing the excitation degrees of freedom is shown in Equation (1). \[\left( \begin{array}{cc}\mathbf{M}_{ss}&\mathbf{M}_{sb}\\ \mathbf{M}_{bs}&\mathbf{M}_{bb}\end{array} \right)\left( \begin{array}{c}\ddot{ \mathbf{Y}}_{s}\\ \ddot{\mathbf{Y}}_{b}\end{array} \right)+\left( \begin{array}{cc}\mathbf{C}_{ ss}&\mathbf{C}_{sb}\\ \mathbf{C}_{bs}&\mathbf{C}_{bb}\end{array} \right)\left( \begin{array}{c}\dot{ \mathbf{Y}}_{s}\\ \dot{\mathbf{Y}}_{b}\end{array} \right)+\left( \begin{array}{cc}\mathbf{K}_{ ss}&\mathbf{K}_{sb}\\ \mathbf{K}_{bs}&\mathbf{K}_{bb}\end{array} \right)\left( \begin{array}{c} \mathbf{Y}_{s}\\ \mathbf{Y}_{b}\end{array} \right)=\left( \begin{array}{c}\mathbf{0}\\ \mathbf{0}\end{array} \right). \tag{1}\]  
The matrices \(\mathbf{M}\), \(\mathbf{C}\), \(\mathbf{K}\) are the mass, damping, and stiffness matrices, respectively, for different degrees of freedom; The subscript \(s\) and \(b\) refer to the structural degrees of freedom and the excitation degrees of freedom; \(\mathbf{Y}\) stands for the absolute displacement of degrees of freedom and \(\mathbf{0}\) stands for the zero matrices. In RMM, the responses of a structure are separated into dynamic responses and pseudo-static responses as shown in Equation (2). \[\left( \begin{array}{c}\mathbf{Y}_{s}\\ \mathbf{Y}_{b}\end{array} \right)=\left( \begin{array}{c}\mathbf{Y}_{s}^{d}\\ \mathbf{0}\end{array} \right)+\left( \begin{array}{c}\mathbf{Y}_{s}^{p}\\ \mathbf{Y}_{b}\end{array} \right). \tag{2}\]  
The pseudo-static responses satisfy the following equation:  
\[\left( \begin{array}{cc}\mathbf{K}_{ss}&\mathbf{K}_{sb}\\ \mathbf{K}_{bs}&\mathbf{K}_{bb}\end{array} \right)\left( \begin{array}{c} \mathbf{Y}_{s}^{p}\\ \mathbf{Y}_{b}^{p}\end{array} \right)=\left( \begin{array}{c}\mathbf{0}\\ \mathbf{0}\end{array} \right). \tag{3}\]  
Define \(\mathbf{R}=-\mathbf{K}_{ss}^{-1}\mathbf{K}_{sb}\) then the pseudo-static responses can be solved by Equation (4). \[\mathbf{Y}_{s}^{p}=\mathbf{RY}_{b}. \tag{4}\]  
The dynamic responses of the structure can be solved by Equation (5). \[\mathbf{M}_{ss}\ddot{\mathbf{Y}}_{s}{}^{d}+\mathbf{C}_{ss}\dot{\mathbf{Y}}_{s} {}^{d}+\mathbf{K}_{ss}\mathbf{Y}_{s}{}^{d}=-(\mathbf{M}_{ss}\mathbf{R}+\mathbf{ M}_{sb})\ddot{\mathbf{Y}}_{b}-(\mathbf{C}_{ss}\mathbf{R}+\mathbf{C}_{sb})\dot{ \mathbf{Y}}_{b}. \tag{5}\]  
According to [PERSON] et al. [1], the damping term at the right-hand side of Equation (5) is much smaller than the inertial term and thus is negligible and the dynamic responses of the structure can be solved by Equation (6). Most of the studies considering multiple support excitation adopted Equation (6) as the EOM of the dynamic responses. \[\mathbf{M}_{ss}\ddot{\mathbf{Y}}_{s}{}^{d}+\mathbf{C}_{ss}\dot{\mathbf{Y}}_{s }{}^{d}+\mathbf{K}_{ss}\mathbf{Y}_{s}{}^{d}=-(\mathbf{M}_{ss}\mathbf{R}+\mathbf{ M}_{sb})\ddot{\mathbf{Y}}_{b}. \tag{6}\]  
The EOM of LMM is shown in Equation (7), where \(\mathbf{\widetilde{M}}_{bb}\) stands for the added large masses. \[\left( \begin{array}{cc}\mathbf{M}_{ss}&\mathbf{M}_{sb}\\ \mathbf{M}_{bs}&\mathbf{M}_{bb}+\mathbf{\widetilde{M}}_{bb}\end{array} \right) \left( \begin{array}{c}\ddot{\mathbf{Y}}_{s}\\ \ddot{\mathbf{Y}}_{b}\end{array} \right)+\left( \begin{array}{cc}\mathbf{C}_{ ss}&\mathbf{C}_{sb}\\ \mathbf{C}_{bs}&\mathbf{C}_{bb}\end{array} \right)\left( \begin{array}{c} \dot{\mathbf{Y}}_{s}\\ \dot{\mathbf{Y}}_{b}\end{array} \right)+\left( \begin{array}{cc}\mathbf{K}_{ ss}&\mathbf{K}_{sb}\\ \mathbf{K}_{bs}&\mathbf{K}_{bb}\end{array} \right)\left( \begin{array}{c} \mathbf{Y}_{s}\\ \mathbf{Y}_{b}\end{array} \right)=\left( \begin{array}{c}\mathbf{0}\\ \mathbf{\widetilde{M}}_{bb}\end{array} \right)\ddot{\mathbf{Y}}_{b} \tag{7}\]Extract the first line of Equation (7) and ignore the inertial and damping contribution of the support. The EOM of ADM is shown in Equation (8).