# Error Caused by Damping Formulating in Multiple Support Excitation Problems
## 3 Error in the Frequency Response Function of Different Modeling Methods
### Error in LMM  
Properly formulating a support damping matrix can eliminate the virtual damping error. [PERSON] et al. [8] derived the support damping matrix from Equation (7) with the modal damping method. In the modal damping method, the modal damping matrix \(\left( \begin{array}{cc}\mathbf{C}_{\text{ss}}&\mathbf{C}_{\text{sb}}\\ \mathbf{C}_{\text{bs}}&\mathbf{C}_{\text{bb}}\end{array} \right)\) is calculated with \(\left( \begin{array}{cc}\mathbf{M}_{\text{ss}}&\mathbf{M}_{\text{sb}}\\ \mathbf{M}_{\text{bs}}&\mathbf{M}_{\text{bb}}+\overline{\mathbf{M}}_{\text{bb} }\end{array} \right)\)and \(\left( \begin{array}{cc}\mathbf{K}_{\text{ss}}&\mathbf{K}_{\text{sb}}\\ \mathbf{K}_{\text{bs}}&\mathbf{K}_{\text{bb}}\end{array} \right)\); [PERSON] et al. [8] showed that when the large masses approach infinity, \(\mathbf{C}_{\text{ss}}\) in \(\left( \begin{array}{cc}\mathbf{C}_{\text{ss}}&\mathbf{C}_{\text{sb}}\\ \mathbf{C}_{\text{bs}}&\mathbf{C}_{\text{bb}}\end{array} \right)\)converges to the modal damping matrix calculated with \(\mathbf{M}_{\text{ss}}\) and \(\mathbf{K}_{\text{ss}}\), at the same time, the support damping matrix \(\mathbf{C}_{\text{sb}}\) converges to  
\[\mathbf{C}_{\text{sb}}\rightarrow\mathbf{C}_{\text{ss}}\mathbf{K}_{\text{ss}} ^{-1}\mathbf{K}_{\text{sb}}, \tag{9}\]  
where \(\rightarrow\) stands for convergence. When \(\mathbf{C}_{\text{sb}}=\mathbf{C}_{\text{ss}}\mathbf{K}_{\text{ss}}^{-1} \mathbf{K}_{\text{sb}}\),  
\[\left( \begin{array}{cc}\mathbf{C}_{\text{ss}}&\mathbf{C}_{\text{sb}}\end{array}  \right)\left( \begin{array}{cc}\mathbf{\Phi}_{\text{sb}}\\ \mathbf{\Phi}_{\text{bb}}\end{array} \right)=\mathbf{0}, \tag{10}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Capital Superscript** & **Modeling Method** \\ \hline \(L\) & LMM \\ \(IL\) & LMM when the largest masses approach infinity \\ \(R\) & RMM \\ \(A\) & ADM \\ \(U\) & The method used for uniform excitation problem in relative coordinates \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of the capital superscripts. where \(\left( \begin{array}{cc}\mathbf{\Phi}_{sb}T&\mathbf{\Phi}_{bb}T\end{array} \right)^{\rm T}\) stand for the rigid modes and the modes related to the relative movement of the supports. The damping matrix in Equation (7) satisfies Equation (10), which is to say, in LMM when the modal damping method is used and when the large masses approaches infinity the virtual damping equal to zero. \(\mathbf{C}_{sb}\) can be seen as a damping compensator that can eliminate virtual damping from the modal of the upper structure. It can work with the other support damping in soil-structure interaction problems. [PERSON] et al. [8] introduce an integrated EOM as shown in Equation (11) by extracting the first line of Equation (7) and assuming \(\mathbf{C}_{sb}=\mathbf{C}_{sb}\mathbf{K}_{ss}^{-1}\mathbf{K}_{sb}\). \[\mathbf{M}_{ss}\tilde{\mathbf{Y}}_{s}+\mathbf{C}_{ss}\dot{\mathbf{Y}}_{s}+ \mathbf{K}_{ss}\mathbf{Y}_{s}=-\mathbf{C}_{sb}\dot{\mathbf{Y}}_{b}-\mathbf{K}_ {sb}\mathbf{Y}_{b}, \tag{11}\]  
This EOM is equivalent to LMM when the large masses approach infinity and has a simpler expression compared with RMM. In the next subsection, this equation is proved to be equivalent to RMM. In the following context of this section, the EOM derived by the LMM-M would be used as a reference to show the damping truncation error in the LMM-R method. In linear systems, the responses of a structure can be expressed as the linear combinations of all the modal responses. The damping truncation error will be studied in the frequency domain of modal coordinates. The modes of the structure in Equation (7) is divided into two parts. The first part contains the modes having zero support motion, called structural modes in this paper; the others are called excitation modes. Transform Equation (7) into modal coordinates, the dynamic equation for each modal participation factor is shown in Equation (12).