# Error Caused by Damping Formulating in Multiple Support Excitation Problems
## 3 Error in the Frequency Response Function of Different Modeling Methods
### Error in LMM
As a consequence, the error ratio at resonance frequency is a preferable value to represent the error in LMM-R. Table 2 shows examples of the value of \(r_{i}^{Lr}\Big{|}_{\omega=\omega_{i}=\sqrt{\omega_{r_{1}}\omega_{r_{2}}}}\) for different parameters. In Table 2, \(r_{i}^{Lr}\) increases with the increase of \(v\) and \(\zeta\); the influence of \(v\) is more significant than \(\zeta\). The error ratio can be as large as 43.3%. The value in Table 2 is a good reference for estimating the largest damping truncation error of a particular mode. An example is demonstrated in Figure 2 showing the shape and the trace of peaks of \(r_{i}^{Lr}\) when the mode has different natural frequencies. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\mathbf{\
u}\) & \(\mathbf{r_{i}^{Lr}(\zeta=0.02)}\) & \(\mathbf{r_{i}^{Lr}(\zeta=0.04)}\) & \(\mathbf{r_{i}^{Lr}(\zeta=0.06)}\) \\ \hline
2 & 0.064 & 0.073 & 0.085 \\
4 & 0.251 & 0.253 & 0.257 \\
6 & 0.429 & 0.431 & 0.433 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Error ratio \(r_{i}^{Lr}\) for different \(v\) and \(\zeta\). Figure 2: Examples of \(r^{Lr}\) for a particular parameter setting. In Figure 2, the large value of \(r_{i}^{Lr}\) near 0 Hz is aroused by the second term at the right-hand side of Equation (15). As is analyzed, though the error ratio near 0 Hz is very large, the absolute value of error is small. The large value is the main component of virtual damping error in LMM-R. When \(\omega<\omega_{i}\), the error ratio decrease to zero at first. After that, the error ratio increase and reach its peak at the resonance point, showing that the influence of the large value around 0 Hz and the errors around the resonance point are separate. As analyzed before the peak of the error ratio reach is top when \(\omega_{i}=\sqrt{\omega_{r_{1}}\omega_{r_{2}}}\), which is the main feature of the damping truncation error of the Rayleigh damping method. Studies have shown that in the MSE problem, more modes need to be considered [15]. The damping truncation error might not be negligible in the MSE problem. \[\textbf{C}\left( \begin{array}{c}\textbf{\Phi}_{sb}\\ \textbf{\Phi}_{bb}\end{array} \right)=\alpha_{0}\left( \begin{array}{c}\textbf {M}_{ss}\textbf{\Phi}_{sb}+\textbf{M}_{sb}\textbf{\Phi}_{bb}\\ \textbf{M}_{bs}\textbf{\Phi}_{bb}\end{array} \right). \tag{21}\]  
In LMM-R, the damping matrix **C** does not satisfy Equation (10). This means that the damping truncation error of LMM-R contains the virtual damping error. At the end of the next subsection, another Rayleigh damping formula is introduced, that can eliminate virtual damping in LMM-R. In the next subsection, using RMM with the Rayleigh damping method is proved free from virtual damping and has the same form as Equation (11). LMM has its advantage in time history study. The free-from virtual damping version of LMM can be easily derived from Equation (11). In this version, the Rayleigh damping is shown as follows. \[\textbf{C}=\alpha_{0}\left( \begin{array}{cc}\textbf{M}_{ss}&\textbf{M}_{sb }+\widetilde{\textbf{M}}_{sb}\\ \textbf{M}_{bs}&\textbf{0}\end{array} \right)+\alpha_{1}\left( \begin{array}[] {cc}\textbf{K}_{ss}&\textbf{K}_{sb}\\ \textbf{K}_{bs}&\textbf{K}_{bb}\end{array} \right); \tag{22}\]  
where \(\widetilde{\textbf{M}}_{sb}=\textbf{M}_{ss}\textbf{K}_{ss}{}^{-1}\textbf{K}_{ sb}\). With the help of \(\widetilde{\textbf{M}}_{sb}\), the second term at the right-hand side of Equation (15) is canceled. The EOM of the modal participation factor of mode \(i\) is shown as follows. \[\ddot{y}_{i}+\alpha_{0}\dot{y}_{i}+\alpha_{1}\omega_{i}^{2}\dot{y}_{i}+\omega_ {i}^{2}y_{i}=\frac{1}{2}\{\textbf{\Phi}_{ss}{}^{T}\textbf{K}_{sb}\}_{i,*} \widetilde{\textbf{Y}}_{b}; \tag{23}\]  
Except for the damping ratio, the equation has the same form as Equation (12). With Equation (22), the EOM of LMM-R approaches RMM-R when the large mass approach infinity. With the help of Equation (22), the EOM of the LMM-R would have no virtual damping error but has only the damping truncation error inside.