# Error Caused by Damping Formulating in Multiple Support Excitation Problems
## 3 Error in the Frequency Response Function of Different Modeling Methods
### Error in LMM
The mode shape matrix \(\mathbf{\Phi}_{ss}\) is normalized to satisfy \(\mathbf{\Phi}_{ss}^{\rm T}\mathbf{K}_{ss}\mathbf{\Phi}_{ss}=2\mathbf{I}\), where \(\mathbf{I}\) represents the identity matrix. The process of derivation is shown in Appendix A.  
\[\ddot{y}_{i}^{Lm}+2\zeta_{i}\omega_{i}\dot{y}_{i}^{Lm}+\omega_{i}^{2}y_{i}^{Lm }=\frac{1}{2}\big{\{}\mathbf{\Phi}_{ss}{}^{\rm T}\mathbf{K}_{sb}\big{\}}_{i,\star }\ddot{\mathbf{Y}}_{b}. \tag{12}\]  
where \(y_{i}^{Lm}\) is the response of mode \(i\); \(\omega_{i}\) and \(\zeta_{i}\) is the natural frequency and damping ratio for the \(i^{th}\) mode, respectively; \(\big{\{}\mathbf{\Phi}_{ss}{}^{\rm T}\mathbf{K}_{sb}\big{\}}_{i,\star}\) stands for the \(i^{th}\) row of the matrix \(\mathbf{\Phi}_{ss}{}^{\rm T}\mathbf{K}_{sb}\). Equation (12) is actually the same as the dynamic part of Equation (6) in modal coordinates. The responses of the excitation modes are the counterpart of the pseudo-static part of RMM in LMM; and this part can be solved directly from Equation (4) according to Equation (A6) and the conclusion that \(\mathbf{\Phi}_{bs}\rightarrow\mathbf{0}\). This derivation shows in one aspect that LMM is equivalent to RMM when the large masses approach infinity. This conclusion also justifies the assumption of the pseudo-static motion in RMM. Therefore, only the structural modes are studied. When the Rayleigh damping method is used in LMM, the damping matrix should be calculated without the large masses, as shown in Equation (13), otherwise, the large mass proportional damping would affect the accuracy of the support motions. \[\mathbf{C}=\alpha_{0}\Bigg{(} \begin{array}{cc}\mathbf{M}_{ss}&\mathbf{M}_{ sb}\\ \mathbf{M}_{bs}&\mathbf{0}\end{array} \Bigg{)}+\alpha_{1}\Bigg{(} \begin{array}{ cc}\mathbf{K}_{ss}&\mathbf{K}_{sb}\\ \mathbf{K}_{bs}&\mathbf{K}_{bb}\end{array} \Bigg{)}. \tag{13}\]  
\(\alpha_{0}\) and \(\alpha_{1}\) are parameters of the Rayleigh damping method. In the Rayleigh damping method, the damping ratios of the two modes with frequency \(\omega_{r_{1}}\) and \(\omega_{r_{2}}\) are determined at first. \(\alpha_{0}\) and \(\alpha_{1}\) are then calculated to make sure the damping ratio of these two modes being the expected value \(\zeta_{r_{1}}\) and \(\zeta_{r_{2}}\). When these two damping ratios are assumed to be equal, i.e., \(\zeta_{r_{1}}=\zeta_{r_{2}}=\zeta\), \(\alpha_{0}\) and \(\alpha_{1}\) can be solved by Equation (14). \[\alpha_{0}=2\zeta\frac{\omega_{r_{1}}\omega_{r_{2}}}{\omega_{r_{1}}+\omega_{r_{ 2}}},\ \alpha_{1}=2\zeta\frac{1}{\omega_{r_{1}}+\omega_{r_{2}}}, \tag{14}\]  
The damping ratios of the other modes are \(\alpha_{0}/2\omega_{i}+\alpha_{1}\omega_{i}/2\). The first mode of the structure is usually chosen as \(\omega_{r_{1}}\); the modes whose natural frequency larger than \(\omega_{r_{2}}\) would have a damping ratio larger than \(\zeta\) so that the energy of these modes is underestimated. Therefore, \(\omega_{\tau_{2}}\) is chosen carefully that the underestimation of these modes would not significantly affect the result. For this reason, in this paper, only the damping truncation error of the modes whose natural frequencies are between \(\omega_{\tau_{1}}\) and \(\omega_{\tau_{2}}\) are analyzed. The relationship among \(\omega_{\tau_{1}}\), \(\omega_{\tau_{2}}\) and the natural frequency of some of the modes are briefly depicted in Figure 1.  
The smallest possible value is \(\min\!\left(\zeta_{i}^{r}\right)=2\zeta\sqrt{\omega_{\tau_{1}}\omega_{\tau_{2}} }/\left(\omega_{\tau_{1}}+\omega_{\tau_{2}}\right)\), when \(\omega_{i}=\sqrt{\omega_{\tau_{1}}\omega_{\tau_{2}}}\) and \(\zeta_{i}^{r}\) is the damping ratio of the \(i^{th}\) mode. Usually, the smallest value cannot be achieved. Transforming the first row of Equation (7) into modal coordinates, the EOM is shown in Equation (15).