# Error Caused by Damping Formulating in Multiple Support Excitation Problems
## 3 Error in the Frequency Response Function of Different Modeling Methods
### Error in RMM  
The way RMM solves the response of the structure guarantees that the EOMs derived with RMM does not contain virtual damping. In Appendix A, the following equation is proved. \[\textbf{\Phi}_{sb}\rightarrow\textbf{R}\textbf{\Phi}_{bb}, \tag{24}\]  
Equation (24) indicates that the mode shapes related to the relative motion of the supports are not affected by the mass distribution of the structure. This conclusion gives the physical meaning to separate the responses of the structure into pseudo-static responses and dynamic responses in RMM. Substituting Equation (24) and (4) into Equation (2), Equation (25) can be obtained. \[\textbf{Y}_{s}^{d}=\textbf{Y}_{s}-\textbf{\Phi}_{sb}\textbf{\Phi}_{bb}^{-1} \textbf{Y}_{b}, \tag{25}\]  
In Equation (25), the dynamic responses equal to subtracting the influence of the rigid body modes and the modes related to the relative motion of the supports from the total responses. In other words, RMM does not contain virtual damping. To show this property of RMM in another aspect, put all the equations in RMM in a single equation. Adding Equation (4) to Equation (6) and transforming the result into absolute coordinates, the EOM of the absolute displacement is Equation (26). \[\mathbf{M}_{\text{ss}}\ddot{\mathbf{Y}}_{s}+\mathbf{C}_{\text{ss}}\dot{\mathbf{Y }}_{s}+\mathbf{K}_{\text{ss}}\mathbf{Y}_{s}=-\mathbf{C}_{\text{sb}}\dot{\mathbf{ Y}}_{b}-\mathbf{K}_{\text{sb}}\mathbf{Y}_{b}, \tag{26}\]  
\(\mathbf{C}_{\text{sb}}=-\mathbf{C}_{\text{ss}}\mathbf{R}\) is assumed in the RMM. Equation (26) is the same as the one introduced by [PERSON] et al. (2018), which is derived from a modified LMM. The form of \(\mathbf{C}_{\text{sb}}\) in Equation (9) guarantees the establishing of Equation (10). The integrated EOM in Equation (26) is more efficient and can also be used in nonlinear analysis. The above derivation proved that the assumption in RMM that \(\mathbf{C}_{\text{sb}}+\mathbf{C}_{\text{ss}}\mathbf{R}=\mathbf{0}\) will not induce the virtual damping error regardless of the damping formulating method. But using the Rayleigh damping method would induce the damping truncation error in the calculation. Transforming Equation (26) into modal coordinates, the dynamic equation for the modal participation vector is shown in Equation (27). \[\widetilde{y}_{i}^{R}=\frac{-2j\zeta_{i}s_{i}-1}{2(-s_{i}^{2}+2j\zeta_{i}s_{i} +1)}\mathbf{\Phi}_{\text{ss}}^{T}\mathbf{K}_{\text{sb}}\mathbf{Y}_{b}, \tag{27}\]  
where \(s_{i}=\omega/\omega_{i}\), \(y_{i}^{R}\) is the modal participation factor of mode \(i\) and \(\widetilde{y}_{i}^{R}\) is the Fourier transform of \(y_{i}^{R}\); \(\zeta_{i}\) is the damping ratio this mode. the damping truncation error is measured with error ratio \(r_{i}^{Rr}\), the EOM derived with RMM-M is used as the reference. When \(s_{i}=1\),  
\[r_{i}^{Rr}\big{|}_{s_{i}=1}=\Bigg{|}\frac{\widetilde{y}_{i}^{Rr}-\widetilde{y }_{i}^{Rm}}{\widetilde{y}_{i}^{Rm}}\Bigg{|}_{s=1}=\frac{1}{\sqrt{4\zeta^{2}+1 }}\big{|}\eta_{i}-1\big{|}; \tag{28}\]  
The superscripts \(Rm\) and \(Rr\) represent the modelling method RMM-M and RMM-R, respectively. In Equation (28), when \(\zeta\) is small enough, it has little influence on \(r_{i}^{Rr}\big{|}_{s_{i}=1}\); \(r_{i}^{Rr}\big{|}_{s=1}\) increases with \(\eta\); the largest value of \(r_{i}^{Rr}\big{|}_{s=1}\) is obtained in the mode whose natural frequency is the closest to the frequency \(\sqrt{\omega_{\
u_{1}}\omega_{\eta_{2}}}\). When \(s_{i}=2\),  
\[r_{i}^{Rr}\big{|}_{s=2}=\frac{16\big{|}\zeta_{i}^{r}-\zeta\big{|}}{\sqrt{(16 \zeta_{i}^{r2}+9)(16\zeta^{2}+1)}}\approx\frac{16}{3}\zeta\Big{|}\frac{1}{ \eta_{i}}-1\Big{|}; \tag{29}\]  
When \(s\) approaches infinity,  
\[r_{i}^{Rr}\big{|}_{s\sim\infty}\rightarrow\bigg{|}\frac{1}{\eta_{i}}-1\bigg{|}. \tag{30}\]  
The larger \(\eta_{i}\) is, the larger the error is. For a particular mode, the error ratio decreases rapidly from the resonance frequency and then increases gradually as \(s\) approach infinity.