# Error Caused by Damping Formulating in Multiple Support Excitation Problems
## 3 Error in the Frequency Response Function of Different Modeling Methods
### Error in ADM  
In this section, the error caused by ignoring support damping, i.e., the ADM, will be studied. Firstly, virtual damping error is studied. Transforming Equation (8) into modal coordinates, the EOM of ADM is shown in Equation (31). \[\widehat{y}_{i}^{A}=\frac{-1}{2(-s^{2}+2j\zeta_{i}s+1)}\mathbf{\Phi}_{ss}^{T} \mathbf{K}_{sb}\mathbf{Y}_{b}, \tag{31}\]  
Firstly, the error ratio \(r_{i}^{Am}\) is derived. When the modal damping method is used, it is shown in Equation (32)  
\[r_{i}^{Am}=\left|\frac{\widehat{y}_{i}^{Am}(s_{i})-\frac{\gamma_{i}^{Rm}(s_{i })}{\widehat{y}_{i}^{Rm}(s_{i})}}{\widehat{y}_{i}^{Rm}(s_{i})}\right|=\sqrt{1- \frac{1}{4\zeta^{2}s_{i}^{2}+1}}, \tag{32}\]  
where the superscript \(Am\) and \(Lm\) represents the EOM derive by ADM-M and LMM-M respectively. When \(s_{i}=0\), \(r_{i}^{Am}=0\); when \(s_{i}\) is small,  
\[r_{i}^{Am}\big{|}_{s_{i}\in(0,\epsilon)}\approx 2\zeta s_{i}. \tag{33}\]  
When \(s_{i}\) approaches infinity, \(r_{i}^{Am}\to 1\). The error ratio for different damping ratios and different \(s_{i}\) is shown in Table 4.  
Figure 3: Examples of \(r^{Rr}\) for a particular parameter setting. As shown in Table 4 and Equation (33), the errors increase with the increase of frequency, and also increase with the damping ratio of the mode; the error ratio can be 10% at the resonance frequency when \(\zeta=0.05\); when the natural frequency of the mode is smaller than the dominant frequency of the earthquake, the error can also be significant in the higher frequency range. Figure 4 shows examples of \(r^{Am}\) for modes with different natural frequencies, the parameter setting is the same as Figure 2.  
As is shown in Figure 4, the shape of \(r^{Am}\) is not related to \(\omega_{i}\). \(r^{Am}\) increases monotonously with the increase in frequency and finally approach 1 as the frequency approach infinity. The value of \(r^{Am}\) in Figure 4 at the resonance points are marked with squares and the value of them are all 10.0%; the value of \(r^{Am}\) when \(\omega=2\omega_{i}\) are all 19.6%, these values are the same as predicted in Table 4\(r^{Am}\) is caused by neglecting support damping, and is one kind of virtual damping error. The same as LMM-R, in ADM-R, both the virtual damping error and the damping truncation error exist at the same time. The error ratio of ADM-R when \(s_{i}=1\) is shown in Equation (34). \[r_{i}^{Ar}\big{|}_{s_{i}=1}=\left|\frac{\widehat{y}_{i}^{Ar}(s_{i})-\widehat{ y}_{i}^{Rm}(s_{i})}{\widehat{y}_{i}^{Rm}(s_{i})}\right|_{s_{i}=1}=\frac{1}{ \sqrt{4\zeta^{2}+1}}\big{|}\eta_{i}-1\big{|}, \tag{34}\]  
The influence of \(\zeta\) is small, \(r_{i}^{Ar}\big{|}_{s_{i}=1}\) also achieves its possible maximum value when \(\omega_{i}=\sqrt{\omega_{r_{1}}\omega_{r_{2}}}\); when \(s_{i}=2\), \(r_{i}^{Ar}\big{|}_{s_{i}=2}\) increases linearly with \(\zeta\), and increases with the increase of \(\eta\). \[r_{i}^{Ar}\big{|}_{s_{i}=2}=4\zeta\sqrt{\frac{(-4\eta_{i}+1)^{2}+16\zeta^{2}}{ (16\zeta^{2}+9\eta_{i}^{2})(16\zeta^{2}+1)}}\approx\Big{|}\frac{16}{3}-\frac{ 4}{3\eta_{i}}\Big{|}\zeta, \tag{35}\]  
When \(s_{i}\) approaches infinity, \(r_{i}^{Ar}\big{|}_{s_{i}\rightarrow\infty}\) approaches 1. Table 5 shows the value of \(r_{i}^{Ar}\) for different parameters. Figure 4: Examples of \(r^{Am}\) for a particular parameter setting. In Table 5, the value of \(r_{i}^{Ar}\big{|}_{s_{i}=2}\) can be larger than \(r_{i}^{Ar}\big{|}_{s_{i}=1}\), indicating that for structures having natural frequencies smaller than the dominant frequency of the excitation, the error caused by the excitation can be larger than the error caused by resonance response. The influence of \(\
u\) is more significant than \(\zeta\) at resonance frequency, whereas the influence of \(\zeta\) is significant when \(s_{i}=2\). Figure 5 shows the features of \(r^{Ar}\), the parameter setting is the same as Figure 2.  
As is shown in Figure 5, the main features of \(r^{Ar}\) are the combination of \(r^{Ar}\) and \(r^{Kr}\). \(r^{Ar}\) is the joint result of virtual damping error and damping truncation error.