# Error Caused by Damping Formulating in Multiple Support Excitation Problems
## 4 Damping Formulating Error in MSE Problem  
In the previous section, the virtual damping error and the damping truncation error that exist in the FRF of the modal participation factors are studied. In this section, a spectral analysis is used to demonstrate how the effect of MSE affects the accuracy of the structural responses is studied. The spatially variate ground motion is simulated as a random process, this method has been used by many researchers to study the effect of MSE. The elements in the power spectral density (PSD) matrix of the excitation are usually described as follows. \[S_{e,ij}(\omega)=\gamma_{ij}(\omega)\sqrt{S_{e,ii}(\omega)S_{e,jj}(\omega)} \tag{36}\]  
where \(S_{e,ij}\) is the cross power spectral density function between the ground displacement \(x_{i}\) and \(x_{j}\). \(\gamma_{ij}\) is the coherence function for \(x_{i}\) and \(x_{j}\). \(\big{|}\gamma_{ij}\big{|}\), which is the amplitude of the coherence function, usually named as lagged coherence [16] or incoherence effect [17], is represented by \(\gamma_{ij}^{l}(\omega)\) in this study. The phase of \(\gamma_{ij}\) is mainly composed of the wave passage effect, represented as \(\gamma_{ij}^{w}(\omega)\), and site response effect describing the phase difference  
Figure 5: Examples of \(r^{Ar}\) for a particular parameter setting. between different soil conditions, represented as \(\gamma_{ij}^{s}(\omega)\). [PERSON] [17] proposed the following formula for describing \(\gamma_{ij}\). \[\gamma_{ij}(\omega)=\gamma_{ij}^{J}(\omega)\gamma_{ij}^{w}(\omega)\gamma_{ij}^{ s}(\omega)=\gamma^{J}(\omega)\exp[i(\theta_{ij}^{w}(\omega)+\theta_{ij}^{s}( \omega))] \tag{37}\]  
The phases of the coherence function including the wave passage effect and the site-response effect are shown as follows. \[\theta_{ij}^{w}(\omega)=-T_{ij}\omega,\ \theta_{ij}^{s}(\omega)=\theta_{j}( \omega)-\theta_{i}(\omega) \tag{38}\]  
where \(T_{ij}=\eta_{ij}t_{ij}\), describing the time for the ground motion to travel from support \(i\) to \(j\); \(t_{ij}=d_{ij}/v_{p}\) is the time delay parameter, \(d_{ij}\) is the distance between support \(i\) and \(j\), \(v_{p}\) is the apparent propagation velocity of the earthquake; \(\eta_{ij}\) is a coefficient describing the incident angle of the ground motion, \(-1\leq\eta_{ij}\leq 1\); \(T_{ij}\) is a universal coefficient describing the influence of both \(v_{p}\) and \(\eta_{ij}\); \(\theta_{i}(\omega)\) is the phase of the local soil filter at support \(i\). Then, the PSD of the absolute displacement responses in modal coordinates is derived. To better illustrate how different factors of MSE affect the structural responses, the soil conditions at different structure supports are assumed to be the same. The PSD of the concerned responses can be derived as follows [1]. \[S_{y_{i}}(\omega)=S_{g}\big{|}f_{y_{i}}\big{|}\Bigg{[}\sum_{j=1}^{n_{s}}q_{ij} ^{2}+2\sum_{j<k}q_{ij}q_{k}\gamma_{jk}^{J}\cos\Big{(}T_{jk}\omega\Big{)}\Bigg{]} \tag{39}\]  
\(f_{y_{i}}=\frac{-2j\zeta_{ij}s-1}{2(-s_{i}^{2}+2j\zeta_{ijk}+1)}\); \(S_{g}(\omega)\) is the spectral density function of the ground, \(S_{y_{i}}\) is the PSD of the concerned modes \(y_{i}\); \(q_{ij}\) are constant coefficients. When MSE is not considered, the PSD \(\hat{S}_{y_{i}}\) of \(y_{i}\) for the corresponding uniform excitation problem is. \[\hat{S}_{y_{i}}(\omega)=S_{g}\big{|}f_{y_{i}}\Bigg{[}\sum_{j=1}^{n_{s}}q_{ij} ^{2}+2\sum_{j<k}q_{ij}q_{ik}\Bigg{]} \tag{40}\]  
Ripples with decreasing amplitude can be observed in Equation (39) because of the decaying parameter \(\gamma_{jk}^{J}\) of the incoherence effect and the cosine function brought by the wave passage effect. With the help of these ripples, \(S_{y_{i}}(\omega)\) vibrates around \(S_{g}\big{|}f_{y_{i}}\big{|}^{2}\sum_{j=1}^{n_{s}}q_{ij}^{2}\). \(q_{ij}\) can either be positive or negative, \(S_{y_{i}}(\omega)\) can be significantly larger or smaller than \(\hat{S}_{y_{i}}\). When the ridges of the ripples encounter the resonance peak, the amplitude of the resonance response will increase.