# Error Caused by Damping Formulating in Multiple Support Excitation Problems
## 3 Error in the Frequency Response Function of Different Modeling Methods
### Error in LMM
\[\ddot{y}_{i}^{Lr}+\alpha_{0}\dot{y}_{i}^{Lr}+\alpha_{1}\alpha_{i}^{2}\dot{y}_{ i}^{Lr}+\omega_{i}^{2}y_{i}^{Lr}=\frac{1}{2}\big{\{}\mathbf{\Phi}_{ss}{}^{T} \mathbf{K}_{sb}\big{\}}_{i,s}\ddot{\mathbf{Y}}_{b}+\frac{1}{2}\alpha_{0}\big{\{} \mathbf{\Phi}_{ss}{}^{T}\mathbf{K}_{sb}\big{\}}_{i,s}\dot{\mathbf{Y}}_{b}. \tag{15}\]  
To compare Equation (12) with Equation (15), a variable \(z_{i}=y_{i}^{Lr}-y_{i}^{Lm}\) is defined to measure the error when the Rayleigh damping method is used in LMM. Defining another parameter,  
\[\beta=\frac{\left(\alpha_{0}+\alpha_{1}\omega_{i}^{2}\right)-2\zeta\omega_{i }}{\alpha_{0}}=\left(\frac{\omega_{i}}{\alpha_{1}}-1\right)\!\left(\frac{ \omega_{i}}{\alpha_{2}}-1\right). \tag{16}\]  
The error \(z_{i}\) can be expressed in the frequency domain as follows. \[\widetilde{z}_{i}=\left(\frac{1}{2}j\alpha_{0}\omega\frac{(\beta-1)\omega^{2 }+2j\zeta\omega_{i}\omega+\omega_{i}^{2}}{\big{(}-\omega^{2}+2j\zeta\omega_{i }\omega+\omega_{i}^{2}\big{)}\big{(}-\omega^{2}+j(\alpha_{0}+\alpha_{1}\omega _{i}^{2})\omega+\omega_{i}^{2}\big{)}}\right)\!\left(\mathbf{\Phi}_{ss}{}^{T} \mathbf{K}_{sb}\right)_{i,s}\mathbf{Y}_{b}. \tag{17}\]  
where \(\widetilde{z}_{i}\) stand for the Fourier transform of \(z_{i}\). Define \(\mu_{i}=\ \omega_{i}/\omega_{\tau_{1}}\) and \(v=\ \omega_{\tau_{2}}/\omega_{\tau_{1}}\). In the vicinity of zero frequency,  
\[z_{i}|_{(0,\epsilon)}\approx\frac{j\zeta\
u}{\mu_{i}^{2}(1+\
u)\omega_{\tau_{ 1}}}\omega\big{\{}\mathbf{\Phi}_{ss}{}^{T}\mathbf{K}_{sb}\big{\}}_{i,s} \mathbf{Y}_{b}, \tag{18}\]  
where \(\epsilon\) is a small positive number. In Equation (18), the error transfer function at zero frequency is zero; the amplitude of this error increase with frequency \(\omega\), and is negatively correlated to the fundamental frequency of the structure, \(\omega_{\tau_{1}}\); the larger the damping ratio \(\zeta\) is the larger the error is. Define the error ratio \(r_{i}^{Lr}=\left|\left(y_{i}^{Lr}-y_{i}^{Lm}\right)\!\left|\!\left/y_{i}^{Lm}\right|\right.\). The value of \(r_{i}^{Lr}\) at zero frequency is infinite, which attribute to the second term at the right-hand side of Equation (5). As shown in Equation (17),the error has a peak in the vicinity of the resonance frequency. The value at the resonance frequency is derived as follows,  
\[r_{i}^{Lr}\Big{|}_{\omega_{i}}=\sqrt{\left(1-\eta_{i}\right)^{2}+\frac{4\zeta^{2 }v^{2}}{\left(v+\mu^{2}\right)^{2}}}, \tag{19}\]  
where \(\eta_{i}=\zeta/\zeta_{i}^{r}\), \(\eta_{i}\) represents the multiple of the expected damping ratio over the damping ratio of mode \(i\). In Equation (19), for a particular structure, the first term in the square root symbol increases with the parameter \(\eta_{i}\) and the second term decrease with \(\mu\). The possible largest value of \(\eta_{i}\) is shown in Equation (20), and is achieved in the mode having the smallest damping ratio. \[\max(\eta_{i})=\frac{1}{2\sqrt{v}}+\frac{1}{2}\sqrt{v}. \tag{20}\]  
From Equation (19), the behavior of the peak of \(r_{i}^{Lr}\) is complex, the natural frequency of the mode having the largest error is between \(\omega_{r_{1}}\) and \(\sqrt{\omega_{r_{1}}\omega_{r_{2}}}\). The largest value of \(r_{i}^{Lr}\) is larger than the larger value in \(r_{i}^{Lr}\Big{|}_{\omega_{i}=\omega_{r_{1}}}=2\zeta\) and \(r_{i}^{Lr}\Big{|}_{\omega_{i}=\sqrt{\omega_{r_{1}}\omega_{r_{2}}}}=\sqrt{\left( 1-\frac{1}{2\sqrt{v}}-\frac{1}{2}\sqrt{v}\right)^{2}+\zeta^{2}}\). Equation (20) also indicates that, for a particular structure, the larger the value of \(v\) is the larger the error is. The value \(\max(\eta)\) doesn't have an upper limit, as a result, the error ratio \(r_{i}^{Lr}\Big{|}_{\omega_{i}}\) can be a very large value. When \(\omega\) approaches infinity \(r_{i}^{Lr}\Big{|}_{\omega\rightarrow\infty}=0\). It is obvious in Equation (17) that the error ratio would be smaller than \(r_{i}^{Lr}\Big{|}_{\omega=\omega_{i}}\) for the frequencies larger than the resonance frequency.