## 3 Frequency Response Calculation of the CLD Plate Considering Two Kinds of Damping
### Derivation of the Frequency Response  
The modal superposition method is applied to solve the frequency response of the CLD plate under base excitation. For most viscoelastic composite structures, the equivalent viscous modal damping ratio of system is usually less than 0.2, and thus, they can be thought of as a small damping system. In this system, the damping is believed to have no influence on the eigenvector of the system, so the real modal theory can be used to characterize the vibration behavior of the composite structure. In fact, the real modal theory has been used by many researchers [16; 17] to analyze the dynamic behavior of the viscoelastic composite structure. The characteristic equation of solving the real mode can be expressed as:  
\[\Big{[}-\omega_{r}^{2}\mathbf{M}+\mathbf{K}(\omega)\Big{]}\mathbf{\varphi}_{r}=0\quad r= 1,2,\cdots,n \tag{6}\]  
where \(\omega_{r}^{2}\) and \(\mathbf{\varphi}_{r}\) are the \(r\)-th order eigenvalue and eigenvector (or modal shape vector), respectively; and \(n\) is the total dimension of the whole system. Here, the eigenvector \(\mathbf{\varphi}_{r}\) is normal and satisfies \(\mathbf{\varphi}_{r}^{T}\mathbf{M}\mathbf{\varphi}_{r}=1\). In fact, \(\omega_{r}^{2}\) and \(\mathbf{\varphi}_{r}\) are also the function of excitation frequency \(\omega\), but \((\omega)\) is not contained in the expression for the sake of convenience in writing. The obtained modal shape vector can form the modal shape matrix \(\mathbf{\varphi}\). Premultiply Equation (3) by the transpose matrix of the modal shape matrix \(\mathbf{\varphi}^{T}\) and substitute \(\mathbf{X}=\mathbf{\varphi}\mathbf{X}_{N}\) into Equation (3), and this yields:  
\[\mathbf{\varphi}^{T}[\mathbf{K}^{*}(\omega)+\mathrm{i}\omega\mathbf{C}_{1}-\omega^{2} \mathbf{M}]\mathbf{\varphi}\mathbf{X}_{N}=\mathbf{F}_{N} \tag{7}\]  
where, \(\mathbf{F}_{N}\) is the amplitude vector of base excitation expressed by modal coordinates and expressed as:  
\[\mathbf{F}_{N}=\mathbf{\varphi}^{T}\mathbf{F} \tag{8}\]  
From Equation (7), a set of independent equations of motion expressed in modal coordinates \(x_{Nr}(r=1,2,\ldots,n)\) can be obtained and expressed as:  
\[\Big{(}\mathbf{\varphi}_{r}^{T}\mathbf{K}^{*}(\omega)\mathbf{\varphi}_{r}+\mathrm{i}\mathbf{ \varphi}_{r}^{T}\mathbf{C}_{1}\mathbf{\varphi}_{r}\omega+\mathbf{\varphi}_{r}^{T}\mathbf{M} \mathbf{\varphi}_{r}\Big{)}x_{Nr}=\mathbf{\varphi}_{r}^{T}\mathbf{F}=f_{Nr} \tag{9}\]  
The first and third items in the above equation can be expressed as:  
\[\mathbf{\varphi}_{r}^{T}\mathbf{K}^{*}(\omega)\mathbf{\varphi}_{r}=\omega_{r}^{2}\big{[} 1+\mathrm{i}\eta_{r}\big{]},\ \mathbf{\varphi}_{r}^{T}\mathbf{M}\mathbf{\varphi}_{r}=1 \tag{10}\]  
where, \(\eta_{r}\) is the \(r\)-th order modal loss factor that is produced by viscoelastic materials. According to the modal damping assumption, the second item in the brackets of Equation (9) can be written as:  
\[\mathbf{\varphi}_{r}^{T}\mathbf{C}_{1}\mathbf{\varphi}_{r}=2\xi_{r}\omega_{r} \tag{11}\]  
where, \(\xi_{r}\) is the \(r\)-th order modal damping ratio produced by the remaining equivalent viscous damping, which is called the remaining equivalent modal damping ratio. It is important to note that obtaining Equation (11) is based on \(\mathbf{\varphi}_{r}^{T}\mathbf{C}_{1}\mathbf{\varphi}_{r}\), which is a diagonal matrix in the derivation. The remaining damping can be introduced by the measured modal damping ratio of the system before damping treatment. Equation (9) can be transformed into the modal coordinates by substituting Equations (10) and (11) into it, which then yields:\[x_{Nr}^{*}=\frac{f_{Nr}}{\omega_{r}^{2}[1+\mathrm{i}\eta_{r}]+\mathrm{i}2\omega \delta_{r}\omega_{r}-\omega^{2}} \tag{12}\]  
The contribution of each mode \(X_{r}^{*}\) is:  
\[X_{r}^{*}=x_{Nr}^{*}\mathbf{\varphi}_{r} \tag{13}\]  
Finally, the vibration response of the composite structure under base excitation is obtained:  
\[\mathbf{X}=\left|\sum_{r=1}^{n}\mathbf{X}_{r}^{*}\right| \tag{14}\]  
where, \(|.|\) refers to the solution of the modular.