## 3 Frequency Response Calculation of the CLD Plate Considering Two Kinds of Damping
### Derivation of the Mass Matrix and Stiffness Matrix  
In this section, layer-wise theory was applied to derive the stiffness and mass matrices in Equation (3). As shown in Figure 4, the plane CLD plate element with four nodes and seven degrees of freedom of each node is selected to simulate the CLD plate [18]. The seven degrees of freedom are \(u_{c},v_{c},u_{\mathrm{b}},v_{\mathrm{b}},w,\theta_{x},\theta_{y}\), respectively, where, \(u_{c},v_{c}\) are the \(x\) and \(y\) direction displacements in the neutral plane of the constraining layer and \(u_{\mathrm{b}},v_{\mathrm{b}}\) are the \(x\) and \(y\) direction displacements in the neutral plane of the metal substrate, respectively. Symbols \(w,\theta_{x},\theta_{y}\) are the deflection of the composite plate and the rotation angle of the neutral plane of the composite plate around the \(x\) axis and \(y\) axis, respectively. The stiffness and mass matrix of the CLD plate can be derived according to the following procedure. Figure 4: Plane CLD plate element. Figure 3: The procedure of solving the frequency response of the CLD plate. 1. Dividing the composite thin plate into several elements. 2. Solving the contribution of each layer to the stiffness and mass matrices of the whole CLD element. The contribution of each layer to the element stiffness matrix can be expressed as: \[\mathbf{K}_{i}^{\mathrm{e}}=H_{i}\iint\limits_{A}\mathbf{B}_{i}^{\mathrm{T}}\mathbf{D}_{i} \mathbf{B}_{i}\mathrm{d}A+\frac{H_{i}^{3}}{12}\iint\limits_{A}\mathbf{B}^{\mathrm{T}} \mathbf{D}_{i}\mathbf{B}\;\mathrm{d}A,\quad i=\mathrm{b},\mathrm{c}\] (15a) \[\mathbf{K}_{\mathrm{v}}^{\mathrm{*e}}(\omega)=H_{\mathrm{v}}\iint\limits_{A}\mathbf{B }_{\mathrm{v}}^{\mathrm{T}}\mathbf{D}_{\mathrm{v}}\mathbf{B}_{\mathrm{v}}\mathrm{d}A+ \frac{H_{\mathrm{v}}^{3}}{12}\iint\limits_{A}\mathbf{B}^{\mathrm{T}}\mathbf{D}_{ \mathrm{v}}\mathbf{B}\mathrm{d}A+G_{\mathrm{v}}^{*}(\omega)H_{\mathrm{v}}\iint \limits_{A}\mathbf{N}_{10}^{\mathrm{T}}\mathbf{N}_{10}+\mathbf{N}_{11}^{\mathrm{T}}\mathbf{N}_ {11}\mathrm{d}A\] (15b) The contribution of each layer to the element mass matrix can be expressed as: \[\mathbf{M}_{\mathrm{c}}^{\mathrm{e}}=\int\limits_{A}\rho_{\mathrm{c}}\left(\mathbf{N }_{1}^{\mathrm{T}}\mathbf{N}_{1}+\mathbf{N}_{2}^{\mathrm{T}}\mathbf{N}_{2}+\mathbf{N}_{5}^{ \mathrm{T}}\mathbf{N}_{5}\right)\,\mathrm{d}A\] (16a) \[\mathbf{M}_{\mathrm{v}}^{\mathrm{e}}=\int\limits_{A}\rho_{\mathrm{v}}\left(\mathbf{N }_{8}^{\mathrm{T}}\mathbf{N}_{8}+\mathbf{N}_{9}^{\mathrm{T}}\mathbf{N}_{9}+\mathbf{N}_{5}^{ \mathrm{T}}\mathbf{N}_{5}\right)\,\mathrm{d}A\] (16b) \[\mathbf{M}_{\mathrm{b}}^{\mathrm{e}}=\int\limits_{A}\rho_{\mathrm{b}}\left(\mathbf{N }_{3}^{\mathrm{T}}\mathbf{N}_{3}+\mathbf{N}_{4}^{\mathrm{T}}\mathbf{N}_{4}+\mathbf{N}_{5}^{ \mathrm{T}}\mathbf{N}_{5}\right)\,\mathrm{d}A\] (16c) where, \(\mathbf{B}\), \(\mathbf{B}_{\mathrm{b}}\), \(\mathbf{B}_{\mathrm{c}}\), and \(\mathbf{B}_{\mathrm{v}}\) are the strain matrices; \(\mathbf{N}_{1}\), \(\mathbf{N}_{2}\), \(\mathbf{N}_{3}\), \(\mathbf{N}_{4}\), \(\mathbf{N}_{5}\), \(\mathbf{N}_{8}\), \(\mathbf{N}_{9}\), \(\mathbf{N}_{10}\), and \(\mathbf{N}_{11}\) are the shape function vectors of the element; and \(\rho_{\mathrm{c}}\), \(\rho_{\mathrm{v}}\), and \(\rho_{\mathrm{b}}\) refer to the densities of the constraining layer, viscoelastic core layer, and metal substrate, respectively. Symbols \(\mathbf{D}_{\mathrm{c}}\), \(\mathbf{D}_{\mathrm{b}}\), and \(\mathbf{D}_{\mathrm{v}}^{*}\) represent the elastic matrices of the constraining layer, metal substrate, and viscoelastic core layer, respectively, which are given by: \[\mathbf{D}_{i}=\frac{E_{i}}{(1-\mu_{i}^{2})}\left[ \begin{array}{ccc}1&\mu_{i}&0 \\ \mu_{i}&1&0\\ 0&0&\frac{1}{2}(1-\mu_{i})\end{array} \right]i=\mathrm{b},\mathrm{c}\] (17a) \[\mathbf{D}_{\mathrm{v}}=\frac{2G_{\mathrm{v}}^{*}(\omega)}{1+\mu_{\mathrm{v}}} \left[ \begin{array}{ccc}1&\mu_{\mathrm{v}}&0\\ \mu_{\mathrm{v}}&1&0\\ 0&0&\frac{1}{2}(1-\mu_{\mathrm{v}})\end{array}