## Appendix A  
The matrices in Equation (17) are written as  
\[\mathbf{D_{\epsilon}}=\sum_{i=1}^{n}\int_{i}^{i+1}\frac{E_{\epsilon}^{i}{}^{2}}{1 -v_{\epsilon}^{2}}dz\]  
\[\mathbf{M}=\frac{1}{2}\rho_{c}h\int_{0}^{a}\int_{0}^{b}WW^{T}dxdy+\frac{1}{2} \rho_{p}h_{p}\int_{y_{1}}^{y_{2}}\int_{x_{1}}^{x_{2}}WW^{T}dxdy\]  
\[\mathbf{K_{1}} =D_{c}\int_{0}^{a}\int_{0}^{b}\int_{0}^{\left(\frac{2}{2}W^{T} \frac{2}{\alpha^{2}}\frac{\partial^{2}W^{T}}{\partial x^{2}}+v_{c}\frac{ \partial^{2}W}{\  
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