## 2 Vibration Analysis of Composite Cantilever Plate  
Consider the free vibration of a rectangular symmetrically laminated composite plate of size \(a\) (length) \(\times\)\(b\) (width) \(\times\)\(h\) (thickness) with the left end clamped as shown in Figure 1. The fiber angle of the \(i\)th layer is \(\theta_{i}\). Herein, two coordinate systems, namely, reference coordinate system \(x\)-\(y\)-\(z\) and natural coordinate system \(\xi\)-\(\eta\)-\(z\) are used to describe the geometry of the plate. Both \(x\)-\(y\) and \(\xi\)-\(\eta\) are located at the mid-plane of the plate with (\(0\leq x\)\(\leq\)\(a\), \(-b/2\)\(\leq\)\(y\)\(\leq\)\(b/2\)) and (\(0\)\(\leq\)\(\xi\)\(\leq\)\(1\), \(-0.5\)\(\leq\)\(\eta\)\(\leq\)\(0.5\)), respectively. The relation between the two coordinate systems is \(\xi=x/a\) and \(\eta=y/b\). In the SFSDT, the total vertical deflection is the sum of two parts, namely, the bending and through-thickness shear-deformation-induced deflections. Assuming no plate thickness change and neglecting the in-plane displacements, the displacement field of the symmetrically laminated plate under free vibration is written as  
\[ \begin{array}{l}u(x,y,z,t)=-z\frac{\partial W_{b}}{\partial x}\\ v(x,y,z,t)=-z\frac{\partial W_{b}}{\partial y}\\ w(x,y,z,t)=[w_{b}(x,y,t)+w_{s}(x,y,t)]\end{array}  \tag{1}\]  
where \(u\), \(v\), and \(w\) are the displacement components in the \(x\)-\(y\)-\(z\) coordinate system of the plate; \(w_{b}\) is the vertical deflection induced by bending; and \(w_{s}\) is the vertical deflection induced by through-thickness shear deformation. It is noted that herein only two independent displacement components (\(w_{b}\) and \(w_{s}\)) rather than three independent displacement components (two shear rotations plus one vertical deflection) for the FSDT are required to describe the displacement field of the plate. In general, it is common practice to use the natural coordinate system to express the characteristic functions in the Ritz method. Hence, in the free vibration analysis, the vertical displacement components of the plate in the \(\xi\)-\(\eta\) coordinate system are expressed as  
\[ \begin{array}{l}w_{b}(x,y,t)=W_{b}(\xi,\eta)sin\;\omega t\\ w_{s}(x,y,t)=W_{s}(\xi,\eta)sin\;\omega t\end{array}  \tag{2}\]  
where \(W_{b}\) and \(W_{s}\) are the bending and through-thickness shear-deformation-induced deflected shapes, respectively. Referring to the \(\xi\)-\(\eta\) coordinate system, discarding the effect of time, the strain-displacement relations of the plate are expressed as  
\[ \begin{array}{l}\varepsilon_{\xi}=-\frac{z}{a^{2}}\frac{\partial^{2}W_{b}}{ \partial z^{2}}\\ \varepsilon_{\eta}=-\frac{z}{a^{2}}\frac{\partial^{2}W_{b}}{\partial y^{2}} \\ \gamma_{\xi\eta}=\frac{\partial v_{0}}{\partial z}-\frac{z}{a}\frac{\partial^ {2}W_{b}}{\partial z\partial\eta}\\ \gamma_{\eta z}=\frac{1}{\partial W_{b}}\frac{\partial^{2}W_{b}}{\partial y} \\ \gamma_{\xi z}=\frac{1}{a}\frac{\partial^{2}W_{b}}{\partial y}\end{array}  \tag{3}\]  
Figure 1: Configuration of composite cantilever plate. where \(\varepsilon\) and \(\gamma\) are the normal and shear strains, respectively. The stress-strain relations of a composite lamina with arbitrary fiber angle in the \(\xi\)-\(\eta\)-\(z\) coordinate system can be expressed in the following general form [71]:  
\[\left\{ \begin{array}{c}\sigma_{\xi}\\ \sigma_{\eta}\\ \tau_{\eta z}\\ \tau_{\xi\eta z}\\ \tau_{\xi\eta}\\ \tau_{\xi\eta}\end{array} \right\}=\left\{ \begin{array}{ccccc}\overline{ \Omega}_{11}&\overline{\Omega}_{12}&0&0&\overline{\Omega}_{16}\\ \overline{\Omega}_{21}&\overline{\Omega}_{22}&0&0&\overline{\Omega}_{26}\\ 0&0&\overline{\Omega}_{44}&\overline{\Omega}_{45}&0\\ 0&0&\overline{\Omega}_{45}&\overline{\Omega}_{55}&0\\ \overline{\Omega}_{16}&\overline{\Omega}_{26}&0&0&\overline{\Omega}_{66} \end{array} \right\}\left\{ \begin{array}{c}\varepsilon_{\xi}\\ \varepsilon_{\eta}\\ \gamma_{\eta z}\\ \gamma_{\xi\eta}\\ \gamma_{\xi\eta}\end{array}