## 2 Vibration Analysis of Composite Cantilever Plate
\right\} \tag{4}\]  
where \(\sigma\) and \(\tau\) are the normal and shear stresses, respectively, and \(\overline{\Omega}_{ij}\) is the transformed lamina stiffness coefficient, which depends on the material properties and lamina fiber angle. The relations between the transformed and untransformed lamina stiffness coefficients are expressed as  
\[ \begin{array}{c}\overline{\Omega}_{11}=Q_{11}c^{4}+2(Q_{12}+2Q_{66})c^{2}s^ {2}+Q_{22}s^{4}\\ \overline{\Omega}_{12}=(Q_{11}+Q_{22}-4Q_{66})c^{2}s^{2}+Q_{11}\big{(}c^{4}+s ^{4}\big{)}\\ \overline{\Omega}_{22}=Q_{11}s^{4}+2(Q_{11}+2Q_{66})c^{2}s^{2}+Q_{22}c^{4}\\ \overline{\Omega}_{16}=(Q_{11}-Q_{12}-2Q_{66})c^{3}s+(Q_{12}-Q_{22}+2Q_{66})c ^{3}\\ \overline{\Omega}_{26}=(Q_{11}-Q_{12}-2Q_{66})c^{3}s+(Q_{12}-Q_{22}+2Q_{66})c ^{3}s\\ \overline{\Omega}_{66}=(Q_{11}+Q_{22}-2Q_{12}-2Q_{66})c^{2}s^{2}+Q_{66}\big{(} c^{4}+s^{4}\big{)}\\ \overline{\Omega}_{44}=Q_{44}c^{2}+Q_{55}c^{2}\\ \overline{\Omega}_{55}=Q_{55}c^{2}+Q_{44}c^{2}\\ \overline{\Omega}_{45}=(Q_{55}-Q_{44})cs\end{array}  \tag{5}\]  
with  
\[ \begin{array}{c}Q_{11}=\frac{E_{1}}{1-v_{12}v_{21}};\ Q_{12}=\frac{v_{12}E_{2 }}{1-v_{12}v_{21}};\ Q_{22}=\frac{E_{2}}{1-v_{12}v_{21}};\ Q_{44}=G_{23}\\ Q_{55}=G_{13};\ Q_{66}=G_{12};\ c=cos\ \theta_{i^{\prime}};\ s=sin\ \theta_{i}\end{array}  \tag{6}\]  
where \(Q_{ij}\) is the untransformed lamina stiffness constant, \(E_{i}\) is [PERSON]'s modulus in the \(i\)th direction, \(v_{ij}\) is [PERSON]'s ratio, and \(G_{ij}\) is the in-plane shear modulus. For simplicity, the stress-strain relations in Equation (4) for the \(i\)th layer in the plate can be written in matrix form as  
\[\sigma_{i}=\overline{Q}_{i}\varepsilon_{i} \tag{7}\]  
The strain energy \(U\) of the plate can be expressed as  
\[U=\frac{1}{2}\int\sigma^{T}\varepsilon_{i}dV=\frac{1}{2}\int\varepsilon_{i} ^{T}\overline{Q}_{i}\varepsilon_{i}dV \tag{8}\]  
In view of Equations (1), (4), and (8), the maximum strain energy \(U_{max}\) of the plate can be expressed as  
\[U_{max}=\frac{1}{2\alpha a^{2}}\iint\left\{\begin{array}{c}\frac{\partial^{2 }W_{b}}{\partial\xi^{2}}\\ \alpha^{2}\frac{\partial^{2}W_{b}}{\partial\tau^{2}}\\ 2\alpha^{2}\frac{\partial^{2}W_{b}}{\partial\tau^{2}}\\ 2\alpha^{2}\frac{\partial^{2}W_{b}}{\partial\tau^{2}}\\ 2\alpha^{2}\frac{\partial^{2}W_{b}}{\partial\tau^{2}}\\ 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \with  
\[D_{jk}=\sum_{i=1}^{NL}\overline{Q}_{jk}^{\ (i)}\left(\frac{z_{i}^{3}-z_{i+1} ^{3}}{3}\right)(j,k=1,2,6) \tag{11}\]  
and  
\[D_{jk}=K_{p}\sum_{i=1}^{NL}\overline{Q}_{jk}^{\ (i)}(z_{i}-z_{i+1})(j,k=4,5) \tag{12}\]  
where \(\alpha=b/a\), \(K_{p}\) = shear correction factor, and \(NL\) = number of layers. Regarding the determination of \(K_{p}\), as mentioned above, the exact values of \(K_{p}\) for different composite laminates can be calculated using the methods reported in the literature. In fact, for engineering applications, a database collecting the exact values of \(K_{p}\) for different composite laminates can be established. Herein, for illustration purposes and also without loss of generality, \(K_{p}\) is assumed to be 0.8. With the consideration of the rotatory inertia effects, the kinetic energy \(T\) of the plate with density \(\rho\) and the inclusion of rotatory effects is expressed as  
\[T=\frac{1}{2}{\int_{V}}\rho\left(\dot{u}^{2}+\dot{v}^{2}+\dot{w}^{2}\right)dV \tag{13}\]  
In view of Equations (2) and (13), the maximum kinetic energy \(T_{max}\) can be written as  
\[T_{max}=\frac{\rho\omega^{2}h}{2\alpha a^{2}}{\int\!\!\!\int_{A}}\left[\frac{ h^{2}}{12}\left(\frac{\partial W_{b}}{\partial\xi}\right)^{2}+\frac{h^{2}}{12} \left(\frac{\partial W_{b}}{\partial\eta}\right)^{2}+{W_{b}}^{2}+2{W_{b}}{W_ {s}}+{W_{s}}^{2}\right]d\xi d\eta \tag{14}\]  
The Lagrangian to be used in the Ritz method for the plate is defined as  
\[L=T_{max}-U_{max} \tag{15}\]  
According to the Ritz method, each plate displacement can be approximated using a finite series of characteristic functions associated with equal numbers of unknown constants.