## 3 Ritz Method  
Regarding the use of the Ritz method in the above plate vibration analysis, it is noted that the choice of appropriate characteristic functions for approximating the bending and through-thickness shear-deformation-induced vertical deflection components is vital in attaining good approximations of the true modal characteristics of the plate. Two issues should be dealt with in determining the characteristic functions. The first issue is related to the independence between the two sets of characteristic functions for \(W_{b}\) and \(W_{s}\), respectively. In view of the total deflection in Equation (1), the two sets of characteristic functions must be independent so that each deflection component can have its own contribution to the total deflection. It is worth noting that if the two sets of characteristic functions are dependent, then, after summation, only one set of characteristic functions will be left to approximate the total deflection. The consequence of the choice of dependent sets of characteristic functions is that the contribution of \(W_{s}\) to the total deflection will be underestimated. It has been found that the underestimation of the total deflection will lead to incorrect predictions of natural frequencies for thick plates, especially those with a/h less than 50. The second issue is related to the satisfaction of the displacement and force conditions at the fixed edge of the cantilever plate. In view of the displacement boundary conditions at the fixed edge of the cantilever plate, it is required that the vertical displacement and bending rotation at \(\xi=0\) equal to zero  
\[W(0,\eta)=0 \tag{17}\]  
\[\frac{\partial W_{b}}{\partial\xi}(0,\eta)=0 \tag{18}\]  
In view of Equation (1), Equation (17) can be rewritten as  
\[W_{b}(0,\eta)=0 \tag{19}\]  
and  
\[W_{s}(0,\eta)=0 \tag{20}\]  
As indicted in Ref. [72], different displacement boundary conditions at the fixed end of a cantilever beam can be chosen to solve for the deflection of the beam under the conditions that the bending moment and shear force at the fixed end are not zero. For the cantilever plate under consideration in this study, the requirements of nonzero bending moment and shear force at the fixed edge of the plate must be observed to maintain the equilibrium/stability of the plate. It is noted that in view of Equation (3), for having nonzero shear strain/shear stress at the fixed edge of the plate, \(\frac{\partial W_{b}}{\partial\xi}(0,\eta)\) should not be zero. Here, nonzero \(\frac{\partial W_{s}}{\partial\xi}(0,\eta)\) implies the nonconformity of the zero-slope condition at the fixed edge of the plate. Therefore, both shear stress and slope at the fixed edge being zero cannot be satisfied at the same time. Nevertheless, as will be shown in the following examples, the effects of slope nonconformity on the natural frequencies of thick composite cantilever plates are negligible. Hence, in observing the requirements of the above two issues, the characteristic functions for approximating \(W_{b}\) and \(W_{s}\) in the natural coordinate system can be chosen, respectively, as  
\[W_{b}=\sum_{i=2}^{M_{b}}\sum_{j=0}^{N_{b}}A_{ij}\xi^{i}\eta^{j} \tag{21}\]  
and  
\[W_{s}=\sum_{i=1}^{M_{b}}\Biggl{\{}B_{i00}sin\left(\frac{i\pi\xi}{2}\right)+ \sum_{j=1}^{N_{b}}\biggl{[}B_{ij1}sin\left(\frac{i\pi\xi}{2}\right)sin(j\pi \eta)+B_{ij2}sin\left(\frac{i\pi\xi}{2}\right)cos(j\pi\eta)\biggr{]}\Biggr{\}} \tag{22}\]  
where \(A_{ij}\) and \(B_{ijk}\) are unknown constants and \(k\) is 0, 1, or 2. It is noted that the two sets of characteristic functions given in Equations (21) and (22), respectively, are independent. Hence, \(W_{b}\) and \(W_{s}\) can have their own contributions to the total deflection. Furthermore, the polynomials used to approximate \(W_{b}\) in Equation (21) produce zero bending-induced deflection and slope but a nonzero bending moment at the plate fixed end.