## 3 Comprehensive Frequency Parameter and Mode Shape Results
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multicolumn{1}{c}{\multirow{2}{*}{\(R=r_{\rm 00}=\pi_{\rm r0}\)}} & \multirow{2}{*}{\(s\)} & \multicolumn{1}{c}{\multicolumn{1}{c}{\(\overline{\rm{First}}\)} & \multicolumn{1}{c}{\(\overline{\rm{Second}}\)} & \multicolumn{1}{c}{\(\overline{\rm{Third}}\)} & \multicolumn{1}{c}{\(\overline{\rm{Fourth}}\)} & \multicolumn{1}{c}{\(\overline{\rm{Fifth}}\)} & \multicolumn{1}{c}{\(\overline{\rm{Sitth}}\)} & \multicolumn{1}{c}{\(\overline{\rm{Sitth}}\)} & \multicolumn{1}{c}{\(\overline{\rm{Sitth}}\)} & \multicolumn{1}{c}{\(\overline{\rm{Fifth}}\)} & \multicolumn{1}{c}{\(\overline{\rm{Sitth}}\)} & \multicolumn{1}{c}{\(\overline{\rm{Fifth}}\)} \\ \hline  & 30 & 3.6848 & 17.605 & 19.598 & 38.507 & 51.354 & 53.809 & 73.260 & 74.947 & 105.03 & 107.55 \\  & 50 & 3.6849 & 17.606 & 19.599 & 38.508 & 51.355 & 53.810 & 73.262 & 74.949 & 105.03 & 107.56 \\
0.1 & 100 & 3.6851 & 17.606 & 19.599 & 38.510 & 51.356 & 53.811 & 73.263 & 74.950 & 105.04 & 107.56 \\  & 150 & 3.6851 & 17.606 & 19.599 & 38.510 & 51.357 & 53.811 & 73.264 & 74.951 & 105.04 & 107.56 \\  & 200 & 3.6851 & 17.606 & 19.599 & 38.510 & 51.357 & 53.811 & 73.264 & 74.951 & 105.04 & 107.56 \\  & 30 & 6.0770 & 21.436 & 28.816 & 43.352 & 57.077 & 59.670 & 78.939 & 81.131 & 111.95 & 114.49 \\  & 50 & 6.0771 & 21.433 & 28.813 & 43.351 & 57.078 & 59.670 & 78.942 & 81.133 & 111.94 & 114.48 \\
0.8 & 100 & 6.0771 & 21.433 & 28.813 & 43.351 & 57.078 & 59.670 & 78.942 & 81.133 & 111.94 & 114.49 \\  & 150 & 6.0771 & 21.433 & 28.813 & 43.351 & 57.078 & 59.670 & 78.942 & 81.133 & 111.94 & 114.48 \\  & 200 & 6.0771 & 21.433 & 28.813 & 43.351 & 57.078 & 59.670 & 78.942 & 81.133 & 111.94 & 114.48 \\ \hline \end{tabular}
\end{table} 
Table 10: Convergence of the dimensionless frequency parameter \(\phi\) for the RFFF square plates with \(r=r_{\rm 00}=r_{\rm r0}\) equaling 0.1 and 0.8, respectively. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \multicolumn{1}{c}{\multirow{2}{*}{\(R_{\rm 00}\)}} & \multirow{2}{*}{\(s\)} & \multicolumn{1}{c}{\(\overline{\rm{First}}\)} & \multicolumn{1}{c}{\(\overline{\rm{Second}}\)} & \multicolumn{1}{c}{\(\overline{\rm{Third}}\)} & \multicolumn{1}{c}{\(\overline{\rm{Fourth}}\)} & \multicolumn{1}{c}{\(\overline{\rm{Fifth}}\)} & \multicolumn{1}{c}{\(\overline{\rm{Sitth}}\)} & \multicolumn{1}{c}{\(\overline{\rm{Sitth}}\)} & \multicolumn{1}{c}{\(\overline{\rm{Sitth}}\)} & \multicolumn{1}{c}{\(\overline{\rm{Fifth}}\)} & \multicolumn{1}{c}{\(\overline{\rm{Sitth}}\)} & \multicolumn{1}{c}{\(\overline{\rm{Sitth}}\)} & \multicolumn{1}{c}{\(\overline{\rm{Fifth}}\)} \\ \hline  & 30 & 4.3767 & 18.256 & 20.676 & 39.453 & 52.078 & 55.200 & 74.152 & 76.140 & 105.77 & 109.08 \\  
We finally carry out convergence study for the dimensionless frequency parameter \(\phi\) of all the five cases, as illustrated by Tables 7-11, in which the convergent results are precise up to five significant figures and are marked in bold. It should be pointed that all the given solutions converge at a slower rate due to the Fourier series form solution. However, the acquired simultaneous equations are easily handled through using the commercial software such as Mathematica 10.3. For each case, it is obviously seen that most of \(\phi\) are convergent at \(s=150\), which leads to 150 series of terms that are adopted throughout the present work. Excellent agreement is found between the present numerical/graphical results with FEM solutions and other analytical solutions, which confirms the qualification of the present method in solving complex plate problems. Obviously, all the present the non-dimensional natural frequency parameters and the associated vibration mode shapes can be adopted for engineers and scientists for academic and practical applications. The laws indicated by the present parametric analysis enable designers to effectively avoid resonance problems of rectangular thin plate structures.