## 4 Result and Discussion
\begin{tabular}{c} **Plate with Two** \\ **Point Masses** \\ \end{tabular}  } & \multirow{2}{*}{\ \begin{tabular}{c} **\% \(\lambda^{2}\)** \\ **Point Masses** \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} **\% \(\lambda^{2}\)** \\ **Point Masses** \\ \end{tabular}  } \\ \hline (0,0) & 13.49 & 13.46 & 0.223 & 13.45 & 0.296 & 13.35 & 1.033 \\ (0,1) & 13.50 & 13.48 & 0.148 & 13.44 & 0.444 & 13.32 & 1.333 \\ (0,2) & 14.12 & 14.08 & 0.283 & 14.06 & 0.424 & 14.02 & 0.708 \\ (0,3) & 16.67 Figure 5: Comparison of % variation of natural frequency parameter with modes for uniform plate with different combinations of point masses. Figure 6: Comparison of variation of natural frequency parameter with modes for unloaded plate and for plate with four point masses. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multirow{2}{*}{**Mode**} & \multicolumn{4}{c}{**Natural Frequency Parameter, \(\lambda^{2}\)**} \\ \cline{3-6}  & & \(\mathbf{T_{x}=0.00}\) & \(\mathbf{T_{x}=0.25}\) & \(\mathbf{T_{x}=0.50}\) & \(\mathbf{T_{x}=0.75}\) \\ \hline \multirow{4}{*}{I} & (0,0) & 13.4852 & 12.9987 & 12.4701 & 11.9301 \\  & (0,1) & 13.4902 & 12.9765 & 12.4443 & 11.8937 \\  & (0,2) & 14.1025 & 13.6040 & 13.0838 & 12.5412 \\  & (0,3) & 16.6635 & 16.0712 & 15.4619 & 14.8258 \\ \hline \multirow{4}{*}{II} & (0,0) & 13.4842 & 12.8924 & 12.2715 & 11.6202 \\  & (0,1) & 13.4902 & 12.8805 & 12.2418 & 11.5762 \\ \cline{1-1}  & (0,2) & 14.1035 & 13.5102 & 12.8858 & 12.2304 \\ \cline{1-1}  & (0,3) & 16.6638 & 15.9618 & 15.2302 & 14.4612 \\ \hline \multirow{4}{*}{III} & (0,0) & 13.4852 & 13.4820 & 13.4850 & 13.4852 \\ \cline{1-1}  & (0,1) & 13.4906 & 13.4853 & 13.4902 & 13.4902 \\ \cline{1-1}  & (0,2) & 14.1037 & 14.1008 & 14.1022 & 14.1032 \\ \cline{1-1}  & (0,3) & 16.6638 & 16.6612 & 16.6639 & 16.6635 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Numerical comparison of different frequency parameter \(\lambda^{2}\) with different modes of plate with four point masses combinations for different parabolic thickness variation and for different taper parameter (T\({}_{x}\)). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multirow{2}{*}{**Mode**} & \multicolumn{4}{c}{**Natural Frequency Parameter, \(\lambda^{2}\)**} \\ \cline{3-6}  & & \(\mathbf{T_{x}=0.00}\) & \(\mathbf{T_{x}=0.25}\) & \(\mathbf{T_{x}=0.50}\) & \(\mathbf{T_{x}=0.75}\) \\ \hline \multirow{4}{*}{I} & (0,0) & 13.4802 & 12.9904 & 12.4703 & 11.9305 \\  & (0,1) & 13.4942 & 12.9745 & 12.4442 & 11.8918 \\  & (0,2) & 14.1132 & 13.6135 & 13.0936 & 12.5502 \\  & (0,3) & 16.6624 & 16.0733 & 15.4611 & 14.8254 \\ \hline \multirow{4}{*}{II} & (0,0) & 13.4811 & 12.8952 & 12.2718 & 11.6202 \\  & (0,1) & 13.4932 & 12.8774 & 12.2414 & 11.5752 \\  & (0,2) & 14.1134 & 13.5187 & 12.8943 & 12.2384 \\  & (0,3) & 16.6624 & 15.9610 & 15.2277 & 14.4605 \\ \hline \multirow{4}{*}{III} & (0,0) & 13.4812 & 13.4891 & 13.4902 & 13.4904 \\  & (0,1) & 13.4943 & 13.4784 & 13.4808 & 13.4804 \\ \cline{1-1}  & (0,2) & 14.1136 & 14.1109 & 14.1134 & 14.1125 \\ \cline{1-1}  & (0,3) & 16.6625 & 16.6600 & 16.6624 & 16.6618 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Numerical comparison of different frequency parameter \(\lambda^{2}\) with different modes of plate with one point mass combinations for different parabolic thickness variations and for different taper parameters (T\({}_{x}\)).