## 4 Result and Discussion  
Effect of Natural Frequency Parameter (\(\lambda^{2}\)) of Plate with Different Combinations of Point Masses with Different Taper Ratios  
In this paper, the simulation results for the effect of natural frequency parameter (\(\lambda^{2}\)) due to different combination of point mass are considered. The different parabolic thickness variations are taken where the analysis of the plate is done by keeping mass of the plate + point mass constant. We have considered the first four frequency parameter and hence the numerical comparison is done between the uniform unloaded plate and plate with different combination of point masses for uniform thickness at T\({}_{\rm x}\) = 0 as reported in Table 3. We find from Table 3 that the effect of natural frequency parameter for the unloaded plate and the plate with different combination of point masses is almost same. For further simulation results, a comparison of percentage variation of frequency parameter with the modes are investigated for different cases of plate with point masses combination as reported in Figure 5. It is clear from Figure 5 that the (0,1) mode increases for two point mass and four point mass combinations but decreases for one point mass combinations. In our numerical simulation results, we see that there is abrupt decrease of (0, 3) mode for all point masses combinations due to more stiffness associated with these modes. In our numerical simulation we compare \(\lambda^{2}\) with all modes both for unloaded plate and plate with four point masses combinations as reported in Figure 6. We find that the effect of natural frequency parameter due to four point masses shows the little decrease in the frequency parameter. This may happen due to more stiffness associated with this plate. Further, in this numerical simulation process, we compare Tables 4-6 for natural frequency parameter (\(\lambda^{2}\)) of plate with different combinations of point masses combinations for different cases of tapered plate. It is observed from the Tables 4-6 that Case II plate (parabolically decreasing--increasing thickness variation) reports the reduction in natural frequency parameter for all cases of thickness variations with different combinations of point masses in respect to Case I plate (parabolic decreasing thickness variation). This reduction of natural frequency parameter for Case II plate may be due to the less stiffness associated than that of Case I plate. It is found that due to more stiffness associated with Case III plate (parabolic increasing--decreasing thickness variation), it shows the almost equal effect of natural frequency parameter as that of uniform unloaded plate for all combination of point masses. However, for plate with different parabolically thickness variations with all cases of four point mass combinations, alteration of modes are observed at higher taper ratios. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Plate**} & \multirow{2}{*}{**Mode**} & \multicolumn{2}{c}{**Non Dimensional Frequency Parameter, \(\lambda^{2}\)**} \\ \cline{3-4}  & & **[PERSON]. [PERSON] et al. [10]** & **Present Work** \\ \hline Uniform plate & \begin{tabular}{c} (0, 0) \\ (0, 1) \\ (0, 2) \\ (0, 3) \\ \end{tabular} & \
\begin{tabular}{c} 13.61 \\ 13.43 \\ 15.28 \\ 16.81 \\ \end{tabular} &
\begin{tabular}{c} 13.49 \\ 13.50 \\ 14.12 \\ 16.67 \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison and validation of natural frequency parameter \(\lambda^{2}\) of clamped-free uniform annular circular plate with that of [PERSON] et al. [10] at T\({}_{\rm x}\) = 0.00. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Mode**} & \multirow{2}{*}{ \begin{tabular}{c} **Un-Loaded** \\ **Plate** \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} **Plate with One** \\ **Point Mass** \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} **\% \(\lambda^{2}\)** \\ **Point Masses** \\ \end{tabular}  } & \multirow{2}{*}{