## 3 Results and Discussion  
### Validation of Natural Frequency Parameter and Acoustic Power Calculation  
In this paper, the natural frequency parameter of a uniform unloaded annular circular plate is validated with the published result of [PERSON] et al. [10] and is reported in Table 2. In reference [10], [PERSON] et al. provide the solution for the natural frequency parameter of a uniform annular circular plate by Thick and thin plate theories. In our study we have calculated our result using FEM by taking the same dimension of plate as that of [PERSON] et al. From Table 2, it is clearly understand that in this paper the results obtained are almost equal to the published results [10]. For the acoustic power calculation, the computed analytical, numerical and published experimental results [10] are considered as reported in Figure 4. From Figure 4, a good agreement of computed acoustic results is seen to be obtained analytically and numerically in line with published experimentally results [10]. \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}  & & **H. [PERSON] et al. [10]** & **Present Work** \\ \hline Uniform plate & (0,0) & 11.96 & 13.4929 \\ b/a = 0.54 & (0,1) & 13.43 & 13.4946 \\ h/a = 0.21 & (0,2) & 15.28 & 14.1185 \\ \(\lambda^{2}\) & (0,3) & 18.75 & 16.6681 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Validation and comparison of natural frequency parameter \(\lambda^{2}\) of uniform clamped-free annular circular plate obtained in the present work with that of the published result of [PERSON] et al. [10]. Figure 3: Plate with different arrangements of rectangular and concentric stiffener patches with (0, 3) modes. Effect of Natural Frequency Parameter (\(\lambda^{2}\)) of Plate with Different Combinations of Rectangular and Concentric Stiffener Patches with Different Taper Ratios  
In this paper, the effect of the natural frequency parameter (\(\lambda^{2}\)) is investigated for annular plates with different attachment of rectangular and concentric stiffener patches for different positions. The analysis is made for annular plates with different cases of parabolic thickness variations keeping the mass of the plate plus patches constant. Table 3 compares the first four natural frequency parameters numerically of a uniform unloaded plate for taper ratio, \(\rm T_{x}=0.00\), with different attachments of rectangular and concentric stiffener patches along with the percentage variation of \(\lambda^{2}\). It is clear from Table 3 that the plate with different arrangements of rectangular stiffener patches has the same effect of natural frequency parameters as that of the unloaded plate for taper ratio, \(\rm T_{x}=0.00\). However, for concentric stiffener patches, the natural frequency parameter decreases with addition of patches and minimum for 4 concentric stiffener patches. In Table 3, the negative variation of frequency parameter is calculated by \(\frac{P^{2}(\frac{iiif\text{\emph{e}ner}}{12(original)})-I^{2}(original)}{I^{2}( original)}*100\). Figures 5 and 6 show the comparison of negative % variation of natural frequency parameter with different modes for a plate with taper ratio, \(\rm T_{x}=0.00\) and with different attachment of rectangular and concentric stiffener patches. From Figure 5 it is shown that due to less stiffness associated with these modes, the (0, 2) mode of plate with 4 rectangular stiffener patches and (0, 0) mode of plate with 1 rectangular stiffener patch show the lowest percentage variation of \(\lambda^{2}\). However, due to greater stiffness associated with the (0, 1) mode of plate with 1 rectangular stiffener patch, it showed the highest percentage variation of \(\lambda^{2}\).