# An Alternative Series Solution for Free Vibration Analysis of Asymmetric L-Shaped Membranes
## 3 Results
### Convergence for the Asymmetric Case  
This section examines the convergence behavior of the first ten eigenvalue squares for the asymmetric L-shaped membrane, as outlined in Table 4. The analysis spans truncation indices \(N\) of 20, 80, 160, 240, 420, and 520. The data indicate a clear convergence pattern as \(N\) increases, with eigenvalues stabilizing at higher truncation indices. Notably, \(k_{1}^{2}\) and \(k_{2}^{2}\) show rapid convergence, reaching stable values at lower \(N\). For example, \(k_{1}^{2}\) stabilizes at 12.5435 starting from \(N=240\), illustrating the efficient convergence for lower-order eigenvalues.  
Higher-order eigenvalues, such as \(k_{10}^{2}\), exhibit greater sensitivity, with values adjusting slightly even at \(N=420\). This heightened sensitivity underscores the need for a larger \(N\) to  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\(k_{i}^{2}\)} & \multicolumn{6}{c}{\(N\)} \\ \cline{2-7}  & **20** & **80** & **160** & **240** & **420** & **520** \\ \hline
1 & 12.5462 & 12.5439 & 12.5436 & 12.5435 & 12.5435 & 12.5435 \\
2 & 20.9193 & 20.9149 & 20.9143 & 20.9142 & 20.9141 & 20.9141 \\
3 & 33.6475 & 33.6362 & 33.6349 & 33.6346 & 33.6343 & 33.6343 \\
4 & 39.0199 & 39.0129 & 39.0121 & 39.0119 & 39.0117 & 39.0117 \\
5 & 44.0337 & 44.0335 & 44.0335 & 44.0335 & 44.0335 & 44.0335 \\
6 & 51.4015 & 51.3983 & 51.3978 & 51.3977 & 51.3976 & 51.3976 \\
7 & 57.7097 & 57.7091 & 57.7090 & 57.7089 & 57.7089 & 57.7089 \\
8 & 63.7629 & 63.7609 & 63.7606 & 63.7606 & 63.7605 & 63.7605 \\
9 & 77.7272 & 77.7215 & 77.7210 & 77.7208 & 77.7207 & 77.7207 \\
10 & 84.7964 & 84.7879 & 84.7869 & 84.7867 & 84.7864 & 84.7864 \\ \hline \hline \end{tabular} Note: The asterisk (*) means that all the values given in the literature are squared.  
\end{table}
Table 4: The square of the first ten eigenvalues versus truncation indices \(N\) for the asymmetric case.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(k_{i}^{2}\) & **Present** & **[PERSON] and [PERSON] [33]** & **[PERSON] [31]** & **[PERSON] et al. [30] * & **[PERSON] et al. [32] * \\ \hline
1 & 9.6397 & 9.6397 & 9.6397 & 9.6814 & 9.6870 \\
2 & 15.1973 & 15.1973 & 5.1973 & 15.2022 & 15.1960 \\
3 & 19.7392 & 19.7392 & \(2\pi^{2}\) & 19.7474 & 19.7394 \\
4 & 29.5215 & 29.5215 & 29.5215 & 29.5316 & 29.5218 \\
5 & 31.9127 & 31.9126 & 31.9126 & 32.0560 & 32.0277 \\
6 & 41.4745 & 41.4745 & 41.4745 & 41.5612 & 41.5625 \\
7 & 44.9485 & 44.9485 & 44.9485 & 44.9798 & 44.9423 \\
8 & 49.3480 & 49.3480 & \(5\pi^{2}\) & 49.3773 & 49.3492 \\
9 & 49.3480 & 49.3480 & \(5\pi^{2}\) & 49.4195 & - \\
10 & 56.7096 & 56.7096 & 56.7096 & 56.8878 & - \\ \hline \hline \end{tabular} Note: The asterisk (*) means that all the values given in the literature are squared.  
\end{table}
Table 3: Comparison of the first ten eigenvalue squares for symmetric cases with established results.  
capture the complexity of higher-order modes accurately. For instance, \(k_{10}^{2}\) changes from 84.7964 at \(N=20\) to 84.7864 at \(N=520\).  
Overall, the convergence behavior is uniform across the eigenvalues, assuring the numerical stability and robustness of the method used. The consistent results for higher truncation indices validate the approach, demonstrating its capability to effectively handle the free vibration analysis of asymmetric L-shaped membranes. This detailed investigation into the convergence and sensitivity of eigenvalues highlights the importance of selecting appropriate truncation indices to ensure accurate and reliable computational results.