## 4 Reference Case Study
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **PLATE CASE** & & **MEMBRANE CASE** \\ \hline \(\mathrm{f_{2 ANP}/f_{1 ANP}}\) & 2.081 & \(\mathrm{f_{2 ANM}/f_{1 ANM}}\) & 1.594 \\ \(\mathrm{f_{3 ANP}/f_{1 ANP}}\) & 3.414 & \(\mathrm{f_{3 ANM}/f_{1 ANM}}\) & 2.136 \\ \(\mathrm{f_{4 ANP}/f_{1 ANP}}\) & 3.893 & \(\mathrm{f_{4 ANM}/f_{1 ANM}}\) & 2.296 \\ \(\mathrm{f_{5 ANP}/f_{1 ANP}}\) & 4.996 & \(\mathrm{f_{5 ANM}/f_{1 ANM}}\) & 2.653 \\ \(\mathrm{f_{6 ANP}/f_{1 ANP}}\) & 5.954 & \(\mathrm{f_{6 ANM}/f_{1 ANM}}\) & 2.918 \\ \(\mathrm{f_{7 ANP}/f_{1 ANP}}\) & 6.819 & \(\mathrm{f_{7 ANM}/f_{1 ANM}}\) & 3.156 \\ \(\mathrm{f_{8 ANP}/f_{1 ANP}}\) & 8.279 & \(\mathrm{f_{8 ANM}/f_{1 ANM}}\) & 3.501 \\ \(\mathrm{f_{9 ANP}/f_{1 ANP}}\) & 8.722 & \(\mathrm{f_{9 ANM}/f_{1 ANM}}\) & 3.600 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Ratio \(\mathrm{f_{n}/f_{1}}\) for the first nine natural frequencies for plate and membrane case. The response in this transition zone was studied by [PERSON] [30]. If Equation (3) given by [PERSON] is used to calculate the natural frequencies (f\({}_{\text{WAH}}\)), the values of Table 4 are obtained for the different strain values provided by this author. The strain values are related to the dimensionless parameter \(\phi\), also given in Table 4. The numerical frequencies obtained with FEM (f\({}_{\text{FEM}}\)) and the ratio between both (f\({}_{\text{WAH}}\)/f\({}_{\text{FEM}}\)) are also included in Table 4.  
The dimensionless parameter \(\phi\) is given for values of nodal diameters (n) and nodal circles (s) between 0 and 2. For modes with n and/or s greater than 2, the frequencies cannot be obtained with the values provided by [PERSON]. Due to this, among the first nine modes the 5 th and 7 th cannot be  
\begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline  & \multicolumn{6}{c}{**STRAIN**} \\ \cline{2-10}  & \multicolumn{2}{c}{**0**} & \multicolumn{6}{c}{**9.410 \(\boldsymbol{\times}\)10\({}^{-7}\)**} & \multicolumn{6}{c}{**1.882 \(\boldsymbol{\times}\)10\({}^{-6}\)**} \\ \cline{2-10}  & \multicolumn{2}{c}{**f\({}_{\text{WAH}}\)**} & \multicolumn{2}{c}{**f\({}_{\text{FEM}}\)**} & \multicolumn{2}{c}{**f\({}_{\text{WAH}}\)**} & \multicolumn{2}{c}{**f\({}_{\text{FEM}}\)**} & \multicolumn{2}{c}{**f\({}_{\text{WAH}}\)**} & \multicolumn{2}{c}{**f\({}_{\text{FEM}}\)**} & \multicolumn{2}{c}{**f\({}_{\text{WAH}}\)**} \\ \hline Mode 1 & 10.203 & 10.179 & 1.002 & 11.319 & 11.346 & 0.998 & 12.398 & 12.395 & 1.000 \\ Mode 2 & 21.176 & 21.185 & 1.000 & 22.565 & 22.581 & 0.999 & 23.917 & 23.890 & 1.001 \\ Mode 3 & 34.921 & 34.754 & 1.005 & 36.468 & 36.261 & 1.006 & 37.932 & 37.704 & 1.006 \\ Mode 4 & 39.674 & 39.634 & 1.001 & 41.085 & 41.141 & 0.999 & 42.547 & 42.595 & 0.999 \\ Mode 6 & 60.157 & 60.622 & 0.992 & 61.804 & 62.196 & 0.994 & 63.245 & 63.732 & 0.992 \\ Mode 8 & 84.521 & 84.296 & 1.003 & 86.146 & 85.915 & 1.003 & 87.735 & 87.503 & 1.003 \\ Mode 9 & 88.795 & 88.810 & 1.000 & 90.417 & 90.429 & 1.000 & 92.003 & 92.019 & 1.000 \\ \hline  & \multicolumn{6}{c}{**3.764 \(\boldsymbol{\times}\)10\({}^{-6}\)**} & \multicolumn{6}{c}{**5.646 \(\boldsymbol{\times}\)10\({}^{-6}\)**} & \multicolumn{6}{c}{**7.528 \(\boldsymbol{\times}\)10\({}^{-6}\)**} \\ \cline{2-10}  & \multicolumn{2}{c}{**f\({}_{\text{WAH}}\)**} & \multicolumn{2}{c}{**f\({}_{\text{FEM}}\)**} & \multicolumn{2}{c}{**f\({}_{\text{WAH}}\)**} & \multicolumn{2}{c}{**f\({}_{\text{FEM}}\)**} & \multicolumn{2}{c}{**f\({}_{\text{WAH}}\)**} & \multicolumn{2}{c}{**f\({}_{\text{FEM}}\)**} & \multicolumn{2}{c}{**f\({}_{\text{WAH}}\)**} \\ \hline Mode 1 & 14.214 & 14.244 & 0.998 & 15.870 & 15.861 & 1.001 & 17.312 & 17.313 & 1.000 \\ Mode 2 & 26.315 & 26.302 & 1.001 & 28.486 & 28.496 & 1.000 & 30.505 & 30.523 & 0.999 \\ Mode 3 & 40.763 & 40.429 & 1.008 & 43.236 & 42.972 & 1.006 & 45.508 & 45.365 &calculated with the Wah equation, so their values do not appear in Table 4. Nevertheless, regardless these limitations, the comparison with these results is useful to validate our numerical results.