# Linear Vibration Analysis of Shells Using a Seven-Parameter Spectral/_hp_ Finite Element Model
## 6 Numerical Results
### Convergence Study
#### 6.1.2 Shells
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**B.C.**} & \multirow{2}{*}{\(\textbf{b/}h_{0}\)} & \multirow{2}{*}{**Mesh Size**} & \multicolumn{10}{c}{**Dimensionless Frequency Parameter**} \\ \cline{3-13}  & & & \(\lambda_{1}\) & \(\lambda_{2}\) & \(\lambda_{3}\) & \(\lambda_{4}\) & \(\lambda_{5}\) & \(\lambda_{6}\) & \(\lambda_{7}\) & \(\lambda_{8}\) & \(\lambda_{9}\) & \(\lambda_{10}\) \\ \hline SSSS & 20 & 4 & 1.4395 & 3.4873 & 3.5268 & 5.4898 & 6.7130 & 6.9748 & 8.0993 & 8.5927 & 8.7670 & 8.7840 \\ (LTV) & 16 & 1.4317 & 3.4744 & 3.5049 & 5.4478 & 6.6048 & 6.7935 & 8.0989 & 8.5979 & 8.5620 & 8.6508 \\  & & 36 & 1.4304 & 3.4720 & 3.5028 & 5.4424 & 6.6003 & 6.7904 & 8.0899 & 8.5599 & 8.5920 & 8.6431 \\  & & 64 & 1.4301 & 3.4714 & 3.5023 & 5.4411 & 6.5991 & 6.7898 & 8.0989 & 8.5581 & 8.5920 & 8.6412 \\ \hline SCCC & 20 & 4 & 2.0928 & 3.8518 & 4.8478 & 5.9369 & 6.4744 & 6.9981 & 8.9493 & 9.4589 & 10.4564 & 10.6023 \\ (LTV) & 16 & 2.0773 & 3.8307 & 4.7784 & 5.9253 & 6.3929 & 6.8794 & 8.4673 & 9.2928 & 10.0036 & 10.5687 \\  & & 36 & 2.0730 & 3.8265 & 4.7678 & 5.9292 & 6.3816 & 6.8740 & 8.4780 & 9.2778 & 9.9834 & 10.5607 \\  & & 64 & 2.0710 & 3.8247 & 4.7629 & 5.9219 & 6.3768 & 6.8721 & 8.4696 & 9.2735 & 9.9751 & 10.5578 \\ \hline CCCC & 10 & 4 & 3.0223 & 5.8861 & 5.8973 & 8.3730 & 10.0989 & 10.2055 & 12.2775 & 12.3962 & 12.5543 & 12.5610 \\ (LTV) & 16 & 2.9998 & 5.8341 & 5.8493 & 8.3089 & 8.9615 & 9.9473 & 12.0541 & 12.0657 & 12.5249 & 12.5317 \\  & & 36 & 2.9836 & 5.8057 & 5.8155 & 8.2721 & 9.8170 & 9.9016 & 12.0050 & 12.0183 & 12.5057 & 12.5126 \\  & & 64 & 2.9804 & 5.8001 & 5.8098 & 8.2649 & 9.8086 & 9.8930 & 11.9957 & 12.0090 & 12.5042 & 12.5112 \\ \hline CCCC & 20 & 4 & 2.3831 & 4.4976 & 4.7480 & 6.7065 & 7.7330 & 8.4742 & 9.9484 & 10.2233 & 12.6720 & 12.8649 \\ (PTV) & 16 & 2.3655 & 4.4543 & 4.6832 & 6.6233 & 7.5201 & 8.0129 & 9.7064 & 9.7584 & 11.3271 & 12.1759 \\  & & 36 & 2.3533 & 4.4315 & 4.6585 & 6.5960 & 7.4785 & 7.9704 & 9.6596 & 9.7117 & 11.2627 & 12.1013 \\  & & 64 & 2.3508 & 4.4267 & 4.6539 & 6.5896 & 7.4706 & 7.9628 & 9.6507 & 9.7029 & 11.2503 & 12.0891 \\ \hline \end{tabular}
\end{table} 
Table 2: Convergence of dimensionless frequency parameter for square plates of variable thickness. Figure 5: Revolution profiles for (**a**) hemispherical, and (**b**,**c**) cylindrical shells. Note that this expression works for a thickness ratio \(h_{1}/h_{0}=3\), where \(h_{1}=h\left(\varphi_{1}\right)\). Table 3 contains the results of the convergence study, using elements with \(p=4\), for the truncated hemispheres with constant and variable thicknesses. In both cases, the errors for the last increment are in general below of 0.43%. Thus, the study shows that a good convergence can be achieved with a mesh size of 49 elements. The geometrical parameters of the cylinder with constant thickness are taken from [PERSON] et al. [34], which correspond to \(L=0.502\,\mathrm{m}\), \(r=0.0635\,\mathrm{m}\), and \(h=0.00163\,\mathrm{m}\), while the material properties are \(E=210\,\mathrm{GPa}\), \(\
u=0.28\), and \(\rho=7800\,\mathrm{kg/m^{3}}\). The study of convergence is realized for two types of boundary conditions, Cases V and VI given by Equations (43) and (44), respectively, and using elements with polynomial order \(p=8\). The results are presented in Table 4. In both cases, the relative errors are below of 0.1 order for the last increment. For that reason, it is concluded that a convergence can be achieved with a mesh size of nine elements. The cylinders with linear and linear symmetric thickness variations have the geometrical parameters reported in the work by [PERSON] & [PERSON] [30], which are: \(h_{m}=0.508\,\mathrm{mm}\), \(h_{0}=2h_{m}/3\), \(r/h_{m}=100\) and \(L/r=2\). For these cases, the material properties are: \(E=203.5\,\mathrm{GPa}\), \(\
u=0.285\), and \(\rho=7846\,\mathrm{kg/m^{3}}\). Additionally, the boundary conditions applied correspond to Case VI, given by Equation (44), where the edge with the maximum thickness is clamped.