# Linear Vibration Analysis of Shells Using a Seven-Parameter Spectral/_hp_ Finite Element Model
## 6 Numerical Results
### Convergence Study
#### 6.1.2 Shells  
Here, two cases for truncated half hemispheres and three cases for cylinders are studied. In Figure 5, the revolution profiles of (a) a truncated hemisphere, (b) a cylinder with linear variation, and (c) a cylinder with linear symmetric variation [30] are shown. The important geometric parameters are shown in these figures. Note that an angle \(\theta\), not shown, lies in the \(xy\)-plane for the hemisphere and in the \(yz\)-plane for the cylinder. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**B.C.**} & \multirow{2}{*}{**b/h**} & \multirow{2}{*}{**Mesh Size**} & \multicolumn{8}{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 SSS & 10 & 4 & 1.8850 & 4.6013 & 4.6013 & 7.0535 & 8.1524 & 8.7411 & 8.7411 & 8.8824 & 8.8935 & 9.2255 \\  & & 16 & 1.8768 & 4.5881 & 4.5881 & 7.0230 & 8.1452 & 8.6764 & 8.6764 & 8.7754 & 8.7886 & 9.2255 \\  & & 36 & 1.8758 & 4.5866 & 4.5866 & 7.0193 & 8.1452 & 8.6764 & 8.6764 & 8.7731 & 8.7864 & 9.2255 \\  & & 64 & 1.8757 & 4.5864 & 4.5864 & 7.0187 & 8.1452 & 8.6764 & 8.6764 & 8.7728 & 8.7861 & 9.2255 \\ \hline SCSC & 10 & 4 & 2.7475 & 5.0590 & 5.9661 & 6.2146 & 8.2245 & 9.1435 & 10.6854 & 10.9426 & 10.9804 & 11.7858 \\  & & 16 & 2.7289 & 5.0420 & 5.9548 & 6.1604 & 8.1746 & 9.0403 & 10.6534 & 10.7096 & 10.9684 & 11.7122 \\  & & 36 & 2.7236 & 5.0379 & 5.9524 & 6.1481 & 8.1632 & 9.0368 & 10.6470 & 10.6881 & 10.9667 & 11.7026 \\  & & 64 & 2.7212 & 5.0362 & 5.9515 & 6.1425 & 8.1584 & 9.0357 & 10.6450 & 10.6795 & 10.9663 & 11.6989 \\ \hline CCCC & 10 & 4 & 3.4000 & 6.5599 & 6.5599 & 9.2440 & 11.0969 & 11.2003 & 12.5671 & 12.5671 & 13.4031 & 13.4031 \\  & & 16 & 3.3754 & 6.5062 & 6.5062 & 9.8116 & 10.8716 & 10.9702 & 12.5399 & 12.5399 & 13.2043 & 13.2043 \\  & & 36 & 3.3676 & 6.4928 & 6.4928 & 9.1648 & 10.8503 & 10.9478 & 12.5351 & 12.5351 & 13.1818 & 13.1818 \\  & & 64 & 3.3638 & 6.4864 & 6.4864 & 9.1569 & 10.8415 & 10.9384 & 12.5340 & 12.5340 & 13.1721 & 13.1721 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Convergence of dimensionless frequency parameter for square plates of ratio \(b/h=10\). The dimensionless frequency parameter for the hemispheres is denoted by \(\Lambda\) (see [PERSON] et al. [35])  
\[\Lambda=2\pi fr\left(\frac{\rho}{G}\right)^{1/2}, \tag{41}\]  
where \(G\) is the shear modulus. In addition, for these kinds of geometries, the following boundary conditions are considered:  
* All free edges (FF) \[\mathbf{u}(r,\varphi_{0},\theta)\
eq 0,\ \ \ \mathbf{u}(r,\varphi_{1},\theta) \
eq 0.\] (42) On the the other hand, for cylinders, the boundary conditions are:
* All clamped edges (CC) \[\mathbf{u}(0,y,r)=0,\ \ \ \mathbf{u}(L,y,r)=0.\] (43)
* A clamped edge and a free edge (CF) \[\mathbf{u}(0,y,r)=0,\ \ \ \mathbf{u}(L,y,r)\
eq 0.\] (44)  
The first case studied is a truncated hemisphere with constant thickness, where its geometrical parameters are taken from the work by [PERSON] and [PERSON] [32], and those correspond to \(\varphi_{0}=\pi/6\), \(\varphi_{1}=\pi/2\) and \(r/h=5\). The second case analyzed is a truncated hemisphere with linear variable thickness, and its geometric parameters are given by \(\varphi_{0}=\pi/6\), \(\varphi_{1}=\pi/2\) and \(r/h_{m}=5\), where \(h_{m}\) corresponds to the mean thickness defined as \(h_{m}=\left(h_{0}+h_{1}\right)/2\). The thickness variation is determined as [35]  
\[h\left(\varphi\right)=h_{0}\left[1+2\left(\frac{\varphi-\varphi_{0}}{\varphi_ {1}-\varphi_{0}}\right)\right]. \tag{45}\]