# Linear Vibration Analysis of Shells Using a Seven-Parameter Spectral/_hp_ Finite Element Model
## 6 Numerical Results
### Convergence Study
#### 6.1.1 Plates  
In the convergence study, the material properties considered are: \(\
u=0.30\), \(E=210\) GPa, and \(\rho=7850\) kg/m\({}^{3}\). The natural frequencies for rectangular plates, with dimensions \(a\times b\), are reported using a dimensionless parameter defined in reference [35] as  
\[\lambda=\frac{2b^{2}f}{\pi}\left[\frac{Eh^{2}}{12\rho\left(1-\
u^{2}\right)} \right]^{1/2}, \tag{34}\]  
where \(f\) is the natural frequency expressed in Hz and \(h\) represents the thickness of the geometry. In the case of a plate with variable thickness, \(h\) takes the value of the smallest thickness.  
First, the present formulation is tested to model square plates with ratio \(a/b=1\), and constant thickness. For this geometry, the convergence study is performed for the following boundary conditions:  
1. All edges simply-supported (SSSS) \[u_{3}(x,0)=0,\ \ \ u_{3}(0,y)=0,\ \ \ u_{3}(x,b)=0,\ \ \ u_{3}(a,y)=0.\] (35)
2. Two opposite edges simply-supported and the other two clamped (SCSC) \[u_{3}(x,0)=0,\ \ \ \ u(0,y)=0,\ \ \ u_{3}(x,b)=0,\ \ \ \mathbf{u}(a,y)=0.\] (36)
3. All edges clamped (CCCC) \[\mathbf{u}(x,0)=0,\ \ \ \mathbf{u}(0,y)=0,\ \ \ \mathbf{u}(x,b)=0,\ \ \ \mathbf{u}(a,y)=0.\] (37)  
Table 1 contains results using different mesh sizes for Cases I, II, and III. In all the studies, the relative error for the last mesh refinement (from 36 to 64 elements) is less than 0.12%. For that reason, a mesh size of 36 elements is used on the verification section to model plates with constant thickness.  
In addition, the present formulation is tested to model plates with variable thickness in one direction along its mid-surface. Thickness variation is considered as follows:  
\[h\left(y\right)=h_{1}\left[1-\chi\ \frac{y}{2b}\right], \tag{38}\]  
where \(h_{1}\) is the maximum thickness, \(\chi\) is the tapered ratio, and \(b\) is the length of the side along \(y\) axis. Thus, the dimensionless frequency parameter \(\lambda\), given by Equation (34), is calculated using the maximum thickness. Additionally, for this variation, the minimum thickness in the plate can be calculated setting \(y=b\) in Equation (38), which gives  
\[h_{0}\equiv h\left(b\right)=h_{1}\left[1-\frac{\chi}{2}\right]. \tag{39}\]  
Table 2 contains the frequency parameter values obtained in the convergence study performed by applying each case of the boundary conditions given in Equations (35)-(37) to a plate with variable thickness. From those values, it is concluded that the results obtained using 36 elements are converged, and this mesh size will be used for the numerical verification.  
In addition to the plates with linear variable thickness, a square plate with parabolic thickness variation is analyzed. The thickness variation used is the same presented by [PERSON] et al. [35], and it is given as follows:  
\[h\left(y\right)=h_{1}\left[1+\frac{1}{2}\left(\frac{y}{b}\right)^{2}-\frac{y} {b}\right]. \tag{40}\]  
This plate is subject to the boundary conditions given in Case III, Equation (37), with a ratio \(b/h_{0}=20\) and \(h_{0}/h_{1}=0.5\). Table 2 also presents the convergence study of plates with linear and parabolic thickness variation, LTV and PTV denote a linear and parabolic thickness variation, respectively. The values obtained for the last refinement are in the order of 0.1%. For that reason, a mesh with 36 elements is used for the numerical validation.