## 6 Numerical Results and Discussion  
In this work we analyze the free vibration problem of FG spherical, hyperbolic paraboloidal, and circular cylindrical shells with porosities. For the sake of completeness, these geometries are schematically depicted in Figure 3.  
The FGM is considered to be the composition of metal (SUS304) and ceramic (Al\({}_{2}\)O\({}_{3}\)), whose properties are provided in Table 1.  
In order to verify the accuracy of the proposed formulation, a comparative investigation is carried out against the existing results in the literature. For this reason, a simply supported isotropic spherical shell has been considered. In this case, we consider \(a=b=1.0118\) m, \(h=0.0191\) m, \(R=R_{1}=R_{2}=1.91\) m, \(E=1\) Pa, \(\rho=1\)kg/m\({}^{3}\), and \(\
u=0.3\)[30]. Table 2 shows the main results in terms of natural frequencies for different modal shapes and wave numbers. As shown in Table 2, there is a very close agreement between results based on our proposed formulation and the predictions by [PERSON] et al. [28] and [PERSON] et al. [29]. Hereafter, we define the frequencies in the dimensionless form as  
\[\hat{\omega}=\omega\frac{a^{2}}{h}\sqrt{\frac{\rho_{c}}{E_{c}}}. \tag{50}\]  
Tables 3-5 illustrate the dimensionless frequencies of FG porous spherical, hyperbolic paraboloidal, and circular cylindrical shells by using different values of the power-law index p \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material** & \(E\) (**Gpa**) & \(\
u\) & \(\rho\) (kg/m\({}^{3}\))** & **Reference** \\ \hline SUS304 & 201.04 & 0.3262 & 8166 & [27; 36; 38] \\ Al\({}_{2}\)O\({}_{3}\) & 349.55 & 0.24 & 3800 & [27; 36; 38] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties of metal (SUS304) and ceramic (Al\({}_{2}\)O\({}_{3}\)). \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Frequency**} \\ \cline{2-4}
**Mode (\(m,n\))** & **Present Study** & **[PERSON] et al. [28]** & **[PERSON] et al. [29]** \\ \hline (1,1) & 0.52692 & 0.52864 & 0.50211 \\ (2,1) & 0.58120 & 0.58954 & 0.56247 \\ (1,2) & 0.58120 & 0.58954 & 0.56248 \\ (2,2) & 0.66757 & 0.68370 & 0.65706 \\ (3,1) & 0.73173 & 0.75974 & 0.73915 \\ (1,3) & 0.73173 & 0.75974 & 0.74035 \\ (3,2) & 0.87284 & 0.88680 & 0.86359 \\ (2,3) & 0.87284 & 0.88680 & 0.86360 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of the circular frequencies (rad/s) for an isotropic simply-supported (SSSS) spherical shell. Figure 3: Geometries of shell structures considered in the applications: (**a**) Spherical shell; (**b**) hyperbolic paraboloidal shell; (**c**) circular cylindrical shell. and porosity volume fraction \(\beta\) for the FGM-types 1 and 2 power-law distributions. In particular, Table 3 shows the dimensionless frequency for a simply supported spherical shell with \(a=b=1\) m, \(R_{1}=R_{2}=1\) m and thickness \(h=0.01\) m. Four different wave numbers \((m,n)\) are considered, together with various \(p\) and \(\beta\) for two different ceramic volume fraction distributions. Table 4 shows similar results for the simply supported FG porous hyperbolic paraboloidal shell with \(a=b=1\) m, thickness \(h=0.01\)m, and curvatures \(R_{1}=-R_{2}=1\) m.  
In Table 5 we report the dimensionless frequencies of FG porous as functions of \(p\) and \(\beta\) for the circular cylindrical shell with simply supported boundary condition and square planform of \(a=b=1\) m, thickness \(h=0.01\) m, and curvatures \(R_{1}=1\), \(R_{2}=\infty\). Based on the results in Tables 3-5, the dimensionless frequency of FG porous shell structures can be easily modified by varying the power-law index \(p\) and porosity volume fraction \(\beta\). In more detail, the dimensionless frequencies for a FGM-type 1 are always higher than those related to an FGM-type 2. Similar behavior can be observed for hyperbolic paraboloidal and circular cylindrical shells (see Tables 4 and 5).