## 4 Numerical Investigation
### Parametric Study
Moreover, Figure 10 shows the effect of an even porosity volume fraction on the dimensionless excitation frequency of a BD-FG porous cylindrical shell. Note that the dimensionless excitation frequency decreases for increasing values of \(n_{x}\) and \(n_{z}\). In addition, for a fixed value of \(n_{x}\), \(n_{z}\), the excitation frequency tends to converge to a common point. As expected, at the intersection point, the effect of an even porosity volume fraction, \(\zeta\), on the dimensionless excitation frequency is almost negligible. However, for different values of \(n_{x}\), \(n_{z}\), the effect of even porosity between the ceramic and metal phases can change significantly, such that before the intersection point, an increased even porosity volume fraction increases the dimensionless excitation frequency, and the contrary occurs  
Figure 8: Effect of the transverse power index on the DIR for a BD-FG cylindrical shell with \(R=0.5\) m, \(h/R=0.02\), \(L/R=1\), \(n_{x}=0\), \(\alpha=0.3\). Figure 9: Effect of the longitudinal and transverse power indexes on the DIR for a FG cylindrical shell with \(R=0.5\) m, \(h/R=0.02\), \(L/R=1\), \(\alpha=0.3\). after the intersection point, with a gradual decrease in the excitation frequency for an enhanced \(\zeta\). The additional Figures 11 and 12 show the different response for different values of \(\zeta\), while assuming the same value for \(n_{x}\) and \(n_{z}\). In detail, under the assumption \(n_{x}=n_{z}=0.15\) (Figure 11), it is visible that an increased porosity \(\zeta\) moves the DIR towards higher excitation frequencies. A reversed behavior occurs in Figure 12 under the assumption \(n_{x}=n_{z}=1.5\), since an increased value of \(\zeta\) causes a general shift of the DIR to lower excitation frequencies. This confirms the effect of either the even porosity volume fraction \(\zeta\) and the power indexes \(n_{x}\) and \(n_{z}\) on the stability response of the structure. Figure 4:0 Effect of the even porosity volume fraction \(\zeta\) on the DIR for a BD-FG porous cylindrical shell with \(R=0.5\) m, \(h/R=0.02\), \(L/R=1\), \(\alpha=0.3\), \(n_{x}=n_{z}=0.15\). Figure 4:1 Effect of the even porosity volume fraction \(\zeta\) on the dimensionless excitation frequency versus the longitudinal and transverse power indexes for a BD-FG porous cylindrical shell with \(R=0.5\) m, \(h/R=0.02\), \(L/R=1\), \(\alpha=0.3\), \(\beta=0\). In Figures 13-15 we repeat the parametric analysis to evaluate the effect of an uneven porosity between two phases of the second ceramic and second metal. In detail, Figure 13 shows the variation of the dimensionless excitation frequency versus \(n_{x}\), \(n_{z}\), for different values of \(\xi\). Figure 14 is devoted to check for the influence of \(\xi\) on the DIR of the structure for \(n_{x}=n_{z}=0.15\). Additionally, in this case, we can observe as the DIR moves to the right side by increasing \(\xi\) and it takes place at higher excitation frequencies. Nevertheless, by assuming \(n_{x}=n_{z}=1.5\), a different trend is noticed for the DIR in Figure 15, since an increased value of \(\xi\) yields the DIR to occur at lower excitation frequencies and its width gets smaller. Figure 4:2 Effect of the even porosity volume fraction \(\zeta\) on the DIR for a BD-FG porous cylindrical shell with \(R=0.5\) m, \(h/R=0.02\), \(L/R=1\), \(\alpha=0.3\), \(n_{x}=n_{z}=0.15\). Figure 4:3 Effect of the uneven porosity volume fraction \(\xi\) on the dimensionless excitation frequency versus the longitudinal and transverse power indexes for a BD-FG porous cylindrical shell with \(R=0.5\) m, \(h/R=0.02\), \(L/R=1\), \(\alpha=0.3\), \(\beta=0\). The last parametric analysis considers the possible sensitivity of the response to the elastic foundation. For this reason, Figures 16 and 17 plot the variation of the DIR with the Winkler or the Pasternak foundation coefficients, respectively. A noteworthy increase in stiffness emerges from both figures, where the origin of the DIR moves towards higher values of frequency.