## 4 Numerical Investigation
### Parametric Study
Based on the plots of Figure 7, it is worth noting that a C-C boundary conditions yields higher values of the dimensionless excitation frequencies than those ones provided by a S-S or C-S supports, due to an increased stiffness of the structure. Furthermore, the origin of the instability region tends to get away from the origin. Once the dynamic load factor \(\beta\) reaches the unit value, the width of the DIR for S-S boundary condition becomes smaller, compared to the other boundary conditions. This means that, for lower values of dimensionless excitation frequency, a BD-FG cylindrical shell with S-S supports tends to become more unstable compared to the other boundary conditions. Figure 5: Effect of the static load factor on the DIR for a BD-FG cylindrical shell with \(R=0.5\) m, \(h/R=0.02\), \(L/R=1\), \(n_{x}=n_{z}=1\), \(\alpha=0.3\). Figure 6: Effect of boundary conditions on the DIR for a BD-FG cylindrical shell with \(R=0.5\) m, \(h/R=0.02\), \(L/R=1\), \(n_{x}=n_{z}=1\), \(\alpha=0.3\). A further investigation is devoted to study the influence of the power law index along the length \(n_{x}\) on the dynamic buckling behavior of one-directional FG cylindrical shell, as depicted Figure 7. In such a case, the DIR takes place at lower frequencies owning to an increased magnitude of the power law index. The effect of an increased dimensionless excitation frequency related to the origin of the DIR is meaningful within the range \(0.2\leq n_{x}\leq 5\). For greater values of \(n_{x}\), the variation in frequency corresponding to the origin of DIR becomes less remarkable. A one-directional FG cylindrical shell with \(n_{x}=10\) or \(n_{x}=8\) is more sensitive to the dynamic instability for lower excitation frequencies compared with those ones with \(n_{x}\leq 5\). For a conventional FG cylindrical shell, we also investigate the effect of the transverse power index \(n_{z}\) on the DIR, as plotted in Figure 8. It can be mentioned that for a constant value of the dynamic load factor, the enhancement of \(n_{z}\) yields a reduction width of the dynamic instability region, especially for lower excitation frequencies. In addition, the origin DIR moves to a lower dimensionless excitation frequency. Comparing the results from Figures 7 and 8, it can be concluded that, a double increase of both \(n_{x}\), and \(n_{z}\), leads to a reduction in the excitation frequency. Nevertheless, \(n_{x}\) plays an important role in the reduction of the excitation frequency and in the increase of the structural instability. For \(\beta=0\), for example, we note a reduction of the excitation frequency equal to 49.3% and 44.49%, for and increasing value of \(n_{x}\) and \(n_{z}\) from 0.2 up to 10, respectively. Figure 9 shows the effect of the transverse and longitudinal volume fraction indexes on the DIR. In detail, for an increase of these two parameters, the origin of DIR moves to higher dimensionless excitation frequency, and the DIR width declines. In conclusion, the double increase of \(n_{x}\) and \(n_{z}\) leads a metal phase reinforcement, with a overall decrease of the structural stiffness and an increase in the structural instability. BD-FG cylindrical shells with lower values of \(n_{z}\) and \(n_{x}\), are less sensitive to the dynamic instability, due to their higher stiffness. The contrary occurs for higher values of \(n_{x}\) and \(n_{z}\) due to an increased deformability of the structure. The importance of applying BD-FG materials is highlighted when the variation of material properties is considered in a single direction or more directions simultaneously. This issue can be beneficial for the fabrication and design purposes of modern FG structures. Subsequently, the dynamic stability of BD-FG cylindrical shells can be controlled selecting appropriate power indexes corresponding to the desired direction. Figure 7: Effect of the longitudinal power index on the DIR for a FG cylindrical shell with \(R=0.5\) m, \(h/R=0.02\), \(L/R=1\), \(n_{z}=0\), \(\alpha=0.3\).