## 5 Conclusions  
This work investigates the dynamic stability of BD-FG cylindrical shells embedded in an elastic foundation, including possible effects related to porosity. The material properties of BD-FG porous cylindrical shells are computed according to a modified BD power law model. Using the Hamilton's principle, we determine the governing equations of the problem, under the classical TSDT assumptions. The aforementioned equations are rewritten into a system of Mathieu-Hill equations, according to a GDQ approach. The work is also devoted to determine the DIR of the structure while applying the Bolotin method. After a preliminary validation of the proposed formulation, with respect to the  
Figure 16: Effect of the Winkler coefficient on the DIR for a BD-FG cylindrical shell with \(R=0.5\) m, \(h/R=0.02\), \(L/R=1\), \(\alpha=0.3\), \(K_{g}=0\), \(n_{x}=n_{z}=1.5\). Figure 17: Effect of the Pasternak coefficient on the DIR for a BD-FG cylindrical shell with \(R=0.5\) m, \(h/R=0.02\), \(L/R=1\), \(\alpha=0.3\), \(K_{w}=0\), \(n_{x}=n_{z}=1\). available literature, we perform a large numerical investigation to check for the sensitivity of the response both in terms of excitation frequencies and DIRs, for different thickness-to-radius ratios, length-to-radius ratios, boundary conditions, transverse and longitudinal power law indexes, even and uneven porosity volume fractions, and foundation parameters. Based on the systematic numerical investigation, the main conclusions can be summarized as follows
* An increased thickness-to-radius ratio causes a general shift of the DIR origin towards higher excitation frequencies. Moreover, the DIR gets wider at a certain value of the dynamic load factor. * An increased length-to-radius dimensionless ratio moves the DIR origin towards lower excitation frequencies, whereas the DIR gets smaller. * A simultaneous increase of longitudinal and transverse power indexes yields an overall decrease in the excitation frequencies associated with the DIR origin. * The control of the dynamic instability for a BD-FG cylindrical shell, is convenient for an appropriate selection of the power indexes. * BD-FG cylindrical shells with a simply support at both ends are more unstable than the clamped-clamped or clamped-simply supported structures, because of their higher deformability. * The effect of coefficients and type of porosity on the structural DIR depend on the extent of the longitudinal and transverse power law indexes. There exists a certain value for these indexes, for which the excitation frequencies corresponding to the DIR can invert their behavior. * A general increase in the elastic foundation coefficients yields higher excitation structural frequencies especially when a Pasternak foundation is assumed instead of a Winkler foundation. Anyway, the presence of an elastic foundation makes the structure stiffer and more stable. Conceptualization, F.A., H.T., R.D. and F.T. ; Formal analysis, F.A., H.T., R.D. and F.T. ; Investigation, F.A., H.T. and F.T. ; Validation, H.T., R.D. and F.T. ; Writing--Original Draft, F.A., H.T., R.D. and F.T. ; Writing--Review & Editing, R.D. and F.T. All authors have read and agreed to the published version of the manuscript. This research received no external funding. The authors declare no conflict of interest. Here below are the equations of motion in their final form  
\[\begin{array}{l}\left(\frac{\mathrm{d}A_{11}(x)}{\mathrm{d}x}\right)\left( \frac{\mathrm{d}U(x)}{\mathrm{d}x}\right)+A_{11}(x)\left(\frac{\mathrm{d}^{2} U(x)}{\mathrm{d}x^{2}}\right)-\frac{n^{2}A_{66}(x  
\[