# Aerothermoelastic Analysis of Conical Shell in Supersonic Flow
## 4 Numerical Results and Discussions
### Aerothermoelastic Stability Analysis  
The influence of the structural parameters of a conical shell, such as the ratio of length to radius, ratio of thickness to radius, the semi-vertex angle, as well as thermal and aerodynamic loads, on its aerothermoelastic stability is carried out. Figure 5 depicts the influence of the ratio of length to radius of the conical shell on the thermal buckling stability at different volume fraction indices. As is shown in Figure 5, the critical buckling temperature change \(\Delta T_{cr}\) of the structure slightly increases with the increase in the ratio of length to radius. Variations in the critical buckling temperature change with the ratio of thickness to radius under different volume fraction indices are shown in Figure 6. It can be found  
Figure 4: Natural frequency versus free-stream static pressure for a conical shell with different modes (\(p=1\)). Figure 5: Influence of the length-to-radius ratio of the conical shell on critical buckling temperature change. that with the increase in the thickness-to-radius ratio of conical shell, the critical buckling temperature change of the structure will gradually increase, and the structure is less prone to thermal buckling. As can be observed from the comparison between Figures 5 and 6, the ratio of thickness to radius has a more significant influence on thermal buckling stability than the ratio of length to radius. In addition, it can be found from Figures 5 and 6 that the \(\Delta T_{cr}\) of the conical shell is proportional to the volume fraction of the functionally graded material. As shown in Figure 7, the variation of \(\Delta T_{cr}\) is analyzed when the semi-vertex angle \(\alpha\) increases from \(10^{\circ}\) to \(60^{\circ}\). It can be clearly observed that when the semi-vertex angle of the structure is small (\(\alpha\leq 15^{\circ}\)), the critical buckling temperature change is little affected by the semi-vertex angle; when the semi-vertex angle is large (\(\alpha\geq 15^{\circ}\)), the critical buckling temperature change will gradually decrease with the increase in the semi-vertex angle. It can be shown that the smaller the semi-vertex angle of the structure, the better the thermal buckling stability of the conical shell. Figure 6: Influence of the thickness-to-radius ratio of the conical shell on critical buckling temperature change. Figure 7: Influence of the semi-vertex angle of the conical shell on critical buckling temperature change. Figure 11: Influence of Mach number on critical free-stream static pressure (\(p=1\)). Figure 12: Influence of temperature change on critical free-stream static pressure (\(p=1\)). Figure 10: Influence of semi-vertex angle of conical shell on critical free-stream static pressure (\(p=1\)). Figure 8 analyzes the effect of the length-to-radius ratio of the conical shell on the critical free-stream static pressure. From the figure, it can be found that the increase in the length-to-radius ratio leads to a decrease in the critical flutter pressure. In addition, with the increase in the length-to-radius ratio, the natural frequencies of the conical shell decrease. Figure 9 describes the effect of the thickness-to-radius ratio of the conical shell on the critical free-stream static pressure. It can be found from the figure that the critical free-stream static pressure and the natural frequencies increase as the ratio of radius to thickness increases. The influence of the semi-vertex angle of the conical shell on the critical free-stream static pressure is given in Figure 10. It can be observed that with the increase in the semi-vertex angle, the critical free-stream static pressure decreases accordingly. Furthermore, the natural frequency is inversely proportional to the semi-vertex angle. Figure 11 describes the variation of the critical free-stream static pressure of conical shell with Mach number. It can be observed from Figure 11 that as the Mach number increases, the critical free-stream static pressure decreases.