# Aerothermoelastic Analysis of Conical Shell in Supersonic Flow
## 4 Numerical Results and Discussions
### Natural Vibration Characteristics Analysis  
In this section, the effects of temperature change and free-stream static pressure on the fundamental frequency of the composite conical shell are analyzed. As shown in Figure 3, the influence of temperature change on the natural frequency of a conical shell under different volume fraction indices is studied. It can be observed from the figure that in a thermal environment, the natural frequency solved based on the linear temperature hypothesis is basically consistent with the natural frequency calculated based on the polynomial series representation of the temperature distribution. It can be concluded that the linear temperature distribution is reasonable for thin-walled conical shells. Additionally, as the temperature difference between the upper and lower surfaces increases, the natural frequency of the conical shell first decreases slowly, then rapidly, and finally becomes zero. The temperature change corresponding to the zero natural frequency is the critical buckling temperature change of the structure, represented by \(\Delta T_{cr}\). In addition, it can be also found that the fundamental frequency of the conical shell increases as the volume fraction increases, and the critical buckling temperature change is proportional to the index of volume fraction.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Property** & \(\mathbf{P_{-1}}\) & \(\mathbf{P_{0}}\) & \(\mathbf{P_{1}}\) & \(\mathbf{P_{2}}\) & \(\mathbf{P_{3}}\) \\ \hline \(E\) (pa) & SUS304 & 0 & \(201.04\times 10^{9}\) & \(3.079\times 10^{-4}\) & \(-6.534\times 10^{-7}\) & 0 \\  & Si\({}_{3}\)N\({}_{4}\) & 0 & \(348.43\times 10^{9}\) & \(-3.070\times 10^{-4}\) & \(2.130\times 10^{-7}\) & \(-8.946\times 10^{-11}\) \\ \(\mu\) & SUS304 & 0 & 0.31 & 0 & 0 & 0 \\  & Si\({}_{3}\)N\({}_{4}\) & 0 & 0.24 & 0 & 0 & 0 \\ \(\rho\) (kg/m\({}^{3}\)) & SUS304 & 0 & 8166 & 0 & 0 & 0 \\  & Si\({}_{3}\)N\({}_{4}\) & 0 & 2370 & 0 & 0 & 0 \\ \(\beta\) (1/K) & SUS304 & 0 & \(12.330\times 10^{-6}\) & \(8.086\times 10^{-4}\) & 0 & 0 \\  & Si\({}_{3}\)N\({}_{4}\) & 0 & \(5.8723\times 10^{-6}\) & \(9.095\times 10^{-4}\) & 0 & 0 \\ \(k\) (W/m\({}^{\circ}\)K) & SUS304 & 0 & 15.379 & 0 & 0 & 0 \\  & Si\({}_{3}\)N\({}_{4}\) & 0 & 13.723 & 0 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Material properties of a functionally graded conical shell.  
Figure 3: The influence of temperature change on the natural frequency of a conical shell under different volume fraction indices.  
The influence of free-stream static pressure on the natural frequency of the conical shell is studied in Figure 4. As is shown in the figure, for the conical shell under free-clamped boundary conditions, with the increase in \(p_{\infty}\), the natural frequencies of the first mode decrease gradually and finally become zero, the natural frequencies of second mode decrease gradually but remain basically stable, the natural frequencies of third order increase gradually, and the natural frequencies of fourth order decrease gradually. When the free-stream static pressure \(p_{\infty}\) is further increased, the natural frequencies of the third and fourth modes merge into a single mode. When two adjacent natural frequencies converge with the increase in \(p_{\infty}\), the conical shell loses stability and flutters.