# Research on Panel Flutter Considering the Effect of Convective Active Cooling
## 3 Panel Stability
Taking the case with \(P_{Gr}=1.0\) as an example, three critical dynamic pressures exist on the curve when \(-R_{0}/\pi^{2}=2\), which means three bifurcations and at least four panel response forms, which are buckling motion, flutter (anharmonic periodic motion or chaotic motion), buckling motion, and flutter(LCO) in turn. When \(-R_{0}/\pi^{2}\in(2.8,5)\), the panel bifurcates from buckling motion to flutter after which no multiple bifurcations or more complex bifurcation behaviors occur. As the in-plane load increases from 0.0, the critical dynamic pressure of panel flutter decreases as a whole. The stability curves tend to be horizontal with high \(P_{Gr}\) and high \(-R_{0}/\pi^{2}\). A smaller \(P_{Gr}\) results in a larger area of stability and buckling region, which means that the panel is more likely to stabilize. Figure 10: Stability regions under different pressure drop ratios. Figure 9: Bifurcation diagrams of deformation extreme: (**a**) \(P_{Gr}=2.0\); (**b**) \(P_{Gr}=5.0\). The coefficient of head-panel pressure \(\alpha\) determines the pressure difference on the surface of the panel, which is another important factor affecting the stability of the panel. Figure 11 plots the peak of deformation versus freestream dynamic pressure \(\lambda\) under different coefficients of head-panel pressure \(\alpha\). It can be seen from Figure 11 that as \(\alpha\) increases, the panel bifurcates from buckling motion or stable state to LCO. The buckling deformation and limit cycle amplitude increase gradually and the critical dynamic pressure decreases first and then increases. Figure 12 shows the stability regions of the panel under an in-plane load for different coefficient of head-panel pressure \(\alpha\). With the increase in \(\alpha\), the area of the stable and buckling region decreases first and then increases. The minimum value occurs at \(\alpha=1.4\) when \(P_{Gr}=2.0\), and at \(\alpha=1.0\) when \(P_{Gr}=\infty\) among all cases. When \(\alpha=0.2\) and \(\alpha=1.8\), the upper and lower surfaces of the panel have the same static pressure difference in the opposite direction. However, the area of the stable and buckling region is larger when \(\alpha=0.2\) than that when \(\alpha=1.8\). The same result is concluded when \(\alpha=0.4\) and \(\alpha=1.4\). As the in-plane load increases, the critical dynamic pressure of the panel flutter decreases, and meanwhile, three bifurcations and more complex bifurcation behaviors occur. Figure 11: Peak of deformation versus dynamic pressure under different coefficients of head-panel pressure: (**a**) \(P_{Gr}=2.0\); (**b**) \(P_{Gr}=\infty\). Figure 12: Stability regions under different coefficients of head-panel pressure: (**a**) \(P_{Gr}=2.0\); (**b**) \(P_{Gr}=\infty\). Figure 13 shows the relationship between the critical dynamic pressure and the buckling deformation of the panel when the bifurcation occurs. The value \(\alpha\) increases from 0.2 to 2.0 along the curves and positive ordinate. The figure mainly reflects two points:  
1. Contradiction between the panel deformation and flutter critical dynamic pressure. It is not difficult to see that the panel deformation is increased with the increase in the critical dynamic pressure. Therefore, not only the flight safety but also the strength safety of the material should be considered in the structural design process. 2. The safety envelope and flutter envelope of the panel. The \(W\sim\lambda\) plane is divided into three regions by all curve clusters, labeled I, II, and III in Figure 13. No flutter occurs regardless of the \(P_{G}\), and \(\alpha\) values in zone I. Flutter always occurs in zone II. Whether flutter occurs or not depends on the values of \((P_{G},\alpha)\). The delimitation curves of zones I and III, and zones I and II form the safety envelope and the flutter envelope of the panel, respectively. The coolant injection direction is another important factor in the analysis of panel stability.