# Research on Panel Flutter Considering the Effect of Convective Active Cooling
## 3 Panel Stability  
There are five main factors that contribute to the effect of coolant on panel flutter, including the coefficient of head-panel pressure \(\alpha\),the pressure drop ratio \(P_{Gr}\), the coolant injection direction, inertial resistance \(C_{c1}\) and viscous resistance \(C_{c2}\). The effect of the pressure drop ratio \(P_{Gr}\) on the panel stability, which changes the differential pressure distribution between the upper and lower surfaces of the panel, is discussed first in this section. Figure 4 shows the time history of the panel response at different pressure drop ratios \(P_{Gr}\), when \(\lambda\) equals 300.0 and 350.0, respectively. In Figure 4a, the panel is buckled, but the buckling deformation of the panel decreases as the pressure drop ratio increases. When the pressure drop ratio is infinite, the panel is in a flat and stable state. In Figure 4b, the motion morph changes with different pressure drop ratios. It is buckled when \(P_{Gr}\) equals 1.0 and 2.0 and LCO when \(P_{Gr}\) equals 5.0, 10.0, and \(\infty\). The amplitude of the LCO increases, and the equilibrium position moves towards 0. Figure 4: Time history of panel response under different pressure drop ratios: (**a**) \(\lambda=300\); (**b**) \(\lambda=350\). Figure 3: Comparison of the peaks of LCO. Figure 5 shows the curves of the peak of deformation \(W_{p}\) versus dynamic pressure \(\lambda\) under different pressure drop ratios \(P_{Gr}\). It can be seen that the panel bifurcates from the stable state to LCO, and the critical dynamic pressure \(\lambda_{cr}\) at which bifurcation occurs decreases with increasing \(P_{Gr}\). Some cases are shown. The panel bifurcates from the buckling state to LCO at \(\lambda=420.0\) when \(P_{Gr}=1.0\); moreover, \(\lambda=349.0\) corresponds to \(P_{Gr}=5.0\), and \(\lambda=344.0\) corresponds to \(P_{Gr}=\infty\) under which the panel bifurcates from flat and stable to LCO. Figure 6 shows local bifurcation diagrams with several bifurcations of deformation extreme value versus dynamic pressure when \(R_{0}=-2\pi^{2}\) and \(R_{0}=-3.5\pi^{2}\). In Figure 6a, three bifurcations occur: from the buckling state to the anharmonic periodic motion at \(\lambda=211.7\); from the anharmonic periodic motion to the buckling state at \(\lambda=248.6\); and from the buckling state to the LCO at \(\lambda=263\). In Figure 6b, more than three bifurcations occur, and these are not detailed here. Figure 5: Peak of deformation versus dynamic pressure under different pressure drop ratios. Set \(R_{0}=-4\pi^{2}\). Figure 7 shows the time history and phase diagram of the panel deformation when \(\lambda\)= 150.0. The panel is in buckling motion and gathers as a point on the phase plane at \(P_{Gr}=2.0\), while the panel performs chaotic motion and the phase diagram is disordered, reflecting strong nonlinearity at \(P_{Gr}=5.0\). Figure 8 shows the results when \(\lambda\)= 350.0, where anharmonic periodic motion occurs. Bifurcation diagrams with several bifurcations of deformation extreme value versus dynamic pressure are shown in Figure 9. For \(P_{Gr}=2.0\), the first bifurcation of the panel occurs at \(\lambda=186.0\) from the buckling state to the chaotic motion. The second bifurcation occurs at \(\lambda=196.0\) from chaotic motion to anharmonic periodic motion. The third bifurcation occurs at \(\lambda=469.0\) from anharmonic periodic motion to LCO. There are other bifurcations, but they are not mentioned here. They all show the complex and strongly nonlinear nature of the panel dynamic response. For \(P_{Gr}=5  
Based on the above results, the stability regions of the panel with an in-plane load at different pressure drop ratios are shown in Figure 10. These curves divide the \(\left(-R_{0}/\pi^{2}\right)-\lambda\) plane into two regions: the stable and buckled region and the flutter region. The flutter region has complex motion forms, including LCO, anharmonic periodic motion, chaotic motion, etc. The curves indicate a nonfunctional relationship between \(-R_{0}/\pi^{2}\) and \(\lambda\).