# Research on Panel Flutter Considering the Effect of Convective Active Cooling
## 2 Theoretical Analysis
The non-dimensionalised equation of motion is  
\[\begin{array}{l}\frac{\partial^{2}W}{\partial t^{2}}+\frac{\partial^{4}W}{ \partial\xi^{4}}-6\big{(}1-\
u^{2}\big{)}\left[\int_{0}^{1}\left(\frac{ \partial W}{\partial\xi}\right)^{2}d\xi\right]\frac{\partial^{2}W}{\partial \xi^{2}}-R_{0}\frac  
the panel deformation [4,14]. Substitute Equation (10) into Equation (9), multiply by \(\sin(r\pi\xi)(r=1,2,\cdots,N)\), and then weighted integrate over the panel length. The result is  
\[ \begin{array}{l}\frac{(rn)^{4}}{2}a_{r}+6\big{(}1-v^{2}\big{)}\bigg{[}\!\sum_{ n}\frac{(n\pi)^{2}}{2}a_{n}\!^{2}\bigg{]}\frac{(r\pi)^{2}}{2}a_{r}+R_{0}\frac{(rn)^{2} }{2}a_{r}+\frac{1}{2}\frac{d^{2}a_{r}}{d\tau^{2}}\\ \\ +\lambda\bigg{\{}\frac{1}{2}\frac{M_{\infty}^{2}-2}{M_{\infty}^{2}-2}\bigg{(} \frac{\mu}{\lambda\sqrt{M_{\infty}^{2}-1}}\bigg{)}^{\frac{1}{2}}\frac{da_{r}} {dt}+\sum\limits_{n}n\pi a_{n}\delta_{1n}+\frac{\gamma+1}{4}M_{\infty}\frac{ \mu}{\lambda}\sum\limits_{m}m\pi a_{m}n\pi a_{n}\delta_{2\text{nm}}\bigg{\}}\\ \\ =\Big{[}\frac{1-(-1)^{r}}{r\pi}(\alpha-1)+\frac{(-1)^{r}}{r\pi}\frac{a}{R_{Cr }}\Big{]}P_{\infty}+\frac{1}{2}C_{c1}\frac{d^{2}a_{r}}{d\tau^{2}}+\frac{1}{2}C _{c2}\frac{da_{r}}{d\tau}\ \ (r=1,2,\cdots,N)\end{array}  \tag{11}\]  
where  
\[\delta_{1n}=\int_{0}^{1}\cos(n\pi\xi)\sin(r\pi\xi)d\xi\]  
\[\delta_{2\text{nm}}=\int_{0}^{1}\cos(n\pi\xi)\cos(n\pi\xi)\sin(r\pi\xi)d\xi\]  
Equations (11) are a set of 2-order nonlinear differential equations in time. Important effecting parameters contain \(R_{0},\lambda,M_{\infty},\mu,\alpha,P_{Gr},C_{c1},C_{c2}\). Setting \(y_{r}=\frac{da_{r}}{d\tau},\frac{dy_{r}}{d\tau}=\frac{d^{2}a_{r}}{d\tau^{2}}\), the equations become a set of 1-order differential equations, which are solved by the 4 Runge-Kutta method. In this paper, the material properties and geometrical dimensions are \(E=71.7\) GPa, \(\
u=0.33,\ a=1.0\) m, \(a/h=200\). The aircraft flies at an altitude of 30 km. A time step of \(\triangle\tau=0.0001\) and initialization value of \(a_{1}=0.0001\) are adopted. All figures are plots at a typical point \(\xi=0.75\). Figure 2 shows the close agreement between the stability regions given in [4,14] and the results of the present model where \(M_{\infty}=5.0,\ \mu=0.05,\ \alpha=1.0,\ P_{Gr}=\infty,C_{c1}=0.0\), \(C_{c2}=0.0\). Under the coupling effect of freestream and in-plane load, the dynamic characteristics of the panel are complex, including flat and stable, buckled but dynamically stable, Limit Cycle Oscillation (LCO), anharmonic periodic motion, and chaotic motion. In fact, the regions flat and stable and buckled but dynamically stable belong to panel stabilization, whereas the regions LCO and anharmonic periodic motion and chaotic motion belong to panel flutter. The transition process of the motion state of the panel is called bifurcation. Bifurcation is an important nonlinear phenomenon, which means the global behaviors of nonlinear mathematical systems have changed suddenly. Figure 3 compares the peak of LCO from the present model to [4,6,14] when in-plane load \(-R_{0}/\pi^{2}=0,1,2,3\), which shows the close agreement with other reference results. More research on panel flutter with coolant effect is carried out in the following sections. Figure 2: Stability regions. In this section, the dynamic analytical model of panel flutter under the effect of coolant was established. The panel dynamics equations were solved by [PERSON]'s method in space domain and the 4\({}^{\text{th}}\) Runge-Kutta method in time domain. The nonlinear modified piston theory was used to predict the unsteady aerodynamic loads. The first six modes were adopted to approximate the panel deformation. Through analysis and derivation, there were eight important effecting parameters: \(R_{0},\lambda,M_{\infty},\mu,\alpha,P_{Gr},C_{c1},C_{c2}\), respectively, which affected the characteristics of the panel flutter. In the end, the accuracy of the solution method was verified.