Today, I've tried whether I can derive the fluttering speed if an aeroplane wing is treated as a continuous beam-rod model (i.e. torsion and bending). The equations of motion are:
$ \frac{\partial^2}{\partial y^2} \left( EI \frac{\partial^2 h}{\partial y^2} \right) + m \frac{\partial^2 h}{\partial t^2} + m x_\alpha \frac{\partial^2 \alpha}{\partial t^2}+L=0$
$ -\frac{\partial}{\partial y} \left( GJ \frac{\partial \alpha}{\partial y} \right) + I_\alpha \frac{\partial^2 \alpha}{\partial t^2} + mx_\alpha \frac{\partial^2 h}{\partial t^2} - M = 0$
Let $L=qca_0(\alpha+\alpha_0)$ and $M=qcea_0(\alpha+\alpha_0)$. Here, apart from $h(y,t)$ and $\alpha(y,t)$ which are functions of time $t$ and coordinate $y$, the remaining variables are constants. Let $h(y,t)=s(y)e^{pt}$ and $\alpha(y,t)=k(y)e^{pt}$.
The goal is to solve for $q$ which $\alpha \rightarrow \infty$. How can I solve for $q$ to have $p>0$?
Thanks in advance. Your help is greatly appreciated.