I recently bumbped into a rather basic but interesting question on aeroelasticity. I've learned to derive the fluttering critical speed from Pines's theory but it involves some spring stiffness like for normal spring $K_h$ and torsion spring $K_t$.
However, I want to see if I can derive the same thing if the wing is now 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$
For simplicity, let $h(y,t)=0$ and $\alpha=s(y)e^{pt}$ so we can focus entirely on the torsion dynamic response. Let $L=qca_0(\alpha+\alpha_0)$ and $M=qcea_0(\alpha+\alpha_0)$. But now things get out of my control as I don't know how to solve for $p$ (to let its real part positive), given $s(y)$ unsolved too.