2.1 Formulation du modèle de Scanlan

Le modèle non linéaire de Scanlan

L’équation non linéaire du modèle de R. Scanlan est:

\[ \boxed { \begin{array}{l}J_0\ddot\alpha +C(\alpha -\alpha _0)=f(\alpha ,\dot\alpha ),\alpha (0)=0,\dot\alpha (0)=0, \\ f(\alpha ,\dot\alpha )=\displaystyle \frac{\varrho SL}{2}V^2(1-2\displaystyle \frac{\dot\alpha L}{V}\sin (\alpha )+(\displaystyle \frac{\dot\alpha L}{V})^2)c_{m_0}(\alpha _ a), \\ \alpha _ a=\hbox{arctan}(\tan (\alpha )-\displaystyle \frac{L\dot\alpha }{V\cos (\alpha )}). \end{array}} \]

On a, pour le critère de Poincaré-Bendixson:

\[ \hskip-8.53582677165pt\boxed {\begin{array}{l}\displaystyle \frac{\partial f}{\partial \dot\alpha }(\alpha ,\dot\alpha )=\varrho SL^2 Vp\\ p=[c_{m_0}(\alpha _ a)(\displaystyle \frac{\dot\alpha L}{V}-sin(\alpha ))-\displaystyle \frac{\partial c_{m_0}}{\partial \alpha }(\alpha _ a)\displaystyle \frac{\cos ^2(\alpha _ a)}{2\cos (\alpha )}(1-2\displaystyle \frac{\dot\alpha L}{V}+(\displaystyle \frac{\dot\alpha L}{V})^2))\end{array}} \]

et pour celui de l’énergie: $\boxed {q(\alpha ,\dot\alpha )=\displaystyle \frac{f(\alpha ,\dot\alpha )-f(\alpha ,0)}{\dot\alpha }.}$