1.1 Le modèle non linéaire de Den Hartog

Le modèle non linéaire de Den Hartog

L’équation de l’aéroélasticité de Den Hartog pour le pilonnement et pour une incidence nominale $\alpha =0$, est:

\[ \hskip-56.905511811pt\boxed {\begin{array}{l}m\ddot z+kz=f(\dot z),z(0)=z_0,\dot z(0)=z_1,\\ \hbox{avec: } f(\dot z)=\displaystyle \frac{\varrho S V_ a^2}{2}[c_ x(\alpha _ a)\sin (\alpha _ a)+c_ z(\alpha _ a)\cos (\alpha _ a)]\\ \hbox{et: } V_ a^2=V^2(1+(\displaystyle \frac{\dot z}{V})^2),\alpha _ a=-\hbox{arctan}(\displaystyle \frac{\dot z}{V}).\end{array}} \]\[ \hskip-28.4527559055pt\begin{array}{l}\hbox{ Posons: } \boxed {\begin{array}{l} q_ c(\dot z)=2\dot zc_ z(\alpha _ a)-V(\displaystyle \frac{\partial c_ z}{\partial \alpha }(\alpha _ a)+c_ x(\alpha _ a)),\\ q_ s(\dot z)=2\dot zc_ x(\alpha _ a)+V(c_ z(\alpha _ a)-\displaystyle \frac{\partial c_ x}{\partial \alpha }(\alpha _ a)), \\ \displaystyle \frac{\partial f}{\partial \dot z}(\dot z)=\displaystyle \frac{\varrho S}{2}[q_ c(\dot z)\cos (\alpha _ a)+q_ s(\dot z)\sin (\alpha _ a)].\end{array}}\end{array} \]

 

  crit. P-B=>

 

\[ \boxed {\begin{array}{l}D^+=\{ \dot z\vert ~ \frac{\partial f}{\partial \dot z}(\dot z){>}0\} ,\\ D^-=\{ \dot z\vert ~ \frac{\partial f}{\partial \dot z}(\dot z){<}0\} . .\end{array}} \]

  crit. Ener=>

 

\[ \boxed {\begin{array}{l}E^+=\{ \dot z\vert ~ \frac{f(\dot z)-f(0)}{\dot z}{>}0\} ,\\ E^-=\{ \dot z\vert ~ \frac{f(\dot z)-f(0)}{\dot z}{<}0\} .\end{array}} \]