I had came across a simplified simplified rocket lateral dynamics model seen in this image below:
. It has vanes at the exit which generate lift force and can control the rocket orientation- the lift force is actually a ''side force'' which can impose moments about the rockets centre of gravity. The link :https://github.com/build-week/hover-jet/blob/feature/start-design-scripts/design-scripts/jet_vane_speed.ipynb contains more infomation by the author. In it, linear and angular momentum equations are present for the current orientation.
Fj: engine thrust
Lv: lift force
Dv: drag force
rv: Distance from vanes to rocket centre of mass
alpha: vane angle of attack
theta: pitch over angle of the rocket
angular momentum:
$$ L_v r_v = I \ddot{\theta} $$
Linear momentum:
$$ -(F_j - D_v) \sin\theta + L_v \cos\theta = m \ddot{x} $$
I don't seem to understand the sign convention for the lift force generated by the vanes in these equations; at the current angle of attack seen in the image, the lift force Lv would be in a south-west direction. In the equations, it seems the author took it in the north-east direction. In other words, should the equations instead read as:
$$ -L_v r_v = I \ddot{\theta} $$
$$ -(F_j - D_v) \sin\theta - L_v \cos\theta = m \ddot{x} $$ Does anyone happen to know why he made the signs by Lv positive instead? Any advice is appreciated