Simplified rocket lateral dynamics model sign convention

Question

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

Answer 1

The equations the author gives are correct.

Notice that with the angular momentum equation: $$L_vr_v = I\ddot{\theta}$$ We would expect a positive $L_v$ to increase $\theta$. Similarly with the linear momentum equation, a positive $L_v$ should increase $x$.

Now for the confusion: The author has drawn an $L_v$ vector on his/her diagram pointing in the negative direction! They show the positive direction with that dashed line. This would then be carried through the equations with a built-in negative, giving the same result as your rewritten equations.

This is actually a common convention for body-centered coordinate systems for aircraft: forward (through the nose) and starboard (through the right wing) are taken as positive.