TITLE: Calculate the position of rocket acted upon simultaneously by multiple thrusters?
QUESTION [0 upvotes]: I'm looking for an equation that will let me predict the position of a rocket after a period of time given that it is acted upon by multiple forces.
By multiple forces I mean the main thruster force and a second attitude thruster, one that affects the direction of the rocket.
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>|<  attitude thrusters
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^^^ 
main thrusters

I can easily find equations to predict the position of the rocket if only the main thruster is firing, but when we introduce the attitude thruster because it rotates the ship it consequently modifies the direction that the main thruster is acting in. I've looked into this a fair bit trying to combine equations of torque with linear motion equations but I'm still confused?
fyi. My maths level is not particularly high I achieved an A at alevel in mechanics
Thanks
Piers

REPLY [0 votes]: If both thrusters operate simultaneously , the $\theta (t)$ equation would be unchanged since the main thrusters provide zero torque. Coupling the equations for the centre of mass co-ordinates gives ..
$$ \frac{d^2}{dt^2}\left ( x_1(t)\right) = \frac{T_m}{M} \cos(\theta_0 + \omega _0 t + k_a t^2) $$
$$ \frac{d^2}{dt^2}(x_2(t)) = \frac{T_m}{M} \sin(\theta_0 + \omega _0 t + k_a t^2) $$
where $k_a = \frac{6 T_a l }{M L^2 }$
At this point, I think we have to give up on finding an analytic solution. This system would need to be solved numerically by taking tiny time increments and keeping track of $x_1(t)\; v_1(t)\;x_2(t)\; v_2(t)\; \theta (t)\; \text{ and } \omega(t).$
We may as well include the effect of the attitude thrusters on the centre of mass co-ordinates,so that the DEs to be solved numerically are ...
$$ \frac{d^2}{dt^2}\left ( x_1(t)\right) = \frac{T_m}{M} \cos(\theta_0 + \omega _0 t + k_a t^2) - \frac{T_a}{M} \sin(\theta_0 + \omega _0 t + k_a t^2) $$
$$ \frac{d^2}{dt^2}(x_2(t)) = \frac{T_m}{M} \sin(\theta_0 + \omega _0 t + k_a t^2)  +  \frac{T_a}{M} \cos(\theta_0 + \omega _0 t + k_a t^2)$$
You might want to consider an scheme in which the attitude thrusters are only active for a short period of time $\Delta t$ during which they fire in one direction for half the time and the opposite direction for the other half of the time so that $\omega (t) = 0 $ unless the attitude thrusters are firing, ( surely you don't want your rocket to be constantly spinning, it would be very wasteful to fire the main thrusters while the rocket spins. )
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