ramana wrote:A question: Do we know the spacing of the 500N engines on the lander?
I am assuming they are mounted in a square configuration.
Also whats the rotational moment of inertia of the lander along its principal axes?
I want to calculate how much the engine has to over perform in order to tumble the lander?
The spacing might be obtainable from public sources, but the moment of inertia is not something easy to obtain.
But - we don't really need the exact values for either of these.
Technically, even a tiny force will eventually cause a large velocity change and a large displacement. So if a 10 ton object is subject to a nano-newton of force, it will eventually accelerate to great speeds, and cover large distances (if there are no opposing forces).
In the same way, even a minute over-performance by one engine will eventually tumble the lander head-over-heels, just a question of time. It could be a year, or a million years, if the over-performance was like 1 ppm or 1 pp-trillion.
So we can use this to get a good estimate of tumbling time.
As below - assume a mass for the lander, assume a "diameter" (with cylindrical lander assumption), and calculate the maximum moment of inertia (which is the value if all of the lander's mass was situated at half the diameter away from its rotational axis).
I_max=m*r^2 (I_max=max. moment of inertia, m=mass, r=radius)
Then calculate the tumble times for various values of over-performance, like 1%, 2%, 5%, etc. Finally, apply a fudge factor for the moment of inertia. And we get a ball-park value of the tumble time.
Recall, (Turning Moment)=(Moment of Inertia)*(Angular Acceleration), just like (Force)=(Mass)*(Linear Acceleration).
If we have the angular acceleration, and we have the tumbling angle (assume 180 degrees, full head-over-heels), we get the tumble time from (Angle)=0.5*(Angular Acceleration)*(Time)^2,
just like (Distance)=0.5*(Linear Acceleration)*(Time)^2.
So in this vein, please see the calcs. below:
In the above, the tumble times are calculated for two different values of mass - 2000 kg and 5000 kg. Radius is assumed to be half a meter. As seen, even a 1% over-performance by one engine, with a moment arm of 0.125 m (or 5 inches), will tumble a 5 ton lander head-over-heels within 2 minutes, even if the moment of inertia were at its maximum possible value. With a realistic fudge factor on the moment of inertia like 0.5 or so, a 5 ton lander will tumble in 80 seconds with a 1% over-performance on one 500 N engine.
Hope there are no mistakes in the above.