WOBBLE STABILISATION OF ROTATING BODIES — AN EXPLORATION
If any of you have watched The Expanse, then you must know that Tycho station exists. Sure, it looks like a cool space station from the outset, having all the common features which floating outposts must — rotating rings, docking platforms, correction thrusters. But, of course, it was hard for me to ignore a significant part of the Tycho station — it was, well, rotating. And a rotating body in free space cannot escape from the plague that is random wobbles.
Granted, it is a big station, and has a high moment of inertia. However, it is a free-floating body in space, under the absence of any external torques, and thus, is prone to the phenomenon of torque free precession. The angular momentum of a closed system free from the action of external torques must always be conserved, and thus, ideally, one would expect a rotating body in space to maintain its axis of rotation. However, this is not always true — a solution arises from the Euler equations of motion in which a torque-free body can still undergo rotations about all of its principal axes while still conserving angular momentum.
TORQUE FREE PRECESSION — AN INTUITIVE LOOK
Consider an oblong rotating body, as shown below:
This is a uniform ellipsoid, rotating with angular speed ω about one of its principal axis, here denoted as e3. Ideally, one would expect it to continue with this rotation forever. However, there is the possibility of wobbles in the absence of external torques (given the right initial conditions). The situation is demonstrated below:
As you can see, the angular momentum vector of the system has rotated — something that shouldn’t occur in free space. Thus, to compensate, the system generates angular momentum Lc about a third axis:
Thus, the body begins to wobble, or precess, about the main axis e3 — not ideal if you want a space station.
THE EULER EQUATIONS:
The Euler equations of motion turn out as follows, if ω is the rotation speed of the oblong body, I0 is the moment of inertia about e3, I are the moments of inertia about e1, e2, which must be identical for steady precession to occur, and ω1, ω2 are the angular velocities about the other two axes respectively. Thus,
From the third equation, ω is a constant. Thus, it can be shown that:
Where
The angular velocities essentially oscillate such that the overall vector moves in a circle — precession. This isn’t the main focus of this article, however — we would like to predict the behavior of the system under some kind of external driving torque.
WHAT WOULD TYCHO DO?
This was a fun exploration — let us model the output behaviour of the system when under the influence of some kind of sinusoidal driving torque. Thus, from simplifying the Euler equations, we get:
Now, notice that it is impossible for both ω1 and ω2 to be zero — thus, we can set one of them to zero, giving us controlled rotation about one axis only — better than precession. Setting ω2 to zero, we get:
Thus, from (1), we get:
And,
Thus, it is easy to see that the two driving torques have to be a quarter cycle out of phase with each other. Further, the ratio of the two torques has to be:
Thus, the two torques have identical magnitudes when the driving torque oscillates at the same rate that the body precesses. Now, what is the most general steady state response? (Transients exist, of course, but we assume them to be damped in some fashion — really reaching here, am I not?) Let:
Thus,
Solving for C, D, we get:
It’s easy to see that, as α increases, both C and D would tend to zero, assuming, of course, that both A and B were real (were B imaginary, then it would be possible for D to be zero. Here, both driving torques are in phase). Further, note that both the precession rate and driving frequency can’t be the same, or the amplitude becomes unbounded — it would thus be difficult for passive control to occur. However, by arbitrarily increasing α, it would be possible for both C and D to be zero.
As you can see, the plot looks a lot like that of a resonant oscillator — not a coincidence, given what we’re going to see in the next section.
PHYSICAL MECHANISM FOR SINGLE AXIS CONTROL
Alright, α and Ω cannot be the same. The only requirement is for the torques to be out of phase. However, even if we look at active control, how can this be done physically? Well, one can mount four springs arranged in a square, as shown below, to give the required driving torques.
The springs are 90 degrees apart, and thus exert torques that are a quarter cycle out of phase. The only restriction is that the springs cannot have the same spring constant, or else we once again hit the α = Ωroadblock. Further, the springs have to be driven with a frequency α. Not that Tycho would need the extra bills — it has a high moment of inertia anyways.
FINAL REFLECTIONS
An asteroid is most definitely not a symmetric body. However, in space, a rotating asteroid is still subject to the same wobbles, which are bound to be periodic as the total energy of the system is a constant. So, let us say that you were an engineer from the recent season of For All Mankind, and wanted to capture an asteroid and reduce its wobble while tugging it. The angular velocity along the three axes isn’t going to be a pure sinusoid — instead, it’ll be some kind of sum. Thus, to “cancel” these wobbles, the control torques (which could be provided by external thrusters) would also need to account for this complexity.
And the input thrust would also have to be a similar Fourier integral. So, that’s it for today!
