Conservation of Angular momentum for an irregular body rotating about three axes

I am having trouble understanding how conservation of momentum can be used to calculate final angular velocity for a body rotating in multiple axis with an asymmetrical MoI.

Suppose a satellite is tumbling at different angular speeds about its three principle axis of rotation. At a certain point, the solar panels and antennas deploy, which increases the principle moments of inertia ($$I_{xx}$$, $$I_{yy}$$ and $$I_{zz}$$ all change). Since angular momentum is conserved, the spacecraft's spin rate will change as well. I can't seem to figure out how to determine the final angular velocity about each axis. Can I just treat each axis independent of eachother, and calculate the final angular speed about each axis with the conservation of momentum formula? For example, here is how I would calculate the final angular speed about the x-axis: $$ω_{x}^\prime = \frac{I_{xx}ω_{x}}{I_{xx}^\prime}$$

Or, since the satellite is spinning in three axis and each axis has a different principle MoI, I need to calculate the final angular velocities like so:

$$\vec{I}\times\vec{ω} =\vec{I}^\prime\times\vec{ω}^\prime$$

I hope this makes sense. It seems as though many worked examples just consider conservation of angular momentum about one axis and I am having trouble understanding it within a 3-dimensonal system.

• I don't see why all three principle axes would remain the same after deploying panels and antennas. Assuming electric motors are used, the center of mass would not move, but unless the body of the satellite has some obvious geometric symmetries in all three dimensions that are unaffected by the structural change - the principle axes may change. Maybe the reason the examples show just one axis is because just one axis has a symmetry that keeps it principle, but the other two principle axes are rotated, as well as having their principle moments change. Nov 21, 2023 at 21:01
• Hi Chad, I am saying all three principle MoI values would change due to the deployment, which I why I have I and I^prime on either side of the equations above. Nov 21, 2023 at 21:05
• I understand. What I don't understand is why the principle axes have to remain the same due to the deployment, i.e. their direction - not just the principle MoI values. Nov 21, 2023 at 21:07
• I see. I suppose they don't and now that you have explained this I can't imagine they would stay the same after the deployment. I suppose it adds more complexity to the question I am asking. With a change in principle axes and MoI, would solving the problem be easier with a simulation, or could it be calculated relatively easily analytically? Nov 21, 2023 at 21:25

The angular momentum vector is not constant in the body frame.1 Instead, you would have to use Euler's equations, which are (assuming no external torques) $$\frac{d\vec{L}}{dt} + \vec{\omega} \times \vec{L} = 0,$$ or, in terms of the inertia tensor, $$\frac{d}{dt} (\mathbf{I} \vec{\omega}) +\vec{\omega} \times (\mathbf{I} \vec{\omega}) = 0.$$ As the satellite deploys, its inertia tensor will change; but if the inertia tensor is known as a function of time in the body frame, then (in principle) one can solve this equation for $$\omega$$ as a function of $$t$$ and find the final angular velocity of the satellite. For a realistic scenario, numerical techniques are probably the way to go; the problem boils down to solving a system of 3 coupled nonlinear time-dependent ODEs in $$\omega_x$$, $$\omega_y$$, and $$\omega_z$$, which is the sort of thing that Runge-Kutta solvers were designed to do.

As was pointed out in the comments, there is no particular guarantee that the principal axes of the body will remain the same during the process. One can imagine a nice symmetric situation in which they do; in which case the equations can be written out in terms of the components along these fixed principal axes as \begin{align*} \frac{d}{dt} (I_{xx} \omega_x) &= (I_{yy} - I_{zz}) \omega_y \omega_z \\ \frac{d}{dt} (I_{yy} \omega_y) &= (I_{zz} - I_{xx}) \omega_z \omega_x \\ \frac{d}{dt} (I_{zz} \omega_z) &= (I_{xx} - I_{yy}) \omega_x \omega_y \end{align*} But even in this symmetric case, it will not be the case that $$\omega_x I_{xx} = \omega'_x I'_{xx}$$. As you can see from the above equations, if $$\omega_y$$ and $$\omega_z$$ are non-zero at any time during the process, then $$I_xx \omega_x$$ will change (and similarly for $$I_{yy} \omega_y$$ and $$I_{zz} \omega_z$$.)2

1 It's possible to show that the norm of $$\vec{L}$$ is constant in the body frame, but its direction will vary unless it points along one of the body's principal axes.

2 Note that this is true even if the inertia tensor of the body doesn't change. This is the well-known phenomenon of "torque-free precession", in which a body's rotation axis changes with time when it's not rotating around a principal axis.

First off: I recommend against attempts to think in terms of a superposition of three rotations, around three axes.

In linear mechanics the three degrees of freedom are independent, but in the case of rotation the three axes of rotation aren't.

I recommend always thinking in terms of a single instantaneous axis of rotation.

General discussion

For an object with three different moments of inertia: as it rotates there are internal stresses that transfer momentum internally.

To get an idea of the resulting motion I recommend the following two resources:
Youtube video by David Brown Dzhanibekov effect
Article by Nicholas Mecholsky Analytic formula for the geometric phase of an Asymmetric top

In the case an object with three different moments of inertia: The orientation of the angular momentum vector is constant (conservation of angulal momentum), but the direction and magnitude of the instantaneous angular velocity of the object are not constant.

To find a final angular velocity is quite an intractable problem. It is a fully deterministic process, but an exhaustive calculation would require very complex equations indeed; not feasible.

The effect of deploying antenna's and solar panels depends on the direction of the instantaneous angular velocity. That is, the effect it will have depends on what instantaneous axis the object happens to be rotating around at the instant of moment-of-inertia change

About the two resources that I recommended. It's not that I am suggesting you should understand all of the mathematics there. Rather, my suggestion is that you use the information there to get an idea of the intricacies of the problem.

If you change the ratio's of the moments of inertia then the before and after motion pattern will be different.

Incidentally, in order to have perfect conservation of energy the rotating body must be perfectly rigid. Well, a spacecraft with deployed solar panels and deployed antenna's is of course not perfectly rigid.

The higher the rigidity, the less opportunity there is to dissipate energy. Conversely: any non-rigidity gives opportunity to dissipate energy. When there is dissiplation of energy: eventually the rotational motion will settle on rotation around the axis with the largest moment of inertia. When that state is reached there is still rotational kinetic energy, but there is no longer opportunity to dissipate that kinetic energy.