# Rotational motion about a centre of mass

1. Nov 5, 2008

### quitequick

Consider a free floating spacecraft in space sufficiently far from any gravitational or other external forces. The spacecraft is equipped with attitude control thrusters. The thrusters fail and fire randomly and intermittently but symmetrically about the craft's centre of mass i.e. the craft is subject to a collection of random torque forces in magnitude and direction. Eventually, the thrusters all run out of fuel. The craft is now 'spinning randomly' about it's centre of mass.

Am I correct in thinking that this spin is in fact a rotation about a constant single arbitrary axis through the centre of mass and with a constant angular velocity? Intuitively to me, it feels like it should because to move the spin axis and therefore the mass of the craft would require some acceleration or force. But I know my intuition can be wrong!

If I am correct, I believe the implication is that if a comms satellite wants to have a certain constant orientation to the earth e.g. to have an antenna always pointing to the surface, the satellite needs only initially induce a spin of 360 degrees per orbit of the earth. There would be no requirement for constant attitude correction (assuming no solar, atmospheric etc. external influences).

What do you think?

2. Nov 5, 2008

### D H

Staff Emeritus
That's more or less what they do -- except of course they never can attain exactly the same rotation rate as the Earth (any error will lead to a drift if not corrected), the comm sats are not exactly geostationary (the orbit as viewed by an Earth-bound observer looks like a figure 8 rather than a singular point), and there are external torques (Earth gravity gradient torque and solar radiation torque).

3. Nov 5, 2008

### m.e.t.a.

Your intuition seems to be correct on both points. A body cannot rotate about more than one axis without an external force. As for the geostationary satellite, some small thrust might have to be periodically applied to counteract the satellite's precession -- which might differ to the Earth's own precession -- over the course of the year. But don't hold me to that.

- m.e.t.a

4. Nov 5, 2008

### KLoux

Can you please elaborate? It seems to me that the craft should be able to rotate about more than one axis, if that's the state it was left in when the boosters ran out of fuel. Why can it not be rolling about its own fore-aft axis, for example, while it is also rotating about some other constant axis? The axis that it is rolling about is constantly changing direction, so the rotations cannot be viewed as a rotation about some other fixed axis, right?

-Kerry

5. Nov 5, 2008

### D H

Staff Emeritus
Euler's rotation theorem, aka Euler's fixed point theorem, says that any motion of a rigid body in which some point on the body has a zero instantaneous velocity is equivalent to an instantaneous rotation about an axis passing that point. Thus the motion of a rigid body can be separated into the translation of some selected point on the body and the rotation of the body about an axis passing through that point. Something quite amazing happens if you choose the center of mass as that special point: The translational and rotational equations of motions decouple.

Newton's second law of motion has a rotational equivalent:

$${\boldsymbol F} = \frac d{dt}(m\boldsymbol v}) \;\;\to\;\; {\boldsymbol N} = \frac d{dt}({\mathbf I}{\boldsymbol{\omega}})$$

There's one problem with this formulation: The inertia tensor and angular velocity must be represented in inertial coordinates, and that means the inertia tensor will be time-varying. A way around this is to use a frame rotating with the body. Just as inertial forces arise when investigating translation in non-inertial frames, an inertial torque arises when investigating rotation in the rotating body frame. This is easily addressed with the transport theorem, which says that for any vector quantity q,

$$\left.\frac d{dt}\right._{{inertial}} {\boldsymbol q} = \left.\frac d{dt}\right._{{rotating}} {\boldsymbol q} \;+\; {\boldsymbol{\omega}} \times {\boldsymbol q}$$

Applying this to the inertial frame rotational equations of motion yields

$${\boldsymbol N} = \mathbf I \frac{d}{dt}({\boldsymbol{\omega}}) + {\boldsymbol{\omega}{\times ({\mathbf I}{\boldsymbol{\omega}})$$

where now the inertia tensor I and angular velocity omega are expressed in the body frame. If there are no external torques N present, the body will only be subject to a torque free precession caused by this inertial torque term. For small rotation rates (one rotation per day is small), the inertial torque will be very, very small.

6. Nov 5, 2008

### KLoux

Wow, thanks for the explanation.

I know this isn't applicable to the original post, since it was given that the craft was rotating about its center of mass, but what if the craft is rotating about the center of the Earth, as well as about its center of mass? There is no point on the craft with V=0 in the Earth reference frame. Does this mean there is no point you can choose that will decouple the rotational equations from the translational equations?

Thanks,

Kerry

7. Nov 5, 2008

### D H

Staff Emeritus
Any motion of a rigid body can always be described as a sum of a translation of the body's center of mass plus a rotation about some axis passing through the body's center of mass. Imagine that you are some distance away from some rigid body and are moving at exactly the same velocity as the body's center of mass. The body will appear to you to have a stationary center and to be rotating about some axis passing through the center of the body.

8. Nov 5, 2008

### quitequick

Thanks for all the answers - really useful.

I'm thinking that gyroscopic forces can explain this as well somehow. But I'm not sure how. Again, intuitively, if there were more than one axis of rotation (and I suppose I could imagine either 1 or 3, anything else would seem arbitrary) how would the gyroscopic forces interact - or not. Does this somehow explain that can be only one axis of rotation?

Is there an everyday real life example that demonstrates this one axis answer?

9. Nov 5, 2008

### D H

Staff Emeritus
quitequick,
I think I see where your confusion is coming from. You are seeing three rotation axes associated with a gyroscope as being distinctly separate rotations. Think of them instead as vectors.

I'll make an analogy in the translation domain because translation is a lot easier to visualize than rotation. Suppose the spacecraft is a cube with six thrusters, one per cube face. Each thruster is fully throttleable, is individual controlled, is oriented normal to the center of its cube face and is located at the center of its cube face. The thrusters operated individually only let the spacecraft move in +x, -x, +y, -y, +z, and -z directions. Operated collectively, however, the thrusters can obtain motion in any desired direction. The reason this works is that acceleration, like position and velocity, is a vector.

End of analogy. The reason for making the analogy: angular velocity (and angular momentum) are also vector quantities (to be very picky, they are psedovectors.)