RSE #41 RSE #48

Spin, Spin, Spin

Rocket Science for Earthlings number 48

a continuing series for the gravitationally impaired

There seem to be two types of explanations for gyroscopic motion, really really simple or, matrix calculus which is as difficult as it sounds. Simple explanations do not allow you to make the calculations necessary to design spacecraft and the matrix calculus explanations, while they might be absolutely correct and elegant - well it would take me two years of study just to understand the math. So, I've come up with a much simpler system which does make calculations possible but that I can work with. Most explanations of gyros explain the circular precession of a gyroscope due to the Earth's gravitational field, but in Space a spinning spacecraft acts as a free gyroscope. Only the impulse of rocket thrust, either as short bursts of as continuous thrust can act on the spacecraft.

First, lets us pay homage to the law of conservation of momentum which makes it possible to understand gyros. All hail to Newton - or is it Galileo? When a mass is in motion and is acted upon by an impulse of outside force, it's vector direction of motion (momentum) is changed. The vectors add and the mass leaves on the net result vector. The best example of this is the motion of the puck on an air hockey table. See the drawing below.

After the impulse the puck is on a new course, with just a little more speed. Now consider the example of a spacecraft in orbit. An impulse (small of course) applied at a right angle to its course will not cause it to fly off into deep space but will change the plane of it's orbit. The same is true of the gyro. Take the drawing above, cut it out and wrap the momentum vector and results vectors around a cylinder such that they each cover 90 degrees, you now see that the impulse has changed the plane of the gyros rotation - this is precession.

Now let's give an example from a historical spacecraft. Syncom was a spinning disk shaped spacecraft which had full attitude and translational control with just two thrusters as opposed to the usual twelve needed for a non spinning spacecraft. A very neat and simple design that had a lot going for it. Both thrusters were pulse fired, the attitude thruster could precess the spacecraft into any attitude, and the translation thruster would push at right angles to the spin axis. You just had to pick the timing of the firing pulse to change the attitude or push the spacecraft anywhere you wanted it to go. The firing dwell could be up to 60 degrees of rotation.

But how to calculate? The equations, yes math! The equations for rotational math are exactly like linear math with different symbols;

You need two things, the moment of inertia around the spin axis, and the moment of inertia at right angle to the spin axis (this has to be averaged if the masses inside the spacecraft are not distributed equally) these are different. Once the spacecraft is spun up (leave this to the booster rocket) the moment of inertia about the spin axis times the rotational velocity (one revolution is 6.28 radii or radians) gives you the rotational momentum. This is the momentum vector. Now divide the torque of the thruster impulse by the moment of inertia at right angles to the spin axis times the duration of the thrust times the inertial moment at right angles to the spin axis, this gives you the impulse momentum vector. Do the vector addition and measure the angle of the precession. This is basically a graphical or trigonometry approach but much easier than matrix calculus. I hope it works, as usual I working without a net here, this method will probably work if there is a large difference between the spin and impulse momentums.

Now, how do you measure the moment of inertia? There are formulas for basic types of objects, see a good college physics book, but spacecraft don't fit into those models, so you must measure the moment with a trifilar pendulum. Take two triangles, one fixed to the ceiling, one hung below on three wires. This system will rotate around a vertical axis with a period defined (so I've been told) by; time = 2 * pi * radius of gyration * square root of (the hang length divided by gee times the moment of inertia). I'm going to have to build one of these and try it out. First add a small mirror to the bottom platform to give you a reflected light beam spot on the wall to measure the timing, then measure the empty oscillation rate, then put several test items on the platform to get a good idea of how the system reacts. Simple point masses with radius from the rotational axis r have an inertial moment defined as I= M*r^2. Soon you should have enough information to design your own spinning spacecraft!