I have a problem which got me thinking, but I'm unable to solve to my satisfaction. The problem involves a gyro attached to a platform which in turn is attached to a flywheel. (See image below) The constraints are as follow: The platform & flywheel are solidly attached to each other and can only rotate about the global X-axis. The gyro gimbal has one axis of freedom about the local Y-axis. There is no friction. The system has the following properties: The gyro has an angular moment of inertia of 0.25 slug-ft^2 and is spinning at 1000 rad/sec (angular momentum of 250 lbf-ft-sec). The platform/flywheel has an angular moment of inertia of 80 slug-ft^2. The initial conditions are that the gyro is spun up with its gimbal is locked in place. The platform is then spun about the x-axis to an angular velocity of 1 rad/sec. The gyro's gimbal is released when the gyro's spin axis is exactly vertical (aligned with the z-axis). What happens to the system? I know that the gyro will immediately precess. But I'm having trouble with conserving both angular momentum and total energy. As I understand it, if the gyro precesses, it will take angular momentum away from the flywheel so that the total angular momentum along the x-axis is conserved. However, if any momentum is taken from the flywheel, it's energy drops. Where does this energy go? I seem to remember that precession doesn't change the angular momentum of the gyro, which means the gyro didn't gain the energy lost from the flywheel when it's momentum changed. What am I overlooking? How do I conserve both angular momentum and energy?
Hang on - the state of precessing motion is a state where the total angular momentum is changing all the time. During gyroscopic precession the gyroscope wheel is subject to a torque, and that torque is continuously changing the total angular momentum. The thing that doesn't change in the friction-free case is the magnitude of the total angular momentum. In the case of friction-free gyroscopic precession the magnitude of the spin rate does not change and the magnitude of the precession rate does not change. I'm just responding to this detail. I haven't looked whether this affects the outcome of your reasoning.
One way to think about this is what would happen if the flywheel were rotated but the gyro was not spun up at all....
If the gyro was not spinning, then nothing of interest would happen. The flywheel would just continue to spin without any changes. The direction of the angular momentum vector is changing, but it's magnitude remains constant. The direction of the angular momentum vector is changing because the precession is producing a torque reaction which is being countered by the gimbal mount. However ......... I do think I have solved the problem. If I did the math correctly, it looks like the gyro tilts forward about 18.7 degrees, gains the energy from the flywheel turning it into additional spin momentum, and the flywheel comes to a stop. I know I said that the gyro doesn't gain spin momentum from a precession. And that's true for a simple circular movement. But in this case, the gyro is being forced to move in a somewhat complicated motion which allows torque to be applied to the spin axis.
The gyro tilts forward so its angular momentum along the x-axis replaces the missing momentum from the flywheel.
The gyro angular velocity only increased a little to 1000.16 r/s. The reason is that the flywheel had very little energy compared to the gyro.
Right. An interesting modification is to see whether the flywheel would stop rotating at all combinations of Moments of Inertia and angular velocities?
If I understand what's happening correctly, if the flywheel has more angular momentum than the gyro, the flywheel will continue turning (albeit slower) and the gyro will align itself with the flywheel's rotational axis (x-axis).