I explained precession entirely in laymen terms in my answer to: What determines the direction of precession of a gyroscope?
Necessary and sufficient for an object to be in a state of gyroscopic precession: the object is spinning, and a torque is being exerted. In my explanation I show how that happens.
Nutation can be thought of as a form of overshooting. If you release a spinning top roughly then for sure there will be a nutation superimposed upon the precession. You can avoid nutation by releasing very gingerly. (In effect the careful release dampens extra motions.)
Generally the nutation motion of a spinning object will dampen out fairly quickly, the kinetic energy of the nutation motion dissipates relatively fast.
The nature of nutation
I mentioned overshooting. Let’s say you have a gyroscope set up with its spin axis horizontal, so that the torque from gravity is maximal. Then you release the spinning top to just drop down.
- The center of mass of the spinning top then moves down. That downward motion induces precessing motion (see the linked explanation)
- The precessing motion is so fast that the spinning top climbs again.
- The climbing motion reduces the precession rate.
- Because of the reduced precession rate, the center of mass drops down again.
So, the nutation is a cyclic energy conversion process:
- There is a stage of giving in to the torque that is being exerted.
- During that stage the torque is doing work and the kinetic energy of the spinning object increases, which goes to the precession rate.
- When the spinning object is climbing again its kinetic energy decreases again as it gains potential energy.
When nutation has dampened out then the rate of precession is precisely enough to keep gravity from pulling the center of mass of the spinning top lower.
Gyroscopic precession and nutation are intimately related. Nutation is inherently a perturbation of a state of precessing motion. That is, there is no such thing as nutation without gyroscopic precession.
The Earth's spinning motion is very complicated. There is a torque from the Sun and a torque from the Moon, and the Moon's orbital plane is at an angle with the Earth orbit around the Sun. Probably those influences play a role in the Earth wobble.
It's obvious that the angular momentum of the spinning top is not conserved. You get gyroscopic precession when a torque is exerted. That torque causes change of angular momentum.
Still, stable gyroscopic precession is cyclic, so something is conserved there. The direction of the angular momentum is changing all the time; the thing that remains the same is the magnitude of the angular momentum.