Everyone explains it in terms of addition of angular momentum vectors, such as on Wikipedia. But just what is an angular momentum vector? And how do they add? And how do these vectors correspond to the physical world?
A few years ago, I finally came up with an explanation of the gyro effect that, at least to me, makes intuitive sense. This page is my attempt at explaining it.
The whole thing started with reading that the international space station has a different orbit so that the Russians can also launch to it from Baikonur Cosmodrome in Kazakhstan. Kazakhstan is much further north than Cape Canaveral. Had it been just an American space station, the orbit would have been closer to equatorial. Orbits closer to the equator require slightly less energy to launch to because the rotation of the earth already gives you some speed. If you were to launch from the equator to an equatorial orbit, for example, you already have about 2.7% of the speed you need just standing on the equator. But if you launched to a polar orbit, the rotation of the earth gives you no help. Given how ridiculously much fuel (and multiple rocket stages) is needed, 2.7% makes a big difference.
So then I was thinking, once up there, couldn't you sort of scoot over to a different orbit? How would you change your orbit once up there?
So suppose we have this blue ball-shaped spaceship orbiting around the earth over the equator. How would we tilt the orbit?
We could fire our thrusters to push us north (rocket engine pointed south) every time we are on the "front" of the earth, as depicted here.
Intuitively, you expect that it would cause the orbit to be tipped away from us, but that's not what happens.
With the spacecraft whizzing around the earth clockwise (seen from the top, or north, around the earth), pushing it a bit north when it comes by the front won't so much move it north, as it will accelerate it north by a little bit. We are changing it's direction more than anything. So the new path of the spaceship would be the points marked in red.
So firing the thruster to move north every time we are in front of the earth will actually tilt the orbit to the right, not away from us.
Now, imagine, instead of a single spaceship, we have a whole ring of particles, and we accelerate each one a bit north just a bit as it goes by. The same physics still applies. And if we joined all the spaceships with a ring, that "push" to the one spaceship would just push all the ones nearer to us a little bit, but the orbit would still tilt.
Now, if we have a ring of these, we don't need gravity to hold them together. So now imagine pushing that spinning ring up on the near side, down on the far side, it will still tilt to the right.
Seems strange overall, but that is basically what a spinning wheel does when you try to change its axis of rotation. And that is the gyro effect.
For example, if you spin up a bicycle wheel, and only support it on one side, rather than tip down, like a stationary wheel would, it will slowly change its orientation, essentially "rotating" its axis of rotation, as seen from above.
So, gyro effect explained (at least to me).
The other thing to know is in which direction the wheel will react.
A good mental shortcut is to imagine the torque applied to the wheel is "rotated" 90 degrees by the wheel in the direction that its spinning. So this bike wheel rotates clockwise, as seen by me. Gravity wants to tip it away from me (top edge away from me), but the wheel rotates clockwise, so now it's the right edge that will move away from me. Seen from above, I have to keep turning myself counterclockwise to keep the wheel from hitting my arm.
In terms of physics, this shortcut explanation is not correct. But it's an easy way to remember which way the gyro effect will make a spinning wheel react.