Why does the gyroscope not move upwards? I have been watching a lot of videos on gyroscopes. Everything I see says there are two steps in gyro precession:


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*Gravity pulls down imparting a torque

*The gyroscope moves forward


It seems to me that it should go on to say that the forward motion imparts a force that makes the gyroscope go up, but the gyroscope does not go up because the same gravity that is starting the process is the gravity that is capping the upward movement causing the gyro to stay at one level and precess around. It seems to me it's a three part conversion of gravity to torque. 


*

*Gravity Pulls down

*Gyroscope moves forward

*Forward movement makes the gyroscope go up


To me, the process starts with gravity force and ends with gravity force because it is the same force. It also seems to be a one to one conversion, for if it was not the gyroscope would go up or down depending on whether there were losses or gains in the conversion.
You can test it by slowing down or speeding up forward precession. Losses in forward speed would lower the weight of the gyroscope, and gains would raise the gyroscope. You can also see the losses caused by friction. Think of a gyroscope as a transistor. If you spin it up there is now a voltage potential, and as soon as you drop it a constant bias current begins to flow holding the gyroscope at a certain level. Adding or removing from the forward precession is like adding or removing voltage from the bias. The transistor allows current to flow up or down. The gyroscope lifts the weight of the gyroscope up or down.
 A: In order to address your question I need to define some terms:

I define three axes:


*

*Roll axis - the gyroscope wheel spins around the roll axis.

*Pitch axis - motion of the red frame. 
As you can see, the gimbal mounting ensures the pitch axis is perpendicular to the roll axis.

*Swivel axis - motion of the yellow frame.


As you describe, in demonstrations (especially in the case of the gyroscope wheel spinning very fast), when a torque is applied the response that you see with your eyes is that you get a turn at a 90 degrees angle with respect to the torque. Summerizing: if the applied torque is one that tends to pitch the wheel, the response is that a swiveling motion starts.
Here's the thing: while a torque is necessary to get to precession in the first place, the factor that makes it all work is motion. 
When you apply a torque to the spinning gyroscope wheel (say, a downpitching torque), the initial response is that the gyroscope wheel does pitch a bit. The effect of that pitching motion is that a swiveling motion starts. 
The mechanics of this motion transfer is explained with diagrams in an answer that I wrote in 2012, to a question titled What determines the direction of precession of a gyroscope 
Keep in mind: if the gyroscope wheel spins very fast the duration of pitching down will be only a fraction of a second, too short for the naked eye to see. It looks as if the response of swiveling instead of pitching is instantaneous. However, it cannot be instantaneous, that would violate the laws of motion. It's just that it can happen too fast to see with the naked eye.
Back to the pitching and swiveling:
As long as there is still some pitching motion the conversion of pitching motion to swiveling motion will continue. The swiveling motion comes with a tendency to pitch up. The uppitching tendency counteracts the downpitching torque. The downpitching stops when the precession rate is the rate that fully counteracts the downpitchnig torque. This makes it a self-adjusting process.

Some things you can try for yourself:
Take a small platform that you can rotate by hand. (A common name for such thing is 'lazy susan' )
You need a gimbal mounted gyroscope. (The gyroscope that is depicted in the diagram is a gimbal mounted gyroscope.)
With the gimbal mounted gyroscope spinning and precessing, while positioned on the center of the platform:  


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*Rotate the platform to match the swiveling rate. The precessing motion will last longer then, because there will be much less swiveling friction.  

*Rotate the platform in the same direction as the precessing motion, but a little faster. This will make the gyroscope wheel swivel faster than the precession rate needed to counteract the torque, and consequently it will pitch up.



Many attempts at explanantion of gyroscopic precession are worded something like this: "The gyroscope precesses because the angular momentum vector responds in such-and-such a manner to an applied torque." The problem is: that is not an explanation, that is merely restating the observation in different words. I recommend that you judge attempts at explanation by that criterium. Is an actual explanation offered, or is there merely a restatement of the observation in different words?
A: I'm taking the unusual step of posting an additional answer, just to explain why a particular widespread explanation falls short.
Here is how I understand this type of attempted explanation.
To establish a pattern:
Uniform circular motion due to a centripetal force:
With uniform circular motion the acceleration vector is perpendicular to the momentum vector. The momentum vector turns at a constant rate in the direction of the acceleration vector, and thus uniform circular motion is sustained.
The counterpart of that in angular mechanics is depicted in the diagram that was supplied by Aaron Stevens.

A torque vector (green arrow) represents a particular force pair. Here there is the downward force of gravity (red arrow) and there is the upwards force from the triangular block that supports the gyroscope. The combined effect is represented with the torque vector.
In the diagram the torque vector and the angular momentum vector are perpendicular to each other. The suggested precession law is then as follows: the angular momentum vector turns at a constant rate in the direction of the torque vector, and thus gyroscopic precession is sustained.
I will now discuss why this way of representing gyroscopic precession is flawed.
First let me recapitulate the actual behavior of a gyroscope in a range of circumstances.
A If the gyroscope wheel is not spinning then when a force is applied the gyroscope will simply fall in the direction of the force.
B If the gyroscope wheel is spinning slowly then when a force is applied the gyroscope will fall the same as in case A, the difference will be negligable.
C An intermediate spin rate such that you do get a noticable difference with case A, but not enough to look like gyroscopic precession.
D When the gyroscope wheel is spinning sufficiently fast you get that striking effect that the gyroscope swivels rather than pitching down.
To anyone reading this: I very much encourage you to obtain a gimbal mounted gyroscope and start trying out the above range of circumstances.
The range from A to D stretches across a transition. An explanation of gyroscopic precession must explain the behavior for all of the range from A to D in one unified representation. You cannot dismiss cases A, B and C. An attempted explanation that seems to work for case D, but that fails for A, B and C is inadmissible.
With the behavior recapitulated I return to the suggested precession law:
"The angular momentum vector turns at a constant rate in the direction of the torque vector, and thus gyroscopic precession is sustained."
Question:
How should the suggested precession law be applied in case B and C?
Case B: the gyroscope wheel is spinning slowly, so the angular momentum vector is non-zero. Is the suggested precession law only valid above some particular spin rate, or is it valid for every spin rate, no matter how small.
If the suggested precession law is only valid above some particular spin rate, then what is the explanation of that?

There is another aspect:
Let's look at the following hypothetical case:
Let's assume that suggested precession law is as valid as Newton's laws of motion are.
That would imply that the start of gyroscopic precession is instantaneous. It would imply that truly the gyroscope wheel starts swiveling instead of doing any pitching.
That raises the question: where does the angular momentum of the precessing motion come from? The torque (green arrow) is perpendicular to the angular momentum vector of the precessing motion, so the angular momentum of the precessing motion cannot come directly from the torque (green arrow).
In the Feyman lectures on physics Feynman discusses how exactly gyroscopic precession starts, and he points out that the transition to precessing motion does not happen instantaneously. It cannot happen instantaneously, that would violate the laws of motion.
Feynman lectures, chapter 20 Rotation in space

Some people like to say that when one exerts a torque on a gyroscope,
it turns and it precesses, and that the torque produces the
precession.
[...]
When the motion settles down, the axis of the gyro
is a little bit lower than it was at the start. Why? (These are the
more complicated details, but we bring them in because we do not want
the reader to get the idea that the gyroscope is an absolute miracle.
It is a wonderful thing, but it is not a miracle.) If we were holding
the axis absolutely horizontally, and suddenly let go, then the simple
precession equation would tell us that it precesses, that it goes
around in a horizontal plane. But that is impossible! Although we
neglected it before, it is true that the wheel has some moment of
inertia about the precession axis, and if it is moving about that
axis, even slowly, it has a weak angular momentum about the axis.
Where did it come from? If the pivots are perfect, there is no torque
about the vertical axis. How then does it get to precess if there is
no change in the angular momentum? The answer is that the cycloidal
motion of the end of the axis damps down to the average, steady motion
of the center of the equivalent rolling circle. That is, it settles
down a little bit low. Because it is low, the spin angular momentum
now has a small vertical component, which is exactly what is needed
for the precession. So you see it has to go down a little, in order to
go around. It has to yield a little bit to the gravity; by turning its
axis down a little bit, it maintains the rotation about the vertical
axis. That, then, is the way a gyroscope works.

Svilen Kostov and Daniel Hammer have done a table-top experiment to verify Feyman's assertions. The article describing their results is named after the key observation in Feynman's discussion 'It has to go down a little, in order to go around'
