Your question is a bit confusing to me, so I’m going to answer the question I think you are asking, “What are the equations that determine the motion of a gyroscope?”
In my notation here, underlining indicates a vector, ‘x’ indicates a vector cross-product.
Angular momentum of a particle about a point:
[ul][li]r is the vector from the point to the particle[/li][li]p is the momentum vector, equal to mv (mass * velocity).[/li][li]Angular momentum L = r x p.[/li][/ul]
So, for a given gyroscope, you can calculate it’s angular momentum by integrating all the little dLs for all the little chunks defining a dr and a dp. The simplest case of a gyroscope would be a perfectly thin ring of radius r and mass m, rotating at an angular velocity of [sym]w[/sym] in the x-y plane, in a counterclockwise direction as viewed from above (z > 0.) Such a gyroscope would have L = [sym]w[/sym]r[sup]2[/sup]mk, where k is a unit vector in the positive z direction.
Note that the direction of L is perpendicular to the plane of the disk of the gyroscope - remember that a vector cross product uses the “right-hand rule” - r x p means point the fingers of your right hand in the direction of r, curl your right fingers in the direction of p, and your extended thumb points in the direction of the result. This page at Montana State gives a slightly different definition which gives the exact same result, complete with a picture.
Now, given the angular momentum of the gyroscope, you can calculate how torques effect the motion of the gyroscope.
Torque about a point:
[ul][li]r is the vector from the point to the location where a force is being applied[]F is the Force vector being applied[]Torque N = r x F[/li][/ul]
Now, in linear motion we have F = ma, where a is acceleration. Using calculus notation of [sup]dx[/sup]/[sub]dt[/sub] to mean “the time derivative of x”, this equation can be written as
[ul][li]F = m[sup]dv[/sup]/[sub]dt[/sub] OR,[/li][li]F = [sup]dp[/sup]/[sub]dt[/sub][/li][/ul]
Similarly, in rotational motion we have N = [sup]dL[/sup]/[sub]dt[/sub].
So, when you apply a torque to the gyroscope, the angular momentum changes in the direction of the torque vector. The weirdness of a gyroscope comes from the right-hand rule of what the torque vector actually is - it’s at right angles to the force you are applying. For my intuitive explanation, see my post in the Basic Science you don’t understand thread. For a diagram including vectors (but with slightly different notation than I use) see they hyperphysics page on precession. Be sure to scroll down to see all the diagrams. The hyperphysics site is great for exploring this kind of question, BTW.
Next, you want to know what determines the speed at which th gyroscope precesses, and that would be a matter of applying the above to a notional gyroscope. I’m out of time for today, but perhaps another person will be along to discuss this further, if anyone cares. Also, stopping now gives you a chance to tell me how far off I am from your actual question. 