Motion of a gyroscope

There are two external forces acting on the system: the weight W = Mg of the flywheel and its reaction R on the pivot. We will choose this pivot as the reference point for torques. The torque of the reaction R is zero since R is along the axial z-axis. The net torque is then the one created by the weight w, so Στ = net torque = r x w. The related total angular momentum (always in the same direction as the net torque τ) is given by the formula:

1. Flywheel not spinning

When the flywheel does't spin, L_i = 0. The variation dL of the system produced by the torque remains in the initial direction and form the final angular momentum L_f = Σ dL_i. The flywheel fulls and stops at a certain time. (figure a)

2. Flywheel spinning

The flywheel starts to spin at a constant angular speed ω = ω_s about x-axis. The related initial angular momentum is L_i = L_s, which is the spinning angular momentum of the flywheel. When the flywheel is released, at a short time dt, the torque τ w x r = Mg x r gives rise to dL. dL and τ stay parallel in the plane (x,y). Their resultant poduces a new angular momentum L = dl + L_i. Hence the flywheel is shifted to spin about the rotated x-axis along the resultant angular momentum L = dl + L_i.


Since dL and L are always perpendicular, the dispalcement of all dL will be along a circle of radius r. Hence, L_i and all L_i + ΣdL that is L_f are equal in magnitude, but not in direction. Therefore, the rotation of L_f about the axial z-axis gives rise to the angular momentum L_p of angular velocity Ω about the axial z-axis. L_f spins and rotates around the axial of vector unit k in the plane (x,y), we can then write L_f = L_s i + dl j. The rusultant L_r of L_s, dL and L_f is written as: L_r = L_s i + dL j + L_p k.

In a gyroscopic motion systems like a flywheel, the angular velocity ω_p is much smallar the the spinning angular velocity ω_s (ω_p <<ω_s), so the resultant L_r becomes L_r = L_s i + dL j, that is L_p; L_p = L_s i + dL j.

The rotation associated to L_p is called precession.

3. Expression of precession angular speed

The equation τ = dL/dt gives
r x w = Mg r = dL/dt, then;
dL/dt = L_s dφ/dt = L_s ω_p = Mgr
Since L_s = I_s ω_s, where I_s is the moment of inertia of the spinning flywheel, we can write:
ω_p = Mgr/L_s = Mgr/I_s ω_s R is the distance between the axial z-axis to the center (of mass) of the flywheel.

Remark that the precession angular speed ω_p becomes larger when the spinning ω_s weakens. This is what happens at the end when friction reduces the rotation of L_r .

If the flywheel is considered as a solid cylinder of moment of inertia I_cm = I_s = (1/2) M R_o², the above expresion becomes:
ω_p = Mgr/(1/2) M R_o² ω_s = 2gr/R_o² ω_s.

That is independent of the mass of the flywheel.

Finally, note that the change in the positin vector r exhibits a nutational motion producing a translation of the wheel while spinning, along its axis (x-axis).