Dynamic Gyroscope Precession - 2D Side View, Axle-Tip Path and Nutation Graphs

Presets

Speed

Live readouts

Graphs

Numerical checks

running...

How to use

The left view shows the rotor on its axle, pivoting at the grey support; the right view looks down on the path of the axle tip. Try: (1) Fast gyro - the tip circles at a steady rate equal to M g l / (I3 w3); (2) Released at rest - the axle first falls, then nutates in cycloid loops about the mean precession; (3) Slow spin - a lower spin gives faster precession and bigger loops; (4) Sleeping top - stays upright above a critical spin rate and wobbles away below it; (5) raise the spin and watch the precession rate fall in the third graph. Space pauses, R restarts.

The science

Heavy symmetric top. With Euler angles theta (tilt), phi (precession) and psi (spin), the Lagrangian is L = I1 (theta-dot^2 + phi-dot^2 sin^2 theta)/2 + I3 (psi-dot + phi-dot cos theta)^2 / 2 - M g l cos theta where I1 is the moment of inertia about the pivot perpendicular to the axle and I3 about the axle. The angles phi and psi are cyclic, so the momenta p_psi = I3 w3 and p_phi = I1 phi-dot sin^2 theta + I3 w3 cos theta are conserved, leaving one equation for theta:

I1 theta-ddot = I1 phi-dot^2 sin theta cos theta - I3 w3 phi-dot sin theta + M g l sin theta

Fast top. When I3 w3 is large the gravity torque M g l sin theta is balanced by the change of angular momentum direction, giving steady precession Omega = M g l / (I3 w3) independent of tilt and inversely proportional to the spin: a faster spin gives slower precession. Slower precession solutions follow from I1 cos(theta) Omega^2 - I3 w3 Omega + M g l = 0.

Nutation. Release at rest and the axle oscillates about the mean tilt at about I3 w3 / I1 (fast top), tracing loops in the top view.

Sleeping top. A vertical top is stable if w3 > 2 sqrt(I1 M g l) / I3 - the condition used in the 4th preset.

Checks. (1) measured precession matches M g l/(I3 w3) within 3%; (2) energy conserved to 1e-6 by the RK4 integrator; (3) a spin 1.4 times the critical value stays upright, 0.7 times falls; (4) pure-precession start has no nutation.

Simplifications. Frictionless pivot, no air drag or bearing losses (real gyros slowly leak spin and the axle rises), rotor modelled as a uniform disc on a massless axle, rigid body, point pivot at the axle end.