Question

A rigid body has 3 different principle moments of inertia I₁>I₂>I_3, Show that rotations about 1 and 3 axes are stable, while rotation about the 2 axis is unstable. Define what you mean by stability.

Answer 1

This can be understood mathematically,

The fact is that the motion around the axis 2 is not stable,while around axis 1 and axis 3,the motion is stable.This can be proved by linearized stability analysis.

A steady state in which the angular velocity vector has only one non vanishing constant component is a solution of the equations of the motion.

For example,

₁==constant ; ₂=0 ; ₃ =0 is a solution describing rotation around the first axis.

Also, ₁=0 ; ₂ = =constant ;₃ =0 is also a solution describing rotation around the second axis.

Now,the stability of small perturbations around these solutions can be analyzed. A perturbation of the first solution is given by :

Repeat this analysis for the rotation around second axis,

And here, Stability means stable oscillation,which must be harmonic like a mass on a spring. There is a restoring force proportional to displacement.