Tennis Racket Theorem

#physics
Stemming from our discussion in Eulers Equations of motion, we attempt to perform a stability analysis of Euler’s equations, we investigate how a rigid body behaves when its rotation around a principal axis is slightly disturbed. This leads to the Intermediate Axis Theorem (or the "Tennis Racket Theorem").

Consider a rectangular block rotating about $z$ axis with a angular velocity ${\omega }_z$.
Our goal is to examine small perturbations in rotation along with $x$ and $y$ axis, such that ${\omega }_x\ll {\omega }_z$ and ${\omega }_y\ll {\omega }_z$.
Consider the inertia of rotations about these axes as $I_{xx},I_{yy},I_{zz}$

Assume that a small impulsive moment actually provides ${\omega }_x$ and ${\omega }_y$, and no force exists after this.
Therefore, the Euler's equations of motions become $$\begin{align} 0 &= I_{xx}\dot{\omega}{x} - (I - I_{zz})\omega_{y}\omega_{z} \ 0 &= I_{yy}\dot{\omega}{y} - (I - I_{xx})\omega_{z}\omega_{x} \ 0 &= I_{zz}\dot{\omega}{z} - (I - I_{yy})\omega_{x}\omega_{y} \end{align}$$
Analyzing the last equation, $$I_{zz}\dot{\omega}{z} = (I - I_{yy})\omega_{x}\omega_{y}$$
Since we considered ${\omega }_x$ and ${\omega }_y$ to be very small, we treat ${\dot{\omega }}_z=0$, therefore, ${\omega }_z=\omega$, or some constant.
This is known as a linearization step (research to be done)

Substituting this now constant value of ${\omega }_z$ in the first two equations, and attempting to eliminate any of ${\omega }_x$ or ${\omega }_y$ from the resulting equations, we get (assuming we eliminate ${\omega }_y$), $$I_{xx}\ddot{\omega}{x} - \frac{(I - I_{zz})(I_{zz} - I_{xx})}{I_{yy}}\omega^2\omega_{x} = 0 $$ or, $$\ddot{\omega}{x} - \mathbf{A}\omega = 0$$where $\mathbf{A}$ represents the constant factor.

This is a standard linear ODE, the solution of which is (if $\mathbf{A}>0$)$$\omega_{x}(t) = Be^{\sqrt{ \mathbf{A}t }} + Ce^{-\sqrt{ \mathbf{A}t }}$$
And it's oscillatory motion if $\mathbf{A}<0$ (imaginary powers of $e$, $\sin$ and $\cos$ terms).

Therefore, if $\mathbf{A}>0$ we will have exponential divergence of this, and ${\omega }_x$ and ${\omega }_y$ will grow without bounds (atleast as predicted by small perturbation analysis), and the motion is unstable.
Conversely if $\mathbf{A}<0$, oscillatory motion of ${\omega }_x$ and ${\omega }_y$ will ensue, and the motion is stable.

Examining the signed terms in $\mathbf{A}$, the ones responsible for deciding the sign, $$(I_{yy} - I_{zz})(I_{zz} - I_{xx})$$
we see that for $\mathbf{A}$ to be positive, $I_{zz}$ must be intermediate between $I_{xx}$ and $I_{yy}$.

We therefore conclude that the motion of a body rotating about an axis whose MOI is intermediate of the other two axes, is unstable.
Also rotation about the smallest or largest MOI axes are stable.

Rotating Frames and Fictitious Forces