Rigid Body Stability
A rigid object undergoing force-free rotation, i.e. no external force or torque acting on the object, obeys the following Euler Equations of rigid body rotation:
(I₂ - I₃)ω₂ω₃ - I₁ω₁˙ = 0
(I₃ - I₁)ω₃ω₁ - I₂ω₂˙ = 0
(I₁ - I₂)ω₁ω₂ - I₃ω₃˙ = 0
where, I₁, I₂, I₃ are the moments of inertia and ω₁, ω₂, ω₃ are the components of angular velocity about the principal axes ŵ(e₁), ŵ(e₂), ŵ(e₃) respectively.
Consider a rigid body in space with no force acting on it, which has an angular velocity vec(ω) = (ω₁, 0, 0). The object then receives a small perturbation to change its angular velocity to vec(ω) = (ω₁, ω₂, ω₃), where ω₂, ω₃ ≪ ω₁. The rigid-body returns to its force-free rotation after the perturbation.
a) Given that ω₂ and ω₃ are small, show using the first Euler equation above that ω₁˙ ≈ 0 and therefore ω₁ is approximately constant.
b) Solve for ω₂˙ and ω₃˙ using the other two Euler equations.
c) Find ω₂¨ in terms of ω₃˙.
d) By appropriate substitution, find ω₂¨ in terms of ω₂.
e) Now describe the behavior of ω₂(t) for the different scenarios below:
i) I₃ > I₂ > I₁
ii) I₁ > I₂ > I₃
iii) I₂ > I₁ > I₃ (Note: it can lead to the Dzhanibekov effect where the object flips periodically)