Intermediate-Axis Linear Stability

Worked Example

Consider a rigid body with principal moments of inertia I₁ = 1, I₂ = 2, and I₃ = 3. Suppose the body is initially spinning with an angular velocity Ω = 1 about its intermediate axis (Principal axis 2). We introduce a small initial perturbation ε = 0.001 along the first principal axis (so x(0) = 0.001, y(0) = 0). We want to find the perturbation at time t = 2.

First, we calculate the coefficients for the linearized Euler equations around the intermediate axis 2. The perturbation coordinates are x = δω₁ and y = δω₃.

1. Calculate a:
   a = (I₂ - I₃)Ω / I₁ = (2 - 3)(1) / 1 = -1
2. Calculate b:
   b = (I₁ - I₂)Ω / I₃ = (1 - 2)(1) / 3 = -1/3 ≈ -0.3333
3. Find the eigenvalue squared:
   λ² = a × b = (-1) × (-1/3) = 1/3 ≈ 0.3333.
   Since λ² > 0, the perturbation exhibits exponential instability with a growth rate λ = √(1/3) ≈ 0.5774.
4. Evaluate the perturbation at t = 2:
   x(2) = ε × cosh(λ × 2) = 0.001 × cosh(2/√(3)) ≈ 0.001744
   y(2) = (b / λ) × ε × sinh(λ × 2) = (-1/3) / √(1/3) × 0.001 × sinh(2/√(3)) = -√(1/3) × 0.001 × sinh(2/√(3)) ≈ -0.000825

Expected Results:

Eigenvalue squared: 0.3333333333333333
Growth rate: 0.5773502691896257
Perturbation x: 0.001744112480153778
Perturbation y: -0.0008250107763797518
Classification: Exponential instability

Experiment

You can verify this calculation and visualize the perturbation trajectories in the Intermediate-Axis Linear Stability module.

1. Set the Inertias to One two three.
2. Set the Principal axis to 2 (the intermediate axis).
3. Ensure the Spin rate is set to 1.
4. Set the Time to 2.
5. Leave the Perturbation at 0.001.
6. Observe the exponentially diverging trajectory in the phase portrait canvas and the resulting numeric values.
7. Change the Principal axis to 1 or 3 to observe closed, oscillatory orbits (oscillatory stability).

Limitations

This analysis utilizes a small perturbation linear theory. It accurately predicts the local onset of instability or oscillation for infinitesimal initial deviations from steady rotation. However, it does not describe the full nonlinear rigid body dynamics, such as the complete asymptotic damping (which requires energy dissipation models) or the full "tennis-racket flip" (Dzhanibekov effect), where nonlinear terms dominate as the trajectory departs significantly from the intermediate axis.

References

Marsden, J. E., and Ratiu, T. S. (1999). Introduction to Mechanics and Symmetry. Springer. Available at: https://pages.jh.edu/rrynasi1/PhysicalPrinciples/textbooks/Marsden%2BRatiu1999IntroductionToMechanics%2BSymmetry.pdf