Physics Friday 4

Stability of rotation axes:

For an arbitrary rigid body and axis, the relationship between angular velocity and angular velocity L, and between the torque and change in momentum, is given as in the simpler cases by the moment of inertia: , where now the moment of inertia I is not a scalar but a rank-2 symmetric tensor. As I can be represented as a real and symmetric 3×3 matrix, it can be diagonalized, with the eigenvalues as diagonal elements, by suitable rotation of the coordinate axes. These axes are the principal axes, and the corresponding elements of I are the principal moments of inertia. Note that the principle axes are fixed with respect to the rigid body; thus they move in the lab frame.

In the reference frame of the rotating body, we have the Euler equations:



where , , and are the principal moments of inertia; , , and are the components of the angular velocity; and , , and are the components of the applied torque. Note that even if the torque is zero, we still have non-trivial solutions; this is torque-free precession.

We examine the equilibria and stability of the torque-free case. Let us have distinct values for the principal moments of inertia, and label them such that . We have:


Now, let us look for the equilibria, that is to say, cases where is constant (in the body frame). This gives equations:



And as the principal moments of inertia are distinct, their differences are nonzero, and the above equations are met only if two of the components of are zero, which means the equilibria are when the rotation is along one of the principal axes.

Now, we consider the stability; the rotation is stable if a small perturbation of the axis leads to small oscillations about the equilibrium. Let us first consider rotation about the first axis, with the smallest moment of inertia. We perturb the axis by small amounts λ and μ: , with ; and we condsider only terms up to first order in λ and μ. Our equations then become:


From the first (approximate) equation, we have that is a constant, and the other two equations become coupled linear equations for λ and μ:


Decoupling to obtain an equation for λ:


Remember that the solution to is simple harmonic oscillation with frequency Ω, so for λ, we have oscillations of frequency:
(note that the term in the square root is positive as ).
Similarly, we find that μ has a similar oscillation of different amplitude and a π/2 difference in phase, so that the axis of rotation thus performs an elliptical oscillation, or precesses, about the principal axis, and thus it is stable for this axis.

Note that Euler’s equations are symmetric to cyclic permutation of the indicies, thus allowing us to easily find similar results for the other two axes.
For the axis of largest moment of inertia, , we obtain

or oscillations with frequency , with μ again at the same frequency and π/2 difference in phase, so this axis is also stable.

For , we obtain

This is of the form , which has exponential instead oscillatory solutions. From this*, we see that this is a saddle point, and thus unstable. To find the full motion requires the full solution of the equations, and can be visualised with Poinsot’s Construction.

Thus the rotation is stable at the principal axes with the smallest or largest moment of inertia, but not the one with moment of inertia between these. One can demonstrate this by tossing a book or other object of approximately rectangular prism shape: two axes produce a nice “flipping” motion, while the third produces more complex “tumbling” motion. Thus the importance of moment of inertia measurements for satellites to be placed in orbit.

 
 
————————————————————————-
*
From:


we find

Or in terms of initial values:

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