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Free Precession of a Rotating Rigid Body

The free precession of a rotating rigid body is a classic application of Euler's equations. Since the precession is free, there is no torque, and the angular momentum is constant in the space frame. In the body frame, however, the body axes , , (indicated by the red, green, and blue spheres) are fixed, and the angular momentum and space axes (indicated by the red, green, and blue arrows) will change with time. For an axisymmetric ellipsoid, the angular velocity and the angular momentum precess about the symmetry axis . This precession can be visualized by the rotation of a space cone (centered about ) around the body cone (centered about ), with the angular velocity at their point of contact. This Demonstration shows the free precession as a function of time in both the body and space frames. The body's height and radius can be varied, as can the initial angular velocity .

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Euler's equations for a rotating rigid body can be exactly solved for the free motion (with zero torque) of an axisymmetric body, with principal moments of inertia . These give the angular velocity and angular momentum in the body frame, and will exhibit free precession if the angular velocity is not initially along one of the symmetry axes. The body frame and space frame can be related by a product of two rotation matrices. This is used to transform , , and the space and body axes and ), and to visualize their dynamics as a function of time. In the body frame, both and precess about , while in the space frame, and precess about .
The solution to Euler's equations can be found in the following references. For more information, see the Wikipedia entries for "Euler's equations (rigid body dynamics)" and "Precession".
References
[1] J. R. Taylor, "Rotational Motion of Rigid Bodies," Classical Mechanics, Herndon, VA: University Science Books, 2005 pp. 367–416.
[2] S. T. Thornton and J. B. Marion, "Dynamics of Rigid Bodies," Classical Dynamics of Particles and Systems, Pacific Grove, CA: Brooks/Cole, 2004 pp. 411–467.
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