Torque-free
Torque-free precession occurs when the axis of rotation differs slightly from an axis about which the object can rotate stably: a maximum or minimum principal axis. Poinsot's construction is an elegant geometrical method for visualizing the torque-free motion of a rotating rigid body. For example, when a plate is thrown, the plate may have some rotation around an axis that is not its axis of symmetry. This occurs because the angular momentum (L) is constant in absence of torques. Therefore, it will have to be constant in the external reference frame, but the moment of inertia tensor (I) is non-constant in this frame because of the lack of symmetry. Therefore, the spin angular velocity vector about the spin axis will have to evolve in time so that the matrix product remains constant.
When an object is not perfectly solid, internal vortices will tend to damp torque-free precession, and the rotation axis will align itself with one of the inertia axes of the body.
The torque-free precession rate of an object with an axis of symmetry, such as a disk, spinning about an axis not aligned with that axis of symmetry can be calculated as follows:
where is the precession rate, is the spin rate about the axis of symmetry, is the angle between the axis of symmetry and the axis about which it precesses, is the moment of inertia about the axis of symmetry, and is moment of inertia about either of the other two perpendicular principal axes. They should be the same, due to the symmetry of the disk.
For a generic solid object without any axis of symmetry, the evolution of the object's orientation, represented (for example) by a rotation matrix that transforms internal to external coordinates, may be numerically simulated. Given the object's fixed internal moment of inertia tensor and fixed external angular momentum, the instantaneous angular velocity is . Precession occurs by repeatedly recalculating and applying a small rotation vector for the short time ; e.g., for the skew-symmetric matrix . The errors induced by finite time steps tend to increase the rotational kinetic energy, ; this unphysical tendency can be counter-acted by repeatedly applying a small rotation vector perpendicular to both and, noting that .
Another type of torque-free precession can occur when there are multiple reference frames at work. For example, the earth is subject to local torque induced precession due to the gravity of the sun and moon acting upon the earth’s axis, but at the same time the solar system is moving around the galactic center. As a consequence, an accurate measurement of the earth’s axial reorientation relative to objects outside the frame of the moving galaxy (such as distant quasars commonly used as precession measurement reference points) must account for a minor amount of non-local torque-free precession, due to the solar system’s motion.
Read more about this topic: Precession