Gyrodynamics

Gyrodynamics is the study of spinning rigid bodies. It has many applications in modeling of aerospace vehicles. Just consider the gyroscopic devices in inertial navigation systems, gimbaled spin-stabilized sensors, dual-spin satellites, spin- stabilized projectiles and rockets, Magnus rotors, propellers, and turbojets.

The study of the Earth as a spinning object captured the interest of famous dynamicists like Poinsot, Klein, and others in the last centuries. During their time, it was the only practical application. Earth science and astronomy are benefiting to this day from their research.

Technical applications dominate today’s interest. Millions of dollars are spent either improving the performance of gyroscopes or lowering their cost for mass production. They are an integral part of any INS, affecting the accuracy of its nav­igation solution. Wherever a body spins in machinery, technical problems surface because of imperfections. Tires wobble, motor bearings fail, and Hubble gyro­scopes wear out and must be replaced.

For technical details, I refer you to the many excellent texts that are available. An early classic is the theoretical book by Klein and Sommerfeld.6 One of the best treatments, both theoretical and practical, is given by Magnus.7 Unfortunately, these books are written in German. The standard English reference is by Wrigley et al.8 An older account is given by C. S. Draper et al.9 Here, I will cover only some of the fundamental dynamic characteristics of gyroscopes. The mystery that surrounds the precession and nutation modes of fly wheels will be debunked. From the kinetic energy theorem we learn how a spinning body responds to external moments, and we will derive two integrals of motion for force-free bodies.