A Torque-Free Flexible Model Gyro

Abstract

The present investigation is concerned with the behavior of an elastic, oblate, torquefree gyro model. The model has been devised such that it represents accurately arbitrarily large attitudes and arbitrarily large deformations. All simplifying assumptions are incorporated into the model before the theory is applied. The subsequent theoretical development is consequently exact; i.e., the expressions for inertia moments, angular moments, kinetic and elastic energies are all exact. Also, the mass center of the model gyro does not shift within the gyro. Equations remain tractable and the practicing engineer can readily get a feel for the phenomena uncovered. The model is composed of a rigid massless rod connected elastically to a rigid massive disk. At the tip of each rod there is a point mass. The nonlinear equations of quasistatic motion are derived using Euler’s law, and a floating coordinate frame. Following the analysis, various numerical examples are investigated and the results are plotted. The total mechanical energy of the system is determined, and the condition for existence of an energy trap state (minimum energy state at an attitude other than zero) is obtained. When trapped, the gyro is in effect rigid, has a stable attitude, and rotates around the principal axis of maximum inertia, which in turn is collinear with the space-fixed angular momentum vector.