On the decoupling and solution of the Euler kinematic equations governing the motion of a Gyro

Abstract

The decoupling and solution of the three strongly non-linear ordinary differential equations governing the motion of an arbitrary rigid body which is free to rotate about a fixed point (gyro) are presented. The theory developed is based on the assumption that the components of the instantaneous angular velocity are known and are smooth arbitrary functions of time. The resulting equivalent differential equation is second order with respect to the angle of the nodding motion. Furthermore, through a quantitative analysis, analytical solutions of this equation were obtained, which determine the law of motion of the gyro for some general conditions in accordance with the physical problem. Finally, two theoretical applications are investigated, the solutions of which indicate the potential value and effectiveness of the theory proposed.