Corrected Mathematical Models for Motions of the Gyroscope with one Side Free Support

Abstract

The recent publications about gyroscope effects explained their physics and described them by mathematical models based on the action of forces and inertial torques of classical mechanics. This new analytical approach finally solved the old problem of the dynamic of rotating objects and showed their kinetic energy is the base of gyroscopic effects. Gyroscopic effects result from the action of the two sets of interrelated inertial torques acting about two axes. Each set contains torques generated by the centrifugal, Coriolis forces, and the change in the angular momentum. Detailed study of the inertial torque of the centrifugal force showed its expression derived with an error of mathematical processing. This error gives a less value for the angular velocity of the slow rotation of the gyroscope about one axis that cannot be measured. The angular velocity of the fast rotation of the gyroscope about the other axis is measured but remains of the same value that gives the expression of the torque with error. This manuscript presents the corrected mathematical model for the motion of the gyroscope suspended from the flexible cord.

Principles of Classical Mechanics Describe Gyroscopic Effects

“…The physics of the gyroscopic effects has expressed the principle of the conservation of mechanical energy for spinning objects. The fundamental principles of gyroscopic effects and expressions of mathematical models for gyroscopic inertial torques were published and represented in Table 1 [13][14][15][16]. The action of the external and inertial torques and motions of the spinning disc is presented at the Cartesian 3D coordinate system Σoxyz in Fig.…”