Gyroscopic Torques Generated by a Spinning Ring Torus

The known publications related to the gyroscope theory consider only several geometries of the spinning rotors, like the disc, bars, rings, spheres, and others. The geometries of the spinning objects in engineering can have many designs that generate different values of inertial torques. A computing of inertial torques produced by the object’s rotating masses depends on the radii of their locations. Gyroscopic devices with a spinning torus or ring like other objects should be computed for the quality of their operation. The simplified expression for the radius of the mass disposition for the ring is presented by its middle radius which does not give the correct results for inertial torques and motions of gyroscopic devices. The radius of distributed masses of a ring is bigger than its middle radius because of the differences in the external and internal semirings. A torus is a ring whose exact expression of the radius for distributed mass should be derived for the correct solution for the inertial torques generated by the rotating mass. The precise formulas for inertial torques acting on the spinning torus enable finding its optimal size for the gyroscopic devices. This paper presents the method and derivation of mathematical models for the inertial torques produced by the spinning torus and its interrelated angular velocities about the axes of rotations that manifest gyroscopic effects.