Corrected Inertial Torques of Gyroscopic Effects

Abstract

The published manuscripts in the area of gyroscope theory were presented mainly by the simplified approaches in which mathematical models contain many uncertainties. New research in machine dynamics opened breakthrough directions in gyroscopic effects of rotating objects that give the correct solutions. The pioneering work meets many problems when solving the scientific innovations that are accompanied by successes and omissions. New mathematical models for the gyroscopic inertial torques were derived with incorrect processing of the integral equations that give distorted results. The gyroscopic devices in engineering manifest gyroscopic effects as the action of the inertial torques which computing is crucial for mathematical describing of their motions. The corrected mathematical processing of the equations for the inertial torques acting in a gyroscope is presented in this manuscript.

The Correlation of Gyroscope Axial Velocities

“…The first publications related to gyroscopic effects contain unpleasant mistakes of the pioneering work for complex problems. The mistakes are presented by the incorrect mathematical processing of a complex integral equation for the inertial torque originated by the centrifugal forces of the rotating mass of the revolving disc [14] . This mathematical mistake yielded the incorrect solution for the correlation of gyroscope axial rotations.…”

“…These principles are the centrifugal, and Coriolis forces, the change in the angular momentum, and kinetic energy conservation which were developed over three hundred years. The gyroscopic effects are manifestation of the operation of principles pointed out above that is presented by the action of the sets of inertial torques and correlated rotations of the revolving disc [14] . One set presents four inertial torques operating about one axis, which are generated by two centrifugal and one Coriolis force, and the torque of the change in the angular momentum.…”

Inertial Forces and Torques Acting on a Spinning Annulus

“…The method for the analytical solution for all inertial torques acting on the spinning disc is described in several publications [21,22]. The inertial torques are generated by the distributed mass elements of the spanning disc disposed on the circle of 2/3 of its radius.…”

“…The latest studies of the physics of the gyroscopic effects yield mathematical models for the inertial torques and the interrelated dependency of the angular velocities of the spinning disc motions about axes of rotation. The solution to gyroscopic effects is based on the principle of mechanical energy conservation [20][21][22]. The method of deriving mathematical models for inertial torques shows the action of the system of the centrifugal, Coriolis forces generated by the rotating mass, and the change in the angular momentum of the spinning disc.…”

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.…”