Inertial forces acting on a gyroscope

Gyroscopic devices for navigation and control systems are widely applied in various industries, such as shipping and aerospace. A remarkable property of gyroscopes is that their axes can be maintained within a particular space. This interesting property of a spinning disc mounted on an axle is represented by a mathematical model formulated based on L. Euler's principle of change in angular momentum. Nevertheless, numerous publications and analytical approaches in known gyroscope theories do not correspond to practical tests on gyroscopes. A simple rotating disc creates problems that do not have long-term solutions. Recent investigations in this area have demonstrated that the origin of gyroscope properties is more sophisticated than that described in known hypotheses. Researchers have not considered the action of inertial forces produced by the mass elements and center mass of the spinning rotor that create internal resistance and precession torques. Resistance torque is established through the actions of centrifugal and Coriolis forces. Precession torque is established through the actions of common inertial forces and a change in angular momentum. These internal torques act simultaneously and interdependently on two axes and represent the fundamental principles of gyroscope theory. Equations for internal inertial torques of a spinning disc have been formulated through mathematical analysis and differential and integral equations. These calculus methods provide a basis for understanding the rates of change in inertial forces acting on a gyroscope and include the use of functions, their derivatives, and integrals in modeling the physical processes in gyroscopes. This paper presents mathematical models for several internal inertial torques generated by the load torque acting on a spinning rotor. These models can describe all gyroscope properties and represent their novelty for machine dynamics and engineering.