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Mathematical Problems in Engineering
Volume 2018, Article ID 6132891, 10 pages
https://doi.org/10.1155/2018/6132891
Research Article

A Mathematical Model for Top Nutation Based on Inertial Forces of Distributed Masses

Kyrgyz State Technical University, Bishkek, Kyrgyzstan

Correspondence should be addressed to Ryspek Usubamatov; moc.oohay@1070kepsyr

Received 8 June 2017; Revised 8 November 2017; Accepted 8 January 2018; Published 5 March 2018

Academic Editor: Filippo Cacace

Copyright © 2018 Ryspek Usubamatov and Albina Omorova. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The main property of gyroscopic devices is maintaining the axis of a spinning rotor, a mathematical model formulated on the principle of the change in the angular momentum. This principle is used for mathematical modeling of the motions of a top at known publications. Nevertheless, practical tests of gyroscopic devices do not correspond to this analytical approach. Recent investigations have demonstrated that the origin of gyroscope properties is more complex than that represented in known publications. The applied torque on a gyroscope produces internal torques of the spinning rotor based on the action of the several inertial forces. These forces are the centrifugal, Coriolis, and common inertial forces as well as the change in the angular momentum generated by the mass elements and center-mass of the spinning rotor. The action of these torques manifests itself in the resistance and precession torques of the gyroscopic devices. These inertial torques act simultaneously and interdependently around two axes and represent the fundamental principles of the gyroscope theory. The new inertial torques enable deriving mathematical models for the motions of well-known top that is the simplest form of gyroscopic devices. The novelty of the work is mathematical models for the motions of the top based on action of eight inertial forces acting around its two axes. The obtained mathematical models for the top nutation and self-stabilization are represented in terms of machine dynamics and vibration analysis. The new analytical approach for motions of the well-balanced top and top with eccentricity of the center-mass definitely responds to the practical results.