Mathematical Problems in Engineering

Volume 2018, Article ID 6132891, 10 pages

https://doi.org/10.1155/2018/6132891

## 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.

#### 1. Introduction

L. Euler first laid out the mathematical foundations for the gyroscope theory in his work on the dynamics of rigid bodies back in 1765. Since Industrial Revolution time, I. Newton, J-L. Lagrange, L. Poinsot, J. L. R. D’Alembert, P-S. Laplace, L. Foucault, and other brilliant scientists have investigated, developed, and added new interpretations of the gyroscope effects, which are in full display in the rotor’s persistence in maintaining its plane of rotation. The applied theory of gyroscopes emerged mainly in the twentieth century in numerous publications that described the gyroscope effects [1]. The gyroscope properties feature many engineering mechanisms and devices with rotating parts, which thus need to be computed for their proper functioning [2, 3]. Gyroscope effects are used in numerous gyroscopic devices in aerospace engineering, as well as on ships and other industries [4, 5]. The most basic textbooks of classical mechanics have chapters on the gyroscope theory [6, 7] and consider motions with vibrations analysis [8]. Numerous and valuable publications have dedicated themselves to gyroscope effects and their applications in engineering [9–12]. Intuitively and without mathematical models, some researchers have noted that, on a gyroscope, the inertial forces that manifest the gyroscope effects are acting. However, in known publications, mathematical models for the gyroscope effects do not seem to match their practical applications in gyroscopic devices [13–17]. Therefore, researches have spawned artificial terms such as gyroscope resistance and gyroscope couple, as well as fantastical properties that contradict rules of classical mechanics. It is for this reason that the gyroscope theory still attracts many researchers who seek to discover true gyroscope theory [16, 18].

Mechanically, a gyroscope is a spinning disc in which the axle is free to assume any orientation. The simplest gyroscope is represented by a top toy that is one of the most remarkable and widely recognized toys in the world. They still attract attention through their astonishing behavior and unusual gyroscopic properties. The motions of top toy have been described analytically in numerous publications and with complex numerical modeling [19] based on Lagrangian dynamics on Euler coordinates that are solved with computer software [20].

However, all publications contain approximations, assumptions, and simplifications and explain gyroscope effects by the physical principle of the spinning rotor’s angular momentum that generates the precession torque and motions. Gyroscope effects still represent a problem that remains to be solved. The origin of gyroscope effects is more complex than those represented in the theories known to date. Recent investigations into the physical principles of gyroscope motions demonstrate four classical inertial forces acting upon a spinning rotor to generate the gyroscope effects. Research shows that centrifugal, common inertial, and Coriolis forces, in addition to change in the angular momentum of spinning rotors, are the basis for all gyroscope effects and properties [21–24]. Applied to the gyroscope, an external torque generates several torques based on the action of the aforementioned forces. In turn, the centrifugal and Coriolis forces of the rotating mass elements generate a resistance torque that counteracts the inclination of the rotor’s location. Common inertial forces, rotating mass elements, and change in the angular momentum of a spinning rotor generate a precession torque. All torques are interrelated and occur simultaneously. New mathematical models for acting torques accurately describe the physics of gyroscope effects and are validated by tests [21]. Table 1 represents four different torques generated by the inertial forces of the mass elements and center-mass of the spinning rotor under the action of the external torque that was applied to the gyroscope. The action of these torques should be considered for mathematical models of the gyroscopic devices and the top’s motions and nutation.