ANALYTICAL SOLUTION OF SOME PROBLEMS OF APPLIED THEORY OF GYROSCOPES

A. Tyurehodzhayev

2012

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DOI 10.21608/amme.2012.37257
Data confidence 70 medium
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Abstract

Motion of solid body with a fixed point is described by a system of nonlineardifferential equations of motion of L. Euler. Gyroscopic instruments represent anaxisymmetric solid body with a fixed point. The first partial solutions of the problemwere obtained in works [1-4]. The subsequent development of Mechanics andMathematics demonstrated that a nonlinear system of motion of L. Euler equationsmay describe a wide class of motions of celestial bodies, stability of motion ofspacecrafts, Earth satellites, ships, aircrafts, monorail trains, etc.The great interest to the problems with a fixed point is due to the gyroscopic effects,which became widespread in the modern technology, navigation and in many otherareas.More than 250 years have passed since the origination of nonlinear equations ofmotion of L. Euler, however, the interest in obtaining the solutions of these equationsdo not weaken. The world holds a huge amount of publications, for example, [5-14].The analytical solutions of some problems characterizing the functioning ofgyroscopes have been obtained based on the method of partial discretization ofnonlinear differential equations, formed by the author of the report.This paper demonstrates an analytical solution of the nonlinear problem on themotion of an axisymmetric solid body with a fixed point at a high velocity rate of selfrotationn. New analytical results have been obtained based on the method of partialdiscretization of nonlinear differential equations.