ABSTRACT
Motion of solid body with a fixed point is described by a system of nonlinear
differential equations of motion of L. Euler. Gyroscopic instruments represent an
axisymmetric solid body with a fixed point. The first partial solutions of the problem
were obtained in works [1-4]. The subsequent development of Mechanics and
Mathematics demonstrated that a nonlinear system of motion of L. Euler equations
may describe a wide class of motions of celestial bodies, stability of motion of
spacecrafts, 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 other
areas.
More than 250 years have passed since the origination of nonlinear equations of
motion of L. Euler, however, the interest in obtaining the solutions of these equations
do not weaken. The world holds a huge amount of publications, for example, [5-14].
The analytical solutions of some problems characterizing the functioning of
gyroscopes have been obtained based on the method of partial discretization of
nonlinear differential equations, formed by the author of the report.
This paper demonstrates an analytical solution of the nonlinear problem on the
motion of an axisymmetric solid body with a fixed point at a high velocity rate of selfrotation
n. New analytical results have been obtained based on the method of partial
discretization of nonlinear differential equations.