ABOUT the STABILITY of TWO RIGID BODIES UNIFORM ROTATIONS

V.F. Gubareva, N.V. Kovalenko

The paper is devoted to the investigation of influence of small dynamic unsymmetry on the stability of uniform rotations around vertical axis of finite rotation of two rigid bodies, connected via ideal spherical joint. There are bodies mass centers on the principal axis; one of the bodies has a fixed point.

Linearized in neighborhood of studied solution, the system of the differential motion equations with periodic coefficients has been presented in the matrix form. Right hand side of the system of differential equations was represented as a matrix series in terms of power of the small parameter of dynamic unsymmetry, using the substitution of variables. It is a Hamilton system, i. e. it has got reciprocal characteristic equation of monodormy matrix. Characteristic exponents expansions in terms of power of the small parameter of dynamic unsymmetry up to the first power inclusively have been proved to be pure imaginary and different.

It is established that if for dynamic symmetric bodies are done the necessary stability conditions, then they do not disturb for bodies system with a small parameter of dynamic unsymmetry.