Evolution of motion of a rigid body similar to Lagrange top under the influence of slowly time varying torques

Abstract

Perturbed rotational rigid body motions similar to the case of Lagrange top subjected to restoring and perturbation slowly time-varying torques of forces are studied in this paper. The restoring torque also depends on the small angle of nutation. The following problem is formulated in the current research: analyzing the solution’s behavior of system of motion equations for the values of small parameter different from zero on a sufficiently large interval of time. An averaging method is used for solving the problem. We have established the terms for feasibility to average (with respect to phase of nutation angle) the equations of rigid body motion related to Lagrange case. The averaged system of equations is received in the first approximation. Asymptotic approach permits to obtain some qualitative results and to describe evolution of motion using simplified averaged equations. In the case of rotational motion of a body immersed in the linear-dissipative medium, the numerical integration of the averaged system of equations is conducted. A new class of rotational motions of the Lagrange top is studied for an unsteady perturbation torque, as well as restoring torque that slowly varies with time and depends on the small nutation angle. These results are an extension of our former works (in the absence of slow time τ in the expressions for μ and M _i or μ, where M _i are dependent on τ only).