The motion of a rotating body in a resisting medium

The motion of a rotating rigid body in a resisting medium under the action of a sinusoidal or biharmonic time-depending restoring torque and small perturbation torques is considered in a nonlinear formulation. A justification of the representation of the perturbations by slowly varying parameters and parameters of small asymmetry is given. The solutions of the equations of non-perturbed motion are presented in terms of Jacobi's elliptic functions. For the case where the nutational torque biharmonically depends on the nutation angle, the equations of non-perturbed motion are represented in terms of the angle-action variables, which can be expressed in terms of complete elliptic integrals. The averaged equations of motion of an axially symmetric body under the action of the biharmonic torque and small damping torques are constructed. The equations of perturbed motion of an asymmetric body are reduced to a standard two-frequency system, and a partially averaged system is constructed. Necessary and sufficient conditions of the stability of nonlinear resonances and those of the Lyapunov stability of the motion in the neighborhood of a stationary point under the action of small perturbations are obtained. A numerical example is given to show that the stability of the resonance does not imply the stability in the neighborhood of the stationary point, and vice versa.