This paper examines the rotational motion of a compound mechanical system comprising a rigid carrier body equipped with internal gyroscopic devices and a point mass that moves along a prescribed trajectory relative to the body. The system undergoes free motion in a uniform gravitational field. We derive the complete equations of motion accounting for the constant gyrostatic torque (GT) generated by internal rotors. Using asymptotic methods, we develop approximate dynamical equations valid under two distinct physical scenarios: (i) when the moving mass is small relative to the carrier mass and executes rapid oscillations and (ii) when the mass oscillates with small amplitude near a fixed location within the body, regardless of mass ratio. The accuracy and validity range of these approximations are rigorously established. For the first scenario, we have approached the idea that gyrostatic coupling fundamentally alters the system’s integrability properties while introducing beneficial stabilization mechanisms. We characterize families of permanent rotational states and analyze their stability using linear perturbation theory. The second scenario reveals that the approximate dynamics correspond to gyrostat motion rather than the classical Euler–Poinsot case. Comprehensive numerical simulations validate theoretical predictions and demonstrate applications to spacecraft attitude control problems. The results provide practical design guidelines for gyrostabilized systems with internal moving components.
Hussein et al. (Mon,) studied this question.