On the Orbital Stability of Pendulum-like Vibrations of a Rigid Body Carrying a Rotor

One of the most notable effects in mechanics is the stabilization of the unstable upper equilibrium position of a symmetric body fixed from one point on its axis of symmetry, either by giving the body a suitable angular velocity or by adding a suitably spinned rotor along its axis. This effect is widely used in technology and in space dynamics.
The aim of the present article is to explore the effect of the presence of a rotor on a simple periodic motion of the rigid body and its motion as a physical pendulum.
The equation in the variation for pendulum vibrations takes the form

$$\frac{d^2\gamma_3}{du^2}+\alpha\left[\alpha \nu^2+\frac{1}{2}+\rho^2 - (\alpha+1)\nu^2sn^2u+2\nu \rho \sqrt{\alpha}cnu \right]\gamma_3=0,$$

in which $\alpha$ depends on the moments of inertia, $\rho$ on the gyrostatic momentum of the rotor and $\nu$ (the modulus of the elliptic function) depends on the total energy of the motion. This equation, which reduces to Lame’s equation when $\rho=0$, has not been studied to any extent in the literature. The determination of the zones of stability and instability of plane motion reduces to finding conditions for the existence of primitive periodic solutions (with periods $4K(\nu)$, $8K(\nu)$) with those parameters. Complete analysis of primitive periodic solutions of this equation is performed analogously to that of Ince for Lame’s equation. Zones of stability and instability are determined analytically and illustrated in a graphical form by plotting surfaces separating them in the three-dimensional space of parameters. The problem is also solved numerically in certain regions of the parameter space, and results are compared to analytical ones.