On the Orbital Stability of Pendulum-Like Vibration of a Rigid Body Carrying a Rotor

Document Type : Original articles

Authors

1 Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

2 Mathematics Department, Faculty of Science, Damietta University, New Damietta, Egypt

Abstract

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; its motion as a
physical pendulum. The equation in the variation for pendulum vibrations takes the form
(d2 γ3)/(du2 )+α(αν2+1/2+ρ2-(α+1) ν2 sn2 u+2νρ sqrt(α) cnu) γ3=0

in which α depends on the moments of inertia, ρ on the gyrostatic momentum of the rotor
and v (the modulus of the elliptic function) depends on the total energy of the motion. This
equation, which reduces to Lame's equation when ρ= 0 was not studied to any extent in the
literature. The determination of the zones of stability and instability of plane motion reduces to
finding conditions on those parameters for existence of primitive periodic solutions (with periods
4K(ν), 8K(ν)). Full analysis of primitive periodic solutions of this equation is performed analogous
to that of Ince for Lame's equation. Zones of stability and instability are determined analytically
and illustrated in a graphics form by plotting surfaces separating them in the three dimensional
space of parameters. The problem is also solved numerically on certain sections of the parameter
space and results are compared to analytical ones.

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