Abstract
This work aims to study the stability of certain motions of a rigid body rotating about its fixed point and carrying a rotor that rotates with constant angular velocity about an axis parallel to one of the principal axes. This motion is presumed to take place due to the combined influence of the magnetic field and the Newtonian force field. The equations of motion are deduced, and moreover, they are expressed as a Lie–Poisson Hamilton system. The permanent rotations are calculated and interpreted mechanically. The sufficient conditions for instability are presented employing the linear approximation method. The energy-Casimir method is applied to gain sufficient conditions for stability. The regions of linear stability and Lyapunov stability are illustrated graphically for certain values of the parameters.
Highlights
A gyrostat is a simple multibody which consists of a rigid body and other bodies which are usually called rotors moving in such a way that their motion does not change the distribution of mass for the gyrostat [1]. e gyrostat is well known in the literature as a dual-spin body due to the motion of the two bodies which compose the gyrostat
Borisov and Mamaev [5] contain most of those integrable problems up to 2001, and some cases were presented by several authors. e second category regards the problem of study periodic solutions, bifurcation, and chaos in some problems of rigid body-gyrostat. e third one is the stability problem of the equilibria in the dynamics of a rigid body-gyrostat moving in an orbit or about its fixed point
E current work is interested in studying the stability of permanent rotations for the motion of a charged gyrostat moving due to the combined influence of the magnetic field and Newtonian force field. is work is regarded as an extension of some previous works
Summary
A gyrostat is a simple multibody which consists of a rigid body and other bodies which are usually called rotors moving in such a way that their motion does not change the distribution of mass for the gyrostat [1]. e gyrostat is well known in the literature as a dual-spin body due to the motion of the two bodies which compose the gyrostat. Most of the works related to the rigid body and its extension to gyrostat can be assorted into three categories. E third one is the stability problem of the equilibria in the dynamics of a rigid body-gyrostat moving in an orbit or about its fixed point (see, e.g., [15,16,17,18,19,20,21,22,23]). E current work is interested in studying the stability of permanent rotations for the motion of a charged gyrostat moving due to the combined influence of the magnetic field and Newtonian force field.
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