Stability of the permanent rotations of an asymmetric gyrostat in a uniform Newtonian field

Abstract

The stability of the permanent rotations of a heavy gyrostat is analyzed by means of the Energy-Casimir method. Su cient and necessary conditions are established for some of the permanent rotations. The geometry of the gyrostat and the value of the gyrostatic moment are relevant in order to get stable permanent rotations. Moreover, the necessary conditions are also su cient, for some configurations of the gyrostat.

On the motion of a gyro in the presence of a Newtonian force field and applied moments

“…Considering the system (5) for ε = 0 and after grasping the meaning of all used variables, we can change the variables P, Q, r, ψ, θ, ϕ and γ in the system (5) and in equalities (9) to the new variables a, b, r, ψ, θ, α and γ . After the application of some simple and easy reductions, the following system can be obtained from equations (11)…”

“…In the work by Cavas and Vigueras [10], the authors investigated the analytical solutions for a gyrostat similar to Lagrange’s case under the action of a Newtonian field. In the work by Iñarrea et al [11], the authors studied the permanent rotations of a gyro beside its stability using the method of Energy–Casimir [12].…”

“…The dynamical motion of a rigid body is considered as a core subject of the field of theoretical mechanics and it has great interest from a scientific and practical point of view. The solution of the rigid body problem has been studied in a great number of scientific works see monographs [1][2][3][4][5][6][7][8][9][10][11]. As known, this problem has three first integrals related with energy, area and geometric integrals [1].…”

The necessary and sufficient condition for the stability of a rigid body

“…The orbital stability of a symmetric rigid body (pendulum oscillations and rotations) is carried out in [12] when the first two principal axes are equal twice of the third one. Recently, the permanent rotation of asymmetric rigid body was investigated in [13] when the body subjected to Newtonian field and a gyrostatic moment. The authors obtained the necessary and sufficient conditions, for some cases of the gyrostat configurations.…”

“…The stability of a rigid body has shed the attention of outstanding researchers during the last seven decades whether if the body contains fluid [1][2][3][4] or under the action of attracting center [5][6][7][8][9][10][11][12][13]. The stability of the rotational motion of a solid containing an ideal incompressible homogenous fluid filled entirely or partially was studied in [1] and [2].…”