Abstract
We consider the motion of a charged rigid body about a fixed point carrying a rotor that is attached along one of the principal axes of the body. This motion occurs under the action of the resultant of the uniform gravity field and the homogeneous magnetic field. The equations of motion are formulated, and they are presented by means of the Hamiltonian function in the framework of the Lie–Poisson system. These equations of motion have six equilibrium solutions. The sufficient conditions for instability for these equilibria are studied by utilizing the linear approximation method, while the sufficient conditions for stability are presented by means of the energy-Casimir method. For certain configuration of the body, the regions of Lyapunov stability and instability are determined in the plane of some parameters. Furthermore, we clarify that the regions of Lyapunov stability are a portion of the regions of linear stability.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Rumiantsev, V.V.: Stability of permanent rotations of a heavy rigid body. Prikl. Math. Mekh. 20, 51–66 (1956)
Pozharitskii, G.K.: On the stability of permanent rotations of a rigid body with a fixed point under the action of a Newtonian central force field. J. Appl. Math. Mech. 23, 1134–1137 (1959)
Irtegov, V.D.: On the problem of stability of steady motions of a rigid body in a potential force field. J. Appl. Math. Mech. 30, 1113–1117 (1966)
Guliaev, M.P.: On the stability of rotations of a rigid body with one fixed point in the Euler case. Prikl. Math. Mech. 23, 579–582 (1959)
Lyapunov, A.M.: The General Problem of Stability of Motion. Obshch, Kharkov (1892)
Routh, E.J.: Dynamics of a System of Rigid Bodies: The Advanced Part. Dover Publications, New York (1955)
Meirovitch, L.: Methods of Analytical Dynamics. McGraw-Hill, New York (1970)
Rumiantsev, V.V.: On the stability of gyrostats. Prikl. Math. Mech. 25, 9–16 (1961)
Kane, T.R., Flower, R.C.: Equivalence of two gyrostatic stability problems. J. Appl. Mech. 37, 1146–1147 (1970)
da Silva, M.: Attitude stability of a gravity-stabilized gyrostat satellite. Celest. Mech. 2, 147–165 (1970)
Rumiantsev, V.V.: On the stability of motion of certain types of gyrostats. J. Appl. Math. Mech. 25, 1158–1169 (1961)
Anchev, A.: On the stability of permanent rotations of a heavy gyrostat. J. Appl. Math. Mech. 26, 26–34 (1962)
Anchev, A.: On the stability of permanent rotations of a quasi-symmetrical gyrostat. J. Appl. Math. Mech. 28, 194–197 (1964)
Kolesnikov, N.I.: On the stability of a free gyrostat. Prikl. Math. Mech. 27, 699–702 (1963)
Leimanis, E.: The General Problem of the Motion of Coupled Rigid Bodies About a Fixed Point. Springer, Berlin (1965)
Volterra, V.: Sur la théorie des variationas des latitudes. Acta Math. 22, 201–357 (1899)
Cochran, J.E., Shu, P.H., Rew, S.D.: Attitude motion of asymmetric dual-spin spacecraft. J. Guid. Control Dyn. 5, 37–42 (1982)
Hall, C.D.: Spinup dynamics of biaxial gyrostats. J. Astronaut. Sci. 43, 263–275 (1995)
Hall, C.D.: Spinup dynamics of gyrostats. J. Guid. Control Dyn. 18, 1177–1183 (1995)
Iñarrea, M., Lanchares, V.: Chaos in the reorientation process of a dual-spin spacecraft with time dependent moments of inertia. Int. J. Bifurc. Chaos 10, 997–1018 (2000)
Lanchares, V., Iñarrea, M., Salas, J.P.: Spin rotor stabilization of a dual-spin spacecraft with time dependent moments of inertia. Int. J. Bifurc. Chaos 8, 609–617 (1998)
Vera, A.J., Vigueras, A.: Hamiltonian dynamics of a gyrostat in the \(n\)-body problem: relative equilibria. Celest. Mech. Dyn. Astron. 94, 289–315 (2006)
Iñarrea, M., Lanchares, V., Pascual, A.I., Elipe, A.: Stability of the permanent rotations of an asymmetric gyrostat in a uniform Newtonian field. Appl. Math. Comput. 293, 404–415 (2017)
Guirao, J.L.G., Vera, J.A.: Equilibria, stability and Hamiltonian Hopf bifurcation of a gyrostat in an incompressible ideal fluid. Phys. D 241, 1648–1654 (2012)
Vera, J.A.: The gyrostat with a fixed point in a Newtonian force field: relative equilibria and stability. J. Math. Anal. Appl. 401, 836–849 (2013)
Anchev, A.: Permanent rotations of a heavy gyrostat having a stationary point. J. Appl. Math. Mech. 31, 48–58 (1967)
Hassan, S.Z., Kharrat, B.N., Yehia, H.M.: On the stability of motion of a gyrostat about a fixed point under the action of non-symmetric fields. Eur. J. Mech. A/Solids 18, 313–318 (1999)
Tsogas, V., Kalvouridis, T.J., Mavraganis, A.G.: Equilibrium states of a gyrostat satellite in an annular configuration of \(N\) big bodies. Acta Mech. 175, 181–195 (2005)
Elipe, A., Lanchares, V.: Two equivalent problems: gyrostats in free motion and parametric quadratic Hamiltonians. Mech. Res. Commun. 24, 583–590 (1997)
Kalvouridis, T.J., Tsogas, V.: Rigid body dynamics in the restricted ring problem of \(n+1\) bodies. Astrophys. Space Sci. 282, 749–763 (2002)
Borisov, A.V., Mamaev, I.S.: Rigid Body Dynamics-Hamiltonian Methods, Integrability. Chaos. Institute of Computer Science, Izhevsk, Moscow (2005). (in Russian)
Yehia, H.M., Elmandouh, A.A.: New conditional integrable cases of motion of a rigid body with Kovalevskaya’s configuration. J. Phys. A Math. Theor. 44, 012001 (2011)
Yehia, H.M., Elmandouh, A.A.: A new integrable problem with a quartic integral in the dynamics of a rigid body. J. Phys. A Math. Theor. 46, 142001 (2013)
Elmandouh, A.A.: New integrable problems in rigid body dynamics with quartic integrals. Acta Mech. 226, 2461–2472 (2015)
Elmandouh, A.A.: New integrable problems in the dynamics of particle and rigid body dynamic. Acta Mech. 226, 3749–3762 (2015)
Bradbery, T.C.: Theoretical Mechanics. Wiley, New York (1968)
Birtea, P., Caşu, I., Comǎnescu, D.: Hamiltonian-Poisson formulation for the rotational motion of a rigid body in the presence of an axisymmetric force field and a gyroscopic torque. Phys. Lett. A 375, 3941–3945 (2011)
Deprit, A., Elipe, A.: Complete reduction of the Euler-Poinsot problem. J. Astronaut. Sci. 41, 603–628 (1993)
Chetaev, N.G.: The Stability of Motion. Pergamon Press, New York (1961)
Holm, D., Marsden, J.E., Ratiu, T.S., Weinstein, A.: Nonlinear stability of fluid and plasma equilibria. Phys. Rep. 123, 1–116 (1985)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Elmandouh, A.A. On the stability of the permanent rotations of a charged rigid body-gyrostat. Acta Mech 228, 3947–3959 (2017). https://doi.org/10.1007/s00707-017-1927-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-017-1927-z