Stability of the uniform rotation of a gyrostat round the vertical main axis on an absolutely smooth horizontal plane
type: Item
instance of: scholarly article
title: Stability of the uniform rotation of a gyrostat round the vertical main axis on an absolutely smooth horizontal plane
zbMATH Open document ID: 0618.70002
DOI: 10.1016/0021-8928(86)90057-2
author: S. A. Belikov
published in: Journal of Applied Mathematics and Mechanics
publication date: 1986-01-01T00:00:00Z
review text: The motion of a gyrostat on an absolutely smooth plane is discussed. A Hamilton function which gives the canonical equations of motion is obtained. This admits of particular solutions, namely uniform rotations round a vertical axis which are identical with that of the uniform rotations of the rotor. A transition to a system with two degrees of freedom is realized, and the expansion of the Hamiltonian in the vicinity of the corresponding position of equilibrium, with an accuracy to within fourth-order terms, is obtained. In the region of admissible values of the parameters the domain of the necessary stability conditions, and the domains where the Hamiltonian functions are of fixed sign and alternating, are examined. In those cases where the Hamiltonian is not of fixed sign, its normalization is performed, both a non-resonance situation and resonances of the first, second and fourth order being considered. The sufficient conditions for stability of uniform gyrostat rotation in terms of constraints on the coefficients of normal forms are obtained. For a clear interpretation of the results, special cases where the values of all the parameters except two are fixed, are given. The plane domain of the necessary stability conditions and resonance curves are constructed, and using computer results stability on the curves is discussed.
Mathematics Subject Classification ID: 70E05
Mathematics Subject Classification ID: 70-08
Mathematics Subject Classification ID: 70H15
Mathematics Subject Classification ID: 70K20
zbMATH DE Number: 4003567
zbMATH Keywords: gyrostat
zbMATH Keywords: absolutely smooth plane
zbMATH Keywords: Hamilton function
zbMATH Keywords: canonical equations
zbMATH Keywords: uniform rotations
zbMATH Keywords: system with two degrees of freedom
zbMATH Keywords: expansion of the Hamiltonian
zbMATH Keywords: position of equilibrium
zbMATH Keywords: accuracy
zbMATH Keywords: fourth- order terms
zbMATH Keywords: necessary stability conditions
zbMATH Keywords: resonances
zbMATH Keywords: sufficient conditions for stability
zbMATH Keywords: uniform gyrostat rotation
zbMATH Keywords: constraints
MaRDI profile type: Publication
full work available at URL: 0021-8928(86)90057-2
OpenAlex ID: W2029252149
cites work: On realizing nonholonomic constraints by viscous friction forces and Celtic stones stability
cites work: On stability of steady motions of a heavy solid body on an absolutely smooth horizontal plane
cites work: On permanent rotations of a heavy solid body on an absolutely rough horizontal plane
cites work: On the stability of motion of gyrostats
cites work: On the dynamics of a solid on an absolutely rough plane
cites work: Q3326901
cites work: On the stability of steady motions
cites work: Q4174830
cites work: Q4184384
cites work: Q4401088
cites work: Q5151953
cites work: On stability of an autonomous Hamiltonian system with two degrees of freedom under first-order resonance
cites work: Q3964064
cites work: Q4153661
cites work: Stability of the Lagrange solutions of the restricted three-body problem for the critical ratio of the masses
description: scientific article; zbMATH DE number 4003567