Stability analysis of resonant rotation of a gyrostat in an elliptic orbit under third-and fourth-order resonances. (English) Zbl 1516.34065
Summary: This paper presents the stability of resonant rotation of a symmetric gyrostat under third- and fourth-order resonances, whose center of mass moves in an elliptic orbit in a central Newtonian gravitational field. The resonant rotation is a special planar periodic motion of the gyrostat about its center of mass, i. e., the body performs one rotation in absolute space during two orbital revolutions of its center of mass. The equations of motion of the gyrostat are derived as a periodic Hamiltonian system with three degrees of freedom and a constructive algorithm based on a symplectic map is used to calculate the coefficients of the normalized Hamiltonian. By analyzing the Floquet multipliers of the linearized equations of perturbed motion, the unstable region of the resonant rotation and the region of stability in the first-order approximation are determined in the dimensionless parameter plane. In addition, the third- and fourth-order resonances are obtained in the linear stability region and further nonlinear stability analysis is performed in the third- and fourth-order resonant cases.
MSC:
| 34C20 | Transformation and reduction of ordinary differential equations and systems, normal forms |
| 34C25 | Periodic solutions to ordinary differential equations |
| 70E17 | Motion of a rigid body with a fixed point |
| 70E50 | Stability problems in rigid body dynamics |
Keywords:
Hamiltonian systems; normal form; gyrostat; stability analysis; symplectic maps; periodic motionReferences:
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