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.

本文介绍了对称陀螺仪在三阶和四阶共振下的谐振旋转稳定性，其质心在牛顿中心引力场的椭圆轨道上移动。共振旋转是陀螺仪绕其质心的一种特殊的平面周期运动，即物体在其质心的两个轨道公转期间在绝对空间中进行一次旋转。陀螺仪的运动方程导出为具有三个自由度的周期性哈密顿系统，并使用基于辛映射的构造算法来计算归一化哈密顿量的系数。通过分析扰动运动线性化方程的 Floquet 乘数，在无量纲参数平面中确定共振旋转的不稳定区域和一阶近似中的稳定区域。此外，在线性稳定区域获得了三阶和四阶共振，并在三阶和四阶共振情况下进行了进一步的非线性稳定性分析。