Abstract This paper focuses on the motion of a rigid body, near to Lagrange's case, about a fixed point in which the ellipsoid of inertia is closed to the ellipsoid of rotation. The body is subjected to a gyrostatic moment vector, about the principal axis of rotation, in which the first two components are null. It is presupposed that the body spins rabidly around one of the major or the minor principal axis of the ellipsoid of inertia. The asymptotic solutions of the governing system of motion for the case of nonzero basic amplitude are achieved using Krylov-Bogoliubov-Mitropolski (KBM) method and its modifications. Computer codes are utilized to represent these solutions in some plots besides the phase plane diagrams in order to reveal the good impact of the gyrostatic moment on the dynamical motion. Moreover, Runge-Kutta fourth order is used to obtain the numerical solutions of the original system of motion. The comparison between the approximate solutions and the numerical one displays a good consistency between them which reveal the good accuracy of the used perturbation method. The importance of the work due to the gyro applications in calculating aircraft turns in various axis of operation in which is called roll and pitch yaw.

中文翻译：

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椭球惯性接近旋转椭球的情况下刚体的动力学运动

摘要 本文关注刚体的运动，接近拉格朗日的情况，关于一个不动点的运动，在这个点上惯性椭球闭合于旋转椭球。主体受到绕主轴旋转的回转静力矩矢量，其中前两个分量为零。假设物体围绕惯性椭球的长轴或短主轴之一疯狂旋转。使用 Krylov-Bogoliubov-Mitropolski (KBM) 方法及其修改获得非零基本振幅情况下运动控制系统的渐近解。除了相平面图外，还使用计算机代码在一些图中表示这些解决方案，以揭示回转静力力矩对动力学运动的良好影响。而且，Runge-Kutta 四阶用于获得原始运动系统的数值解。近似解与数值解之间的比较表明它们之间具有良好的一致性，这表明所使用的微扰方法具有良好的准确性。由于陀螺仪应用在计算飞机在各种操作轴上的转弯时，这项工作的重要性被称为滚转和俯仰偏航。