Abstract This work outlines on the three dimensional motion of a rigid body about a fixed point according to Lagrange’s case under the action of a gyrostatic moment and a Newtonian force field. It is considered that the center of mass of the body is shifted slightly with respect to the principal axis of dynamic symmetry. Equations of motion are derived using the principal equation of the angular momentum and are solved using the Poincare method of small parameter to achieve the asymptotic solutions for the case of irrational frequencies. Euler’s angles characterizing the position of the body at any instant are obtained. The diagrammatic representations of the obtained solutions and Euler’s angles are represents through some plots which reflect the good effect of the applied moments on the motion and its impact on the stability of the body. The numerical solutions are obtained using Runge-Kutta algorithms from fourth order. The comparison between the asymptotic solutions and the numerical ones reveal high consistency between them which reveal the good accuracy of the used perturbation method.

中文翻译：

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不合理频率情况下陀螺仪的动力学运动

摘要 本文根据拉格朗日情形，概述了刚体在回旋力矩和牛顿力场作用下绕定点的三维运动。认为物体的质心相对于动态对称的主轴略有偏移。运动方程是利用角动量的主方程推导出来的，并用小参数庞加莱方法求解，以获得无理频率情况下的渐近解。获得表征任何时刻身体位置的欧拉角。得到的解和欧拉角的图形表示是通过一些曲线来表示的，这些曲线反映了施加的力矩对运动的良好效果及其对身体稳定性的影响。数值解是使用 Runge-Kutta 算法从四阶获得的。渐近解与数值解之间的比较表明它们之间具有高度的一致性，这表明所使用的微扰方法具有良好的准确性。