Serret-Andoyer变量描述的刚体定点运动

叙述刚体动力学中Serret-Andoyer变量的基本概念. 导出用其表达的刚体定点运动动力学方程,叙述其在航天器姿态动力学和陀螺力学中的应用.     

The evolution of a rotational motion of an elastic- rigid body about a fixed point in a gravitational field

The rotational motion of a rigid dynamical system including a three dimensional elastic part about a fixed point is studied. The system is under the action of a general gravitational field. The stability anlysis is carried out by expanding the displacement of the deformable part in terms of orthonormalized functions with the independent coefficients. Results are obtained easily based on Routh's equation and Andoyer variables. The system is found to be stable provided that the dissipation energy is minimum corresponds to the case of axially symmetric deformation. On the other hand, the system is unstable, when the angular momentum vector is parallel to the equatorial plane.     

New solvable problems in the dynamics of a rigid body about a fixed point in a potential field

We determine the general form of the potential of the problem of motion of a rigid body about a fixed point, which allows the angular velocity to remain permanently in a principal plane of inertia of the body. Explicit solution of the problem of motion is reduced to inversion of a single integral. A several-parameter generalization of the classical case due to Bobylev and Steklov is found. Special cases solvable in elliptic and ultraelliptic functions of time are discussed.     

On the motion of a heavy dynamically symmetric rigid body with vibrating suspension point

The motion of a dynamically symmetric rigid body in a homogeneous field of gravity is studied. One point lying on the symmetry axis of the body (the suspension point) performs high-frequency periodic or conditionally periodic vibrations of small amplitude. In the framework of approximate equations of motion obtained earlier, we find necessary and sufficient conditions for the stability of the body rotation about the vertical symmetry axis and study the existence and stability of regular precessions of the body in the coordinate system translationally moving together with the suspension point.     

Quasi-Euler-Poinsot motion of a rigid body

The phase-plane method of nonlinear oscillation is used to discuss the influence of the small dissipation upon the Euler-Poinsot motion of a rigid body about a fixed point. The equations of phase coordinates are applied instead of Eulerian equations, and the global characteristics of the motion of rigid body are analysed according to the distribution and the type of the singular points. A Chaplygin's sphere on a rough plane, a rigid body in viscous medium and one with a cavity filled with viscous fluid are discussed as examples. It is shown that the motions of rigid bodies dissipated by various physical factors have a common qualitative character. The rigid body tends to make a permanent rotation about the principal axis of the largest moment of inertia. The transitive process can change from oscillatory to aperiodic with the decrease in dissipation.     

On the rotational motion of a gyrostat about a fixed point with mass distribution

The perturbed rotational motion of a gyrostat about a fixed point with mass distribution near to Lagrange’s case is investigated. The gyrostat is subjected under the influence of a variable restoring moment vector, a perturbing moment vector, and a third component of a gyrostatic moment vector. It is assumed that the angular velocity of the gyrostat is sufficiently large, its direction is close to the axis of dynamic symmetry, and the perturbing moments are small as compared to the restoring ones. These assumptions permit us to introduce a small parameter. Averaged systems of the equations of motion in the first and second approximations are obtained. Also, the evolution of the precession angle up to the second approximation is determined. The graphical representations of the nutation and precession angles are presented to describe the motion at any time.     

On the dynamics of the rigid body with a fixed point: periodic orbits and integrability

The aim of the present paper is to study the periodic orbits of a rigid body with a fixed point and quasi-spherical shape under the effect of a Newtonian force field given by different small potentials. For studying these periodic orbits, we shall use averaging theory. Moreover, we provide information on the $\mathcal{C}^{1}$ -integrability of these motions.     

刚体在固定面上纯滚时接触点的速度和加速度

在微小的时间间隔内,应用加速度合成定理和刚体上一点的加速度公式,推导出了接触点的加速度公式. 加速度方向为运动刚体表面在该处的法线方向,大小与动接触线和定接触线的曲率半径和刚体运动的角速度有关. 利用接触点的运动性质,不仅有助于解答圆球在固定圆球面或圆柱在固定圆柱面上的纯滚问题,而且为求解一般形状刚体在任意固定曲面上的纯滚问题提供了方便.     

再论刚体在固定面上纯滚动时接触点的加速度

应用点的速度合成定理和加速度合成定理,推导出刚体在固定曲面上作平面纯滚动时接触点的速度和加速度的计算公式.该方法易于理解.     

Application of the integral manifold method to the analysis of the spatial motion of a rigid body fixed to a cable

We analyze the spatial motion of a rigid body fixed to a cable about its center of mass when the orbital cable system is unrolling. The analysis is based on the integral manifold method, which permits separating the rigid body motion into the slow and fast components. The motion of the rigid body is studied in the case of slow variations in the cable tension force and under the action of various disturbances.We estimate the influence of the static and dynamic asymmetry of the rigid body on its spatial motion about the cable fixation point. An example of the analysis of the rigid body motion when the orbital cable system is unrolling is given for a special program of variations in the cable tension force. The conditions of applicability of the integral manifold method are analyzed.     

刚体的力螺旋运动

对力螺旋矢作用下的刚体力螺旋运动进行了教学上的叙述,与力矢作用下的刚体平动和力偶矢作用下的刚体定轴转动并列．本文对力螺旋矢以及与力螺旋矢相关的力螺旋运动和力螺旋动力学给出教学上的编排和简要的论述与例题．