基于高斯原理的多体系统动力学建模

基于高斯最小拘束原理,以释放中的绳系卫星为背景,建立地球引力场内变长度大变形柔索联系的多体系统动力学模型. 利用基尔霍夫动力学比拟方法将柔索的变形转化为刚性截面沿中心线的转动,使包含刚性分体与变形体的刚柔耦合系统转化为由柔索的刚性截面与刚性分体组成的广义多刚体系统. 由于刚性截面的局部小变形沿弧坐标的积累不受限制,适合描述柔索的超大变形. 文中对此刚柔耦合多体系统导出其在地球引力场中的拘束函数,考虑各分体在空间中相对位置的几何约束条件,利用拉格朗日乘子构成以条件极值问题为特征的数学模型. 将高斯原理用于多体系统动力学的建模,其特点是以寻求函数极值的变分方法代替微分方程,通过数值计算直接得出运动规律. 其形式统一,不随系统拓扑结构的变化而改变,也无需区分树系统或非树系统.对于带控制的多体系统,动力学分析还可根据技术需要与系统的优化结合进行. 

DYNAMIC MODELING OF MULTI-BODY SYSTEM BASED ON GAUSS'S PRINCIPLE

Based on the Gauss's principle of least constraint, the dynamic modeling of a multi-body system connected by an elastic cable with varied lengths and large deformation in gravitational field of the earth was proposed in this paper. The practical background of the topic is the release process of a tethered satellite. The Kirchhoff's method was applied to transform the deformation of the elastic cable to rotation of rigid cross section along the centerline of the cable. Since the local small deformation of the cable can be accumulated limitlessly along the arc-coordinate, the Kirchhoff's model is suitable to describe the super-large deformation of elastic rod. In present paper the Gauss's constraint function of the system of rigid-flexible bodies in gravitational field of the earth was derived, and the geometric constraint conditions concerning relative position of bodies in space were considered using the Lagrange's multipliers. Therefore the dynamical model of the system was established in the form of conditional extremum problem. Applying the approach of Gauss's principle the real motion of the system can be obtained by the variation method directly through seeking the minimal value of constraint function without differential equations. The unified form of the model does not changed for different topologic constructions of the system, and it is unnecessary to distinct the tree system or system with loops. In the case of multi-body system with automatic control, the dynamic analysis can be combined with the optimization for different technique objectives.