变质量力学系统的广义高斯原理及其对高阶非完整系统的推广

THE GENERALIZED GAUSS PRINCIPLE FOR MECHANICAL SYSTEM WITH VARIABLE MASS AND ITS GENERALIZATION TO HIGHER ORDER NONHOLONOMIC SYSTEMS

变分原理具有极大的概括性,可分为微分的和积分的, Gauss原理是微分形式的变分原理.在现有的微分变分原理中,只有Gauss原理具有极值特性,它可表示为拘束函数的Gauss变分等于零.利用Gauss原理可以直接通过求函数极值的方法获得质点系的运动规律,因此Gauss原理在复杂系统的动力学建模以及近似计算等方面发挥其独特作用,例如机器人的设计与分析、非线性振动方程的近似解以及多体系统动力学等.本文研究变质量力学系统的广义Gauss原理及其对高阶非完整力学的推广.首先,建立变质量力学系统的Gauss最小拘束原理,并通过构建修正的拘束函数,将原理推广到二阶线性非完整约束系统.其次,提出变质量力学系统任意阶情形下的广义Gauss原理,在此基础上建立广义Gauss最小拘束原理,并通过构建广义拘束函数,将原理推广应用于变质量高阶非完整约束系统.研究表明:对具有双面理想约束的变质量高阶非完整力学系统,在每一瞬时k次加速度空间所有与约束相容的可能加速度之中,真实运动的加速度使广义拘束函数在k次Gauss变分下取得极小值.文末应用广义Gauss最小拘束原理导出沿粗糙水平面作惯性运动的燃烧匀质圆球和变质量Hamel问题的运动微分方程.

Variational principle has great generality, which can be divided into differential and integral, Gauss principle is the variational principle with differential form. Among the existing differential variational principles, only the Gauss principle has the extreme value characteristics, which can be expressed as the Gauss variation of the compulsion function equals to zero. Gauss principle can be used to obtain the motion law of a particle system directly by finding the extreme value of function. Therefore, Gauss principle plays a unique role in the dynamics modeling and approximate calculation of complex systems, such as the design and analysis of robots, approximate solutions of nonlinear vibration equations and dynamics of multi-body systems. This paper deals with the generalized Gauss principle for mechanical systems with variable mass and its extension to higher order nonholonomic mechanics. Firstly,Gauss’ s principle of least compulsion for mechanical system with variable mass is established, and extended to second order linear nonholonomic constrained systems by constructing modified compulsion function. Secondly, the generalized Gauss principle of mechanical system with variable mass for arbitrary order cases is proposed, and generalized Gauss’ s principle of least compulsion is established, and the generalized compulsion function is constructed to extend the principle to high order nonholonomic constrained systems with variable mass. It is shown that for variable-mass mechanical system with bilateral ideal high-order nonholonomic constraints, the acceleration of real motion minimizes the generalized compulsion function under the k-thGauss variation in every instant among all the possible accelerations compatible with the constraints in the k-thacceleration space. At the end of this paper, the differential equations of motion of a burning uniform sphere moving along a rough horizontal plane and the variablemass Hamel problem are derived by applying the generalized Gauss’ s principle of least compulsion.