受圆柱面约束螺旋杆伸展为直杆的动力学分析

DYNAMICAL ANALYSIS OF STRETCHING PROCESS OF HELICAL ROD TO STRAIGHT ROD UNDER CONSTRAINT OF CYLINDER

以大型空间结构的可伸展机械臂从折叠状态被释放的伸展过程为工程背景,分析受圆柱面单面约束的弹性螺旋杆在惯性力作用下恢复为直杆的动力学过程.对弹性杆空间大变形的分析不允许利用小变形假设进行简化.Kirchhoff动力学比拟理论是研究细长弹性杆超大变形的有效工具.但由于圆柱面约束的存在,不能直接利用无分布力的Kirchhoff模型,而必须在方程中增加分布的约束力.以表述截面姿态的欧拉角为变量,建立受圆柱面约束弹性杆的动力学方程.在圆柱面约束条件下,认为弹性杆在伸展过程中仍维持半径不变的螺旋线形态,仅螺旋线倾角和杆的扭率随时间变化.对简化后的非线性微分方程导出解析积分,以描述伸展运动的动力学过程,导出螺旋杆伸展速度的变化规律,以及从初始状态伸展为直杆所需时间的简明的解析形式计算公式.

The deformation of an elastic helical rod under the unilateral constraint of a cylinder is discussed as a simplified model of the stretching process of an extendable space mast in astronautic technique. The deformation of the rod in stretching process is sufficiently large, and the hypothesis of small deformation can not be applied in analysis. The Kirchhoff＇s dynamical analogy theory is an effective approach in research of super-large deformation of thin elastic rod. Considering the existence of constraint force of the cylinder the Kirchhoff＇s equations of a rod without distributed force can not be applied directly, and the distributed constraint force should be added. In present paper the nonlinear differential equations of dynamics of an elastic rod constrained by a cylinder are established with the Euler angles as attitude variables of the cross section. Assume that the rod maintains the helical shape with unchanged radius in the stretching process as the result of cylindrical constraint, and only the variation of pitching angle of helix and the twisting of the rod are considered. Analytical solutions of the simplified differential equations can be obtained to describe the dynamical process of stretching motion, and simple formulas of stretching velocity and stretching time are given in analytical form.