弹性杆盘绕折叠的力学分析  

Mechanical Analysis of Coiling Process of an Elastic Rod

作　　者：刘延柱[1] 薛纭[2] 

机构地区：[1]上海交通大学工程力学系,上海200030 [2]上海应用技术学院机械与自动化学院,上海200233

出　　处：《力学季刊》2008年第3期343-348,共6页Chinese Quarterly of Mechanics

基　　金：国家自然科学基金(10472067)

摘　　要：以可伸展空间结构元件的盘绕折叠过程为工程背景,分析受圆柱面单面约束的弹性直杆变形为螺旋杆,最终压缩为叠放的平面圆环的变形过程。对此空间大变形的分析不允许利用小变形假设进行简化。由于约束力的存在,也不能直接利用忽略分布力的Kirchhoff弹性杆方程。本文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平衡的非线性方程。利用方程的初积分计算杆截面的内力和力矩。忽略盘绕过程的惯性效应,将参数连续改变的螺旋线状态作为杆盘绕过程中的准平衡状态。导出为实现盘绕过程需要施加的轴向压力和扭矩随螺旋角的变化规律。根据一次近似稳定性理论分析得出:两端铰支弹性杆当相对扭率为零时不能保证螺旋线平衡的稳定性。若杆端支承允许存在相对扭转,则轴向压力和扭矩按文中确定的规律变化时可以保证盘绕过程的稳定性。The deformation of an elastic rod under the unilateral constraint of a cylinder was discussed as a simplified model of the coiling process of an element of extendable space mast in astronautic technique. The process from a straight rod to a helical rod, at last to connected planar rings is a spatial large deforma- tion of the rod. Therefore the hypothesis of small deformation cannot be applied in analysis. Considering the existence of constraint force, the Kirchhoff equations of a rod without distributed force cannot be applied too. The nonlinear differential equations of equilibrium of a rod constrained by a cylinder were established with Euler angles as attitude variables of the cross section. Some first integrals were applied to determine the internal forces and torques. Assume that the coiling process is slowly and the inertial effects are neglected. Consequently the helical configuration with continuously variable parameters can be treated as the quasi - equilibrium state in the coiling process. The necessary axial compressive force and torque in the coiling process were derived as the functions of the pitching angle of helix. The stability of the helical rod in coiling process was analyzed in first approximation. It is shown that the stability conditions cannot be satisfied when the rod has hinged constraints on both ends and the relative twist is restricted to zero. The stable helical equilibrium in whole coiling process can be realized only when the relative twist of the rod is admissible by the constraints. The necessary compressive force and torque to ensure the stability of coiling process were suggested.

关 键 词：空间伸展臂 弹性细杆 圆柱面约束 Kirchhoff理论 

分 类 号：O317[理学—一般力学与力学基础] O343[理学—力学]