滑轮绳索系统中动态节点绳索单元

ROPE ELEMENTS WITH MOVING NODES IN ROPE-PULLEY SYSTEMS

解除了传统有限元方法中单元节点与物质点固结的假设,建立了空间点的速度和加速度与相应物质点的速度和加速度之间的数学关系,强调了虚功率原理中出现的速度和加速度皆为物质速度和物质加速度.在此基础上构造了单元节点既不与空间坐标固定也不与物质坐标固定的动态节点绳索单元.根据滑轮绳索系统的特点,选取绳索出入绳点的弧长坐标、出入绳角、面外摆角以及拉伸应变等空间参数描述了滑轮绳索系统的运动.将绳索与滑轮以及绳索与卷筒之间的相互作用合理简化为物质速度间的约束条件,避免了传统方法中接触力计算不收敛、效率低等缺点.所提方法可精细求解绳索与滑轮间接触边界点位置和卷筒入绳点在卷筒上的运动、滑轮的中心和其连体基的运动、绳索出入滑轮和卷筒时空间方位的变化以及绳索上任意点的拉力变化等细节.可为含绳索机械系统的力学分析提供新的理论基础.所用的解除单元节点与物质点绑定的理论具有一定的普适性,可为有限元方法的理论和应用提供参考.

In traditional finite element methods,a common hypothesis is the nodes of an element being fixed with material points,which place restrictions on the description of a system.In this paper,this hypothesis is released and a new kind of rope element with nodes being no longer fixed with either spatial coordinates or material coordinates is presented.The theory of obtaining material velocity and acceleration of a point from the corresponding spatial velocity and acceleration of the point is established.The velocities and accelerations in the principle of virtual power should being material velocities and accelerations respectively is emphasized and applied.According to the feature of a rope-pulley system,its movement can be described by a new group of variables,such as arc-lengths,azimuthal angles and strains of ropes at entrance and exit contacting points.Instead of traditional methods,conditions of contact between rope and pulley are modeled as the material velocity of a point on the contacting rope being equal to the corresponding point on the pulley.By means of the presented methods,some obstacles,such as frequently divergence and high time consuming that resulting from traditional finite element methods can be removed.Motion equation of the rope-pulley system is derived by the principle of virtual power.Arc-length coordinates,positions of contact boundary points,the movement of pulley centers and their body-fixed reference coordinate systems,shape and positions changing of ropes as well as the tension forces in every point of a rope,can be obtained high precisely.The presented method can provide a new theoretical basis for analysis of mechanical systems with ropes and pulleys.The presented theory of using spatial points as describing variables to replace traditional material points is of applicable widely.It can be a reference for theory and application of finite element methods.