﻿ 含非对称摩擦平面运动刚体动力学LCP方法
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1. 常州工学院机电学院, 常州 213002;
2. 北京航空航天大学航空科学与工程学院, 北京 100191

An LCP method for dynamics of planar-motion rigid-body with non-symmetric friction
WANG Xiaojun1,2, WANG Qi2
1. Department of Electron and Machine, Changzhou Institute of Technology, Changzhou 213002;
2. School of Aeronautic Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100191, China
Abstract: A modeling and numerical calculation method for the dynamics of the rigid-body in planar motion with contact, impact, symmetric and non-symmetric Coulomb's dry friction was presented based on the theory of contact mechanics and the arithmetic of linear complementarity problem (LCP). The local deformations in contact bodies were taken into account although bodies were assumed to be rigid. The normal contact forces were expressed as nonlinear functions of penetration depth and its speed. The friction law adopted symmetric or non-symmetric Coulomb's dry friction model. Firstly, by using the friction saturation, the positive and negative parts of the relative acceleration of contact points, the complementarity conditions and formulations about the friction law were given in order to determine state transitions of stick-slip in numerical method. Then, based on the event-driven scheme, the problems of detecting stick-slip state transitions and solving frictional forces in stick situation were formulated and solved as a linear complementarity problem. Finally, the numerical example of a planar-motion rigid-body was given to analyze its dynamical behaviour affected by symmetric or non-symmetric Coulomb's dry friction and shown that the proposed method was effective.
Key words: non-symmetry     Coulomb's dry friction     contact forces     linear complementarity problem (LCP)     non-smooth

20世纪末和21世纪初,人们利用多种接触力模型研究物体的接触与碰撞问题,如Kelvin-Voigt接触力模型、Hertz接触力模型、Hunt-Crossley接触力模型和Lankarani-Nikravesh(L-N)接触力模型等[1].目前被广泛使用的摩擦模型有Coulomb干摩擦模型、修正的Coulomb摩擦模型、粘性摩擦模型、Stribeck摩擦模型[2]和Dahl摩擦模型[3]等,Coulomb摩擦模型被认为是较简单的摩擦模型[1, 4].

1 接触力学模型 1.1 法向接触力模型

 图 1 两物体间的接触模型 Fig. 1 Contact model between two bodies

1.2 切向接触力模型

2 非光滑动力学方程及其数值求解算法 2.1 非光滑动力学方程

2.2 Coulomb摩擦定律的互补关系

2.3 动力学方程的求解算法

3 算 例 3.1 算例模型

 图 2 算例中的箱体 Fig. 2 Slider in this example

3.2 数值仿真分析

 图 3 FN1和FN2的时间历程图 Fig. 3 Time history diagram of FN1 and FN2

 图 4 δ1和δ2的时间历程图 Fig. 4 Time history diagram of δ1 and δ2

 图 5 x、y和θ的时间历程图 Fig. 5 Time history diagram of x、y and θ

 图 6 x·和Fτ的时间历程图 Fig. 6 Time history diagram of $\dot x$ and Fτ

 图 7 当A=11,12,18N时x和$\dot x$的时间历程图 Fig. 7 Time history diagram of x and $\dot x$ with A=11,12,18N

4 结 论

1) 将物体间的法向接触力表示成物体间嵌入量和嵌入速度的函数,无需引入互补量,更易于分析物体间的接触与分离.

2) 建立了非对称摩擦余量互补关系,将原适用于含对称Coulomb干摩擦的LCP算法推广到非对称Coulomb干摩擦系统,使其具有更广泛的适用性.

3) 数值仿真算例表明,该算法易于分析非光滑系统中物体间的接触与分离和stick-slip现象.

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文章信息

WANG Xiaojun, WANG Qi

An LCP method for dynamics of planar-motion rigid-body with non-symmetric friction

Journal of Beijing University of Aeronautics and Astronsutics, 2015, 41(11): 2023-2028.
http://dx.doi.org/10.13700/j.bh.1001-5965.2014.0765