﻿ 含摩擦与碰撞平面多刚体系统动力学线性互补算法
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 力学学报  2015, Vol. 47 Issue (5): 814-821  DOI: 10.6052/0459-1879-15-168 0

### 引用本文 [复制中英文]

[复制中文]
Wang Xiaojun, Wang Qi. A LCP METHOD FOR THE DYNAMICS OF PLANAR MULTIBODYSYSTEMS WITH IMPACT AND FRICTION[J]. Chinese Journal of Ship Research, 2015, 47(5): 814-821. DOI: 10.6052/0459-1879-15-168.
[复制英文]

### 文章历史

2015-05-11 收稿
2015-06-24 录用
2015–07–07 网络版发表

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

1 系统的描述及接触力模型

1.1 法向接触力模型

 $F_N = K\,\delta ^n + c\delta \dot{\delta }$ (1)

1.2 切向接触力模型

 $F_\tau = \left\{ {{\begin{array}{ll} - \mu F_N {\rm sgn}(\dot{g}_\tau ),&\dot{g}_\tau \ne 0\\ - \mu _0 F_N \mbox{Sgn}(\ddot{g}_\tau ),&\dot{g}_\tau = 0\\ \end{array} }} \right.$ (2)

 $\label{eq1} \mbox{ }(\dot{g}_\tau ) = \left\{ {\begin{array}{ll} + 1,&\dot{g}_\tau > 0 \\ - 1,& \dot{g}_\tau < 0 \\ \end{array}} \right.$ (3)

$\mbox{Sgn}(\cdot)$为多值函数[3, 15],其定义如下

 $\mbox{Sgn}(\ddot{g}_\tau ) = \left\{ {{\begin{array}{ll} + 1,&\ddot{g}_\tau > 0\\{} [- 1,+ 1],&\ddot{g}_\tau = 0\\ - 1,&\ddot{g}_\tau < 0\\ \end{array} }} \right.$ (4)

2 非光滑动力学方程及其算法 2.1 非光滑动力学方程

 ${{ \varPhi }}^\ast ({{ q}}) = {{ 0}}$ (5)

 ${{\tilde{ \varPhi }}}({{ q}},t) = {{ 0}}$ (6)

 ${{ \varPhi }}({{ q}},t) = [{{ \varPhi }}^\ast ,{{\tilde{ \varPhi }}}]^{\rm T} = {{ 0}}$ (7)

 $\left. {\begin{array}{l} {{ M\ddot{ q}}} = {{ \varPhi }}_{{ q}}^{\rm T} {{ \lambda }} + {{ Q}} + {{ Q}}_f \\ {{ \varPhi }}({{ q}},t) = {{ 0}} \\ \end{array}} \right\}$ (8)

2.2 库伦摩擦定律的互补关系

 $\left. {\begin{array}{l} {{ F}}_\tau ^ + = {{ \mu }}_0 {{ F}}_N + {{ F}}_\tau ^\ast \\ {{ F}}_\tau ^ - = {{ \mu }}_0 {{ F}}_N - {{ F}}_\tau ^\ast \\ \end{array}} \right\}$ (9)

 ${{ F}}_\tau ^ + = 2{{ \mu }}_0 {{ F}}_N - {{ F}}_\tau ^ -$ (10)

 $\left. \begin{array}{l} \ddot{g}_{\tau i}^ + = \dfrac{1}{2}(\left| {\ddot{g}_{\tau i}} \right| + \ddot{g}_{\tau i} ) \\[2mm] \ddot{g}_{\tau i}^ - = \dfrac{1}{2}(\left| {\ddot{g}_{\tau i}} \right| - \ddot{g}_{\tau i} ) \\ \end{array} \right\} \quad (i = 1,2,\cdots ,H)$ (11)

 ${{\ddot{ g}}}_\tau ^ + - {{\ddot{ g}}}_\tau ^ - = {{\ddot{ g}}}_\tau$ (12)

 $\left. {\begin{array}{l} {{\ddot{ g}}}_\tau ^ + \geqslant {\bf 0},\quad {{ F}}_\tau ^ + \geqslant {\bf 0},\quad {{\ddot{ g}}}_\tau ^{\rm +T} {{ F}}_\tau ^ + = {\bf 0} \\ {{\ddot{ g}}}_\tau ^ - \geqslant {\bf 0},\quad {{ F}}_\tau ^ - \geqslant {\bf 0},\quad {{\ddot{ g}}}_\tau ^{ \rm -T} {{ F}}_\tau ^ - = {\bf 0} \\ \end{array}} \right\}$ (13)

2.3 动力学方程的算法

 ${{\ddot{ \varPhi } + }}\alpha {{\dot{ \varPhi } + }}\beta \varPhi = 0$ (14)

 ${{ \varPhi}}_{{ q}} {{\ddot{ q} + \dot{ \varPhi }}}_{{ q}} \dot{ q} + \alpha {{ \varPhi }}_{{ q}}\dot{ q} + \beta {{ \varPhi }} + \dot{ \varPhi}_t + \alpha {{ \varPhi }}_t = 0$ (15)

 ${{\ddot{ q}}} = {{ M}}^{ - 1}{{ \varPhi }}_{{ q}}^{\rm T} {{ \lambda }} + {{ M}}^{ - 1}{{ Q}} + {{ M}}^{ - 1}{{ Q}}_f$ (16)

 ${{ \lambda }} = - {{ D}}^{ - 1}{{ A}} - {{ D}}^{ - 1}{{ \varPhi }}_{{ q}} {{ M}}^{ - 1}{{ Q}}_f$ (17)

 ${{\ddot{ q}}} = {{ B}}_1 {{+ B}}_2 {{ Q}}_f$ (18)

 ${{ Q}}_f = {{ G}}_1 {{ F}}_\tau ^s + {{ G}}_2 {{ F}}_\tau ^\ast$ (19)

 ${{\ddot{ q}}} = {{ B}}_1 + B_2 {{ G}}_1 {{ F}}_\tau ^s + {{ B}}_2 {{ G}}_2 {{ F}}_\tau ^\ast$

 ${{\ddot{ q}}} = {{ B}}_3 + {{ B}}_2 {{ G}}_2 {{ F}}_\tau ^\ast$ (20)

 ${{\ddot{ g}}}_\tau = {{ W\ddot{q}}} + {{ w}}_H$ (21)

 ${{\ddot{ g}}}_\tau ^ + - {{\ddot{ g}}}_\tau ^ - = {{ W\ddot{ q}}} + {{ w}}_H$ (22)

 ${{\ddot{ g}}}_\tau ^ - = {{\ddot{ g}}}_\tau ^ + - {{ W B}}_3 - {{ W B}}_2 {{ G}}_2 {{ F}}_{\tau }^{\ast } - {{ w}}_H$ (23)

 $\label{eq2} {{\ddot{ g}}}_\tau ^ - = {{ W B}}_2 {{ G}}_2 {{ F}}_\tau ^ - + {{\ddot{ g}}}_\tau ^ + - {{ W B}}_3 - {{ W B}}_2 {{ G}}_2 {{ \mu }}_0 {{ F}}_N - {{ w}}_H$ (24)

 $\begin{array}{l} \left[ {\begin{array}{*{20}{c}} {\ddot g_\tau ^ - }\\ {F_\tau ^ + } \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {W{B_2}{G_2}1}\\ { - 10} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {F_\tau ^ - }\\ {\ddot g_\tau ^ + } \end{array}} \right] + \\ \qquad \left[ {\begin{array}{*{20}{c}} { - W{B_3} - W{B_2}{G_2}{\mu _0}{F_N} - {w_H}}\\ {2{\mu _0}{F_N}} \end{array}} \right] \end{array}$ (25)

3 算例

 图 1 具有定常和非定常约束的滑块摆杆机构 Fig. 1 Slider-bar Mechanism with scleronomic and rheonomic constraints

 $\left\{ {\begin{array}{l} \phi _1^\ast = x_{C2} - x_{C1} - e\cos \theta _2 - d_1 = 0 \\ \phi _2^\ast = y_{C2} - y_{C1} - e\sin \theta _2 - d_2 = 0\quad \\ \tilde{\phi } = \theta _2 - \theta _1 - \omega {\kern 1pt} {\kern 1pt} t = 0 \\ \end{array}} \right.$

 $\left\{\begin{array}{l} d_1 = (A_x - C_x )\cos \theta _1 - (A_y - C_y )\sin \theta _1 \\ d_2 = (A_x - C_x )\sin \theta _1 + (A_y - C_y )\cos \theta _1 \\ \end{array}\right.$

 图 2 滑块上O1点水平位移图 Fig. 2 Horizontal displacement of point O1 on slider
 图 3 滑块上O1点铅垂位移图 Fig. 3 Vertical displacement of point O1 on slider
 图 4 滑块角位移图 Fig. 4 Angular displacement of slider

 图 5 作用在滑块上的法向接触力 Fig. 5 Normal contact force on slider
 图 6 铰链A的约束力 Fig. 6 Constraint force of joint A
 图 7 驱动摆杆力矩 Fig. 7 Driving bar moment

 图 8 $\left\| {{ \varPhi }} \right\|$的时间历程图 Fig. 8 Time history of $\left\| {{ \varPhi }} \right\|$

 图 9 当$e = 0.1$ m时$F_\tau , \dot{g}_\tau$的时间历程图 Fig. 9 Time history of $F_\tau$ and $\dot{g}_\tau$ with $e = 0.1$ m
 图 10 当$e = 0.15$ m时$F_\tau , \dot{g}_\tau$的时间历程图 Fig. 10 Time history of $F_\tau$ and $\dot{g}_\tau$ with $e = 0.15$ m
4 结 论

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A LCP METHOD FOR THE DYNAMICS OF PLANAR MULTIBODYSYSTEMS WITH IMPACT AND FRICTION
Wang Xiaojun, Wang Qi
1. Department of Electron and Machine, Changzhou Institute of Technology, Changzhou 213002, China;
2. School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China
Fund: The project was supported by the National Natural Science Foundation of China (11372018).
Abstract: This paper is presented to show the modeling and numerical method for the dynamics of the planar multi-rigid-body system with contact, impact and Coulomb's dry friction. The multibody system consists of the rigid bodies which are linked with ideal joints and driving motors, so the system constraint equations included two parts, scleronomic constraint equations and rheonomic constraint equations. Based on the theory of contact mechanics, the local deformations in contact bodies are taken into account and the normal forces of contact surfaces are expressed as nonlinear functions of relative penetration depth and its speed during impact between two bodies. The Coulomb dry friction model is used to describe the tangential frictional forces of contact surfaces. Using the concept of friction saturation and the relative acceleration of the contact point in the tangential direction, the complementarity conditions and formulations about the friction law are given. The problems of detecting stick-slip transitions of contact points and solving frictional forces in stick situation are formulated and solved as a linear complementarity problem (LCP) by the event-driven scheme. The dynamical equations of the system are obtained by Lagrange's equations of the first kind and Baumgarte stabilization method in order to reduce the constraint drift and solve the system motion, normal contact forces and tangential friction forces as well as ideal joint constraint forces and driven constraint forces in the system. Finally, the numerical example of a planar multi-rigid-body like flat beater is given to analyze its dynamical behavior and show the availability of the method in this paper.
Key words: multibody system    Coulomb dry friction    contact force    linear complementarity problem    non-smooth