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Critical angle of revolute pin joint and its application in a four-bar mechanism
ZHANG Jie , WANG Dan , CHEN Wuyi
School of Mechanical Engineering and Automation, Beijing University of Aeronautics and Astronautics, Beijing 100083, China
Received: 2015-12-08; Accepted: 2016-01-08; Published online: 2016-09-20 09:46
Foundation item: National Natural Science Foundation of China (51305013)
Corresponding author. WANG Dan, Tel.:010-82317471, E-mail:b11034@buaa.edu.cn
Abstract: Revolute pin joint (revolute pair) possesses one degree of freedom, i.e. rotation movement, through relative sliding between contact pairs. Existence of friction and clearance in revolute joints has crucial effects on the overall precision of mechanism. Using the concept of critical angle, the marginal value in which revolute joints could keep in balance state is discussed in this study. Firstly, frictional angle and self-lock caused by friction and clearance in revolute pin joints are investigated, which contribute to not only length error but also angle error in mechanism with such joints. Then, critical angles are calculated with and without contact deformation respectively. Moreover, a four-bar mechanism with one revolute joint studied above is chosen as a case study to evaluate the impact of clearance of joint, contact deformation and friction on the overall precision. Analytical results indicate that the clearance in revolute joint and elastic deformation of mechanism play the most important roles in the system, and the contact deformation of joint could be neglected in the overall precision analysis.
Key words: revolute pin joint (revolute pair)     critical angle     frictional angle     four-bar mechanism     precision analysis

1 临界角计算方法及R副结构 1.1 临界角计算方法

 图 1 临界角成立条件 Fig. 1 Conditions for critical angle

1.2 R副结构

R副结构如图 2(a)所示。以运动副截面进行观察，可使用如图 2(b)所示的简化结构表示，主要包括底座、转动体以及与转动体相固连的连杆三部分。由于间隙和接触面间摩擦的共同作用，R副存在一个临界角，使得转动体在任意临界角范围内的载荷条件下可实现自锁。

 图 2 R副结构示意图 Fig. 2 Schematic diagram of structure of revolute pair

R副的临界角也应同时满足使得转动体在底座内表面不滑动和不翻转2个条件。以下将分别从不考虑接触变形和考虑接触变形2种情况进行讨论。

2 R副临界角计算方法 2.1 不考虑接触变形条件

 图 3 不考虑接触变形的R副临界角分析示意图 Fig. 3 Schematic diagrams of critical angle for revolute pair without contact deformation

1)当与运动副相连的连杆另一端为摩擦可忽略的理想运动副时，连杆为二力杆不承受弯矩作用，连杆方向可随转动体发生自由摆转。则转动体所受作用力F方向亦随连杆的摆动而变化，因此当转动体达到静力平衡时接触点必然位于力F作用线与底座的交点处，支撑反力N与作用力F相平衡：

 (1)

2)当连杆另一端同为摩擦不可忽略运动副时，连杆两端运动副可为连杆提供一弯矩M，使得连杆及转动体的平衡受力状态满足图 3(b)，此时转动体受到连杆约束无法沿底座内表面滚动，夹角θ小于摩擦角φf时接触面间将无相对滑动。此时临界角大小为

 (2)

2.2 考虑接触变形条件

 (3)
 图 4 两圆柱体接触应力分布示意图 Fig. 4 Schematic diagram of distribution of contact stress between two cylinders

 (4)

 (5)
 (6)

 (7)

3 应用于四连杆机构的算例分析 3.1 仅有1个非理想铰链的四连杆机构

 图 5 引入一个虚拟连杆的四连杆机构示意图 Fig. 5 Schematic diagram of a four-bar mechanism with one virtual link

 (8)

LBC为确定连杆机构位姿的实际连杆长度，已知长度的初始连杆L2(LB′C)与实际连杆LBC之间的夹角α(∠B′CB)即为连杆的摆角误差。考虑摩擦因素的影响，最终平衡状态连杆L2L5间夹角θ位于上述分析所得到的临界角θmax范围之内，即连杆将在以LBC为对称轴、以2θmax为圆心角的扇形区域内保持稳定，这使得最终计算出的实际连杆长度LBC在一定范围内具有随机性。LBC的数值范围为

 (9)

 (10)

 (11)
3.2 有2个非理想铰链的四连杆机构

 (12)
 图 6 有2个非理想铰链的四连杆机构的2种典型平衡状态 Fig. 6 Two typically balanced states of four-bar mechanism with two non-ideal links

 (13)

 (14)

 (15)

 (16)

3.3 算例分析

 图 7 算例分析四连杆机构示意图 Fig. 7 Schematic diagram of four-bar mechanism in case study

 (17)

 参数类型 数值 四连杆机构参数条件 LAB=LB′C′=LCD=50 mmLAD=100 mmA杆=6.25π mm2 圆柱铰链参数条件 RB=RC，RB′=RC′(θmax1=θmax2=θmax, L5=L6)l柱销=10 mm 变量 铰链间隙δ连杆压力F接触表面滑动摩擦系数μ

 (18)

 (19)

3.3.1 对ΔLBC的讨论

 图 8 ΔL、ε和δ随F及ΔL和ε随δ的变化示意图 Fig. 8 Variation of ΔL, ε and δ with F and variation of ΔL, ε with δ

3.3.2 对αmax的讨论

η=L5/L2，则αmaxη的变化趋势如图 9所示，其中图 9(a)为摩擦系数取为μ=0.3时αmaxη的变化关系，图 9(b)为在2种比值η条件下，αmax随摩擦系数μ的变化关系。

 图 9 αmax随μ和η的变化趋势 Fig. 9 Variation trend of αmax with μandη

4 结论

1)铰链摩擦角的存在使得连杆在机构平衡状态下同时具有长度和偏角两项误差，连杆在此偏角范围内均可达到受力平衡状态并具有随机性，但可通过已知参数的计算确定其平衡范围。

2)当连杆两端均为有摩擦R副时，不考虑接触变形条件下临界角为0，考虑接触变形条件下临界角取值为

3)在四连杆算例分析中可得出，相比于连杆变形量和R副间隙大小，相同受力条件下R副的接触变形量可忽略，有利于简化复杂连杆机构的精度分析。

4)连杆摆角误差的最大值αmax与滑动摩擦系数μ及连杆长度比值η正相关，通过改善铰链的润滑条件、优化连杆长度可以有效降低临界角对机构的精度影响。

 [1] 邹慧君, 高峰. 现代机构学进展[M]. 北京: 高等教育出版社, 2007 : 93 -94. ZOU H J, GAO F. The progress of modern organizations[M]. Beijing: Higher Education Press, 2007 : 93 -94. (in Chinese) [2] 郭卫东. 机械原理[M]. 北京: 科学出版社, 2010 : 168 . GUO W D. Mechanical principle[M]. Beijing: Science Press, 2010 : 168 . (in Chinese) [3] 黄昔光, 廖启征. 空间6R串联机器人机构位置逆解新算法[J]. 北京航空航天大学学报, 2010, 36 (3) : 295 –298. HUANG X G, LIAO Q Z. A new algorithm position inverse solution of 6-R serial robot mechanism[J]. Journal of Beijing University of Aeronautics and Astronautics, 2010, 36 (3) : 295 –298. (in Chinese) [4] 尚国强, 陈五一, 韩先国, 等. 并联机床的球铰链设计与分析[J]. 机械技术史及机械设计, 2008 : 286 –290. SHANG G Q, CHEN W Y, HAN X G, et al. Spherical hinge design and analysis of parallel machine tool[J]. Mechanical Technology and Mechanical Design, 2008 : 286 –290. (in Chinese) [5] REIS V L, DANIEL G B, CAVALCA K L. Dynamic analysis of a lubricated planar slider-crank mechanism considering friction and Hertz contact effects[J]. Mechanism and Machine Theory, 2014, 74 : 257 –273. DOI:10.1016/j.mechmachtheory.2013.11.009 [6] ZHANG X C, ZHANG X M, CHEN Z. Dynamic analysis of a 3-RRR parallel mechanism with multiple clearance joints[J]. Mechanism and Machine Theory, 2014, 78 : 105 –115. DOI:10.1016/j.mechmachtheory.2014.03.005 [7] BAI Z F, ZHAO Y. A hybrid contact force model of revolute joint with clearance for planar mechanical systems[J]. International Journal of Non-Linear Mechanics, 2013, 48 : 15 –36. DOI:10.1016/j.ijnonlinmec.2012.07.003 [8] JUNG K. Dynamic response of a revolute joint with clearance[J]. Mechanism and Machine Theory, 1995, 31 (1) : 121 –134. [9] PAYANDEH S, FARAZ A. Towards approximate models of coulomb frictional moments in:(I) revolute pin joints and (Ⅱ) spherical-socket ball joints[J]. Journal of Engineering Mathematics, 2001, 35 (403) : 283 –296. [10] 李新友, 陈五一, 韩先国. 基于正交设计的3-RPS并联机构精度分析与综合[J]. 北京航空航天大学学报, 2011, 37 (8) : 979 –984. LI X Y, CHEN W Y, HAN X G. 3-RPS parallel mechanism based on orthogonal design precision analysis and synthesis[J]. Journal of Beijing University of Aeronautics and Astronautics, 2011, 37 (8) : 979 –984. (in Chinese) [11] KHAN A W, CHEN W Y. Systematic geometric error modeling for workspace volumetric calibration of a 5-axis turbine blade grinding machine[J]. Chinese Journal of Aeronautics, 2010, 23 (5) : 604 –615. DOI:10.1016/S1000-9361(09)60261-2 [12] 瓦伦丁·L·波波夫.接触力学与摩擦学的原理及其应用[M].李强, 等, 译.北京:清华大学出版社, 2011:169-170. VALENTIN L P.Contact mechanics and tribology principle application[M].LI Q, et al, translated.Beijing:Tsinghua University Press, 2011:169-170(in Chinese). [13] 王元淳, 沃国纬. 弹性力学[M]. 上海: 上海交通大学出版社, 1998 : 133 -135. WANG Y C, WO G W. Elastic mechanics[M]. Shanghai: Shanghai Jiao Tong University Press, 1998 : 133 -135. (in Chinese) [14] 刘鸿文. 材料力学[M]. 北京: 高等教育出版社, 2005 : 34 . LIU H W. Material mechanics[M]. Beijing: Higher Education Press, 2005 : 34 . (in Chinese)

#### 文章信息

ZHANG Jie, WANG Dan, CHEN Wuyi

Critical angle of revolute pin joint and its application in a four-bar mechanism

Journal of Beijing University of Aeronautics and Astronsutics, 2016, 42(12): 2738-2744
http://dx.doi.org/10.13700/j.bh.1001-5965.2015.0812