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Solution region synthesis theory and method of six-bar linkages with 4-position motion generation
Han Jianyou, Cui Guangzhen, Yang Tong
School of Mechanical Engineering, University of Science and Technology Beijing, Beijing 100083, China
Abstract:According to the Stephenson-Ⅱ six-bar linkage with 4-position motion generation, the concept and equation of generalized Burmester curves based on Burmester curve was proposed. The solution region method was given. Firstly, this method required to get solution curves meeting requirements. Then the six-bar linkage with defects was removed from the solution region using the sign of Jacobin determinant value and mechanism motion continuity. Finally, considering the practical engineering requirements, the feasible solution region was obtained. Analysis example demonstrates that the method is practicable and effective. The proposed method is simple and convenient to be realized in programming. Besides that, the method can provide reference for other multi-bar linkages 4-position motion generation.
Key words: six-bar linkages     motion generation     4-positon synthesis     solution curve     solution region

1 解曲线方程的推导

 图 1 刚体平面运动示意图Fig 1 Diagram of rigid body planar motion

xB1,yB1看作未知数,则式(4)有解的条件为

 图 2 六杆机构示意图 Fig 2 Diagram of a six-bar linkage
 图 3 3R开链 Fig. 3 Diagram of a 3R open chain

 图 4 解曲线和机构Fig. 4 Solution curves and six-bar linkage
3 建立解域

 图 5 六杆机构参数示图Fig. 5 Parameters for six-bar linkage
4 机构运动缺陷判定 4.1 六杆机构位置分析和雅可比矩阵推导

4.2 运动缺陷判断

Δ=0时机构处于奇异位形(死点),故当机构在给定位置对应的Δ值的符号不同时,机构将存在运动缺陷.但当给定位置对应的Δ值的符号相同时,机构也可能会存在运动缺陷,接下来通过表 1图 6来进行说明.表 1给出了图 2所示六杆机构在两个位置的参数,固定铰链点a0和b0的坐标分别为(0,0)和(59.8273,0).图 6是上述六杆机构在可运动范围内的Δ曲线.Δ1Δ2分别为给定位置1和位置2的Δ值.

 六杆机构参数 位置1 位置2 θ1/(°) 10.00 50.00 a (26.78,4.72) (17.48,20.83) b (32.17,-15.26) (35.04,31.78) c (50.02,-23.49) (53.29,24.47) d (21.29,-4.22) (19.02,10.45) e (37.70,-23.82) (41.01,23.51) p (43.58,-0.98) (45.73,46.63) Δ值 7.51422 9.94004
 图 6 Δ随输入角变化的曲线图Fig. 6 Δ changes with the input angle

 位置 a b p 1 (13.5108,50.4228) (48.7123,53.7165) (78.5419,59.6493) 2 (26.1008,45.2079) (56.5364,63.1988) (86.7112,59.3935) 3 (36.9121,36.9121) (65.6906,57.4497) (87.78,36.5438) 4 (45.2079,26.1008) (79.0142,36.4511) (91.3807,8.66497)

 铰链点 b0 c d e 限定范围 (55,-5) (55,15) (10,17) (45,24) (98,42) (88,45) (80,46) (95,50)

 图 7 解曲线03-0和03-3Fig. 7 Solution curves of 03-3 and 03-3

 图 8 解曲线13-1和13-3Fig. 8 Solution curves of 13-1 and 13-3

 图 9 解域图Fig. 9 Solution regions

 图 10 综合得到的机构Fig. 10 Synthesized linkage

 机构参数 K1机构 K2机构 b0 (71.25,37.66) (95.0,0.0) c (57.11,37.88) (84.47,29.82) d (15.50,31.28) (24.5,22.79) e (50.00,35.47) (56.18,26.40) r 0.0931 0.2989
6 结 论

1) 六杆机构四位置运动生成的解域综合方法为多杆机构运动生成的尺寸综合问题提供了新的理论、方法和有效的解决途径.

2) 利用解域综合法将可行机构解的信息在有限的范围内表示出来,避免了机构选择的盲目性.这对进一步研究可行机构解的性能以及可行机构解的优选具有一定的意义.

3) 所提出的机构缺陷判别方法是有效和实用的.

4) 综合示例所得到的结果表明所提出的理论和方法的正确性以及所开发软件的有效性.

#### 文章信息

Han Jianyou, Cui Guangzhen, Yang Tong

Solution region synthesis theory and method of six-bar linkages with 4-position motion generation

Journal of Beijing University of Aeronautics and Astronsutics, 2014, 40(9): 1170-1175.
http://dx.doi.org/10.13700/j.bh.1001-5965.2013.0590