A Series of Special Mechanism with Rank More than Six

doi:10.3901/JME.2021.05.031

Abstract

Abstract: The single loop 8-, 9-, …, 12-bar mechanism can be obtained by removing each of the public side links of the adjacent loops and the rotary pairs at their two ends from the 5-loop, 6-loop, …, 9-loop BNT (Bennett) mechanisms collinear with the rack, respectively. Through the analysis of the track circle of the hinge point of the link before and after the removal of the side link, it can be found that the locus circles of their hinge points are identical, which proves that the kinematic performance in the two mechanisms is completely equivalent. We can summarize that the 8-bar to the 12-bar mechanisms meet the BNT constraints, and the degree of freedom is 1, and their loop ranks are 7 to 11, they are in a family consisting of new GBG (Goldberg) mechanism with more than six ranks. This discovery breaks through the traditional view that the existing loop rank with less than or equal to 6. It is concluded that the degree of freedom of N-bar GBG mechanism is equal to N-3 ring BNT mechanism with collinear frame, and the value is 1. The loop rank of N-bar GBG mechanism is d=N-1. Taking 8-bar GBG mechanism as an example, the angular displacement is θ_i(i=2, 3, …, 8), it is proved that the angular displacement are all univariate functions of the angular displacement θ₁ of the original moving parts. The angular displacement of 8-bar GBG mechanism is measured with solidworks, and the result is identical with the corresponding angular displacement of 5-loop BNT mechanism before dismantling the bar. It is proved that the kinematic performance of the two mechanisms is completely equivalent. It is also proved that the 8-bar GBG mechanism is a mechanism with more than six ranks, the degree of freedom is 1, and the loop rank is 7, the practicability and reliability of BNT constraints are judged by trajectory circle. This research findings widens the use range of the loop rank in the mechanism, it makes people have a new understanding about the diversity of constraint properties in mechanical systems.

Key words: Bennett mechanism, Goldberg mechanism, Goldberg mechanism with ranks more than six, locus circle

CLC Number:

TH112

ZHANG Yitong, LI Yanwen, HU Bo, MU Dejun, ZHAO Meixin. A Series of Special Mechanism with Rank More than Six[J]. Journal of Mechanical Engineering, 2021, 57(5): 31-39.

References

[1] GRÜBLER M. Allgemeine eigenschaften der zwangläufigen ebenen kinematische kette:I[J]. Civilingenieur,1883,29(1):167-200. GRÜBLER M. General characteristics of the two-tier kinematic chain:I[J]. Civilingenieur,1883,29(1):167-200.
[2] BAGCI C. Degrees of freedom of motion in mechanisms[J]. Trans. ASME J. Engi. Industry,1971,93B:140-148.
[3] HUNT K H. Kinematic geometry of mechanisms[M]. Oxford:Oxford University Press,1978.
[4] 黄真,赵永生,赵铁石. 高等空间机构学[M]. 北京:高等教育出版社,2006. HUANG Zhen,ZHAO Yongsheng,ZHAO Tieshi. Advanced spatial mechanism[M]. Beijing:Higher Education Press,2006.
[5] LU W J,ZENG D X,HUANG Z. Over-constraints and a unified mobility method for general spatial mechanisms. Part 2:Application of the principle[J]. Chinese Journal of Mechanical Engineering,2016,29(1):1-10.
[6] GOGU G. Chebychev-Grübler-Kutzbach's criterion for mobility calculation of multi-loop mechanisms revisited via theory of linear transformations[J]. European J. Mechanics-A/Solids,2005,24(3):427-441.
[7] GOGU G. Structural synthesis of parallel robots:Part 1:Methodology[M]. Dordrecht:Springer,2008.
[8] YANG Tingli,SUN Dongjin. A general DOF formula for parallel mechanisms and multi-loop spatial mechanisms[J]. ASME Journal of Mechanisms and Robotics,2012,4(1):011001-1-17.
[9] ZHANG Y T,MU D J. New concept and new theory of mobility calculation for multi-loop mechanisms[J]. Sci. China Tech. Sci.,2010,53(6):1598-1604.
[10] ZHANG Y T,LI Y W,WANG L Y. A new formula of mechanism mobility based on virtual constraint loop[J]. Sci. China Tech. Sci.,2011,54(10):2768-2775.
[11] ZHANG Y T,LU W J,MU D J,et al. Novel mobility formula for parallel mechanisms expressed with mobility of general link group[J]. Chinese Journal of Mechanical Engineering,2013,26(6):1082-1090.
[12] MOROSKINE Y F. General analysis of the theory of mechanisms[J]. Moscow:Akad. Nauk,SSSR,1954,1:1.
[13] 张启先. 机构组成学的新探讨[J]. 机械工程学报,1961,9(1):7-32. ZHANG Qixian. Study on structural theory of spatial mechanisms[J]. Chinese Journal of Mechanical Engineering,1961,9(1):7-32.
[14] BENNETT G T. A new mechanism[J]. Engineering,1903,76:777-778.
[15] MYARD F E. Contribution à la géométrie des systèmes articulés[J]. ociete Mathématiques de France,1931,59,183-210. MYARD F E. Contribution to joint system geometry[J]. ociete Mathématiques de France,1931,59,183-210.
[16] GOLDBERG M. New 5-bar and 6-bar linkages in three dimensions[J]. SME J. Mech.,1943,65:649-661.
[17] BAKER J E. The Bennett,Goldberg and Myard linkages in perspective[J]. Mech. Mach. Theory,1979,14:239-253.
[18] BAKER J E. An analysis of Goldberg's anconoidal linkage[J]. Mech. Mach. Theory,1983,18(5):371-376.
[19] CHEN Yan,YOU Zhong. Spatial 6R linkages based on the combination of two Goldberg 5R linkages[J]. Mech. Mach. Theory,2007,42:1484-1498.
[20] 鹿玲,张一同,牟德君,等. Bennett机构交错角区间和机构类型的关系[J]. 机械工程学报,2020,56(9):9-17. LU Ling,ZHANG Yitong,MU Dejun,et al. Relationship between stagger angle interval and mechanism type of the Bennett mechanism[J]. Journal of Mechanical Engineering,2020,56(9):9-17.