Research of Mapping Visualization Among the Bennett Mechanism，Enveloping Hexahedron and J-hyperboloid of One Sheet

Lele Bai; Lubin Hang

doi:10.16578/j.issn.1004.2539.2021.06.002

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Research of Mapping Visualization Among the Bennett Mechanism，Enveloping Hexahedron and J-hyperboloid of One Sheet

Theory·Reserach | 更新时间：2022-11-10

- Research of Mapping Visualization Among the Bennett Mechanism，Enveloping Hexahedron and J-hyperboloid of One Sheet
- Journal of Mechanical Transmission Vol. 45, Issue 6, Pages: 8-15(2021)
- 作者机构：
  
  1.上海工程技术大学机械与汽车工程学院，上海 201620
- 作者简介：
- 基金信息：
- DOI：10.16578/j.issn.1004.2539.2021.06.002
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白乐乐,杭鲁滨.Bennett机构、包络六面体与J-单叶双曲面映射可视化研究[J].机械传动,2021,45(06):8-15.

Bai Lele,Hang Lubin.Research of Mapping Visualization Among the Bennett Mechanism，Enveloping Hexahedron and J-hyperboloid of One Sheet[J].Journal of Mechanical Transmission,2021,45(06):8-15.
白乐乐,杭鲁滨.Bennett机构、包络六面体与J-单叶双曲面映射可视化研究[J].机械传动,2021,45(06):8-15. DOI： 10.16578/j.issn.1004.2539.2021.06.002.

Bai Lele,Hang Lubin.Research of Mapping Visualization Among the Bennett Mechanism，Enveloping Hexahedron and J-hyperboloid of One Sheet[J].Journal of Mechanical Transmission,2021,45(06):8-15. DOI： 10.16578/j.issn.1004.2539.2021.06.002.

摘要

基于Bennett机构构造的包络六面体，推导了机构构型与其运动副轴线对应的单叶双曲面之间瞬态相对位姿关系，可视化探究了Bennett机构、机构包络六面体与单叶双曲面之间动态位姿映射；分析了Bennett机构对角线相等时包络六面体构型的几何特征。由特定结构参数Bennett机构对应单叶双曲面腰部空间形态及腰截面投影可视化研究发现，机构在特定构型包络六面体底面投影与单叶双曲面腰椭圆短轴重合，引发探究该构型机构压力角的启示。研究发现，当输入角等于, $n π + π / 2 (n \in N)$ ,http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17186141&type=,http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17186138&type=,26.07733345,3.47133350,，任一Bennett机构包络六面体底面投影线与腰椭圆短轴重合，机构出现最小压力角且最小压力角等于输出杆件运动副扭转角。从特定结构参数机构与其关联特征几何体、曲面、曲线间映射的可视化研究获得启示，可为进一步揭示运动学普遍规律提供研究思路。

Abstract

Based on the enveloping hexahedron constructed by Bennett mechanism，the instantaneous relative position and orientation relation between the Bennett linkage configuration and the hyperboloid defined by its joint axis are derived. The dynamic mapping between the Bennett mechanism，the enveloping hexahedron，and the hyperboloid is visually explored，the geometric characteristics of the enveloping hexahedron configuration of the Bennett linkage with equal diagonals are analyzed. It is found that the projection of the enveloping hexahedron's bottom surface of the Bennett linkage at the given configuration coincides with the short axis of the waist ellipse through visualization of the projection of the hyperboloid on waist cross-section，which sparks inspiration of exploring the pressure angle of the mechanism. The research discovers that when the input angle is equal to, $n π + π / 2 (n \in N)$ ,http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17184716&type=,http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17184713&type=,26.07733345,3.47133350,，the configuration of the Bennett linkage happens the minimum pressure angle，which equals the twist angle of the output link. The visualization research on the mechanism with specific structural parameters and its associated geometry characteristics，such as geometric surfaces and curves，get inspiration and can provide ideas to reveal the general rules of kinematics.

关键词

Bennett机构运动学单叶双曲面瞬态映射压力角

Keywords

Bennett mechanismKinematicsHyperboloid of one sheetInstantaneous mappingPressure angle

references

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SONG X K，DENG Z Q，GUO H W，et al.Networking of bennett linkages and its application on deployable parabolic cylindrical antenna［J］.Mechanism and Machine Theory，2016，109：95-125.

BENNETT G T.A new mechanism［J］.Engineering，1903，76：777-778.

BENNETT G T.The skew isogram mechanism［J］.Proceeding of London Mathematics Society，1914，13：151-173.

BAKER J E.The bennett，goldberg and myard linkages-in perspective［J］.Mechanism and Machine Theory，1979，14（4）：239-253.

YU H C.The Bennett linkage，its associated tetrahedron and the hyperboloid of its axes［J］.Mechanism and Machine Theory，1981，16（2）：105-114.

BAKER J E.The Bennett linkage and its associated quadric surface［J］.Mechanism and Machine Theory，1988，23（2）：147-156.

YU H C，BAKER J E.On generation of new linkages from Bennett loops［J］.Mechanism and Machine Theory，1981，16（5）：473-485.

BAKER J E.The composition of Bennett's hyperboloid from the loop itself［J］.Journal of Mechanical Design，2004，126（5）：875-880.

BAKER J E.Kinematic investigation of the deployable Bennett loop［J］.Journal of Mechanical Design，2007，129（6）：602-610.

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