具有三阶幻方拓扑构型的厚板折纸机构运动相容性分析  

Foldability Analysis of Thick-panel Origami Mechanism with Third-order Magic Square Topology

作　　者：畅博彦[1,2] 张浩楠 梁栋 金国光[1,2] CHANG Boyan;ZHANG Haonan;LIANG Dong;JIN Guoguang(School of Mechanical Engineering,Tiangong University,Tianjin 300387;Tianjin Key Laboratory of Advanced Mechatronics Equipment Technology,Tianjin 300387)

机构地区：[1]天津工业大学机械工程学院,天津300387 [2]天津市现代机电装备技术重点实验室,天津300387

出　　处：《机械工程学报》2023年第11期201-212,共12页Journal of Mechanical Engineering

基　　金：国家自然科学基金(52005368,52175243);天津市高等学校创新团队培养计划(TD13-5037)资助项目。

摘　　要：折纸机构是折展机构的一种,折痕分布对折纸机构的刚性折展运动有直接影响。首先,提出一类具有三阶幻方拓扑构型的折纸机构,运用拓扑学理论对折纸机构的拓扑组成元素进行分析,通过构造网孔环路矢量和折痕分布矩阵,形成一种可用于描述折纸机构折痕分布特征的矩阵表示方法。其次,建立拓扑图旋转、翻转和镜像操作过程的矩阵运算数学模型,对256种三阶幻方折纸机构拓扑构型进行同构判定后,将得到的23种拓扑构型分为方扭型、轴对称型和不规则型。最后,在考虑板件厚度的同时,为了判断机构是否可刚性折展,将三阶幻方中每一个网孔环路处的单顶点四折痕机构等效为Bennett机构,通过引入分位角参数矩阵、转角特征矩阵和分位角特征矩阵,建立运动相容性判定方程的通用表达式,提出运动相容性系数的概念,并将其应用于厚板机构的刚性折展性分析,发现23种拓扑构型中有19种可用于厚板折展单元的设计,以折展单元与其镜像体进行模块化扩展为例,说明了此类折展单元在构造大尺度折展机构时的应用前景。Rigid foldability of the origami mechanism is directly affected by the crease pattern.Firstly,components of the origami crease pattern with third-order magic square topology are examined with topology theory.To represent the crease assignment features of the origami topology,the mesh circuit vector and the crease assignment matrix are constructed.Secondly,the origami topology is identical before and after the operations of rotating,flipping,and mirroring for two hundred and fifty-six topology configurations.Following isomorphism determination,twenty-three topological configurations are obtained and classified as three types named square-twist type,axisymmetric type,and irregular type.Finally,to determine whether the thick-panel origami mechanism can achieve rigid folding,the single-vertex four-crease mechanism is equivalent with the Bennett linkages kinematically.A general formula of kinematic compatibility is established by introducing the sector angle parameter matrix,the dihedral angle eigen matrix,and the sector angle eigen matrix.The coefficient of kinematic compatibility is proposed and it can be used to determine whether a thick-panel origami mechanism is rigidly foldable or not.Nineteen of the twenty-three topological configurations are available for rigidly foldable thick-panel modules.These modules can be used to construct large scale deployable mechanisms with their mirrored counterpart which have a great prospect.

关 键 词：可展机构 折纸机构 拓扑学 刚性可折性 运动相容性 

分 类 号：TH112[机械工程—机械设计及理论]