基于群集理论的高阶对称结构可动性判定

Mobility of highly symmetric structures based on group theory

基于群集理论,对由转动副及刚性连杆单元构成的高阶对称过约束体系进行了可动性分析。由于该类结构不同于铰接杆系结构,且自应力模数较高,很难用常规可动性判定方法来研究结构的几何稳定性。本文首先从单根单元的广义力平衡方程出发,建立了整体结构的平衡矩阵及位移协调矩阵,并从其零空间分别得到了机构位移模数及自应力模数。随后,根据对称群的不可约子空间及特征标,得到了位移模数及自应力模数的对称表示,从而根据二者的对称属性判定结构的可动性。算例结果表明,本文所述方法合理可行,且讨论的三个高阶对称结构皆被证明是单自由度可动的。

Based on the group theory,a novel method for determining the mobility of highly symmetric over-constrained assemblies is proposed.Different from the pin-jointed assemblies,these structures consist of revolving joints and rigid links,and have many modes of self-stresses.Thus,it is difficult to investigate their geometric stabilities using the conventional criteria for the mobility.By the equilibrium equation about the internal forces and external loads of each element,this study gives the equilibrium matrix and the compatibility matrix for these structures.The modes of mechanism displacement and self-stresses are obtained from the null-spaces of the matrices.Using the subspaces of irreducible representations and the characters from a symmetry group,the symmetry adapted modes of mechanisms and self-stresses could be accordingly expressed,and they will reveal the mobility.Some typical examples based on the geometries of polyhedra are discussed.It could be concluded that the proposed technique is convenient and feasible,and each of the illustrative structures has a single degree of freedom and is validated to be foldable.