A New Group Theoretic Method For The Analysis Of Symmetric Structures

Cyclic symmetry structures are very common in both life and engineering applications.Structures with periodic symmetry often have good mechanical and aesthetic properties.Recent years,with the development of computers,the platform for the study of the periodic structure has become more and more extensive,and more effective and time-saving calculation methods in the exploration of structural calculations have also received increasing attention.Using the group theoretic method to the symmetric structures can lead to lower memory and higher efficiency,at present,the commonly used method is to use the characteristic label table and projection operator theory to establish this base vector.However,traditional group theory methods are not easy to master.In response to these problems,this paper proposes a new group theory approach.The new group theoretic method involves only the basic group definition,and the advanced mathematical theories,such as character table,irreducible representation,projection operator,are disused.This paper describes that for a symmetric structure,the base vector can be obtained by avoiding mathematical concepts such as the signature table and the projection operator.That is,the feature vector of the group representation matrix is a base vector.The eigenvectors of the matrix representation matrix can be used to diagonalize the stiffness matrix,which provides a new approach for the application of the group theory in the analysis of symmetry structures.The truss structure model is established and the static and dynamic conditions are verified.Its computational scheme is easy and flexible.Numerical examples show that the proposed method is based on a reliable theory and can give correct computational results.For the truss structure with multi-layer repeat units,the physical meaning is deduced according to the Kronecker product's expression,and it is applied to concrete examples,and the static and dynamic problems are further calculated and verified.When the symmetric structure has multiple different types of symmetries,different group representation matrices can be combined so that the system matrix can be diagonalized as much as possible.The results show that the results of applying the Kronecker product to the group theory are accurate and reliable.