A Procedure for Improving Discrete Substructure Representation in Dynamic Synthesis

Abstract

N important problem in the dynamic analysis for flexible structures is that of determining the natural frequencies and associated natural modes of vibration, particularly the lower ones. Complex structures are often represented by mathematical models, such as finite-element models, possessing a very large number of degrees of freedom, where the number can reach into the tens of thousands. Because of the large number of degrees of freedom, a direct iterative method for obtaining a partial eigensolution, know as subspace iteration, was developed. The subspace iteration method has a long history. An extensive bibliography is given in Ref. 1 and the current status of the method is discussed in Refs. 2-4 in the context of structural dynamics. Even the subspace iteration method, however, can be overwhelmed by the large number of degrees of freedom in the mathematical model, so that a method for prior reduction of the number of degrees of freedom is desirable. Complex structures are often modeled by breaking the structure into a number of simpler components or substructures. The substructure models are then coupled together to form the whole structure model. The technique is known as substructure synthesis and its origin can be found in Refs. 5 and 6. The idea of Refs. 5 and 6 is to represent the motion of each substructure by a set of substructure normal modes, obtained by solving an eigenvalue problem for each substructure. Then each substructure is represented in the synthesis process by a reduced number of lower substructure modes. The synthesis leads to an eigenvalue problem for the assembled structure of a substantially smaller order than that of the original formulation. The price paid in reducing the order of the eigenvalue problem is that its solution is only an approximation of the actual eigensolution of the original structure.