Efficient method for dynamic responses, modal and buckling analysis of cylindrical periodic structures based on the group theory and Woodbury formula

Abstract

This paper proposes an efficient method that combines group theory, the Woodbury formula, and condensation technology to solve the governing equations related to dynamic responses, modal and buckling analysis of cylindrical periodic structures. Efficient and accurate solution of linear algebraic equations is pivotal for dynamic responses, modal and buckling analysis. Based on the cyclic periodic property of the structures and group theory, the linear algebraic equations can be transformed into a series of independent subequations corresponding to one-dimensional (1D) periodic structures. The degrees of freedom (DOFs) of the unit cell corresponding to the 1D periodic structures can be divided into two boundary-surface DOFs and internal DOFs. To reduce the scale of the 1D periodic structures, we use condensation technique to condense the internal DOFs into the two boundary-surface DOFs. Thus, the scales of these subequations can be significantly reduced. Since the coefficient matrices of the small-scale subequations resemble block-circulant matrices, the Woodbury formula is used to obtain their solutions by solving some newly formed linear algebraic equations whose coefficient matrices are block-circulant. These newly formed linear algebraic equations are then efficiently solved using group theory. Numerical examples are presented to confirm the high accuracy and efficiency of the proposed method.