Extending Zhong-Williams scheme to solve repeated-root wave modes

The eigenvalue scheme proposed by Zhong and Williams in 1995 is a well-conditioned and efficient approach to solve the wave propagation characteristics of periodic structures. However, the original scheme is valid only for single-root wave modes and cannot be directly employed to solve repeated-root wave modes. This limits the application of Zhong-Williams scheme when the cross-section of the structure has certain kind of symmetry (such as cylinders and square-section beams). In this paper we extend Zhong-Williams scheme and make it applicable for structures with repeated-root wave modes. The improvements are two-fold. First, the infinite values caused by repeated roots while solving the generalized symplectic eigen-problem are eliminated by a more general expression. Second, the ill-conditioning caused by repeated roots while recovering the wave modes is resolved by a novel approach based on the inverse power method. A square-section beam is used to verify the proposed method on the propagating waves. Forced response and energy flow analysis of a cylindrical shell and a stiffened cylindrical shell are conducted using wave modes, to verify the proposed method on the overall wave basis. Results show that the proposed improvements of Zhong-Williams scheme are obligatory for the analysis of structures with repeated-root wave modes.