On wave propagation in repetitive structures: Two forms of transfer matrix

Two forms of dynamic transfer matrix are derived for a one-dimensional (beam-like) repetitive pin-jointed structure with point masses located at nodal cross-sections, the displacement-force transfer matrix G, and the displacement-displacement transfer matrix, H. Similarity matrices are introduced to relate G and H, together with their respective metrics. Symplectic orthogonality relationships for the eigenvectors of both G and H are derived, together with relationships between their respective sets of eigenvectors. New expressions for the group velocity are derived. For repetitive structures of finite length, natural frequency equations are derived employing both G and H, including phase-closure and the direct application of boundary (end) conditions. Besides an exposition of the theory, some familiar but much new, the focus of the present paper is on the relationships between the two forms of transfer matrix, including their respective (dis)advantages. Numerical results, together with further theory necessary for interpretation, are presented in companion papers.