On a Derivation of the Transmission Matrices for the Longitudinal, Torsional, and Bending Vibrations of Families of Tapered Rods

The transmission-matrix representation of structural elements finds useful application in the description of multielement systems. Many practical systems use tapered elements for which at present transmission matrices are not readily available. The present paper discusses a particularly useful approach to the derivation of transmission matrices for tapered rods. The transmission-matrix formulation is demonstrated to represent a particularly important fundamental set of the general solution to the governing differential equation. When viewed in this way, the coefficients of the matrix are shown to possess certain properties with respect to each other and the eigenvalue problem. Transmission matrices are derived from known general solutions for certain functional forms of taper and presented in tabular form.