Exact transient vibration of stepped bars, shafts and strings carrying lumped masses

A new analytical method is developed for transient vibration analysis of stepped systems composed of distributed components like elastic bars, flexible shafts and taut strings, and lumped masses. The method, with a distributed transfer function formulation and a residue formula for inverse Laplace transform, gives the exact transient response of a stepped system with any number of components and subject to arbitrary external, boundary and initial excitations. The proposed method does not depend on system eigenfunctions, is able to accurately predict jumps in stress and strain distributions, and is numerically efficient as its utility only requires simple operations of two-by-two matrices.