Analytical and closed-form frequency responses of the longitudinal vibration of N-stepped rods as well as the coupled system of multi-stepped rods and lumped elements with non-classical boundary conditions are proposed in this paper. The closed-form frequency equation for these systems is also obtained. In this article, a new two-way state-flow model based on the dynamics of the rod is developed. In addition to the state-flow model developed for lumped elements, concentrated mass, and spring and damper, the exact frequency response of both stepped rod and the coupled system can be directly written out according to the graph theory. Since the results are analytical and closed-form, they can be directly implemented for analytical and numerical analysis without resorting to complex operations, such as inversion, recursive calculation or iteration computation. Moreover, the derived equations are feasible not only for the multi-stepped and multi-stage rods but also for the coupled system of multi-stepped rods and lumped elements. Finally, the results were checked using multi-stepped rods with classical and non-classical boundary conditions as well as a combination of stepped rods and lumped elements to demonstrate the efficiency of this method.
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