Abstract: When a rod’s shape and mass vary with certain functions such as polynomial, exponential and trig functions, the closed form solutions for the elastic impact problems are presented.(This article presents the closed form solutions for the elastic impact problems when a rod’s shape and mass vary. with such functions as polynomial, exponential and trig functions.) Using direct mode superposition method (DMSM), the impact problems of non-uniform rods and structures with lumped masses and spring supports are discussed. (It also discusses the impact problems of non-uniform rods and structures with lumped masses and spring supports using direct mode superposition method (DMSM). The impacting mass and the impacted object are considered as one integrated vibration system;thus the impact problem is transformed into a vibration problem. The longitudinal displacement of every step of the rod is supposed according to the closedform solution of a free vibration problem, and then the frequency equation and generalized mass of the impacted system are obtained, and the dynamic response is also yielded in the form of Duhamel integration. Compared with the conventional finite element method, the proposed method can reduce the total number of elements and CPU running time. The numerical examples show the advantages of the present method.

Key words: multi-step rod, rod with lumped masses and spring supports, impact, dynamic response, direct mode superposition method