الفهرس 
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Abstract
A finite element formulation is developed in this study to analyze impact-induced waves in a rod collide with a striking mass. The mathematical formulation is established based on St. Venant’s contact theory. The contact force between the rigid mass and the rod tip, the displacement and the velocity of the tip and the striking mass are produced. The effects of mass ratios, the geometrical shape of the rod, and boundary conditions, whether fixed, free, or elastic, are investigated. Both uniform rods and non-uniform rods are considered, including conical rods and rods that have a discontinuity due to a change in material or in a cross-sectional area. The proposed formulation is extended to analyze the stress waves generated in a uniform stationary elastic rod attached to a stationary rigid mass at one end and struck longitudinally by another moving mass at the other end. The effect of the stationary rigid mass on the dynamic response of the system is investigated. A shortcoming in traditional finite element models is to consider the rod and the struck mass as separate identities that gives inaccurate results for the discontinuity at the arrival of the impact wave. Instead, the proposed model considers the rod and the struck mass as one system that gives extremely accurate corresponding results. Wave propagation, transmission, and reflection from the constraint end and at the discontinuous interface are thoroughly analyzed. Succession collision, which is a second impact that may occur between the struck mass and the rod, is also investigated. The finite element results are obtained using the Newmark time integration method, and whenever it is possible, they are compared with analytical solutions, which are obtained with either the direct mode superposition method or wave theory. The analysis and the illustrations presented enrich the understanding of the physical phenomena of impact and wave propagation, including the reflection of the waves. The excellent agreement between the finite element results and the analytical solutions demonstrates that the proposed finite element technique is very effective and provides a convenient, accurate, and applicable means to compute the dynamic responses of the elastic rods subjected to longitudinal impacts. In general, the proposed methodology can be used in practical applications in wave analysis and simulation.