In this paper, the eigenfunction expansion method combined with local contact models are presented to analyze the repeated impact behaviors between the sphere and the beam. The simulations are verified with the experimental results of a simply-supported beam struck 91 times by a sphere. In order to clarify the validity of the eigenfunction expansion method to solve the repeated transverse impacts on beams, the simulation results of the spring-mass method are also compared with the experiments. It shows that with appropriate contact models, the eigenfunction expansion method can predict agreeable results with the experimental results, especially the impact behaviors (including the accumulated permanent indentation, the coefficient of restitution and the separation velocity of the sphere) representing energy dissipations, due to the consideration of structural vibrations, local impact behaviors, as well as the wave propagations. The comparisons between the eigenfunction expansion method and the spring-mass model show that the wave propagations are important for energy dissipations, which cannot be neglected in flexible structure impacts. The studies are important for the analysis of repeated impact response of beam structures.

Considering the elastic–plastic deformation, the wave propagations and energy transmissions of the one-dimensional three-segment composite granular chain are studied. The axial symmetry model for elastic-perfectly plastic materials is built by using the finite element method. Six materials with different yield strengths are selected for the adjustable segment. The results show that the repeated loading and unloading behaviors, as well as the wave propagations in the elastic–plastic granular chain, are complex and significantly different from those in the purely elastic granular chain. The yield strength of the granular materials in the adjustable segment has significant effects on energy dissipation and wave velocity, which could be used to design the impact buffer. The studies show that taking lower yield strength for the adjustable part than the non-adjustable part, the energy dissipation could be increased, and the wave velocity could be reduced, then the arrival time of the impact waves could be delayed. These characteristics of the elastic–plastic granular chain could be used to design metamaterials for impact absorbers in impact protection.