Model and Simulations of Contact Between a Vibrating Beam and an Obstacle Using the Damped Normal Compliance Condition

This work constructs a new mathematical model for the vibrations of a Bernoulli beam that can come in contact with a reactive obstacle situated below its right end. The obstacle reaction is described by the Damped Normal Compliance (DNC) contact condition. This condition, unlike the usual Normal Compliance (NC) contact condition, takes into account the energy dissipation during the contact process. The steady states of the model are described and the model is studied computationally for different values of obstacle stiffness and damping. The computational scheme is shown numerically to converge with a rate higher than 1. The numerical simulations illustrate how the beam’s end penetration and vibrations differ in soft vs. stiff obstacle environments, and how damping modifies the dynamic behavior. The results may be useful for vibration control and material interaction in settings when collisions or repetitive contacts occur. By providing computational and analytical insights, the study is an addition to the currently maturing Mathematical Theory of Contact Mechanics (MTCM).