Interplay Between Dissipation and Modal Truncation in Ball-Beam Impact

Abstract

We study a ball-beam impact in detail; and in particular, we study the interplay between dissipation and modal truncation. With Hertzian contact between a solid ball and an Euler–Bernoulli beam model, we find using detailed numerical simulations that many (well above 60) modes are needed before convergence occurs; that contact dissipation (either viscous or hysteretic) has only a slight effect; and that contact location plays a significant role. However, and more interestingly, we find that as little as 2% modal damping speeds up convergence of the net interaction so that only about 25 modes are needed. We offer a qualitative explanation for this effect in terms of the many subimpacts that occur in the overall single macroscopic impact. In particular, we find that in cases where the overall interaction time is long enough to damp out high modes yet short enough to leave lower modes undissipated, modal truncation at about 25 modes gives good results. In contrast, if modal damping is absent so that higher mode vibrations persist throughout the interaction, final outcomes are less regular and many more modes are needed. The regime of impact interactions studied here occurs for reasonable parameter ranges, e.g., for a 3–4 cm steel ball dropped at speeds of 0.1–1.0 m/s on a meter-long steel beam of net mass 1 kg. We are unaware of any prior similarly detailed numerical study which clearly offers the one summarizing idea that we obtain here.