An optimization approach to establish dynamical equivalence for soft and rigid impact models

This paper studies a computational approach aimed at establishing equivalent dynamical responses within oscillatory impacting systems subject to soft and rigid constraints. The proposed method incorporates an adaptive differential evolution algorithm with the Metropolis criterion to determine the stiffness and damping parameters of the soft constraint for a prescribed coefficient of restitution governing the rigid constraint. The proposed algorithm aims to establish an equivalent dynamical response of the two models based on constraints regarding energy dissipation and contact time duration. Upon examining the dynamical responses of the two impact cases, they exhibit nearly identical outcomes in the two-parameter bifurcation diagrams when subjected to a large restitution coefficient. However, discrepancies arise between the results of the two models when the restitution coefficient is low. Detailed numerical tests, conducted using the proposed method, demonstrate enhanced effectiveness compared to previous techniques, such as the prediction formulas for the different related soft impact model outlined by Okolewski and Blazejczyk-Okolewska [Chaos 31(8), 083110 (2021)]. This method not only finds application in experimentally identifying the physical properties of an impact surface but also provides convenience in employing soft models within impacting systems, which could then avoid potential inaccuracies in handling discontinuities by some integrator during velocity jumps before and after impacts.