Repeated Binary Direct Collinear Impacts Under Incremental Contact Laws With Permanent Indentation: A Hybrid Systems Formulation

Abstract

Incremental contact laws specify the normal contact force through a differential equation carrying an internal state, driven by the indentation and its rate. In some, the force is extinguished at a nonzero indentation, whether by plastic deformation or by an elastic aftereffect, so that a residual deformation remains at the separation. Such laws sit uneasily within rigid body dynamics, which admits no deformation. The tension is tolerable when the indentation is small relative to the bodies, so that it may be carried constitutively rather than geometrically. Even then, the contact law alone does not determine the interaction of the bodies. Because force and indentation no longer vanish together, conditions for the commencement and termination of contact must be supplied separately. So must the fate of the deformation and internal state at separation, neither of which the equations of motion contain. This article formulates the repeated direct collinear impact of two convex bodies under external forces as a hybrid dynamical system. The contact interface is modeled as a massless element carrying the contact law and its state, coupled to the bodies through relative velocity and an interaction force dictated by the contact state. Consequently, all switching and resets are confined to the interface model, leaving the geometry and the inherent equations of motion of the bodies unaltered. The principal analytical properties of the resulting formulations are established, among them passivity, completeness, and non-uniqueness of the solutions. The framework is demonstrated through simulations of the complete two-body system incorporating two contact laws based on the Bouc-Wen model of hysteresis.