Nonlinear phenomena in vibro-impact dynamics: central collisions and energy jumps between two rolling bodies

Abstract

Central collisions and energy jumps between two rolling bodies as singular nonlinear phenomena are investigated in vibro-impact system dynamics. For that purpose, an extension of theory of collision of two rigid bodies, in rolling dynamics, is presented. Under the authors’ use of the Petroviċ’s elements of mathematical phenomenology, especially mathematical analogy between kinetic parameters of central collision of two bodies in translator motions and central collision of two rolling different axial symmetrically bodies, new original expressions of two post-collision outgoing angular velocities for each of rolling bodies after collision are defined. Presented results are the generalization of author’s previous published results. Using this new and original result for collisions, the vibro-impact dynamics of two rolling heavy thin disks with different radiuses on the rotating circle trace in vertical plane in the period of series of collisions is investigated. Using the series of the elliptic integrals, new nonlinear equations for obtaining the angles, which defined the disks positions of successive collisions, are defined. Phase trajectories of the disks in vibro-impact dynamics are analytically and graphically presented. Two cases of vibro-impact dynamics when phase portraits contain trigger of coupled singularities and homoclinic phase trajectory in the form of number “eight,” as well as in the case without that trigger of coupled singularities, are discussed. Phase trajectory branches of dynamics of both of the rolling disks in the period from initial positions to first collision between them are presented. By using the pre-collisions impact (arrival) angular velocities and post-collision (outgoing) angular velocities of the rolling bodies, the rates of the total mechanical energy change for each disk in comparison under kinetic state that corresponds to pre- and post-collision are expressed, and also of both of the rolling bodies as system pre-collision and post-collision kinetic states. From the energy analysis, some conclusions are formulated. Full theory and new methodology for the investigation of the vibro-impact system dynamics with rolling bodies is found.