Collision time in the inelastic bouncing ball model of granular materials

We modified the inelastic bouncing ball model (IBBM) to account for the role of collision time \({\tau }_{c}\) ${\tau }_c$ in defining the dynamics of vertically vibrated confined granular systems. Although \({\tau }_{c}\) ${\tau }_c$ was surmised to be consequential for dissipative systems, previous studies on the accuracy of IBBM did not formally incorporate \({\tau }_{c}\) ${\tau }_c$ as a dynamical variable of the model, focusing instead on other factors during flight, such as air friction. We utilized the discrete element method (DEM) to study the role of \({\tau }_{c}\) ${\tau }_c$ in the granular dynamics, and to cross-validate the efficacy of our reformulation of IBBM to account for the effect of collisions. When the \({\tau }_{c}\) ${\tau }_c$ value is greater than that of \({t}_{0}\) $t_0$ , which is the first instance that the container acceleration exceeds the gravitational acceleration \(g\) $g$ , the time-of-flight decreases, and the location of the bifurcation point shifts in the bifurcation diagram (time-of-flight versus dimensionless acceleration). We model \({\tau }_{c}\) ${\tau }_c$ as representing the range of uncertainty in the occurrence of \({t}_{0}\) $t_0$ . Assuming a separation of timescale between the dynamics of the collision between the center-of-mass (CM) of the granular system and the container, and the time-of-flight of the CM itself, we propose a supporting but separate model for the dependence of \({\tau }_{c}\) ${\tau }_c$ on Γ. The time-of-flight duration is determined when \({\tau }_{c}\) ${\tau }_c$ is known in the modified IBBM that now produces bifurcation diagrams which are in closer agreement with the DEM simulation results.