Modified inelastic bouncing ball model for vertically vibrated granular materials: Transmissibility and its dependence on particle diameter
The modified Inelastic Bouncing Ball Model (mIBBM) has been used to describe the dynamics of a vibrated granular material (GM) by introducing transmissibility Tr to the Inelastic Bouncing Ball Model (IBBM). The Tr is a parameter that compactly describes the properties of granular material and accounts for the container-to-grains force transmission efficiency. Here, we relate Tr behavior to the diameter d of the individual grains. We simulate vertically vibrated monodisperse GM using the discrete element method (DEM) implemented using LIGGGHTS. Then, we measure Tr in three different ways: using the two key features (bifurcation point Γb and critical acceleration Γc) of the bifurcation diagram of the mIBBM and the phase delay ϕ between the acceleration of the GM and that of its container. The bifurcation diagram plots the flight times 𝒯 of the center-of-mass of the GM versus the dimensionless acceleration Γ of the container. The phase delay method, which determines the input-output force ratio while accounting for the input-output delay, effectively serves as the experimental reference. Tr slowly decreases with d when measured using Γc while rapidly increases with d when measured via Γb. The different Tr measurements obtained from both features show the GM's nonlinear behavior, which is consistent with the highly non-monotonic dependence of Tr on Γ via the phase delay method. The result suggests that GMs can be described by more than one bifurcation diagram: GMs driven by weak (strong) vibration are described by the Tr value obtained via Γc (via Γb) method.