Dynamics of a Grain-Filled Ball on a Vibrating Plate

We study experimentally how the bouncing dynamics of a hollow ball on a vibrating plate is modified when it is partially filled with liquid or grains. Whereas empty and liquid-filled balls display a dominant chaotic dynamics, a ball with grains exhibits a rich variety of stationary states, determined by the grain size and filling volume. In the collisional regime, i.e., when the energy injected to the system is mainly dissipated by interparticle collisions, an unexpected period-1 orbit appears independently of the vibration conditions, over a wide range. This is a self-regulated state driven by the formation and collapse of a granular gas within the ball during one cycle. In the frictional regime (dissipation dominated by friction), the grains move collectively and generate different patterns and steady modes: oscillons, waves, period doubling, etc. From a phase diagram and a geometrical analysis, we deduce that these modes are the result of a coupling (synchronization) between the vibrating plate frequency and the trajectory followed by the particles inside the cavity.

Figure 1

A sphere partially filled with grains is placed on a vibrating plate. The bouncing time is measured by electrical contact and registered as a function of $\mathrm{\Gamma }$. A high-speed video is taken to visualize the particle dynamics inside the sphere.

Figure 2

BDs of a spherical container: (a) empty, (b) with 7.5 g of oil (300 cSt), and (c) with different masses $m$ of $2\pm 0.2$ mm glass beads ($V_f$ is the ratio of the volume occupied by the grains to the cavity volume). The BDs are reproducible if they are obtained by increasing or reducing $\mathrm{\Gamma }$; a slight hysteresis can appear, but the main features remain unaltered. [(d),(e)] Snapshots taken each 0.01 s and space-time pictures illustrating periodic orbits of a sphere with 7.5 g of (d) liquid, and (e) grains. Note: In (d) we used oil to visualize the jet formation. The dynamics was considerably more chaotic with less viscous fluids (in particular, with water).

Figure 3

Stationary bouncing dynamics of a grain-filled sphere for $V_f=20\%$ ($P_1^1$ orbits): (a) Sphere position $z$ normalized by $A$, as a function of $\omega t$ for $\mathrm{\Gamma }=2.6$ (black), 3.8 (red), 4.9 (green), 7.0 (blue), and 8.0 (orange). The dashed line corresponds to the sinusoidal motion of the plate. (b) Impact phase $\phi$ as a function of $\mathrm{\Gamma }$. (c) Snapshots showing the grains-sphere interaction during a typical impact in the $P_1^1$ state (see text). (d) $h_s$ and (e) $v_s$ vs $t/T$, see text.

Figure 4

BDs displayed by a sphere partially filled with glass particles of different diameters: (a) $4\pm 0.05\mathrm{mm}$, (b) $1.0\pm 0.2\mathrm{mm}$, and (c) $\sim 300\mu \mathrm{m}$. In all cases, $m=7.5\mathrm{g}$. For the smaller particles, a rich variety of vibrating modes was observed: (d) stable oscillons, (e) asymmetric oscillons, (f) doubling period orbits, and (g) waves (the corresponding parametric regions are indicated in (c). (h) Phase diagram summarizing the observed orbits: static material (SM), surface perturbations (SP), oscillons (O), waves (W), static double period (SDP), and sloshing (S). Oscillons only appear in the interval $12\mathrm{Hz}\lesssim f\lesssim 16\mathrm{Hz}$. (i) Particles trajectories for oscillons and waves. (j) Similar BDs for different grain size indicate similar dissipation capacity.