Dynamics of a Grain-Filled Ball on a Vibrating Plate

Abstract

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. A bouncing ball (BB) on a vibrated plate is a fundamental system used as a model in different physical and engineering problems [1]; for instance, it has been used to describe cosmic-ray particles in astrophysics [2], the dynamic stability in human performance [3], and the cantilever motion in atomic force microscopy [4]. The BB dynamics displays a period doubling route to chaos also found in biological, hydrodynamic, optical, and chemical systems [5]. More complex objects such as dimers [6], trimers [7], bouncing droplets [8], and quantum bouncers [9] also show nonlinear behaviors: bifurcations, rotations, etc. Dimers and trimers exhibit self-propulsion, providing models to study collective motions, such as fish schools [10], flocks [11], and bacteria colonies [12]. Clearly, the study of bouncing objects has been relevant to the description of a large variety of nonlinear behaviors.This Letter explores the bouncing dynamics of a hollow sphere partially filled with grains. We aim to understand how the classical BB dynamics (widely studied in the literature [1,5,13,14]) is affected by the fast energy dissipation typical of granular materials. We compared the dynamics of an empty, a liquid-filled, and a grain-filled sphere. The former two cases exhibit chaotic behaviors while the sphere with grains bounces primarily in steady states. Of particular interest was the observation of a period-1 orbit independent of the vibration conditions in a broad range. This state was observed at low filling ratios V f (< 12%) and using large beads (diameter d ¼ 2 mm). When tiny particles were used (300 μm), different patterns appeared depending on the vibration frequency f, for instance, stable oscillons and surface waves similar to those observed in vibrated granular beds [15][16][17][18][19]. The above grain size dependence reveals, as in the case of granular dampers [20][21][22][23][24][25], the relevance of the main dissipation mechanism: interparticle collisio...

Transition from normal to ballistic diffusion in a one-dimensional impact system

“…For the nondissipative version, the system basically behaves like the standard map in a local approximation [2,9], where some of the previous findings concerning the ballistic transport and accelerator modes (AMs) in the standard map serve as the motivation background for this paper [29][30][31][32][33][34][35]. Yet, despite the simple dynamics, interesting applications for this system can be found in dynamic stability in human performance [36], vibration waves in a nanometric-sized mechanical contact system [37], granular materials [38], experimental devices concerning normal coefficient of restitution [39], mechanical vibrations [40,41], anomalous transport and diffusion [42], thermodynamics [43], crisis between chaotic attractors [44], and chaos control [45], among others [46,47].…”

Collision-enhanced friction of a bouncing ball on a rough vibrating surface

“…Whilst valuable, in a quasi-2d experiment the bounce of a ball is linked to the preceding bounce(s) and collision(s), which determine the distribution of incident linear and rotational velocities. This is further complicated by the relative phase of the vertically oscillating surface at which bounces on the surface take place 23,24 .…”

Self-propulsion of a grain-filled dimer in a vertically vibrated channel

“…Moreover, the dimer is able to respond to the plate’s vibration with various distinctive modes of self-propelled motion. This design is inspired by the earlier studies of a grain-filled bouncing ball on a vibrating or a static plate 19 – 21 , in which the coefficient of restitution and its resulting bouncing dynamics demonstrate rich dependence on the mass and size of grains inside the bouncing ball. In our work, we introduce an asymmetry to the dimer by differentiating the size of grains in the two balls, while keeping the total filling mass of grains as well as other mechanical properties in each ball identical.…”