Mechanical excitation of rodlike particles by a vibrating plate

The experimental realization and investigation of granular gases usually require an initial or permanent excitation of ensembles of particles, either mechanically or electromagnetically. One typical method is the energy supply by a vibrating plate or container wall. We study the efficiency of such an excitation of cylindrical particles by a sinusoidally oscillating wall and characterize the distribution of kinetic energies of excited particles over their degrees of freedom. The influences of excitation frequency and amplitude are analyzed.

Figure 1

The figure sketches the setup with the vibrating plate, a mirror, and two cameras with observation axes along $x$ and $y$, respectively, and it defines the coordinate system used. Two vertical white-coated aluminium side walls and two ITO-coated acrylic glass side walls prevented the particle from leaving the plate. These walls are not shown.

Figure 2

Exemplary positions and rotational velocities of a 15 mm jumping rod at 30 Hz excitation. The sign of the rotations is not considered in this plot. Rotation rates are constant within each jump, except in the seventh full jump in this image, where the particle collided with the side wall, causing a drop of the rotation rate. In case of such very rare events, the respective jumps were excluded from data evaluation.

Figure 3

Typical distribution of energies $E_z$ of a jumping 15 mm rod excited at frequency $f=30$ Hz, amplitude $A=2.89$ mm ($v_{max}=0.545$ m/s). The average energy is 15 $\mu \mathrm{J}$. The high-energy tail has an exponential decay [44].

Figure 4

Correlations of energies $E_z$ (a) and $E_{\mathrm{rot}}$ (b) between successive jumps. The vertical velocities show correlations while the rotations are in good approximation uncorrelated.

Figure 5

Selected frequency dependencies at fixed amplitudes $A=2$ mm and $A=2.5$ mm (a) and amplitude dependencies at fixed frequency $f=30$ Hz (b) for the two types of rods. The frequency characteristics can be assumed nearly linear, whereas the amplitude dependence is clearly nonlinear. The ratio of total energies $E$ of long (solid symbols) and short (open symbols) rods is slightly smaller than their mass ratio, i.e., the efficiency of excitation is slightly better for the short pieces. Dashes mark ranges that are not accessible by experiment (see text).

Figure 6

Total energies of the 15 mm rods for different excitation parameters (a) as a function of $A^{3/2}f$ and (b) as a function of the maximum velocities of the moving plate, $v_{max}$. Within experimental scattering, all data can be fitted with common master graphs (see text); only the 15 Hz curve in (a) shows some systematic deviation to less efficient driving, which is mainly a technical problem, see text. The dotted line in (b) marks the kinetic energy of a rod lying on the vibrating plate.

Figure 7

Ratio of the three components and the total mean energy. The “vertical energy” $E_z$ as the sum of kinetic energy of vertical motion and the potential energy relative to the mean plate height is shown in (a); it amounts to about 90% of the total energy. Graph (b) shows the share of rotational energy, which is of the order of 10%, irrespective of the excitation parameters. The kinetic energy of translational motions in the horizontal plane (c) is smaller by more than one order of magnitude, and a systematic decay of its share with larger excitation strengths is evident. Solid symbols: 15 mm rods; open symbols: 7.5 mm rods.

Figure 8

Double impact, image sequence with 240 fps: the rod first touches the plate with the lower end, momentum (upward) and angular momentum (backward) are transferred. The center of mass still moves towards the plate. Then the other end touches the plate, and again momentum upward is transferred, but the angular momentum is opposite to the first impact. Green circles mark the rod ends, dashed circles mark the previous positions. The apparent slight bend of the rod in the last two images and motion blurring is a camera artifact.