Dynamics of a Bouncing Dimer

We investigate the dynamics of a dimer bouncing on a vertically oscillated plate. The dimer, composed of two spheres rigidly connected by a light rod, exhibits several modes depending on initial and driving conditions. The first excited mode has a novel horizontal drift in which one end of the dimer stays on the plate during most of the cycle, while the other end bounces in phase with the plate. The speed and direction of the drift depend on the aspect ratio of the dimer. We employ event-driven simulations based on a detailed treatment of frictional interactions between the dimer and the plate in order to elucidate the nature of the transport mechanism in the drift mode.

Figure 1

(a) Photo of a dimer bouncing on the vibrated plate. (b) A sketch of the system.

Figure 2

The vertical position of the left (○) and right (▵) tips [8] of the dimer as a function of time for the lowest energy modes observed in the experiment. (a) The flutter (F) mode. (b) The jump (J) mode. (c) The drift (D) mode. The solid curve corresponds to the vertical position of the plate ($A_r=3.9$, $f=50\mathrm{Hz}$, $\Gamma =0.9$.)

Figure 3

(a) Horizontal position of the left tip of the dimer as a function of time in the drift mode for $A_r=3.9$ (solid line) and 5.7 (dashed line). (b) The corresponding horizontal tip velocity. The thick solid line indicates the vertical position of the plate ($f=25\mathrm{Hz}$, $\Gamma =0.9$).

Figure 4

(a) The mean horizontal speed $⟨u⟩$ of the dimer scaled by gravity and frequency versus $A_r$ for three values of driving frequency. Positive direction points towards the bouncing end of the dimer. Symbols correspond to experiments, and the line to the event-driven simulations at $f=25\mathrm{Hz}$ (lines corresponding to $f=50,75$ are virtually indistinguishable from this line and are not shown). Parameters of simulations are $\Gamma =0.9$, $ϵ=0.65$, $\mu =0.2$, ${\mu }_s=0.4$, and $\widehat{\mu }=0.1$, ${\widehat{\mu }}_s=0.2$. (b) Phase diagram of the direction of dimer translation in the D mode as a function of plate inclination angle and $A_r$. Robust uphill motion is observed for a broad range of parameters.

Figure 5

Vertical coordinates of left and right tips of the dimer (a) and the horizontal velocity of the left contact point for the short dimer $A_r=3.9$ (b) and long dimer $A_r=6.1$ (c). Different symbols and line styles denote different modes of motion and collisions between the dimer and the vibrating plate: single slide (×), single slip reversal (□), double collision (▵), rolling (thick solid line), and flight (thin solid line). Parameters of simulations are the same as in Fig. 4.