Mode bifurcation of a bouncing dumbbell with chirality

We studied the behavior of a dumbbell bouncing upon a sinusoidally vibrating plate. By introducing chiral asymmetry to the geometry of the dumbbell, we observed a cascade of bifurcations with an increase in the vibration amplitude: spinning, orbital, and rolling. In contrast, for an achiral dumbbell, bifurcation is generated by a change from random motion to vectorial inchworm motion. A simple model particle was considered in a numerical simulation that reproduced the essential aspects of the experimental observation. The mode bifurcation from directional motion to random motion is interpreted analytically by a simple mechanical discussion.

Figure 1

(a) Experimental setup. Left: side view of a chiral asymmetric dumbbell. Right: overall setup. $\alpha$ and $\beta$ are chirality parameters. (b–d) Examples of the characteristic spontaneous motion of a chiral asymmetric dumbbell. The vibration frequency of the plate was fixed at 50 Hz. The scale bar is 10 mm. (b, c) Left panels: Series of snapshots with a time interval of 1 s. Right panels: Trajectories of the center of the dumbbell. (b) Spin + random (SR) mode $\left(\Gamma =1.3\right)$: the dumbbell itself spins, and the centroid moves randomly. (c) Orbital (O) mode $\left(\Gamma =1.4\right)$: orbital motion is accompanied by spinning; the direction of the orbital motion is determined stochastically. (d) Rolling mode $\left(\Gamma =2.5\right)$: series of snapshots with a time interval of 0.3 s. The dumbbell rolls along the axis of the rod with rhythmic switching of the rolling direction.

Figure 2

Angular velocity of spin motion $\Omega$ in the SR mode $\left(\Gamma =1.3\right)$ as a function of $\beta$ for different rod lengths $l$: $12.5$ (squares), $17.5$ (circles), $22.5$ (triangles), and $27.5$ (inverted triangles) cm. Inset: $\Omega$ normalized to the maximum angular velocity ${\Omega }_{\text{max}}$ of the spin motion as a function of $\beta$.

Figure 3

(a) Orbital radius. An increase (decrease) in $\Gamma$ is shown by filled (open) triangles. (b) Kinetic energy of the dumbbell as a function of $\Gamma$. (c) and (d) Phase diagrams of the dumbbell motion: SR mode $\left(red\square \right)$, O mode $\left(green◯\right)$, and Rolling mode $\left(blue▵\right)$. (c): $\Gamma$ vs. the length of the rod $l$, and (d): $\Gamma$ vs. the twist angle $\beta$.

Figure 4

Experimental results for the vertical trajectory of the tips of the dumbbell as a function of the vibration phase. The height of the plate is shown as $z_p\left(\theta \right)$ by the solid blue line. The heights of the lowest points of the disks are shown as $z_1\left(\theta \right)$ and $z_2\left(\theta \right)$ by circles (green) and squares (red), respectively. (a) SR mode and (b) O mode.

Figure 5

Schematic illustration of the chiral symmetric model particle. The two hemispheres have different mass densities: ${\rho }_1$ (blue) and ${\rho }_2$ (green), where ${\rho }_1>{\rho }_2$.

Figure 6

Horizontal trajectories of the tips of the dumbbell at $f=40$ Hz, $l=8$ cm, and ${\beta }^{\prime }=0.01\pi$. (a) SR mode at $\Gamma =1.68$ for $2.7\times {10}^4$ s (inset: trajectory of the center of the dumbbell) and (b) O mode at $\Gamma =1.74$ for $2\times {10}^3$ s.

Figure 7

Vertical trajectories of the tips of the dumbbell as a function of the vibration phase at $f=40$ Hz, $l=8$ cm, and ${\beta }^{\prime }=0.01\pi$. (a) SR mode at $\Gamma =1.68$ and (b) O mode at $\Gamma =1.74$.

Figure 8

Dependence of translational velocity of a symmetric dumbbell on the stiffness $k_n$. The viscous coefficient is chosen so that the restitution coefficient is identical. Inset: the relation between translational velocity $v$ and contact duration $t_c$ for different stiffnesses.

Figure 9

Angular velocity of the spin motion $\Omega$ as a function of ${\beta }^{\prime }$ for different rod lengths $l$: $3.5$ (squares), 4 (circles), 5 (triangles), 6 (inverted triangles), 8 (diamonds) cm. Inset: scaled by the maximum value.

Figure 10

Dependence of the critical acceleration ${\Gamma }_c$ on the restitution coefficient $e^{\prime }$. Lower limit of steady collision in the theoretical analysis (dotted line), experiments (green square), and numerical simulation with ${\beta }^{\prime }=0.01\pi$ (blue triangles), ${\beta }^{\prime }=0.75\pi$ (red circles), and ${\rho }_1={\rho }_2$ (cyan diamonds). Experimental data are given for rubber $\left(e^{\prime }=0.223\pm 0.024\right)$, aluminum $\left(e^{\prime }=0.305\pm 0.008\right)$, and glass plates $\left(e^{\prime }=0.472\pm 0.026\right)$.

Figure 11

Schematic drawing of bouncing dimer model.