Statistics of injected power on a bouncing ball subjected to a randomly vibrating piston

We present an experimental study on the statistical properties of the injected power needed to maintain an inelastic ball bouncing constantly on a randomly accelerating piston in the presence of gravity. We compute the injected power at each collision of the ball with the moving piston by measuring the velocity of the piston and the force exerted on the piston by the ball. The probability density function of the injected power has its most probable value close to zero and displays two asymmetric exponential tails, depending on the restitution coefficient, the piston acceleration, and its frequency content. This distribution can be deduced from a simple model assuming quasi-Gaussian statistics for the force and velocity of the piston.

Figure 1

Experimental setup. Inset: Typical temporal trace of the force signal for a steel ball (ball mass $m=8\text{g}$, restitution coefficient $r=0.923$) at a given control acceleration $\mathrm{\Gamma }=1.09$.

Figure 2

(a) Temporal traces for the force (black) and velocity (red) traces between two impacts, defining $d_n$ for $\mathrm{\Gamma }=1.09$. (b) Temporal traces for the force (black) and velocity (red) traces, around an impact for $\mathrm{\Gamma }=1.1$. The shaded region defines ${\tau }_n$. Arbitrary units are used to place both curves in the same scale. (c) Dependence of ${\tau }_n$ as a function of $\mathrm{\Gamma }$. Error bars stand for the standard deviation for ${\tau }_n$. The bead is made of steel.

Figure 3

Typical trace of force impacts of a steel ball on a piston on repose. Symbols ($\times$) stand for peak detections. Inset: Semilog plot of time lag between impacts $t_n$ vs $n$ for a steel ball. Best fitting slope is ln($-0.93$).

Figure 4

(a) PDF for $F_n$ for $\mathrm{\Gamma }=0.50$, 0.60, 0.68, 0.80, 0.87, 1.0, 1.09, and 1.28 for steel ($r_c=0.770$) and $f_l=0$ Hz. Inset: PDF of the normalized variable $F_n/\sigma \left(F_n\right)$ for $\mathrm{\Gamma }=0.50$, 0.60, 0.68, 0.80, 0.87, 1.00, and 1.09 for steel ($r_c=0.770$) and $f_l=0$ Hz. (b) PDF of $v_n$ for $\mathrm{\Gamma }=0.50$, 0.60, 0.68, 0.80, 0.87, 1.0, and 1.09 for steel ($r_c=0.770$) and $f_l=0$ Hz. Inset: PDF of the normalized variable $v_n/\sigma \left(v_n\right)$ for $\mathrm{\Gamma }=0.50$, 0.60, 0.68, 0.80, 0.87, 1.0, 1.09, and 1.28 for steel ($r_c=0.770$) and $f_l=0$ Hz. Dashed line is a normalized Gaussian. Arrows indicate how $\mathrm{\Gamma }$ increases.

Figure 5

(a) PDF for $F_n$ for steel ($\circ$), brass ($\times$), bronze ($\square$), nylon ($\diamond$), and polyurethane ($+$) for $\mathrm{\Gamma }=1.00$ and $f_l=0$ Hz. Inset: PDF for the normalized variable $F_n/\sigma \left(F_n\right)$ for steel ($\circ$), brass ($\times$), bronze ($\square$), nylon ($\diamond$), and polyurethane ($+$) for $\mathrm{\Gamma }=1.00$ and $f_l=0$ Hz. (b) PDF for $v_n$ for steel ($\circ$), brass ($\times$), bronze ($\square$), nylon ($\diamond$), and polyurethane ($+$) for $\mathrm{\Gamma }=1.00$ and $f_l=0$ Hz. Inset: PDF for the normalized variable $v_n/\sigma \left(v_n\right)$ for steel ($\circ$), brass ($\times$), bronze ($\square$), nylon ($\diamond$), and polyurethane ($+$) for $\mathrm{\Gamma }=1.00$ and $f_l=0\mathrm{Hz}$.

Figure 6

(a) PDF for $F_n$ for $f_l=0(\circ )$, 50 ($\square$), 100 ($\diamond$), and 150 ($*$) Hz for $\mathrm{\Gamma }=1.00$, and $r_c=0.742$ (brass). Inset: PDF for the normalized variable $F_n/\sigma \left(F_n\right)$ for $f_l=0(\circ )$, 50 ($\square$), 100 ($\diamond$), and 150 ($*$) Hz for $\mathrm{\Gamma }=1.00$ and $r_c=0.742$ (brass). (b) PDF for $v_n$ for $f_l=0(\circ )$, 50 ($\square$), 100 ($\diamond$), and 150 ($*$) Hz for $\mathrm{\Gamma }=1.00$ and $r_c=0.742$ (brass). Inset: PDF for the normalized variable $v_n/\sigma \left(v_n\right)$ for $f_l=0(\circ )$, 50 ($\square$), 100 ($\diamond$), and 150 ($*$) Hz for $\mathrm{\Gamma }=1.00$ and $r_c=0.742$ (brass).

Figure 7

(a) PDF for $I_n$ for $\mathrm{\Gamma }=0.50$, 0.60, 0.68, 0.80, 0.87, 1.00, and 1.09 for steel ($r_c=0.770$) and $f_l=0$ Hz. Inset: PDF of the normalized variable $I_n/\sigma \left(I_n\right)$ for $\mathrm{\Gamma }=0.50$, 0.60, 0.68, 0.80, 0.87, 1.00, and 1.09 for steel ($r_c=0.770$) and $f_l=0$ Hz. Dashed lines are exponential fits for both positive and negative tails. (b) PDF for $I_n$ for steel ($\circ$), brass ($\times$), bronze ($\square$), nylon ($\diamond$), and polyurethane ($+$) for $\mathrm{\Gamma }=1.00$ and $f_l=0$ Hz. Inset: PDF for the normalized variable $I_n/\sigma \left(I_n\right)$ for steel ($\circ$), brass ($\times$), bronze ($\square$), nylon ($\diamond$), and polyurethane ($+$) for $\mathrm{\Gamma }=1.00$ and $f_l=0$ Hz. Dashed lines are exponential fits for both positive and negative tails. (c) PDF for $I_n$ for $f_l=0(\circ )$, 50 ($\square$), 100 ($\diamond$), and 150 ($*$) Hz for $\mathrm{\Gamma }=1.00$ and $r_c=0.742$ (brass). Inset: PDF for the normalized variable $I_n/\sigma \left(I_n\right)$ for $f_l=0(\circ )$, 50 ($\square$), 100 ($\diamond$), and 150 ($*$) Hz for $\mathrm{\Gamma }=1.00$ and $r_c=0.742$ (brass). Dashed lines are exponential fits for both positive and negative tails.

Figure 8

PDF of the reduced variable $X/(1-{\langle X\rangle }^2)$ with $X=I_n\sqrt{1-2/\pi }/\left[\sigma \left(F_n\right)\sigma \left(v_n\right)\right]$ for steel ($\circ$), brass ($\times$), bronze ($\square$), nylon ($\diamond$), and polyurethane ($+$) for $\mathrm{\Gamma }=1.00$ and $f_l=0$ Hz. Dots show the approximation $P\left(X\right)\sim exp(cX-|X\left|\right)/\sqrt{\left|X\right|}$ for each value of $r_c$.