Can we obtain the coefficient of restitution from the sound of a bouncing ball?

The coefficient of restitution may be determined from the sound signal emitted by a sphere bouncing repeatedly off the ground. Although there is a large number of publications exploiting this method, so far, there is no quantitative discussion of the error related to this type of measurement. Analyzing the main error sources, we find that even tiny deviations of the shape from the perfect sphere may lead to substantial errors that dominate the overall error of the measurement. Therefore, we come to the conclusion that the well-established method to measure the coefficient of restitution through the emitted sound is applicable only for the case of nearly perfect spheres. For larger falling height, air drag may lead to considerable error, too.

Figure 1

Schematic diagram of the experimental setup.

Figure 2

Coefficient of restitution as a function of the impact velocity. Each of the approximately $280000$ data points corresponds to an impact. The lines shows the median (full line) and the 90 % quantile (dashed lines).

Figure 3

Sound signal measured by the piezoelectric sensor in different resolution. Gray shaded areas in panels (a)–(c) indicate the time interval displayed in the subsequent panel of higher resolution, panels (b)–(d). (a) Full signal corresponding to one drop and approximately 100 subsequent bounces; (b) sound due to a single bounce at $t\approx 8.498\text{s}$. The next impact occurs at $t\approx 8.518\text{s}$; (c) sound at the beginning of an impact; (d) high resolution where discrete sample points are shown with symbols. Dashed lines show the thresholds, $V_{\text{thr}}$, used for the impact recognition. In this case, the impact was recognized about two sample intervals after the physical contact, corresponding to a delay of $4\mu \text{s}$.

Figure 4

Two typical images obtained by long-time exposure. When the sphere crosses the LASER sheet, the sphere's reflection indicates the spacial location of its impacts on the glass plate. A star indicates the position of the first impact. Reflections of the spots from the glass plate were removed by digital post-processing.

Figure 5

Relative error of the coefficient of restitution due to air drag [Eq. (8)], as a function of impact velocity. Full line, with the assumption of Stokes drag, Eq. (10a); dashed line, with the assumption of Newton drag, Eq. (10b).

Figure 6

SEM pictures of the sphere. (a) before the experiment (0 impacts) and (b) after ${10}^5$ impacts. Only very small scratches at the surface are noticeable. Panels (c)–(e) show magnifications of the surface, (c) before the experiment (0 impacts), (d) after about ${10}^5$ impacts, (e) toward the end of the experiment after about ${10}^6$ impacts. Very small scratches are noticeable. The region shown in each image covers about $1.8$% of the contact area for a collision starting at height $0.1\text{m}$.

Figure 7

(a) Replot of Fig. 2 [only median of $\varepsilon \left(v\right)]$, where we analyze the data from the first and last ${10}^4$ impacts separately. (b) Histogram of the measured data for a particular velocity, $v=(0.7\pm 0.05)\mathrm{m}/\mathrm{s}$. In both figures (a, b), no systematic trend is apparent.

Figure 8

Numerical model of a rough sphere [40], where a large number of tiny spheres is attached to random positions at the surface of a large central (perfect) sphere. For visibility, the number of asperities is much smaller than in the simulation. For the same reason, the size of the asperities is not to scale.

Figure 9

Coefficient of restitution for a rough particle plotted against the impact velocity. The data are colored according to the normalized frequency of occurrence.

Figure 10

Standard deviation of the data shown in Fig. 9 computed due to Eqs. (18), as a function of impact velocity.

Figure 11

Sketch of a dumbbell particle consisting of two identical spheres of radius $R$ and distance $L\ll R$. For better visibility, the value of $L$ appears exaggerated.

Figure 12

The coefficient of restitution as a function of velocity for eccentricity $L/R=0.1$. The data are colored according to the normalized frequency of occurrence.

Figure 13

Standard deviation of the numerically determined coefficient of restitution for eccentricity $L/R=0.01$ and $L/R=0.02$, as a function of impact velocity.

Figure 14

Top: Contributions to the total error characterized by the standard deviations as a function of impact velocity. Bottom: Same data but in logarithmic scale. For small impact velocity, imperfections of the shape of the ball dominate the total error. For larger impact velocity, $v\gtrsim 1.2\mathrm{m}/\mathrm{s}$, corresponding to dropping height $h\gtrsim 7\mathrm{cm}$, the error due to air drag becomes considerable (see second horizontal scale). The given numbers are typical but specific for our set of parameters; see Sec. 2.