Forces encountered by a sphere during impact into sand

We describe direct measurements of the acceleration of an object impacting on a loosely packed granular bed under various pressures, using an instrumented sphere. The sphere acts as a noninvasive probe that measures and continuously transmits the acceleration as it penetrates into the sand, using a radio signal. The time-resolved acceleration of the sphere reveals the detailed dynamics during the impact that cannot be resolved from the position information alone. Because of the unobstructed penetration, we see a downward acceleration of the sphere at the moment the air cavity collapses. The compressibility of the sand bed is observed through the oscillatory behavior of the acceleration curve for various ambient pressures; it shows the influence of interstitial air on the compaction of the sand as a function of time.

Figure 1

(a) Schematic of the experimental setup consisting of a container filled with fine, loose sand and a pneumatic release mechanism to controllably drop the instrumented particle from various heights. (b) Picture of the instrumented sphere (taken from [20]) with a diameter of $D=2.5$ cm. The particle contains three accelerometers and the required electronics to transfer the measurement signal to the outside world. (c) The three components of the measured acceleration signal in the frame of reference of the particle.

Figure 2

(a) Typical acceleration signals $a_{M,1},a_{M,2}$, and $a_{M,3}$ of the three sensors for an impact with $\text{Fr}=83$ and $P_0=1$ bar. Accelerations are nondimensionalized with $g$. (b) Dimensionless measured acceleration $|{\mathbf{a}}_M|/g$ as a function of time. The small magenta, green, and red symbols represent an experiment done for the same external conditions to give an indication of the reproducibility of the experiment, which is good in the initial and medium phases. The vertical dashed blue line indicates the position of the peak, which is a signature of the jet formation. The solid red line is the fit of the position obtained using the model of the drag law (3) with ${\kappa }_D=3.5$ N/m and $\alpha =0.16$ kg/m. (c) Particle velocity $v\left(t\right)={\int }_t^{t_{\mathrm{stop}}}a\left(t^{\prime }\right)d{t}^{\prime }$ as a function of time $t$. The lines are the same as in (b). (d) Particle position $z\left(t\right)={\int }_0^{t}v\left(t^{\prime }\right)d{t}^{\prime }$ (nondimensionalized with the particle diameter $D$) as a function of $t$. Note that the detailed information in $a\left(t\right)$ is smoothed out and thus lost in $z\left(t\right)$. The lines are the same as in (b).

Figure 3

Measured accelerations for different impact velocity (or Froude number) at ambient pressure ($P_0=1$ bar): (a) $\text{Fr}=3$, (b) $\text{Fr}=23$, (c) $\text{Fr}=43$, and (d) $\text{Fr}=83$. The red solid line is the phenomenological model (3) obtained from the fit of the trajectory. The blue and magenta symbols represent two experiments done for the same external conditions. The vertical dashed blue lines indicate the position of the peak connected to the jet formation.

Figure 4

Measured accelerations for two Froude numbers $\text{Fr}=3$ and 83 and different ambient pressure: (a) $P_0=476$ mbars, (b) $P_0=173$ mbars, (c) $P_0=74$ mbars, and (d) $P_0=9$ mbars. The red solid lines again result from the phenomenological model (3) obtained from the fit of the trajectory. The blue and magenta symbols represent two experiments. The vertical dashed blue lines indicate the position of the peak connected to the jet formation.