Depth and slip ratio dependencies of friction for a sphere rolling on a granular slope

TL;DR

The study addresses frictional dynamics of a sphere rolling on a deformable granular slope by varying initial velocity, slope angle, and sphere density, and introduces an effective translational friction $\mu_d$ derived from an energy-balance framework. It shows that the sinking depth normalized by radius, $\delta/R$, scales with the density ratio as $\delta/R = C_{\rho}(\rho_s/\rho_g)^{3/4}$ with $C_{\rho}$ around $0.38$ for downhill motion, while translational motion decelerates at a nearly constant rate and the travel distance decreases with steeper slopes. The key finding is a linear relation $\mu_d = \beta(\delta/R) + \mu_0$ with $\beta \approx 0.41$, where the intercept $\mu_0$ decreases with slip ratio $s$, and $\mu_d - \beta(\delta/R) = 0.32 - 0.44s$, indicating two additive dissipation mechanisms: sinking deformation and bump-related friction tied to terrain deformation. This work highlights that the normalized depth and slip ratio together govern the effective friction of rolling on deformable granular beds, with implications for granular-robotics and planetary surface dynamics.

Abstract

We experimentally investigate the dynamics of a sphere rolling down a granular slope by varying the initial velocity, slope angle, and sphere density. The results show that the sphere rolls down with constant deceleration while sinking into the granular bed. $δ/R$ (the sinking depth $δ$ normalized to the sphere radius $R$) is scaled by the sphere density normalized by the bulk density of the granular layer. To evaluate the translational energy dissipation, we introduce an effective friction coefficient $μ_\mathrm{d}$. We demonstrate that $μ_\mathrm{d}$ decreases with increasing the slope angle and the slip ratio. Furthermore, systematic measurements over a wide range of sphere densities reveal that $μ_\mathrm{d}$ increases linearly with $δ/R$ : $μ_\mathrm{d}=β(δ/R)+μ_0$. The value of $μ_0$ is linearly decreasing with slip ratio and its coefficient $β(\simeq0.41)$ does not vary significantly. The results suggest that the normalized depth and slip ratio determine the effective friction of a rolling sphere.

Figures (13)

• Figure 1: (a) Schematic view of the experimental setup. By rolling down the rail made of aluminium, the sphere of the radius $R$ obtains initial translational velocity $v_0$. The rolling posture $\theta$ is measured counterclockwise. The sphere rolls down the granular slope of the angle $\alpha$. $\alpha<0$ and $\alpha>0$ (obtained by Fukumoto et al. Fukumoto_2024) are defined as downhill and uphill, respectively. (b) The sphere moves in the $X$ direction by a distance $L$ and sinks by a depth $\delta$ when it stops. [(a), (b)] The size ratio between the sphere and the glass beads is not to scale.
• Figure 2: (a) The raw image of polyethylene sphere rolling down a slope of $\alpha = -5^{\circ}$. The hemispherical part of the sphere is colored in red to measure the rolling posture. (b) The image after image analysis of (a). The green curve and the blue point indicate the detected circle and the center of the sphere respectively.
• Figure 3: (a) The relation between depth normalized by radius $\delta/R$ and initial velocity $v_0$ for polyacetal at $\alpha\simeq-10^{\circ}$ taken as a representative data. Error bars indicate the standard deviation of 5 experimental runs. [(b)-(e)] The relation between $\delta/R$ and angle $\alpha$ [(b) polyethylene (c) polyacetal (d) glass (e) alumina ceramic]. Error bars indicate the standard deviation of various initial velocity $v_0$ and repeated experimental runs. The dashed lines indicate the average of all data in each sphere.
• Figure 4: The double logarithmic plot of $\delta/R$ vs $\rho_\mathrm{s}/\rho_\mathrm{g}$ for both downhill and uphill. We use the data (uphill) obtained by Fukumoto et al. Fukumoto_2024. Error bars indicate the standard deviation of various $\alpha, v_0$ conditions. The blue and red dashed line indicate the scaling relation [Eq. (\ref{['eq:depth_fitting']})] with $C_\mathrm{\rho}=0.38$ and 0.46, respectively.
• Figure 5: The translational $X$ position as a function of time $t$ for the polyacetal sphere [$v_0 \simeq 0.45~\mathrm{m/s}$] taken as a representative data. $L$ indicates the maximum travel distance at each $\alpha$.