3:30 PM - 3:45 PM
[SCG44-13] Energy dissipation of a sphere rolling up a granular slope
Keywords:getting stuck, granular physics, wheel, sphere, energy dissipation
On loose sand surfaces, vehicles are sometimes getting stuck by the wheels spinning out. For example, Spirit, the Mars rover, was getting stuck in 2009. Finally, it gave up continuing to search the Mars environment. It is important to understand the mechanism of stuck phenomena to develop a safe vehicle. We would like to understand the fundamental aspect of stuck occurrence by employing a simplified experimental system. We verify the friction (drag force) exerted on the sphere (without driving force) rolling on the surface of the granular layer (consisting of glass beads). Previous studies investigate the dynamics and friction of the sphere rolling down a granular slope or rolling on a horizontal granular surface. However, the decelerative dynamics of the sphere rolling up a granular slope has not been examined. In the stuck state, the rolling motion of the sphere remains after the cessation of its translational motion. Therefore, we have to analyze the decelerative dynamics of translational motion and rolling motion independently. In this study, we experimentally investigate the dynamics of the sphere rolling up a granular slope. To characterize the decelerative motion, we distinguish the friction due to slip motion and the friction due to the translational motion (groove formation with shallow sinking). In this experiment, the radius of the spheres we used is R= 6.35 mm. The typical glass bead size is 0.8 mm (0.77 mm - 0.91 mm), and the bulk density of the granular layer ρg is ρg=1570 kg/m3. We systematically varied the slope angle α (0°<α<20° at the increment of 5°), the density of the sphere ρs (ρs=930, 1400, 2600, 3900, and 7900 kg/m3), and the initial velocity v0(0.2 m/s <v0< 0.7 m/s) with which the sphere enters the granular slope. Experimental runs with identical conditions are repeated five times to confirm the reproducibility. The dynamics of the sphere are captured with a high-speed camera (Omron Sentech, STCMBS163U3V) with 200 fps. From the acquired data, the penetration depth δ is analyzed. Previous studies show that δ is scaled as the ratio between ρs and ρg, [Uehara et al., Phys. Rev. Lett. 90, 194301 (2003)]. A similar scaling law is confirmed in this experiment, [δ/R=0.46 (ρs/ρg)3/4 (1)]. In some cases, as the translational motion halts, rolling remains. Namely, we can observe the stuck occurrence in this experiment. The dynamics of the translational and rolling motions show constant deceleration (translational deceleration aX, and angular acceleration dω/dt) before the translational stop. (t is elapsed time from the moment when the sphere enters the granular slope.) By considering the equation of rolling motion, [ Idω/dt=-RμsMgcosα (2)], we measure the friction coefficient μs due to slip motion. (M is the mass of the sphere, I is the moment of inertia of the homogeneous sphere, I=2/5MR2, and g is the gravitational acceleration.) Then, we can compute μs using this equation and measured ω (other parameters are known parameters). The estimated μs is almost independent of α, ρs, and we can regard μs as a constant value (μs=0.26 ± 0.04). By considering energy conservation in translational motion, [ 1/2Mv02=Mg(Lsinα-δ)+μdMgLcosα (3)] (L is the maximum traveling distance), we compute the friction coefficient μd. The computed μd is almost independent of α, and μd is an increasing function of ρs. Moreover, μd is proportional to depth δ, [ μd=0.49(δ/R) (4)]. By using these results, we can predict the motion of sphere rolling up a granular slope, including its stuck behavior. We can compute L, with eqs. (1), (3), and (4). Because aX is constant before stopping translational motion, the velocity of the sphere can be written as v(t)= v0-(v02/2L)t. We can also compute the angular velocity of the sphere ω(t) with eq. (2) and μs. The detailed discussion of the distribution of stress acting on the surface of a sphere rolling up a granular slope is a critical future problem.