Modeling a washboard road: From experimental measurements to linear stability analysis

When submitted to the repeated passages of vehicles unpaved roads made of sand or gravel can develop a ripply pattern known as washboard or corrugated road. We propose a stability analysis based on experimental measurements of the force acting on a blade (or plow) dragged on a circular sand track and show that a linear model is sufficient to describe the instability near onset. The relation between the trajectory of the plow and the profile of the sand bed left after its passage is studied experimentally. The various terms in the expression of the lift force created by the flow of granular material on the plow are determined up to first order by imposing a sinusoidal trajectory to the blade on an initially flat sand bed, as well as by imposing a horizontal trajectory on an initially rippled sand bed. Our model recovers all the previously observed features of washboard road and accurately predicts the most unstable wavelength near onset as well as the critical velocity for the instability.

Figure 1

Sketch of the experimental setup. A plow is dragged at a constant horizontal velocity on a 5-m-long circular track filled with sand.

Figure 2

Sketch of the experimental setups. (a) Plow attached to an articulated arm and free to move vertically. (b) Plow mounted on force sensors attached to a computer-controlled translation stage which allows one to impose the trajectory.

Figure 3

Diagram representing the coupling between the dynamics of the plow and the reshaping of the sand bed due to erosion and deposition.

Figure 4

(a) Plots of the trajectory of the plow $y$ (solid symbols) and the profile left after the passage of the plow, $h_n$ (open symbols), for $k=21{\mathrm{m}}^{-1}$ and $v=0.5\mathrm{m}{\mathrm{s}}^{-1}$. The profile $h_n$ measured after the passage of the plow is translated by the distance to the tip of the plow. (b) and (c) Ratio between the amplitude of the profile $h_n$ and the trajectory, $\chi =h_n/y$ as a function of $k$ (for $v=0.5\mathrm{m}{\mathrm{s}}^{-1}$) and $v$ (for $k=21{\mathrm{m}}^{-1}$).

Figure 5

(a) Absolute value and argument of the transfer function $|\underline{H}|$ for $v=0.3\mathrm{m}{\mathrm{s}}^{-1}$ and $\langle M\rangle$ $=$ 0.58 kg. The solid lines correspond to Eq. (7), the dashed line corresponds to ${\omega }_c$ for which $\underline{H}$ is real and $|\underline{H}|$ is minimal. (b) ${\omega }_c$ as a function of $v$ showing a linear dependence for $\langle M\rangle$ $=$ 0.58 kg.

Figure 6

(a), (c) Coefficients of the additional terms $b$ and $c$ as a function of $\langle M\rangle$ (for $v=0.5\mathrm{m}{\mathrm{s}}^{-1}$) and $v$ (for $\langle M\rangle =1$ kg). $b$ appears to be proportional to $v$ (solid line of slope one) and independent of $M$. $c$ is proportional to $M$ and its dependence on $v$ is given by the expression of ${\omega }_c$ found in Fig. 5c (solid line).

Figure 7

Absolute value and argument of the transfer function $|\underline{G}|$ for $M=1$ kg and $v=1.5\mathrm{m}{\mathrm{s}}^{-1}$. The solid lines correspond to Eq. (8) with parameters inferred from Sec. 6b, showing good agreement with the experimental data.

Figure 8

$h_n$ as a function of the number of rotations for $v=1.1\mathrm{m}{\mathrm{s}}^{-1}$ and $\lambda =195\mathrm{mm}$ ($\square$) and $v=0.8\mathrm{m}{\mathrm{s}}^{-1}$ and $\lambda =216mm$ ($\circ$). Straight lines are exponential fits of the data over the first ten rotations.

Figure 9

(Top) Predictions of equation, growth rate $\sigma$ as a function of $\lambda$ for velocities ranging from 0.4 $\mathrm{m}{\mathrm{s}}^{-1}$ to 1.2 $\mathrm{m}{\mathrm{s}}^{-1}$ from bottom to top: (10). (Bottom) Experimental measurements for $v=0.6\mathrm{m}{\mathrm{s}}^{-1}$ ($\times$), $v=0.8\mathrm{m}{\mathrm{s}}^{-1}$ ($•$) and $v=1.1\mathrm{m}{\mathrm{s}}^{-1}$ ($\circ$). Above a critical velocity $v_c$ there exist positive values of $\sigma$ corresponding to the onset of the instability.

Figure 10

(Top) Critical velocity $v_c$ as a function of the mass of the plow $m$ ($\circ$), the straight line corresponds to the prediction of the model and follows a power law with an exponent 0.35. (Bottom) Experimental wave number as a function of the predicted one. The straight line shows $k_{\exp }=k_{\mathrm{th}}$.