Rolling Ribbons

We present the results of a combined experimental and theoretical investigation of rolling elastic ribbons. Particular attention is given to characterizing the steady shapes that arise in static and dynamic rolling configurations. In both cases, above a critical value of the forcing (either gravitational or centrifugal), the ribbon assumes a two-lobed, peanut shape similar to that assumed by rolling droplets. Our theoretical model allows us to rationalize the observed shapes through consideration of the ribbon’s bending and stretching in response to the applied forcing.

Figure 1

A ribbon rolling at different speeds [or Froude number $\mathrm{Fr}=U^2/(g{R}_0)$]: (a) $\mathrm{Fr}=0$, (b) $\mathrm{Fr}=143$, (c) $\mathrm{Fr}=236$ (d) $\mathrm{Fr}=259$, (e) An unsteady shape observed above the critical touchdown speed. (f) A schematic of the ribbon. (g) Experimentally observed dependence of ribbon shape on Froude number. Ribbon properties: $E=0.26\mathrm{MPa}$, $\rho =1050\mathrm{kg}/{\mathrm{m}}^3$, $h=1.6\mathrm{mm}$, $w=27\mathrm{mm}$.

Figure 2

Aspect ratio of static ribbons, $\delta =\frac{H}{2R_0}$, as a function of their normalized stiffness ${\Gamma }_g=\frac{E{I}_0}{\rho g{S}_0{R}_0^3}$. The line is deduced by integrating Eq. (7) with $\mathrm{Fr}=0$. For the letters on the curve, we present in the inset the corresponding picture. The theoretical shape obtained through numerical integration of Eq. (7) is superposed as a thin white dashed line. Scale bars, 10 mm.

Figure 3

Aspect ratio, $\delta =\frac{H}{2R_0}$, of different rolling ribbons as a function of the speed parameter ${\Gamma }_i=\frac{E{h}^2}{\rho R_0^2{U}^2}$. Curves correspond to those predicted by Eq. (7). Circles, squares, diamonds, and triangles correspond to four different ribbons; specifically, ${\Gamma }_g=0.278$, 0.372, 0.429, 0.488. $E=0.26\mathrm{MPa}$ for the first three ribbons and $E=0.56\mathrm{MPa}$ for the last. Inset: Shape comparison for a single ribbon ($R_0=22.8\mathrm{mm}$, ${\Gamma }_g=0.488$, $E=0.56\mathrm{MPa}$) at different rotation speeds: (a) $Cy=0.09$, (b) $Cy=0.16$, (c) $Cy=0.20$, (d) $Cy=0.25$, where $Cy=\rho U^2/E$. Dashed curves are computed by integration of Eq. (7). Scale bar: 10 mm.

Figure 4

Ribbon elongation as a function of the Cauchy number $C_y=\frac{\rho U^2}{E}$. The black line is the dependence predicted by Eq. (6). Different symbols correspond to different values of normalized stiffness ${\Gamma }_g$. Characteristic error bars are shown for only one set of experiments.