Propelled Strings: Rising from Friction

The shape of closed strings and chains propelled at a constant velocity and launched at an angle relative to gravity is studied experimentally, theoretically, and numerically. At low velocity, strings adopt a shape close to the well-known catenary, while at high velocity, they can rise to a nearly horizontal profile. We show that the latter regime can be counterintuitively attributed to aerodynamic effects, although the ambient air exerts no lift on a string moving longitudinally along its profile. A theoretical approach along with numerical simulations confirms these observations and allows one to predict the shape of any closed string or chain. Moreover, depending of the regime, waves rising from any local perturbation along the string may travel either upstream or downstream and seem to die out at the turning point. We show that these observations can be explained by the tension profile along the string, which strongly depends on the aerodynamic effects relative to the weight, and our theoretical analysis allows us to predict the position of the wave front.

Figure 1

A light closed 2 m long string is propelled by two rotating wheels towards the right, at angle $\alpha (0)=27°$ relative to the horizontal and velocity $v=9.5\mathrm{m}{\mathrm{s}}^{-1}$.

Figure 2

Steady-state shapes of a 2 m long (a) heavy beaded chain and (b) light string for increasing velocities. (c) Propelled string (35-cm long) in a vacuum for constant velocity and propelling angle [$\alpha (0)=0$] for various pressures.

Figure 3

(a) Comparison between the experimental, analytical, and numerical profiles, (b) values of $D/W$ for the best analytical fit to experimental profiles as a function of $v^2$.

Figure 4

(a) Evolution of the steady-state string profile as a function of $D/W$ for $\alpha (0)=0$ and (b) for various propelling angles (for $D/W=2$).

Figure 5

Tension and wave celerity within chains and strings in (a) the weight-dominated regime and (b) the drag-dominated regime.

Figure 6

Series of snapshots showing the propagation of slow waves following an initial perturbation in (a) the weight-dominated regime and (b) the drag-dominated regime. (c) and (d) Time evolution of the wave front shown above.