Coiling of Elastic Ropes

A rope falling onto a solid surface typically forms a series of regular coils. Here, we study this phenomenon using laboratory experiments (with cotton threads and softened spaghetti) and an asymptotic “slender-rope” numerical model. The excellent agreement between the two with no adjustable parameters allows us to determine a complete phase diagram for elastic coiling comprising three basic regimes involving different force balances (elastic, gravitational, and inertial) together with resonant “whirling string” and “whirling shaft” eigenmodes in the inertial regime.

Figure 1

Typical coiling configurations for ropes whose physical properties are listed in Table . The numbers on the scale in each panel indicate cm. (a) rope 5, $H=1.2\mathrm{cm}$, $U=26\mathrm{cm}{\mathrm{s}}^{-1}$; (b) rope 4, $H=30\mathrm{cm}$, $U=2.3\mathrm{cm}{\mathrm{s}}^{-1}$; (c) rope 4, $H=80\mathrm{cm}$, $U=100\mathrm{cm}{\mathrm{s}}^{-1}$.

Figure 2

Experimental measurements of the coil radius $R$. (a) $R$ as a function of $H$ for rope 6 with $U=2\mathrm{cm}{\mathrm{s}}^{-1}$ (open squares), rope 1 with $U=10\mathrm{cm}{\mathrm{s}}^{-1}$ (solid squares), and rope 2 with $U=10\mathrm{cm}{\mathrm{s}}^{-1}$ (circles). (b) $R$ as a function of $U$ for rope 6 with $H=1.3\mathrm{cm}$ (squares), rope 5 with $H=50\mathrm{cm}$ (solid circles), and rope 2 with $H=100\mathrm{cm}$ (open circles).

Figure 3

Main portion: Dimensionless coiling frequency $\widehat{\Omega }\equiv \Omega (d^{2}E/\rho g^4{)}^{1/6}$ as a function of dimensionless fall height $\widehat{H}\equiv H(\rho g/d^{2}E{)}^{1/3}$ for several values of the dimensionless feeding velocity $\widehat{U}=U(\rho /d^2{g}^{2}E{)}^{1/6}$. The curves for $\widehat{U}\ge 1.0$ continue indefinitely to the right (not shown). Portions of the curves labeled $E$ and $G$ and the horizontal bars labeled $I$ correspond to the frequencies (2). The first six “whirling string“ and “whirling shaft“ eigenfrequencies are indicated by dotted and dashed lines, respectively. Inset: Phase diagram for coiling as a function of $\widehat{H}$ and $\widehat{U}$. The frequency $\widehat{\Omega }(\widehat{H},\widehat{U})$ is multivalued everywhere above the solid line. The vertical dashed line indicates the smooth transition between elastic ($E$) and gravitational ($G$) coiling. The dashed line in the inertial ($I$) portion of the diagram indicates a smooth transition between whirling string and whirling shaft modes.

Figure 4

Comparison of experimentally measured (circles) and numerically calculated (lines) coiling frequencies. The dimensionless frequency $\widehat{\Omega }$, height $\widehat{H}$, and feeding velocity $\widehat{U}$ are defined in the text. The symbols $E$, $G$, and $I$ indicate elastic, gravitational, and inertial coiling, respectively, and the subscripts “string” and “shaft” indicate the dominant resonant mode type in the portion of the inertial regime in question. Error bars (omitted) reflecting the composite errors of $\Omega$, $H$, $U$, $\lambda$, $d$, and $E$ rarely exceed twice the symbol size. (a) Rope 7 with $U=2\mathrm{cm}{\mathrm{s}}^{-1}$; (b) Rope 3 with $U=10\mathrm{cm}{\mathrm{s}}^{-1}$; (c) Rope 3 with $H=30\mathrm{cm}$; (d) Rope 6 with $H=1.3\mathrm{cm}$.