How a Soft Rod Wraps around a Rotating Cylinder

The unique characteristics of helical coils are utilized in nature, manufacturing processes, and daily life. These coils are also pivotal in the development of soft machines, such as artificial muscles and soft grippers. The stability of these helical coils is generally dependent on the mechanical properties of the rods and geometry of the supporting objects. In this Letter, the shapes formed by a flexible, heavy rod wrapping around a slowly rotating rigid cylinder are investigated through a combination of experimental and theoretical approaches. Three distinct morphologies—tight coiling, helical wrapping, and no wrapping—are identified experimentally. These findings are rationalized by numerical simulations and a geometrically nonlinear Kirchhoff rod theory. Despite the frictional contact present, the local shape of the rod is explained by the interplay between bending elasticity, gravity, and the geometry of the system. Our Letter provides a comprehensive physical understanding of the ordered morphology of soft threads and rods. Implications of this understanding are significant for a wide range of phenomena, from the recently discovered wrapping motility mode of bacterial flagella to the design of an octopus-inspired soft gripper.

Figure 1

(a) Coiling around another object: (a1) morning glory, (a2) garden hose, (a3) spiral wire hangers, (a4) thread bobbins, and (a5) spaghetti. (b) An elastic rod coiling around a metal cylinder observed in our model experiment. (c) Schematic illustration of a soft rod deflected under its own weight.

Figure 2

Geometric characteristics of wrapping. (a) Typical images of coiling patterns, including tight coiling, helical wrapping, and no wrapping. A coexistence phenomenon is observed. (b) Morphological diagram in the $(\lambda ,L_{gb})$ plane obtained from the physical experiment. The different symbols indicate different radii $a$ and gravitobending lengths $L_{gb}$ as indicated in the legend. The solid and dashed lines represent the theoretical predictions obtained using Eq. (4) and Eq. (6). (c) Experimental pitch length $P$ (averaged per turn and rescaled by $2\pi R$) plotted as a function of a nondimensional quantity comprising the suspended length $\lambda$, $L_{gb}$, and $R$. Data from the first and last helical turns are represented as open symbols, indicating that they tend to scatter more than the data from the inner region (full symbols). The colors of the symbols indicate the values of $a/R_0$.

Figure 3

Geometry and parameters defined in our theoretical model. (a) Elastic rod with a radius of $a$, bending modulus of $B$, and length of $L$ wrapping around a rigid cylindrical bar with a radius of $R_0$. (b) Cross-sectional view of (a). ${\theta }_c$ denotes the angle of separation at $s=s_c$. (c) Rod configurations during its unwrapping (from top to bottom) obtained in our simulation for $L/a=422$, $L_{gb}/R=4.35$, and $\mu =0.1$.

Figure 4

Wrapping behaviors for finite $\mu$ observed in the simulation and experiment. (a) An ideal frictionless ($\mu =0$) rod for $L_{gb}/R=3.01$ and $L/a=422$ shows the T–H transition to form a uniform helical winding. (b) Configurations for different values of $\mu$, as indicated, for $L_{gb}/R=3.78$ and $L/a=422$. (c) Log-log plot, showing the relationship between ${\lambda }_{\mathrm{T},\mathrm{H}}/R$ and $L_{gb}/R$, fit with the predicted cubic power law curves (see the main text), from which $c(\mu )$ is determined in (d). (e) Scaling plot of the same data as in (c) using $c(\mu )\mathrm{s}$ from (d) and Fig. S4 [35]. The dotted curve represents the cubic function with prefactor unity.