A Geometric Model for the Coiling of an Elastic Rod Deployed Onto a Moving Substrate

Abstract

We report results from a systematic numerical investigation of the nonlinear patterns that emerge when a slender elastic rod is deployed onto a moving substrate; a system also known as the elastic sewing machine (ESM). The discrete elastic rods (DER) method is employed to quantitatively characterize the coiling patterns, and a comprehensive classification scheme is introduced based on their Fourier spectrum. Our analysis yields physical insight on both the length scales excited by the ESM, as well as the morphology of the patterns. The coiling process is then rationalized using a reduced geometric model (GM) for the evolution of the position and orientation of the contact point between the rod and the belt, as well as the curvature of the rod near contact. This geometric description reproduces almost all of the coiling patterns of the ESM and allows us to establish a unifying bridge between our elastic problem and the analogous patterns obtained when depositing a viscous thread onto a moving surface; a well-known system known as the fluid-mechanical sewing machine (FMSM).

The molten glass sewing machine

“…All the aforementioned patterns and their order of appearance are consistent with what is observed in both viscous and elastic cases [19][20][21]. This robustness suggests the existence of a common explanation for the formation of these patterns, which was recently given in [18,21] and is detailed in the following section so that it may be adapted to molten glass.…”

“…Embedded in geometry, coiling patterns are indeed found to be generic: they resist changes in the intrinsic mechanical properties of the thread. In particular, similar patterns have been observed when the thread is elastic [19][20][21]. Here, we illustrate how such fluidic instabilities may be applied to further the capabilities of additive manufacturing.…”

Helix Electrohydrodynamic Printing of Highly Aligned Serpentine Micro/Nanofibers

“…The coiling behavior of the electrospun micro/nanofiber is similar to the gravity driven rope coiling, although their characteristic lengths are micro/nanoscale and centimeters/millimeters, respectively. Recently, Brun et al [ 37 , 38 ] developed a quasi-static geometrical model (GM) to capture the various buckling patterns of thin viscous jet or elastic rope, and successfully calculated the bifurcation threshold of different patterns. They found that the jet/collector velocity match coefficient was the key factor for pattern variation, and this was coincidence with the HE-printing results.…”

“…In the geometrical model [ 37 , 38 ], the deposited trace on collector, namely the patterns of meander, alternating loops, translated coiling, etc. is a combination of the orbit of the contact point (regular coiling when collector is motionless) and the movement of the collector.…”

Ultra-Stretchable Piezoelectric Nanogenerators via Large-Scale Aligned Fractal Inspired Micro/Nanofibers

“…Higher voltage or smaller nozzle-to-collector distance correspond to stronger electrical filed, thus a larger falling speed and a smaller jet diameter can be achieved due to the more severe electrical stretching. Large falling speed indicates that large collector velocity is needed to stretch the jet from curve to straight (as shown in Figure 2 d) [ 30 , 31 , 32 ]. Small jet diameter corresponds to small bending/torsional stiffness, thus small buckled patterns deposited on collector [ 27 , 28 ].…”