Research Papers

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

[+] Author and Article Information
Mohammad K. Jawed

Department of Mechanical Engineering,
Massachusetts Institute of Technology,
Cambridge, MA 02139
e-mail: khalidjm@mit.edu

Pierre-Thomas Brun

Department of Mathematics,
Massachusetts Institute of Technology,
Cambridge, MA 02139
e-mail: ptbrun@mit.edu

Pedro M. Reis

Department of Mechanical Engineering,
Massachusetts Institute of Technology,
Cambridge, MA 02139;
Department of Civil and
Environmental Engineering,
Massachusetts Institute of Technology,
Cambridge, MA 02139
e-mail: preis@mit.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received July 23, 2015; final manuscript received August 18, 2015; published online September 18, 2015. Editor: Yonggang Huang.

J. Appl. Mech 82(12), 121007 (Sep 18, 2015) (8 pages) Paper No: JAM-15-1380; doi: 10.1115/1.4031363 History: Received July 23, 2015; Revised August 18, 2015

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).

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Grahic Jump Location
Fig. 1

(a) An injector (1) deploys a thin elastic rod (2) onto a conveyor belt (3). (b) Representative snapshots of the trace of the rod left on the belt for the various coiling regimes: (1) meanders, ϵ=0.25 ; (2) stretched coils, ϵ=0.18 ; (3) alternating loops, ϵ=0.55 ; and (4) translated coils, ϵ=0.75. The gray rectangles illustrate the relative scales of the x and y axes.

Grahic Jump Location
Fig. 2

Periodicity length, Λ, as a function of dimensionless speed mismatch, ϵ. Open (and filled) symbols correspond to the patterns obtained along ϵ=0→1 (and ϵ=1→0) direction. The black solid and dashed lines represent the results from the reduced geometry-based model introduced in Secs. 5 and 6.

Grahic Jump Location
Fig. 3

(a) Representative trace of the rod in the belt frame along (xb, yb) for each pattern. (b) Fourier frequency spectra with amplitude normalized by the largest peak, and frequencies ω, normalized by ωc=v/(2πRc), where Rc is the radius of the coil when the belt is static ε = 0. The patterns are: (1) meanders (ϵ=0.25), (2) alternating loops (ϵ=0.55), (3) translated coils (ϵ=0.75), (4) stretched coils (ϵ=0.18), (5) W pattern (ϵ=0.35), (6) alternating stretched coils (ϵ=0.16), and (7) Y pattern (ϵ=0.24).

Grahic Jump Location
Fig. 4

Traces of secondary patterns from DER: (a) alternating stretched coils (ϵ=0.16), (b) W patterns (ϵ=0.35), and (c) Y patterns (ϵ=0.24). See Fig. 1(b2) for stretched coils. The gray rectangles illustrate the relative scales of the x and y axes.

Grahic Jump Location
Fig. 5

Schematic diagrams for (a) the suspended heel, comprising a catenary region and a boundary layer; and (b) time-lapsed superposition of the rod configurations, over a quarter period of the meandering patterns

Grahic Jump Location
Fig. 6

Phase diagram showing the dependence of the FFT amplitude (adjacent bar) on both the transverse frequencies, ωy, and the dimensionless speed mismatch, ϵ: (a) ϵ=0→1 and (b) ϵ=1→0. At each value of ϵ, the amplitudes are normalized by the maximum amplitude. The dashed (and dotted) lines correspond to (a) ωc (and 0.5 ωc) and (b) ωc* (and 0.5 ωc*). For all the patterns, the lowest peak frequency always falls onto either of these lines.

Grahic Jump Location
Fig. 7

(a) Schematic diagram of the trace of the deposited rod (in the plane of the belt, x–y). The rod contacts the belt at the point C(r,ψ), with radial distance r and polar angle ψ=θ−ϕ. The tangent at C is t and the curvature of the rod there is κ=κ̂(ϕ,t,r)=θ′. (b) Normalized curvature, κRc, as a function of ϕ calculated from DER (solid line) and the dashed lines were obtained by fitting Eq. (5) to the simulation data, in the range 0.7<ϵ<1.0. The data were binned for r/Rc∈[0.7,1.3] in steps of 0.1. (c) Dependence of κ on the normalized radius, r/Rc, at ϕ=π/2.

Grahic Jump Location
Fig. 8

Comparison of traces between (1) DER simulations and (2) the GM: (a) meanders (ϵ=0.14), (b) W patterns (ϵ=0.28), (c) alternating loops (ϵ=0.50), and (d) translated coils (ϵ=0.70)

Grahic Jump Location
Fig. 9

Regimes of stability of the patterns along ϵ for (a) the ESM from the DERs simulations and the GM and (b) the FMSM from the DVRs simulations and the corresponding GM for the viscous system [22]



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