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Journal of Applied Mechanics | Volume 82 | Issue 2 | research-article
Research Papers

Extending the Reach of a Rod Injected Into a Cylinder Through Distributed Vibration

[+] Author and Article Information
Jay T. Miller

Department of Civil and
Environmental Engineering,
Massachusetts Institute of Technology,
Cambridge, MA 02139

Connor G. Mulcahy

Department of Mechanical Engineering,
Massachusetts Institute of Technology,
Cambridge, MA 02139

Jahir Pabon, Nathan Wicks

Schlumberger-Doll Research,
Cambridge, MA 02139

Pedro M. Reis

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

1Present address: Schlumberger-Doll Research, Cambridge, MA 02139.

2Corresponding author.

Contributed by the Applied Mechanics of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received October 9, 2014; final manuscript received November 6, 2014; accepted manuscript posted December 12, 2014; published online December 12, 2014. Editor: Yonggang Huang.

J. Appl. Mech 82(2), 021003 (Feb 01, 2015) (6 pages) Paper No: JAM-14-1471; doi: 10.1115/1.4029251 History: Received October 09, 2014; Revised November 06, 2014; Accepted December 12, 2014; Online December 12, 2014
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We present results of an experimental investigation of a new mechanism for extending the reach of an elastic rod injected into a horizontal cylindrical constraint, prior to the onset of helical buckling. This is accomplished through distributed, vertical vibration of the constraint during injection. A model system is developed that allows us to quantify the critical loads and resulting length scales of the buckling configurations, while providing direct access to the buckling process through digital imaging. In the static case (no vibration), we vary the radial size of the cylindrical constraint and find that our experimental results are in good agreement with existing predictions on the critical injection force and length of injected rod for helical buckling. When vertical vibration is introduced, reach can be extended by up to a factor of four, when compared to the static case. The injection speed (below a critical value that we uncover), as well as the amplitude and frequency of vibration, are studied systematically and found to have an effect on the extent of improvement attained.
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References

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Figures

Grahic Jump Location
Fig. 2

(a) Helix initiation length, lh, versus radial clearance, Δr. Theoretical prediction comes from Eq. (3), with μ = 0.54 ± 0.11 (solid line for the mean; dashed lines for the standard deviation). Inset: Schematic definition of Δr. (b) Reaction force at helix initiation, Ph, versus Δr. Equation (4) is plotted as a solid line. (c) Helix initiation length, lh, as a function of injection speed, v, for two radial clearances, Δr = 4.4 and 9.2 mm. Dashed lines represent predictions from Eq. (3). For parts (a) and (c), data points indicate the mean of 10 runs, with error bars representing a standard deviation of 10 runs. For part (b), data represents measurements made within 0.02 m of lh for 10 runs.

+(a) Helix initiation length, lh, versus radial clearance, Δr. Theoretical prediction comes from Eq. (3), with μ = 0.54 ± 0.11 (solid line for the mean; dashed lines for the standard deviation). Inset: Schematic definition of Δr. (b) Reaction force at helix initiation, Ph, versus Δr. Equation (4) is plotted as a solid line. (c) Helix initiation length, lh, as a function of injection speed, v, for two radial clearances, Δr = 4.4 and 9.2 mm. Dashed lines represent predictions from Eq. (3). For parts (a) and (c), data points indicate the mean of 10 runs, with error bars representing a standard deviation of 10 runs. For part (b), data represents measurements made within 0.02 m of lh for 10 runs.
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(a) Helix initiation length, lh, versus radial clearance, Δr. Theoretical prediction comes from Eq. (3), with μ = 0.54 ± 0.11 (solid line for the mean; dashed lines for the standard deviation). Inset: Schematic definition of Δr. (b) Reaction force at helix initiation, Ph, versus Δr. Equation (4) is plotted as a solid line. (c) Helix initiation length, lh, as a function of injection speed, v, for two radial clearances, Δr = 4.4 and 9.2 mm. Dashed lines represent predictions from Eq. (3). For parts (a) and (c), data points indicate the mean of 10 runs, with error bars representing a standard deviation of 10 runs. For part (b), data represents measurements made within 0.02 m of lh for 10 runs.
Grahic Jump Location
Fig. 1

Injection force, P, versus injected length, l, for a rod (E = 1290 ± 12 kPa, ρ = 1200 kg/m3, and r = 1.55 mm) injected into a glass pipe (D = 12.0 mm and Δr = 4.4 mm) at velocity, v = 0.1 m/s. Inset: profile-view photographs of configurations at different values of l = {0.22, 0.73, 0.79, 0.98, and 1.05} m. I—straight; II—sinusoidal; III, IV—helix initiation at lh (solid line) and lock-up at lc (dashed line); and V—end of test.

+Injection force, P, versus injected length, l, for a rod (E = 1290 ± 12 kPa, ρ = 1200 kg/m3, and r = 1.55 mm) injected into a glass pipe (D = 12.0 mm and Δr = 4.4 mm) at velocity, v = 0.1 m/s. Inset: profile-view photographs of configurations at different values of l = {0.22, 0.73, 0.79, 0.98, and 1.05} m. I—straight; II—sinusoidal; III, IV—helix initiation at lh (solid line) and lock-up at lc (dashed line); and V—end of test.
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Injection force, P, versus injected length, l, for a rod (E = 1290 ± 12 kPa, ρ = 1200 kg/m3, and r = 1.55 mm) injected into a glass pipe (D = 12.0 mm and Δr = 4.4 mm) at velocity, v = 0.1 m/s. Inset: profile-view photographs of configurations at different values of l = {0.22, 0.73, 0.79, 0.98, and 1.05} m. I—straight; II—sinusoidal; III, IV—helix initiation at lh (solid line) and lock-up at lc (dashed line); and V—end of test.
Grahic Jump Location
Fig. 3

Normalized helical initiation length, l¯h, versus normalized injection speed, v¯, for a rod injected into a pipe with D = 21.7 mm and Δr = 9.3 mm. Experimental error bars are the standard deviations of 5 runs per injection speed. The dimensionless acceleration is Γ = 2, with f = {50, 100, and 200} Hz. The dashed horizontal lines represent the range of the experiments with no vibration reported in Fig. 2(c), and the solid vertical line shows the separation between fast(v¯ > 1) and slow(v¯ < 1) injection speeds.

+Normalized helical initiation length, l¯h, versus normalized injection speed, v¯, for a rod injected into a pipe with D = 21.7 mm and Δr = 9.3 mm. Experimental error bars are the standard deviations of 5 runs per injection speed. The dimensionless acceleration is Γ = 2, with f = {50, 100, and 200} Hz. The dashed horizontal lines represent the range of the experiments with no vibration reported in Fig. 2(c), and the solid vertical line shows the separation between fast(v¯ > 1) and slow(v¯ < 1) injection speeds.
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Normalized helical initiation length, l¯h, versus normalized injection speed, v¯, for a rod injected into a pipe with D = 21.7 mm and Δr = 9.3 mm. Experimental error bars are the standard deviations of 5 runs per injection speed. The dimensionless acceleration is Γ = 2, with f = {50, 100, and 200} Hz. The dashed horizontal lines represent the range of the experiments with no vibration reported in Fig. 2(c), and the solid vertical line shows the separation between fast(v¯ > 1) and slow(v¯ < 1) injection speeds.
Grahic Jump Location
Fig. 4

Normalized helical initiation length, l¯h, versus dimensionless acceleration, Γ, for a rod injected into a pipe with D = 21.7 mm and Δr = 9.3 mm. The error bars on the experimental data are the standard deviations of 5 runs per injection speed. For all runs, the injection speed was v = 0.1 m/s (fast injection speed for all cases) and three vibration frequencies were tested, f = {50, 100, and 200} Hz. The dashed lines represent the range of the experiments with no vibration reported in Fig. 2(c).

+Normalized helical initiation length, l¯h, versus dimensionless acceleration, Γ, for a rod injected into a pipe with D = 21.7 mm and Δr = 9.3 mm. The error bars on the experimental data are the standard deviations of 5 runs per injection speed. For all runs, the injection speed was v = 0.1 m/s (fast injection speed for all cases) and three vibration frequencies were tested, f = {50, 100, and 200} Hz. The dashed lines represent the range of the experiments with no vibration reported in Fig. 2(c).
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Normalized helical initiation length, l¯h, versus dimensionless acceleration, Γ, for a rod injected into a pipe with D = 21.7 mm and Δr = 9.3 mm. The error bars on the experimental data are the standard deviations of 5 runs per injection speed. For all runs, the injection speed was v = 0.1 m/s (fast injection speed for all cases) and three vibration frequencies were tested, f = {50, 100, and 200} Hz. The dashed lines represent the range of the experiments with no vibration reported in Fig. 2(c).
Grahic Jump Location
Fig. 5

Normalized helix initiation length, l¯h, versus vibration frequency, f, for glass pipes with different radial clearances (see legend), at Γ = 1.75 and v¯=0.1. Solid lines indicate radial clearances exhibiting resonant peaks (Δr = {3.2, 4.4, and 9.3} mm), while dashed lines indicate radial clearances which exhibit no such peaks (Δr = {1.75, 6.6, and 7.7} mm).

+Normalized helix initiation length, l¯h, versus vibration frequency, f, for glass pipes with different radial clearances (see legend), at Γ = 1.75 and v¯=0.1. Solid lines indicate radial clearances exhibiting resonant peaks (Δr = {3.2, 4.4, and 9.3} mm), while dashed lines indicate radial clearances which exhibit no such peaks (Δr = {1.75, 6.6, and 7.7} mm).
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Normalized helix initiation length, l¯h, versus vibration frequency, f, for glass pipes with different radial clearances (see legend), at Γ = 1.75 and v¯=0.1. Solid lines indicate radial clearances exhibiting resonant peaks (Δr = {3.2, 4.4, and 9.3} mm), while dashed lines indicate radial clearances which exhibit no such peaks (Δr = {1.75, 6.6, and 7.7} mm).

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