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Research Papers

# On an Elastic Rod Inside a Slender Tube Under End Twisting Moment

[+] Author and Article Information
Jen-San Chen1

Department of Mechanical Engineering, National Taiwan University, Taipei 10617, Taiwanjschen@ntu.edu.tw

Hong-Chi Li

Department of Mechanical Engineering, National Taiwan University, Taipei 10617, Taiwan

1

Corresponding author.

J. Appl. Mech 78(4), 041009 (Apr 13, 2011) (9 pages) doi:10.1115/1.4003708 History: Received June 17, 2010; Revised October 03, 2010; Posted February 23, 2011; Published April 13, 2011; Online April 13, 2011

## Abstract

In this paper, we study the deformation of a thin elastic rod constrained inside a cylindrical tube and under the action of an end twisting moment. The ends of the rod are clamped in the lateral direction. Unlike the previous works of others, in which only the fully developed line-contact spiral was considered, we present a complete analysis on the deformation when the dimensionless twisting moment $Mx$ is increased from zero. It is found that the straight rod buckles into a spiral shape and touches the inner wall of the tube at the midpoint when $Mx$ reaches 8.987. As $Mx$ increases to 11.472, the contact point in the middle splits into two, leaving the midpoint floating in the air. As $Mx$ increases to 13.022, the midpoint returns to touch the tube wall and the two-point-contact deformation evolves to a three-point-contact deformation. Starting from $Mx=13.098$, the point contact in the middle evolves to a line contact, and the deformation becomes a point-line-point contact configuration and remains so thereafter. In the case when the line-contact pattern is fully developed, it is possible to predict the spiral shape analytically. The numerical results are found to agree very well with those predicted analytically. Finally, an experimental setup is constructed to observe the deformation evolution of the constrained rod under end twist.

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## Figures

Figure 4

The relation between ϕ,x(0.5) and Mx. Energy method predicts that this relation is very close to a straight line Mx=43ϕ,x.

Figure 5

Distribution of the line-contact force q(x) when Mx=35

Figure 7

The point force RA when Mx changes from 8.987 to 20 in one-point (dotted), two-point (dashed), three-point (chained line in the inset), and line-contact (solid) deformations

Figure 8

The point force RB when Mx changes from 13.022 to 20 in three-point (dashed), and line-contact (solid) deformations. The range near Mx=13.098 is magnified and shown in the inset.

Figure 9

Three-dimensional and end views of the four deformation patterns when Mx is equal to (a) 9 (one-point contact), (b) 12 (two-point contact), (c) 13.05 (three-point contact), and (d) 35 (line contact)

Figure 10

Bending moment distributions for various end twisting moments

Figure 15

Relation between helix angle β and twisting moment Mx after line-contact deformation appears

Figure 1

A clamped-clamped twisted rod constrained in a tube. Solid (B-C) and dashed (O-A and A-B) lines represent line contact and no contact with the tube wall, respectively. C is the middle point of the constrained rod.

Figure 2

l1 and l2 for the line-contact deformation when Mx is reduced from 35. It is observed that l2 approaches 0.5 when Mx approaches 13.098.

Figure 3

(a) The radial distance ρ=y2+z2 and (b) angle ϕ as functions of x when Mx=35

Figure 6

The relation between q(0.5) and Mx. Energy method predicts that this relation is close to q=(27/256)Mx4.

Figure 11

Shear force distributions for various end twisting moments

Figure 12

Photograph of experimental set-up

Figure 13

Relation between end displacement e and twisting moment Mx

Figure 14

Relation between the location of the first contact point l1 and twisting moment Mx

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