Research Papers

Deformations and Stability of an Elastica Subjected to an Off-Axis Point Constraint

[+] Author and Article Information
Jen-San Chen1

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

Wei-Chia Ro

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


Corresponding author.

J. Appl. Mech 77(3), 031006 (Feb 04, 2010) (8 pages) doi:10.1115/1.4000426 History: Received January 09, 2009; Revised July 30, 2009; Published February 04, 2010; Online February 04, 2010

This paper studies, both theoretically and experimentally, the deformation, the vibration, and the stability of a buckled elastic strip (also known as an elastica) constrained by a space-fixed point in the middle. One end of the elastica is fully clamped while the other end is allowed to slide without friction and clearance inside a rigid channel. The point constraint is located at a specified height above the clamping plane. The elastic strip buckles when the pushing force reaches the conventional buckling load. At this buckling load, the elastica jumps to a symmetric configuration in contact with the point constraint. As the pushing force increases, a symmetry-breaking bifurcation occurs and the elastica evolves to one of a pair of asymmetric deformations. As the pushing force continues to increase, the asymmetric deformation experiences a second jump to a self-contact configuration. A vibration analysis based on an Eulerian description taking into account the sliding between the elastica and the point constraint is described. The natural frequencies and the stability of the calculated equilibrium configurations can then be determined. The experiment confirms the two jumps and the symmetry-breaking bifurcation predicted theoretically.

Copyright © 2010 by American Society of Mechanical Engineers
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Figure 1

An elastica constrained by a point at H. The dashed and solid curves represent typical symmetric and asymmetric deformations, respectively.

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Figure 2

Load-deflection curve for h=0.03. The solid and dashed curves represent stable and unstable deformations, respectively. The cross marks × represent the experimental measurements. The last points before the jumps occur are circled.

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Figure 3

The free body diagram of a small element ds constrained by the space-fixed point

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Figure 4

The boundary conditions at the opening A of the feeding channel. (a) In equilibrium position, the material point A′ coincides with point A. When the elastica vibrates, the material point A′ (b) retreats in and (c) protrudes out of the channel.

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Figure 5

ω2 of the first two modes as functions of the end force FA for deformation 2. The cross marks × represent the experimental measurements.

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Figure 6

The first two mode shapes of the constrained elastica when FA=1.5

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Figure 7

Top view of the experimental apparatus

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Figure 8

Photographs of (a) straight configuration before Euler buckling, (b) deformation 2, and (c) deformation 4



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