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Research Papers

Snapping of a Planar Elastica With Fixed End Slopes

[+] Author and Article Information
Jen-San Chen1

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

Yong-Zhi Lin

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

1

Corresponding author.

J. Appl. Mech 75(4), 041024 (May 20, 2008) (6 pages) doi:10.1115/1.2871207 History: Received July 30, 2007; Revised November 08, 2007; Published May 20, 2008

In this paper, we study the deformation and stability of a planar elastica. One end of the elastica is clamped and fixed in space. The other end of the elastica is also clamped, but the clamp itself is allowed to slide along a linear track with a slope different from that of the fixed clamp. The elastica deforms after it is subjected to an external pushing force on the moving clamp. It is observed that when the pushing force reaches a critical value, snapping may occur as the elastica jumps from one configuration to another remotely away from the original one. In the theoretical investigation, we calculate the static load-deflection curve for a specified slope difference between the fixed clamp and the moving clamp. To study the stability of the equilibrium configuration, we superpose the equilibrium configuration with a small perturbation and calculate the natural frequencies of the deformed elastica. An experimental setup is designed to measure the load-deflection curve and the natural frequencies of the elastica. The measured load-deflection relation agrees with the theoretical prediction very well. On the other hand, the measured natural frequencies do not agree very well with the theoretical prediction, unless the mass of the moving clamp is taken into account.

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Copyright © 2008 by American Society of Mechanical Engineers
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Figures

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Figure 1

A beam placed between the two clamps with specified directions. End A of the beam is pushed in a distance along the horizontal direction, while End B is fixed in space.

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Figure 2

The free body diagram of a small element ds

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Figure 3

Elastica deformation with two inflection points

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Figure 4

Load-deflection curve for θB=30deg. The symbols ○, ◻, and △ represent deformations without, with one, and with two inflection points, respectively.

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Figure 5

The first four natural frequencies as functions of δA for θB=30deg

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Figure 6

Magnification of the frequency loci near location G1 in Fig. 5. The mode shapes corresponding to the natural frequencies are also shown.

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Figure 7

Schematic diagram of the experimental setup

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Figure 8

Experimental measurement of load-deflection curve

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Figure 9

Power spectrum when the elastica is free from external pushing force

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Figure 10

The measured natural frequencies are recorded with cross marks. The dashed and solid curves are the theoretical predictions neglecting and including the clamp mass, respectively.

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Figure 11

Load-deflection curves for various values of θB

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Figure 12

The first four natural frequencies as functions of δA for θB=0deg

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Figure 13

(a) Self-contact occurs when δA=0.85. (b) Clamp reverse is required for this deformation configuration when δA=1.

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Figure 14

Critical loads PA(cr)+ and PA(cr)− as functions of θB

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