Research Papers

An Analytical Investigation of the Trapeze Effect Acting on a Thin Flexible Ribbon

[+] Author and Article Information
Jérôme F. Sicard

Department of Aerospace Engineering
and Engineering Mechanics,
The University of Texas at Austin,
Austin, TX 78712
e-mail: jerome.sicard@utexas.edu

Jayant Sirohi

Associate Professor
Department of Aerospace Engineering
and Engineering Mechanics,
The University of Texas at Austin,
Austin, TX 78712
e-mail: jayant.sirohi@mail.utexas.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received July 20, 2014; final manuscript received October 8, 2014; accepted manuscript posted October 10, 2014; published online October 30, 2014. Assoc. Editor: George Kardomateas.

J. Appl. Mech 81(12), 121007 (Oct 30, 2014) (9 pages) Paper No: JAM-14-1328; doi: 10.1115/1.4028781 History: Received July 20, 2014; Revised October 08, 2014

This paper systematically explores the extensional–torsional coupling due to the trapeze effect acting on a thin flexible ribbon subjected to combined tension and torsion. Kinematic relationships as well as expressions for the restoring torque associated with this effect are analytically derived. Additionally, the locus of points about which the cross sections of a twisted ribbon under tension rotate is derived. These points, called torsional centers, are found to be coincident with the centroids of the axial stress field at each station along the ribbon. More generally, it is shown that when a flexible slender member is in tension, combined transverse forces must act at the centroid of the axial stress field to produce pure bending and no twist. As a result, the elastic axis (EA) of the member shifts from the locus of shear centers to the locus of centroids of the axial stress field. A numerical model is developed to investigate the effect of the position of the EA on the prediction of steady-state deformations and natural frequencies of a rotating ribbon with tip mass. By assuming the EA to be the locus of the shear centers, the tip twist is overpredicted by a factor of 2 for small twist angles, and up to 2.5 for large twist deformations. In addition, assuming the EA to be the locus of shear centers results in an error of up to 60% in the predicted natural frequencies at large twist angles.

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Fig. 2

Kinematic foreshortening induced in a twisted trapeze, for various chord over length ratios

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Fig. 1

Undeformed (dashed lines) and deformed (solid lines) shape of a trapeze in torsion

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Fig. 3

Undeformed and deformed shape of a thin ribbon in torsion

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Fig. 4

Restoring torque induced by longitudinal stresses in the fibers

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Fig. 5

Free-body-diagram of the forces and moments applied to a rotating ribbon with tip mass

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Fig. 6

Spanwise locus of centroid of the axial stress field

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Fig. 10

Variation of the tip axial deflection with root pitch angles

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Fig. 11

Spanwise variation of total pitch angles

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Fig. 12

Spanwise variation of axial elongation (EA at centroid of the axial stress field)

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Fig. 13

Natural frequencies of the flexible ribbon rotating at 1500 rpm

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Fig. 14

Natural mode shapes

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Fig. 7

Spanwise variation of internal normal force N and internal bending moment M

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Fig. 8

Spanwise locus of the EA relative to the middle axis of the ribbon

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Fig. 9

Variation of the tip twist with root pitch angles




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