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Research Papers

Modeling of a Rolling Flexible Circular Ring

[+] Author and Article Information
François Robert Hogan

Department of Mechanical Engineering,
McGill University,
Montréal, QC H3A 0G4, Canada
e-mail: francois.hogan@mail.mcgill.ca

James Richard Forbes

Assistant Professor
Department of Mechanical Engineering,
McGill University,
Montréal, QC H3A OG4, Canada
e-mail: james.richard.forbes@mgcill.ca

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received September 24, 2014; final manuscript received July 20, 2015; published online August 12, 2015. Assoc. Editor: Alexander F. Vakakis.

J. Appl. Mech 82(11), 111003 (Aug 12, 2015) (14 pages) Paper No: JAM-14-1447; doi: 10.1115/1.4031115 History: Received September 24, 2014

The motion equations of a rolling flexible circular ring are derived using a Lagrangian formulation. The in-plane flexural and out-of-plane twist-bending free vibrations are modeled using the Rayleigh–Ritz method. The motion equations of a flexible circular ring translating and rotating in space are first developed and then constrained to roll on a flat surface by introducing Lagrange multipliers. The motion equations developed capture the nonholonomic nature of the circular ring rolling without slip on a flat surface. Numerical simulations are performed to validate the dynamic model developed and to investigate the effect of the flexibility of the circular ring on its trajectory. The vibrations of the circular ring are observed to impact the ring's motion.

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Figures

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Fig. 1

Flexible circular ring model: (a) in-plane displacements ue and we and out-of-plane displacement ve and (b) cross section of a circular ring and torsion angle Ωe

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Fig. 2

In-plane mode shapes: (a) ωu1 = 0 ( rad/ s), (b) ωu2 = 27.19 ( rad/ s), (c) ωu3 = 76.91 ( rad/ s), (d) ωu4 = 147.47 ( rad/ s), (e) ωu5 = 238.50 ( rad/ s), and (f) ωu6 = 349.87 ( rad/ s)

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Fig. 3

Out-of-plane mode shapes: (a) ωv1 = 0 ( rad/ s), (b) ωv2 = 26.35 ( rad/ s), (c) ωv3 = 75.69 ( rad/ s), (d) ωv4 = 146.08 ( rad/ s), (e) ωv5 = 237.01 ( rad/ s), and (f) ωv6 = 348.34 ( rad/ s)

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Fig. 4

Position of element mass dm

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Fig. 5

Development of the rolling constraint without slip of the circular ring on a flat surface

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Fig. 6

Position of contact point p

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Fig. 7

Isometric view of an out-of-plane vibrating circular ring rolling along a straight line trajectory simulation

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Fig. 8

Top view of an out-of-plane vibrating circular ring rolling along a straight line trajectory simulation

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Fig. 9

Response of the position of the center of the circular ring rasa and the contact point rapa of an out-of-plane vibrating circular ring rolling along a straight line trajectory

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Fig. 10

Response of the Euler angles and the flexible coordinates of an out-of-plane vibrating circular ring rolling along a straight line trajectory

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Fig. 11

The constraint force fa,3, defined as the force applied by the surface on the circular ring in the a→3 axis (i.e., in the z direction), is positive throughout the simulation

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Fig. 12

Response of the position of the center of the circular ring rasa and the contact point rapa of an in-plane vibrating flexible circular ring rolling along a straight line trajectory

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Fig. 13

Isometric view of an in-plane vibrating flexible circular ring rolling along a straight line trajectory

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Fig. 14

Response of Euler angles and flexible coordinates of an in-plane vibrating flexible circular ring rolling along a straight line trajectory

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Fig. 15

The constraint force fa,3, defined as the force applied by the surface on the circular ring in the a→3 axis (i.e., in the z direction), is positive throughout the simulation

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Fig. 16

Using a modified rolling constraint that neglects the we displacement in rgdc of Eq. (25) ensures a proper surface contact between the circular ring and the flat surface

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Fig. 17

Isometric view of an in-plane and out-of-plane vibrating flexible circular ring rolling along a spiral trajectory

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Fig. 18

Response of the position of the center of the circular ring rasa and the contact point rapa of a vibrating flexible circular ring rolling along a spiral trajectory

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Fig. 19

Response of the Euler angles of a vibrating flexible circular ring rolling along a spiral trajectory

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Fig. 20

Response of the flexible coordinates of a vibrating flexible circular ring rolling along a spiral trajectory

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Fig. 21

The constraint force fa,3, defined as the force applied by the surface on the circular ring in the a→3 axis (i.e., in the z direction), is larger than 1 (N) throughout the simulation

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