Research Papers

Modeling of a Rolling Flexible Spherical Shell

[+] Author and Article Information
François Robert Hogan

Department of Mechanical Engineering,
Massachusetts Institute of Technology,
Cambridge, MA 02139
e-mail: fhogan@mit.edu

James Richard Forbes

Assistant Professor
Department of Mechanical Engineering,
McGill University,
Montreal, Quebec H3A 0C3, Canada
e-mail: james.richard.forbes@mcgill.ca

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received September 16, 2015; final manuscript received May 24, 2016; published online July 4, 2016. Assoc. Editor: Alexander F. Vakakis.

J. Appl. Mech 83(9), 091010 (Jul 04, 2016) (12 pages) Paper No: JAM-15-1502; doi: 10.1115/1.4033720 History: Received September 16, 2015; Revised May 24, 2016

The purpose of this paper is to develop the motion equations of a flexible spherical shell rolling without slip on a flat surface. The motivation for this paper stems from tumbleweed rovers, which are envisioned to roll, deform, and bounce on the Martian surface due to the flexible nature of their thin walls. The motion equations are derived using a constrained Lagrangian approach and capture the rolling without slip nonholonomic constraint. Numerical simulations are performed to validate the dynamic model developed and to investigate the effect of the flexibility of the spherical shell on its trajectory.

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Fig. 1

Axisymmetric vibrations of the surface S about the axis s→3 (a) generating curve R and (b) surface of revolution S

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Fig. 2

Axisymmetric mode shapes of spherical shell (a) n = 1, (b) n = 2, (c) n = 3, (d) n = 4, (e) n = 5, and (f) n = 6

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Fig. 3

Position of element mass dm relative to the origin of Fa

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Fig. 4

Spherical shell rolling without slip on a flat surface

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Fig. 5

Spherical shell rolling without slip along a generic trajectory

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Fig. 12

Top view of the curved trajectory of the rolling flexible sphere and its rigid counterpart

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Fig. 13

Response of the position of the center and the Euler angles of the spherical shell rolling in a curved trajectory

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Fig. 14

Response of the flexible coordinates of the spherical shell rolling in a curved trajectory

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Fig. 11

Curved trajectory of the rolling flexible sphere and its rigid counterpart

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Fig. 6

Straight line trajectory of the rolling flexible spherical shell and its rigid counterpart

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Fig. 7

Response of the position of the center and the Euler angles coordinates of the spherical shell rolling in a straight line trajectory

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Fig. 8

Response of the flexible coordinates of the spherical shell rolling in a straight line trajectory

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Fig. 9

Energy versus time associated with the flexible spherical shell rolling along a straight line. No damping present.

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Fig. 10

Error on the constraint norm Ξq˙ versus time

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Fig. 15

Energy analysis for the generic trajectory



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