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TECHNICAL PAPERS

On the Dynamics of the Dynabee

[+] Author and Article Information
D. W. Gulick, O. M. O’Reilly

Department of Mechanical Engineering, University of California, Berkeley, CA 94720-1740

J. Appl. Mech 67(2), 321-325 (Oct 25, 1999) (5 pages) doi:10.1115/1.1304914 History: Received July 01, 1999; Revised October 25, 1999
Copyright © 2000 by ASME
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References

Mishler, A. L., 1973, “Gyroscopic Device,” U.S. Patent 3726146.
Neimark, J. I., and Fufaev, N. A., 1972, Dynamics of Nonholonomic Systems, English translation by J. R. Barbour, American Mathematics Society, Providence, RI.
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Casey,  J., 1995, “On the Advantages of a Geometric Viewpoint in the Derivation of Lagrange’s Equations for a Rigid Continuum,” J. Appl. Math. Phys., 46, pp. S805–S847.
O’Reilly, O. M., and Srinivasa, A. R., 1999, “On Constraints and Potential Energies in Systems of Rigid Bodies and Particles,” submitted for publication.
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Haberman,  R., Rand,  R., and Yuster,  T., 1999, “Resonant Capture and Separatrix Crossing in Dual-Spin Spacecraft,” Nonlinear Dyn., 18, pp. 159–184.
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Figures

Grahic Jump Location
Schematic of the Dynabee. The precessional motion of a vector normal to the circumferential track is also shown.
Grahic Jump Location
The Euler angles and reference frames used to parameterize the precessional motion of the track
Grahic Jump Location
The Euler angles and reference frames used to parameterize the rotor’s rotation relative to the track
Grahic Jump Location
Rolling contact between the rotor and track as viewed from the negative e2 direction
Grahic Jump Location
Phase plane for the rotor response given a constant precession rate of the track. The solid lines are trajectories for several initial conditions, the shaded area is the basin of attraction of the stable equilibrium, and the dashed lines are contours for the ratio |κ32|. The phase plane is of (27) with ζ=29,η=1.7,ν=0.018, and θ=π/4.
Grahic Jump Location
Cross sections of the basin of attraction of the stable equilibrium for (32) when the track’s precession rate is increasing according to (31) with ζ=29,η=1.7,ν=0.018,θ=π/4, and ε=0.01. A sample trajectory is also shown whose initial conditions are ψ(0)=1,δ(0)=−2.5, and δ(0)=0.1.

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