Nonlinear Dynamics of the 3D Pendulum |
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Authors: | Nalin A Chaturvedi Taeyoung Lee Melvin Leok N Harris McClamroch |
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Institution: | 1. Research and Technology Center, Robert Bosch LLC, Palo Alto, CA, 94304, USA 2. Department of Mechanical and Aerospace Engineering, Florida Institute of Technology, Melbourne, FL, USA 3. Department of Mathematics, University of California at San Diego, La Jolla, CA, 92093-0112, USA 4. Department of Aerospace Engineering, The University of Michigan, Ann Arbor, MI, 48109-2140, USA
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Abstract: | A 3D pendulum consists of a rigid body, supported at a fixed pivot, with three rotational degrees of freedom. The pendulum
is acted on by a gravitational force. 3D pendulum dynamics have been much studied in integrable cases that arise when certain
physical symmetry assumptions are made. This paper treats the non-integrable case of the 3D pendulum dynamics when the rigid
body is asymmetric and the center of mass is distinct from the pivot location. 3D pendulum full and reduced models are introduced
and used to study important features of the nonlinear dynamics: conserved quantities, equilibria, relative equilibria, invariant
manifolds, local dynamics, and presence of chaotic motions. The paper provides a unified treatment of the 3D pendulum dynamics
that includes prior results and new results expressed in the framework of geometric mechanics. These results demonstrate the
rich and complex dynamics of the 3D pendulum. |
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