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Stability of coupled rigid body and geometrically exact rods, block diagonalization and the energy, momentum method

Simo, J. C. and Posbergh, T. A. and Marsden, J. E. (1990) Stability of coupled rigid body and geometrically exact rods, block diagonalization and the energy, momentum method. Physics Reports, 193 (6). pp. 279-360. ISSN 0370-1573. doi:10.1016/0370-1573(90)90125-L.

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This paper develops and applies the energy-momentum method to the problem of nonlinear stability of relative equilibria. The method is applied in detail to the stability analysis of uniformly rotating states of geometrically exact rod models, and a rigid body with an attached flexible appendage. Here, the flexible appendage is modeled as a geometrically exact rod capable of accommodating arbitrarily large deformations in three dimensions; including extension, shear, flexure and twist. The model is said to be `geometrically exact' because of the lack of restrictions of the allowable deformations, and the full invariance properties of the model under superposed rigid body motions. We show that a (sharp) necessary condition for nonlinear stability is that the whole assemblage be in uniform (stationary) rotation about the shortest axis of a precisely defined `locked' inertia dyadic. Sufficient conditions are obtained by appending the restriction that the angular velocity of the stationary motion be bounded from above by the square root of the minimum eigenvalue of an associated linear operator. Specific examples are worked out, including the case of a rod attached to a rigid body in uniform rotation. Our technique depends crucially on a special choice of variables, introduced in this paper and referred to as the bloch diagonalization procedure, in which the second variation of the energy augmented with the linear and angular momentum block diagonalizes, separating the rotational from the internal vibration modes.

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Additional Information:© 1990 Published by Elsevier Science B.V. All rights reserved. Available online 19 September 2002. PHYSICS REPORTS (Review Section of Physics Letters) 193, No.6 (1990) 279-362. We thank Tony Block, P.S. Krishnaprasad, Debbie Lewis, J. Luo, John Maddocks and Tudor Ratiu for their generous and enthusiastic input to this work. Research supported by AFOSR contract numbers 2-DJA-544 and 2-DJA-771 with Stanford University. Research partially supported by DOE contract DE-AT03-88ER-12097 and MSI at Cornell University.
Funding AgencyGrant Number
Department of EnergyDE-AT03-88ER-12097
MSI/Cornell UniversityUNSPECIFIED
Air Force Office of Scientific Research (AFOSR)2-DJA-544
Air Force Office of Scientific Research (AFOSR)2-DJA-771
Department of EnergyDE-AT03-88ER-12097
MSI at Cornell UniversityUNSPECIFIED
Series Name:Physics Reports
Issue or Number:6
Record Number:CaltechAUTHORS:20100916-075512944
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:19977
Deposited By: Ruth Sustaita
Deposited On:16 Sep 2010 21:51
Last Modified:08 Nov 2021 23:56

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