Rotation manifold <Emphasis Type="Italic">SO</Emphasis>(3) and its tangential vectors |
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Authors: | Jari Mäkinen |
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Affiliation: | (1) Department of Mechanics and Design, Tampere University of Technology, P.O.Box 589, 33101 Tampere, Finland |
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Abstract: | In this paper, we prove that incremental material rotation vectors belong to different tangent spaces of the rotation manifold
SO(3) at a different instant. Moreover, we show that the material tangent space as the tangent space at unity is not a possible
definition yielding geometrically inconsistent results, although this kind of definition is widely adopted in applied mechanics
community. In addition, we show that the standard Newmark integration scheme for incremental rotations neglects first order
terms of rotation vector, not third order terms. Finally, we show that the rotation interpolation of extracted nodal values
on the rotation manifold is not an objective interpolation under the observer transformation. This clarifies controversy about
the frame-indifference of geometrically exact beam formulations in their finite element implementations. |
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Keywords: | Finite rotation Rotation manifold Rotation interpolation Objectivity Newmark scheme |
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