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Rotation: A review of useful theorems involving proper orthogonal matrices referenced to threedimensional physical space
Rotation: A review of useful theorems involving proper orthogonal matrices referenced to threedimensional physical space Summary:Rebecca M. Brannon Computational Physics and Mechanics Sandia National Laboratories Albuquerque, NM 87185-0820 Publicly available draft of May 9, 2002 Useful and/or little-known theorems involving proper orthogonal matrices are reviewed. Orthogonal matrices appear in the transformation of tensor components from one orthogonal basis to another. The distinction between an orthogonal direction cosine matrix and a rotation operation is discussed. Among the theorems and techniques presented are (1) various ways to characterize a rotation including proper orthogonal tensors, dyadics, Euler angles, axis/angle representations, series expansions, and quaternions; (2) the Euler-Rodrigues formula for converting axis and angle to a rotation tensor; (3) the distinction between rotations and reflections, along with implications for “handedness” of coordinate systems; (4) non-commutivity of sequential rotations, (5) eigenvalues and eigenvectors of a rotation; (6) the polar decomposition theorem for expressing a general deformation as a sequence of shape and volume changes in combination with pure rotations; (7) mixing rotations in Eulerian hydrocodes or interpolating rotations in discrete field approximations; (8) Rates of rotation and the difference between spin and vorticity, (9) Random rotations for simulating crystal distributions; (10) The principle of material frame indifference (PMFI); and (11) a tensor-analysis presentation of classical rigid body mechanics, including direct notation expressions for momentum and energy and the extremely compact direct notation formulation of Euler’s equations (i.e., Newton’s law for rigid bodies). Computer source code is provided for several rotation-related algorithms. password: ZmQ2ZWQK Please select one mirror to download
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Sponsored LinksRotation: A review of useful theorems involving proper orthogonal matrices referenced to threedimensional physical space Keywordsorthogonal rotation rotations matrices theorems involving angle including distinction axis mechanics rigid referenced review tensor notation euler physical rates orthogonal matrices theorems involving matrices referenced dyadics euler euler angles tensors dyadics orthogonal tensors rotation including angle representations series expansions orthogonal matrices referencedBookmark Rotation: A review of useful theorems involving proper orthogonal matrices referenced to threedimensional physical spaceHyperlink code:
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