Rotation variables

Abaqus contains capabilities such as structural elements (beams and shells) for which it is necessary to define arbitrarily large magnitudes of rotation; therefore, a convenient method for storing the rotation at a node is required. The components of a rotation vector ϕ are stored as the degrees of freedom 4, 5, and 6 at any node where a rotation is required.

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The finite rotation vector, ϕ, consists of a rotation magnitude, ϕ=ϕ, and a rotation axis or direction in space, p=ϕ/ϕ. Physically, the rotation ϕ is interpreted as a rotation by ϕ radians around the axis p. To characterize this finite rotation mathematically, the rotation vector ϕ is used to define an orthogonal transformation or rotation matrix. To do so, first define the skew-symmetric matrix ϕ^ associated with ϕ by the relationships

ϕ^ϕ=0    and    ϕ^v=ϕ×v    for all vectors    v.

ϕ is called the axial vector of the skew-symmetric matrix ϕ^. In matrix components relative to the standard Euclidean basis, if ϕ={ϕ1ϕ2ϕ3}T, then


In what follows, a^ will be used to denote the skew-symmetric matrix with axial vector a.

A well-known result from linear algebra is that the exponential of a skew-symmetric matrix ϕ^ is an orthogonal (rotation) matrix that produces the finite rotation ϕ. Let the rotation matrix be C, such that C-1=CT. Then by definition,


However, the above infinite series has the following closed-form expression:


In components,


where p={p1p2p3}T and ϵijk is the alternator tensor, defined by

ϵ123=ϵ231=ϵ312=1;    ϵ132=ϵ213=ϵ321=-1; all other ϵijk=0.

It is this closed-form expression that allows the exact and numerically efficient geometric representation of finite rotations.