# 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. The following topics are discussed:
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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

$ϕ$ 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

$[ϕ^]=[0-ϕ3ϕ2ϕ30-ϕ1-ϕ2ϕ10].$

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,

$C=exp⁡[ϕ^]=I+ϕ^+12!⁢ϕ^2+⋯.$

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

$(1)C=exp⁡[ϕ^]=cos⁡∥ϕ∥⁢I+sin⁡∥ϕ∥∥ϕ∥⁢ϕ^+(1-cos⁡∥ϕ∥)∥ϕ∥2⁢ϕ⁢ϕ.$

In components,

$Ci⁢j=cos⁡ϕ⁢δi⁢j+(1-cos⁡ϕ)⁢pi⁢pj+sin⁡ϕ⁢ϵi⁢k⁢j⁢pk,$

where $p={p1p2p3}T$ and $ϵi⁢j⁢k$ is the alternator tensor, defined by

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