ProductsAbaqus/StandardAbaqus/Explicit 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 ˆϕ
associated with ϕ
by the relationships
ˆϕ⋅ϕ=0
is called the axial vector of the skew-symmetric matrix
.
In matrix components relative to the standard Euclidean basis, if
,
then
In what follows,
will be used to denote the skew-symmetric matrix with axial vector
.
A well-known result from linear algebra is that the exponential of a
skew-symmetric matrix
is an orthogonal (rotation) matrix that produces the finite rotation
.
Let the rotation matrix be ,
such that .
Then by definition,
However, the above infinite series has the following closed-form expression:
In components,
where
and
is the alternator tensor, defined by
It is this closed-form expression that allows the exact and numerically
efficient geometric representation of finite rotations.
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