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. |