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 and ˆϕ⋅v=ϕ×v for all vectors v. ϕ is called the axial vector of the skew-symmetric matrix ˆϕ. In matrix components relative to the standard Euclidean basis, if ϕ={ϕ1ϕ2ϕ3}T, then [ˆϕ]=[0-ϕ3ϕ2ϕ30-ϕ1-ϕ2ϕ10]. In what follows, ˆa 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 ˆϕ 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[ˆϕ]=I+ˆϕ+12!ˆϕ2+⋯. However, the above infinite series has the following closed-form expression: C=exp[ˆϕ]=cos∥ϕ∥I+sin∥ϕ∥∥ϕ∥ˆϕ+(1-cos∥ϕ∥)∥ϕ∥2ϕϕ. In components, Cij=cosϕδij+(1-cosϕ)pipj+sinϕϵikjpk, where p={p1p2p3}T and ϵijk is the alternator tensor, defined by ϵ123=ϵ231=ϵ312=1; It is this closed-form expression that allows the exact and numerically efficient geometric representation of finite rotations. |