Consider a rotor as observed in a space-fixed frame. Let the initial position of a material point be denoted by X. Since the rotor rotates about its axis, the position of the material point X changes in time, even if the rotor is not undergoing any deformation. The current position xR of the material point X without deformation is thus given by

where Xbp is a point located on the axis of rotation, R(t) is the rotation matrix due to the axial rotation. When also including the deformation, the current position x of the material point X is given by

where u(xR(t), t) is the displacement of the point at xR(t) as observed in the space-fixed frame. The velocity of this point can be expressed as

Here, W is a skew-symmetric tensor corresponding to the angular velocity vector Ω, defined as

The components of the skew-symmetric tensor W in terms of components of angular velocity vector are given as

Thus, the tensor W operating on a vector has the same effect as taking the cross product of Ω with the same vector.