Multibody Mechanics/Angular Velocity

Angular velocity is a vector which describes the rotation rate of a reference frame relative to another reference frame.

Typically the Greek letter $$\omega$$ (omega) is used exclusively to represent angular velocities (and many authors also use $$\Omega$$). In this text the angular velocity of frame "B" relative to "A" is denoted as   $${}^A\vec{\omega}^B$$.

The angular velocity vector is defined such that any vector $$\vec{\zeta}$$ which is fixed (constant) in the local frame "B" (reference frame) has its time derivative in another frame "A" expressed with a cross product as   $$\frac{{}^Ad}{dt}\vec{\zeta} = {}^A\vec{\omega}^B \times \vec{\zeta}$$.

This relationship is not difficult to prove, however doing so is of little practical use and is deferred to the end of this discussion. It is useful to note that since vectors can be expressed as linear combinations of basis vectors, this relationship finds its way into of the expression for a the derivative of a general vector (including angular velocity itself).

Angular velocity is not