Rigid Body Kinematics

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Chapter 3 Rigid Body Kinematics

The goal of this chapter is to introduce the topic of the rigid body kinematics. Our focus is on the characterization of the velocity field \(P\in\cB \mapsto \vel_{P/\cA}\) of a rigid body \(\cB\) relative to a referential (or rigid body) \(\cA\text{.}\) We will find out that this vector field behaves like a field of moments, that is, as a vector field satisfying the property \({\bf u}_Q = {\bf u}_P + \bU \times \br_{PQ}\text{.}\) where \(\bU\) is an invariant of the field. We will show that the velocity field \(P\in\cB \mapsto \vel_{P/\cA}\) is entirely specified at any instant by the knowledge of two quantities:

the velocity of a particular point of \(\cB\) and
a vector independent of position, called angular velocity, characterizing the change of orientation of \(\cB\) relative to \(\cA\text{.}\)

We will learn how to determine the angular velocity in terms of the various methods of parametrization of the orientation of a rigid body, introduced in Chapter 1. The notion of angular velocity is pivotal not only in kinematics but also in all tasks involving the time-differentiation of vectors.