In this chapter, we present and develop some kinematical relations that will be needed later. Most of what is presented in this chapter pertains to the kinematics of points, rigid bodies, and reference frames. Within the scope of kinematics, there is no distinction between rigid bodies and reference frames. The methodology outlined herein is essentially that of Kane and his co-workers, with slight variations and additions; see Kane (1968), Kane et al. (1983), and Kane and Levinson (1985).

The motivation for dealing initially with rigid-body kinematics is that the nonlinear analysis of beams can be heuristically decomposed into two parts. The first is a potentially large rigid-body displacement and rotation of the set of material points that make up the reference cross-section. The second is a relatively small elastic deformation superimposed on the first. We will make no approximations concerning the rigid-body part, which is spatially dependent only on the geometry of a line, called the beam reference line, and a frame, the orientation of which varies along this line. Thus, the approximations only affect the three-dimensional deformation of the material surrounding the reference line.

We will first define some basic terms regarding points, frames, rigid bodies, vectors, dyadics, and finite rotation. We will then deal with angular velocity and differentiation of vectors and, similarly, with virtual rotation and variation of vectors. Having this foundation in place then allows us to cover the primitives related to velocity and virtual displacement. To facilitate a matrix-based treatment, we will make use...