Computational Dynamics, 3rd Edition

Chapter 7. SPATIAL DYNAMICS

In the planar analysis, the rotation of the rigid body can be described using one coordinate, such that the angular velocity of the body is defined as the time derivative of this orientation coordinate. Furthermore, the order of the finite rotation is commutative since the body rotation is about the same axis. Two consecutive rotations can be added and the sequence of performing these rotations is immaterial. One of the principal differences between the planar and the spatial kinematics is due to the complexity of defining the orientation of a body in a three-dimensional space. In the spatial analysis, the unconstrained motion of a rigid body is described using six coordinates; three coordinates describe the translation of a reference point on the body and three coordinates define the body orientation. The order of the finite rotation in the spatial analysis is not commutative and, consequently, the sequence of performing the rotations must be taken into consideration. Moreover, the angular velocities of a rigid body are not the time derivatives of a set of orientation coordinates. These angular velocities, however, can be expressed in terms of a selected set of orientation coordinates and their time derivatives.

In this chapter, methods for describing the orientation of rigid bodies in space are presented. The configuration of the rigid body in a multibody system is described using a set of generalized coordinates that define the global position vector ...