ABSTRACT
This chapter presents a summary of representations of large rotations in three dimensions, with several applications in mind: (1) updating large nodal rotational degrees of freedom, (2) initial orientation of element coordinate systems in the global coordinate system via rotation matrices, and (3) separating large element rigid body motions from small deformations via a corotational (CR) finite element formulation. The rotation matrix R can also be expressed as a matrix exponential known as the exponential map. The chapter describes two methods for orienting element local coordinate systems within the common global 3D Cartesian coordinate system. It also uses Euler angles to determine the rotation matrix that corresponds to its orientation in terms of the global Cartesian coordinates of its nodes. The orientation of an element frame generally depends only on the global coordinates of the nodes to which the element is connected, and is independent of nodal rotations. The chapter addresses the incremental variation of a nodal triad R.
