Formulas for Rotation and Angular Velocity Arising from Shake Table Kinematics and Kirchhoff Rod Model Implementation

This technical note presents a pair of formulas to rotate a unit vector to another unit vector and compute the corresponding angular velocity. These formulas arose independently in two of our recent applications: (1) modeling actuator kinematics in a two-degree-of freedom (2-DOF) shaker, and (2) implementation of Kirchhoff rod theory in three-dimensional (3D) for modeling intracranial aneurysm coiling and analysis of electrical conductor nonlinear dynamics. The first formula (for rotation) can be found in the literature. The second formula (for angular velocity), while simple and straightforward, is not found in the literature and is the contribution in this note. This latter formula, due to its simplicity, is likely useful in a number of other applications in computational mechanics and mechanism/robot kinematics.