Derivative kinematics in relatively rotating coordinate frames: investigation on the Razi acceleration

The Razi acceleration has been discovered as a new acceleration term that appears due to relative motion of multiple moving reference frames. When Newton's laws of motion are transformed between a stationary frame and a rotating frame, additional acceleration terms called inertial accelerations, must be introduced to take into account of the acceleration of the rotating frame itself. Traditionally, there are three types of inertial acceleration that acts on a particle in a rotating frame: the centripetal acceleration, the Coriolis acceleration, and the tangential acceleration. Most of the discussion of kinematics of rotating frames is, however, limited to only two frames and often the attention is restricted to two-dimensional models. Introducing a third relatively-rotating frame, the vector derivative transformation between three frames exhibits a peculiar acceleration term, which is written as a triple product of two angular velocity vectors and a position vector. The acceleration term arises when a displacement vector is twice-differentiated using the angular velocities of the rotating frames and transformed into an inertial reference frame. It appears that the Razi acceleration term is present when one transposes the vector derivatives between multiple rotating frames. This discovery leads to a new perspective of analysing the acceleration acting on rigid bodies in compound rotation, such as a tumbling or nutating motion. Three research questions are formulated in investigating the Razi acceleration: (1) How to improve the theory of vector derivatives to produce a complete equation of motion for a rigid body in multiple rotating coordinate frames; (2) What are the components of acceleration for such rigid body motion, and specifically, is the Razi acceleration a necessary inclusion for a complete rigid body acceleration expression, and; (3) Can the Razi acceleration be observed experimentally. This work aims to answer these questions by revising and extending the classical method of Euler derivative transformation between frames. The modification of the method is done for two purposes: to devise a general derivative kinematics formula for systems with an arbitrary number of rotating frames, and to identify the influence and the characteristics of the Razi acceleration. An experiment is setup using an industrial robotic manipulator to simulate compound rotation motion to investigate the appearance of the Razi acceleration in a mechanical system. It has been demonstrated by both mathematical and experimental methods that the acceleration of a rigid body in complex rotation motion can be resolved into its components to show the appearance of the Razi acceleration