On the Razi Acceleration

Abstract

The Razi acceleration is an acceleration term that appears as a result of applying the vector derivative transformation formula in a three relatively rotating coordinate frames system. The Razi term, \((_{A}^{C}\boldsymbol{\omega }_{B} \times _{B}^{C}\boldsymbol{\omega }_{C}) \times ^{C}\mathbf{r}\) appears when the first and the second derivatives of a position vector are taken from two different coordinate frames. This is technically called mixed double derivative transformation. The resulting expression is distinguishable from other types of inertial accelerations such as the Coriolis acceleration \(2_{A}^{C}\boldsymbol{\omega }_{C} \times _{C}^{C}\mathbf{v}\) and the centripetal acceleration \(_{A}^{C}\boldsymbol{\omega }_{C} \times (_{A}^{C}\boldsymbol{\omega }_{C} \times ^{C}\mathbf{r})\). However, the mixed double derivative expression does not give clear dynamical interpretation of the Razi acceleration. This work presents the finding that the Razi term also appears in rigid body rotation about a point. We show that the Razi acceleration has the property of an inertial acceleration, in which it acts on a body in compound rotation motion. This finding shows that the Razi term is another acceleration, along with the Coriolis, centripetal, and tangential accelerations, that appears due to the relative motion of rotating referential coordinate frames.