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.

Appendices

Appendix 1: Alternative Proof of Derivative Transformation Formula

Angular velocity can be defined in terms of time-derivative of rotation matrix. Consider the two coordinate frames system as shown in Fig. 2.2. The angular velocity of B in relative to G, $_{G}\boldsymbol{\omega }_{B}$ is written as follows

$$\displaystyle\begin{array}{rcl} _{G}^{G}\boldsymbol{\omega }_{ B}& =& ^{G}\dot{R}_{ B}\,^{G}R_{ B}^{T}\qquad \mbox{ if expressed in $G$} \\ _{G}^{B}\boldsymbol{\omega }_{ B}& =& ^{G}R_{ B}^{T}\,^{G}\dot{R}_{ B}\qquad \mbox{ if expressed in $B$}{}\end{array}$$

(2.52)

To prove Eq. (2.52), we start with the relation between a vector expressed in two arbitrary coordinate frames G and B.

$$\displaystyle{ ^{G}\mathbf{r} = ^{G}R_{ B}\,^{B}\mathbf{r} }$$

(2.53)

Finding the derivative of the G-vector from G, we get

$$\displaystyle{ _{G}^{G}\dot{\mathbf{r}} = ^{G}\dot{R}_{ B}\,^{B}\mathbf{r} = ^{G}\dot{R}_{ B}\,^{G}R_{ B}^{T}\,^{G}\mathbf{r} = _{ G}^{G}\boldsymbol{\omega }_{ B} \times ^{G}\mathbf{r} }$$

(2.54)

Here, it is assumed that the vector^B r is constant in B. If it is not constant, we have to include the simple derivative of the vector in B and rotate it to G

$$\displaystyle{ _{G}^{G}\dot{\mathbf{r}} = ^{G}\dot{R}_{ B}\,^{B}\mathbf{r} + ^{G}R_{ B}\,_{B}^{B}\dot{\mathbf{r}} = _{ G}^{G}\boldsymbol{\omega }_{ B} \times ^{G}\mathbf{r} + _{ B}^{G}\dot{\mathbf{r}} }$$

(2.55)

The angular velocities between two arbitrary frames are equal and oppositely directed given that both are expressed in the same frame

$$\displaystyle{ _{G}\boldsymbol{\omega }_{B} = -_{B}\boldsymbol{\omega }_{G} }$$

(2.56)

Using this angular velocity property, rearranging Eq. (2.54) gives

$$\displaystyle{ _{B}^{G}\dot{\mathbf{r}} = _{ G}^{G}\dot{\mathbf{r}} + _{ B}^{G}\boldsymbol{\omega }_{ G} \times ^{G}\mathbf{r} }$$

(2.57)

Appendix 2: Proof of the Kinematic Chain Rule for Angular Acceleration Vectors

To prove Eq. (2.29), we start with a composition of n angular velocity vectors which are all expressed in an arbitrary frame g.

$$\displaystyle\begin{array}{rcl}{}_{0}^{g}\boldsymbol{\omega }_{ n}& =& \sum \limits _{i=1}^{n}{}_{ i-1}^{\quad g}\boldsymbol{\omega }_{ i} ={}_{0}^{g}\boldsymbol{\omega }_{ 1} +{}_{1}^{g}\boldsymbol{\omega }_{ 2} + \cdots +{}_{n-1}^{\quad g}\boldsymbol{\omega }_{ n} \\ & =&{}^{g}R_{ 1}\,{}_{0}^{1}\boldsymbol{\omega }_{ 1} +{}^{g}R_{ 2}\,{}_{1}^{2}\boldsymbol{\omega }_{ 2} + \cdots +{}^{g}R_{ n}\,{}_{(n-1)}^{\quad n}\boldsymbol{\omega }_{ n}\quad {}\end{array}$$

(2.58)

We use the technique of differentiating from another frame in Appendix 1. Differentiating the angular velocities from an arbitrary frame g gives

$$\displaystyle\begin{array}{rcl} \frac{{}^{g}d} {dt}\sum \limits _{i=1}^{n}{}_{ i-1}^{\quad g}\boldsymbol{\omega }_{ i}& =& \frac{{}^{g}d} {dt}\left ({}^{g}R_{ 1}\,_{0}^{1}\boldsymbol{\omega }_{ 1} + ^{g}R_{ 2}\,_{1}^{2}\boldsymbol{\omega }_{ 2} + \cdots + ^{g}R_{ n}\,_{(n-1)}^{\quad n}\boldsymbol{\omega }_{ n}\right ) \\ & =& ({}^{g}R_{ 1}\,_{0}^{1}\boldsymbol{\alpha }_{ 1} + ^{g}\dot{R}_{ 1}\,_{0}^{1}\boldsymbol{\omega }_{ 1}) + ({}^{g}R_{ 2}\,_{1}^{2}\boldsymbol{\alpha }_{ 2} + ^{g}\dot{R}_{ 2}\,_{1}^{2}\boldsymbol{\omega }_{ 2}) + \cdots \\ & & \cdots + ({}^{g}R_{ n}\,_{(n-1)}^{\quad n}\boldsymbol{\alpha }_{ n} + ^{g}\dot{R}_{ n}\,_{(n-1)}^{\quad n}\boldsymbol{\omega }_{ n}) \\ & =& ({}^{g}R_{ 1}\,_{0}^{1}\boldsymbol{\alpha }_{ 1} + ^{g}\dot{R}_{ 1}\,^{g}R_{ 1}^{T}\,_{ 0}^{g}\boldsymbol{\omega }_{ 1}) + ({}^{g}R_{ 2}\,_{1}^{2}\boldsymbol{\alpha }_{ 2} + ^{g}\dot{R}_{ 2}\,^{g}R_{ 2}^{T}\,_{ 1}^{g}\boldsymbol{\omega }_{ 2}) + \cdots \\ & & \cdots + ({}^{g}R_{ n}\,_{(n-1)}^{\quad n}\boldsymbol{\alpha }_{ n} + ^{g}\dot{R}_{ n}\,^{g}R_{ n}^{T}\,_{ (n-1)}^{\quad g}\boldsymbol{\omega }_{ n}) \\ & =& (_{0}^{g}\boldsymbol{\alpha }_{ 1} + _{g}^{g}\boldsymbol{\omega }_{ 1} \times _{0}^{g}\boldsymbol{\omega }_{ 1}) + (_{1}^{g}\boldsymbol{\alpha }_{ 2}) + _{g}^{g}\boldsymbol{\omega }_{ 2} \times _{1}^{g}\boldsymbol{\omega }_{ 2}) + \cdots \\ & & \cdots + (_{(n-1)}^{\quad g}\boldsymbol{\alpha }_{ n} + _{g}^{g}\boldsymbol{\omega }_{ n} \times _{(n-1)}^{\quad g}\boldsymbol{\omega }_{ n}) {}\end{array}$$

(2.59)

The whole expression can be transformed to any arbitrary frame f by applying the rotation matrix from g to f,^f R _g .

$$\displaystyle\begin{array}{rcl} \frac{{}^{g}d} {dt}\left (\sum \limits _{i=1}^{n}{}_{ i-1}^{\quad f}\boldsymbol{\omega }_{ i}\right )& =& \left (^{f}R_{ g}\frac{{}^{g}d} {dt}\sum \limits _{i=1}^{n}{}_{ i-1}^{\quad g}\boldsymbol{\omega }_{ i}\right ) \\ & =& \frac{{}^{g}d} {dt}\left (^{f}R_{ g}\sum \limits _{i=1}^{n}{}_{ i-1}^{\quad g}\boldsymbol{\omega }_{ i}\right ) \\ & =& \sum \limits _{i=1}^{n}{}_{ i-1}^{\quad f}\boldsymbol{\alpha }_{ i} + _{g}^{f}\boldsymbol{\omega }_{ i} \times _{i-1}^{\quad f}\boldsymbol{\omega }_{ i}{}\end{array}$$

(2.60)

2.1 Notations

In this article, the following symbols and notations are being used. Lowercase bold letters are used to indicate a vector. Dot accents on a vector indicate that is a time-derivative of the vector.

• r, v (or $\dot{\mathbf{r}}$), and a (or $\ddot{\mathbf{r}}$) are the position, velocity, and acceleration vectors

• $\boldsymbol{\omega }$ and $\boldsymbol{\alpha }$ (or $\dot{\boldsymbol{\omega }}$) are angular velocity and angular acceleration vectors.

Capital letter A, B, C, and G are used to denote a reference frame. In examples where only B and G are used, the former indicates a rotating, body coordinate frame and the latter indicates the global-fixed, inertial body frame. When a reference frame is introduced, its origin point and three basis vectors are indicated. For example:

$$\displaystyle{ A(O\mathbf{a}_{1}\mathbf{a}_{2}\mathbf{a}_{3})\quad G(OXY Z)\quad B(o\hat{\imath}\hat{\jmath}\hat{k}) }$$

Capital letters R is reserved for rotation matrix and transformation matrix. The right subscript on a rotation matrix indicates its departure frame, and the left superscript indicates its destination frame. For example:

$$\displaystyle{ ^{B}R_{ A} = \mbox{ rotation matrix from frame $A$ to frame $B$} }$$

Left superscript is used to denote the coordinate frame in which the vector is expressed. For example, if a vector r is expressed in an arbitrary coordinate frame A which has the basis vector $(\hat{\imath},\hat{\jmath},\hat{k})$, it is written as

$$\displaystyle\begin{array}{rcl} ^{A}\mathbf{r}& =& r_{ 1}\hat{\imath} + r_{2}\hat{\jmath} + r_{3}\hat{k} {}\\ & =& \mbox{ vector $\mathbf{r}$ expressed in frame $A$} {}\\ \end{array}$$

Left subscript is used to denote the coordinate frame from which the vector is differentiated. For example,

$$\displaystyle\begin{array}{rcl} _{B}^{A}\dot{\mathbf{r}}& =& \frac{{}^{V }d} {dt} {{}^{A}\mathbf{r}} {}\\ & =& \mbox{ vector $\mathbf{r}$ expressed in frame $A$ and differentiated in frame $B$} {}\\ _{BB}^{\quad A}\ddot{\mathbf{r}}& =& \frac{{}^{B}d} {dt} \frac{{}^{B}d} {dt} {{}^{A}\mathbf{r}} {}\\ & =& \mbox{ vector $\mathbf{r}$ expressed in frame $A$ and differentiated twice in frame B}{}\\ \end{array}$$

$$\displaystyle{ _{CB}^{\quad A}\ddot{\mathbf{r}} = \frac{{}^{C}d} {dt} \frac{{}^{B}d} {dt} {{}^{A}\mathbf{r}} }$$

$$\displaystyle\begin{array}{rcl} & =& \mbox{ vector $\mathbf{r}$ expressed in frame $A$ and differentiated in frame B} {}\\ & & \mbox{ and differentiated again in frame $C$} {}\\ \end{array}$$

Right subscript is used to denote the coordinate frame to which the vector is referred, and left subscript is the coordinate frame to which it is related. For example, the angular velocity of frame A with respect to frame B is written as

$$\displaystyle{ _{B}\boldsymbol{\omega }_{A} = \mbox{ angular velocity vector of $A$ with respect to $B$} }$$

If the coordinate frame in which the angular velocity is expressed is specified

$$\displaystyle\begin{array}{rcl} _{B}^{C}\boldsymbol{\omega }_{ A}& =& \mbox{ angular velocity vector of $A$ with respect to $B$} {}\\ & & \mbox{ expressed in $C$} {}\\ \end{array}$$

Key Symbols

$\mathbf{a =}\ddot{\mathbf{r}}$ :

Acceleration vector

$\mathbf{v =}\dot{\mathbf{r}}$ :

Velocity vector

r :

Displacement vector

α :

Angular acceleration vector

ω :

Angular velocity vector

F :

Force vector

m :

Mass scalar

n :

Number of elements in the set of noninertial coordinate frames

i :

ith element in a set

f, g :

Elements in the set of noninertial coordinate frames

$\hat{I},\hat{J},\hat{K}$ :

Orthonormal unit vectors of an inertial frame

$\hat{\imath},\hat{\jmath},\hat{k}$ :

Orthonormal unit vectors of a noninertial frame

O :

Origin point of a coordinate frame

$\dot{\theta }$ :

Angular speed of body (spin)

$\dot{\phi }$ :

Angular speed of body (precession)

ψ :

Nutation angle, angle between spin vector and precession vector

R :

Rotation transformation matrix

 □ :

Arbitrary vector

^A r :

Vector r expressed in A-frame

$_{B}^{A}\mathbf{\dot{r}}$ :

Vector $\dot{\mathbf{r}}$ expressed in A-frame and differentiated in B-frame

$_{CB}^{\,\,\,\,A}\mathbf{\dot{r}}$ :

Vector $\ddot{\mathbf{r}}$ expressed in A-frame, first-differentiated in B-frame, and second-differentiated from C-frame

^A R _B :

Rotation transformation matrix from B-frame to A-frame