Relative Motion Analysis: Acceleration

Section 6.3 Relative Motion Analysis: Acceleration

We can derive a similar relationship between the accelerations of point \(P\) relative to \(\cE\) and \(\cF\text{.}\) We start by taking the time derivative in \(\cE\) of equation (6.2.1) to obtain

\begin{equation*} \ba_{P/\cE} = \left( {d \over dt} \vel_{P/\cF} \right)_{\cE}+ \left( {d\over dt} \vel_{P\in \cF / \cE} \right)_{\cE} \end{equation*}

First we relate the derivative of \(\vel_{P/\cF}\) in \(\cE\) to that in \(\cF\) according to formula (3.1.3):

\begin{equation*} \left( {d \over dt}\vel_{P/\cF} \right)_{\cE}= \left( {d\over dt} \vel_{P/\cF} \right)_{\cF} + \bom_{\cF/\cE}\times\vel_{P/\cF} = \ba_{P/\cF} + \bom_{\cF/\cE}\times\vel_{P/\cF} . \end{equation*}

To find the derivative \((d \vel_{P\in \cF /\cE} / dt )_{\cE}\text{,}\) we differentiate the equation \(\vel_{P\in \cF /\cE}=\vel_{B /\cE} +\bom_{\cF/\cE}\times \br_{BP}\) term by term to find:

\begin{equation*} \left( {d \over dt}\vel_{P\in \cF / \cE} \right)_{\cE} = \ba_{B /\cE} + \bal_{\cF/\cE}\times \br_{BP} + \bom_{\cF/\cE}\times ( \vel_{P/\cF} + \bom_{\cF/\cE}\times \br_{BP}) \end{equation*}

where we have used \((d \br_{BP} / dt )_{\cE} = \vel_{P/\cF} + \bom_{\cF/\cE}\times \br_{BP}\text{.}\) We then recognize the sum

\begin{equation*} \ba_{B /\cE} + \bal_{\cF/\cE}\times \br_{BP} + \bom_{\cF/\cE}\times (\bom_{\cF/\cE}\times \br_{B P}) \end{equation*}

as the transport acceleration \(\ba_{P\in\cF / \cE}\) of point \(P\) in \(\cF\) according to equation (6.1.2). Hence, we have shown that the time-derivative of the transport velocity \(\vel_{P\in\cF / \cE}\) is related to the corresponding transport acceleration \(\ba_{P\in\cF / \cE}\) according to

\begin{equation} \frac{d}{dt} \vel_{P\in \cF / \cE} \Big|_{\cE} = \ba_{P\in\cF / \cE} + \bom_{\cF/\cE}\times\vel_{P/\cF}\tag{6.3.1} \end{equation}

Finally, after regrouping all terms, we find the change of referential formula for accelerations:

Theorem 6.3.1. Change of Referential (Acceleration).

Given two referentials \(\cE\) and \(\cF\) in relative motion, the acceleration \(\ba_{P/\cE}\) of a particle \(P\) relative to \(\cE\) can be found from its acceleration \(\ba_{P/\cF}\) relative to \(\cF\) according to the formula

\begin{equation} \ba_{P/\cE} = \ba_{P/\cF} + \ba_{P\in\cF / \cE} + 2 \bom_{\cF/\cE} \times\vel_{P/\cF}\tag{6.3.2} \end{equation}

where

\(\ba_{P\in \cF / \cE}\) is the transport acceleration of \(P\) by \(\cF\) relative to \(\cE\text{,}\)
\(2 \bom_{\cF/\cE} \times\vel_{P/\cF}\) is referred to Coriolis acceleration of \(P\text{,}\)
the accelerations vectors \(\ba_{P/\cE}\) and \(\ba_{P/\cF}\) are commonly called absolute acceleration and relative acceleration, respectively.

Remark 6.3.2.

The derivative (relative to \(\cE\)) of the transport velocity is not equal to the transport acceleration as clearly shown by formula (6.3.1). Unless of course \(\vel_{P/\cF} = {\bf 0}\text{,}\) that is, when \(P\) is attached to \(\cF\) at all times, in which case we have

\begin{equation*} \ba_{P\in\cF / \cE} = \ba_{P/\cE} = \left( {d \over dt}\vel_{P\in \cF / \cE} \right)_{\cE} \end{equation*}

Remark 6.3.3.

As will be seen in the examples below, the application of the change of referential formulas requires in general more labor than the direct approach of taking the time derivative of \(\br_{OP}\) and \(\vel_{P/\cE}\) to find \(\vel_{P/\cE}\) and \(\ba_{P/\cE}\text{,}\) respectively.

Remark 6.3.4.

In practice, formulas (6.2.1)-(6.3.2) are fundamental to problems of dynamics in which the Earth’s rotation must be taken into account. See Chapter 16 for examples.

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