Section 6.6 Kinematics: Summary
We end this chapter by summarizing the methods we may apply for the determination of the velocity and acceleration of a particular point \(Q\) in motion relative to referential \(\cA\text{.}\) Denote by \(A\) a point attached to \(\cA\) at all times.
Method 1: Assume that Point \(Q\) is attached to a rigid body \(\cB\) at all times. Then, the velocity of \(Q\) is found most efficiently by first resolving the position vector \(\br_{AQ}\) along some unit vectors \(\bu\text{,}\) \(\bv\text{,}\) \(\bw\text{,}\) ... which may not necessarily be attached to \(\cA\) or \(\cB\text{:}\)
\begin{equation*}
\br_{AQ} = U(t) \bu + V(t) \bv + W(t) \bw + \cdots
\end{equation*}
Then the velocity of \(Q\) relative to \(\cA\) is obtained as
\begin{align*}
\vel_{Q\in \cB / \cA} \amp = \vel_{Q/\cA} = \left( {d \br_{AQ} \over dt} \right)_\cA \\
\amp = \dot{U} \bu + U \left( {d \bu \over dt} \right)_\cA +\dot{V} \bv + V \left( \frac{d \bv}{dt} \right)_\cA
+\dot{W} \bw + W \left( \frac{d\bw}{dt} \right)_\cA
+ \cdots
\end{align*}
The time-derivatives \((d\bu /dt)_{\cA}\text{,}\) \((d\bv /dt)_{\cA}\) , \((d \bw /dt)_{\cA}\) , ..., are determined by using the concept of angular velocity:
\begin{equation*}
\left( {d \bu \over dt} \right)_\cA = \bom_{\cU /\cA} \times \bu, \qquad
\left( {d \bv \over dt} \right)_\cA = \bom_{\cV /\cA} \times \bv, \quad \cdots
\end{equation*}
where \(\cU\) (resp. \(\cV\)) is an auxiliary referential relative to which \(\bu\) (resp. \(\bv\)) is fixed. The acceleration \(\ba_{Q\in \cB / \cA}= \ba_{Q/\cA}\) of \(Q\) is then simply found as \((d \vel_{Q/\cA} /dt)_{\cA}\) by following the process just outlined.
Method 2: Point \(Q\) is not attached to body \(\cB\) at all times. Then \(\vel_{Q\in\cB/\cA}\) is the transport velocity of \(Q\) by body \(\cB\text{.}\) It must be viewed as the velocity of a particular point instantaeously attached to \(\cB\text{.}\) This velocity cannot be obtained as \((d\br_{AQ} /dt)_\cA\text{.}\) Instead, it must be obtained by using the kinematic screw \(\{\cV_{\cB/\cA} \}\text{,}\) that is, by relating it to the velocity of another point, say \(B\text{,}\) attached to \(\cB\) at all times:
\begin{equation*}
\vel_{Q\in\cB/\cA}= \vel_{B /\cA}+ \bom_{\cB/\cA} \times \br_{BQ}
\end{equation*}
where \(\vel_{B/\cA}\) is a known quantity (typically determined according to Method 1). Likewise, the transport acceleration \(\ba_{Q\in\cB/\cA}\) can be determined from \(\ba_{B /\cA}\) according to
\begin{equation*}
\ba_{Q\in\cB/\cA} = \ba_{B/\cA} + \bal_{\cB /\cA} \times \br_{BQ}
+ \bom_{\cB/\cA} \times (\bom_{\cB/\cA} \times \br_{BQ})
\end{equation*}
Note that another way to determine transport velocity \(\vel_{Q\in\cB/\cA}\) is to use formula (6.2.1):
\begin{equation*}
\vel_{Q\in\cB/\cA}= \vel_{Q/\cA} - \vel_{Q/\cB}
\end{equation*}
where \(\vel_{Q/\cA}\) and \(\vel_{Q/\cB}\) are determined according to (assuming point \(A\) is fixed in \(\cA\) and point \(B\) is fixed in \(\cB\) at all times):
\begin{equation*}
\vel_{Q/\cA}= \left( {d \br_{AQ} \over dt} \right)_\cA , \qquad
\vel_{Q/\cB}= \left( {d \br_{BQ} \over dt} \right)_\cB
\end{equation*}
where each position vector is differentiated as outlined in Method 1. A similar method can be followed for the transport acceleration by using formula (6.3.2):
\begin{equation*}
\ba_{Q\in\cB/\cA}= \ba_{Q/\cA} - \ba_{Q/\cB} -2 \bom_{\cB/\cA} \times \vel_{Q/\cB}
\end{equation*}
Method 3: (transport) velocity \(\vel_{Q\in\cB/\cC}\) is known relative to some referential \(\cC\) whose motion is also known relative to referentials \(\cA\) and \(\cB\text{.}\) Then \(\vel_{Q\in\cB/\cA}\) can be obtained by applying loop formula (6.5.3):
\begin{equation*}
\vel_{Q\in\cB/\cA}= \vel_{Q\in\cB/\cC} + \vel_{Q\in\cC/\cA }
\end{equation*}
where \(\vel_{Q\in\cC/\cA }\) is determined from the knowledge of \(\{\cV_{\cC/\cA} \}\text{.}\) Note that no such rule can be expressed for the acceleration. However, application of formula (6.3.2) gives
\begin{equation*}
\ba_{Q\in\cB/\cA}= \ba_{Q\in\cB/\cC} + \ba_{Q\in\cC/\cA }+ 2 \bom_{\cC/\cA} \times
\vel_{Q\in \cB/\cC}
\end{equation*}
