Skip to main content Contents Index Prev Up Next \(\newcommand{\val}{Val\text{00E9}ry}
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\)
Section 4.9 Time-derivative of Screws
All vector fields and screws encountered in rigid body mechanics are time-dependent. Thus, it is relevant to examine their time-derivative. Consider screw \(\{\cV\}\) corresponding to time-dependent vector field \(P \mapsto \bvv_P (t)\) satisfying the equation: \(\bvv_Q = \bvv_P + \bV\times \br_{PQ}\) for any two points \(P\) and \(Q\) in motion relative to a referential \(\cE\) (of origin \(O\) ). Time-differentiation gives
\begin{equation*}
\frac{d\bvv_Q}{dt}
=
\frac{d\bvv_P}{dt}
+
\frac{d\bV}{dt}\times \br_{PQ}
+ \bV\times\frac{d\br_{PQ} }{dt}
\end{equation*}
where all time-derivatives are performed relative to \(\cE\text{.}\) After substituting \((d\br_{PQ} /dt) = (d\br_{OQ} /dt)- (d\br_{OP} /dt)\) we obtain
\begin{equation*}
\left( \frac{d\bvv_Q}{dt}+ \frac{d\br_{OQ} }{dt} \times\bV \right)
=
\left( \frac{d\bvv_P}{dt} +\frac{d\br_{OP} }{dt} \times\bV \right)
+
\frac{d\bV}{dt}\times \br_{PQ}
\end{equation*}
This last result shows that the field \(P \mapsto (d\bvv_P /dt) +
(d\br_{OP} /dt)\times\bV\) defines a screw of resultant \(\frac{d\bV}{dt}\text{.}\)
Definition 4.9.1 . Time-derivative of a screw.
The time-derivative of screw \(\{\cV\}\) relative to referential \(\cE\) of origin \(O\text{,}\) denoted \(\{\frac{d}{dt} \cV \}_\cE\) is the screw defined by
\begin{equation}
\Big\{ \frac{d}{dt}\cV \Big\}_\cE =
\left\{
\begin{array}{c}
\displaystyle \frac{d\bV}{dt} \\
\; \\
\displaystyle \frac{d {\bf v}_A }{dt} + \frac{d\br_{OA} }{dt}\times\bV
\end{array}
\right\}_A\tag{4.9.1}
\end{equation}
where all derivatives are performed relative to \(\cE\text{.}\)
