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Angular velocity and rotation matrix

  1. Aug 17, 2011 #1
    Hello. Sorry for my English

    There are [itex]R[/itex] - rotation matrix (that performs transformation from associated coordinate system [itex]IE[/itex] to static coordinate system [itex]OI[/itex]) and [itex]\omega[/itex] - angular velocity. The matrix [itex]R[/itex] depends on parameters [itex]\xi[/itex] (for example, Euler angles). I need to express [itex]\omega[/itex] as function of [itex]\xi[/itex].

    Let [itex]r^e[/itex] - components of vector [itex]r[/itex] in the associated coordinate system: [itex]r=Rr^{e}[/itex] and [itex]r^{e}=R^{T}r[/itex]. Than variation of vector [itex]r[/itex]:
    [itex]\delta r=\sum_{i}\frac{\partial Rr^{e}}{\partial\xi_{i}}\delta\xi_{i}=[/itex] [itex]\sum_{i}\left(\frac{\partial R}{\partial\xi_{i}}\delta\xi_{i}\right)r^{e}=[/itex] [itex]\sum_{i}\left(\frac{\partial R}{\partial\xi_{i}}\delta\xi_{i}\right)R^{T}r[/itex]

    On the other hand if I rotate [itex]r[/itex] about [itex]l[/itex] on angle [itex]\Delta\varphi[/itex] then variation of [itex]r[/itex] equals: [itex]\delta r=\Delta\varphi\left[l\times r\right]=\delta\varphi\times r[/itex], where [itex]\delta\varphi = \Delta \varphi l[/itex].

    Comparing [itex]\delta r=\delta\varphi\times r[/itex] with [itex]\delta r=\sum_{i}\left(\frac{\partial R}{\partial\xi_{i}}\delta\xi_{i}\right)R^{T}r[/itex], we get the following expression for omega:
    [itex]\omega = \frac{\delta \varphi}{d t}[/itex]
    [itex]\left[\omega\times\right]=\left(\begin{array}{ccc}0 & -\omega_{3} & \omega_{2}\\
    \omega_{3} & 0 & -\omega_{1}\\
    -\omega_{2} & \omega_{1} & 0\end{array}\right)=\sum_{i}\left(\frac{\partial R}{\partial\xi_{i}}\dot{\xi_{i}}\right)R^{T}[/itex]

    Is it right?
  2. jcsd
  3. Aug 19, 2011 #2


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  4. Aug 22, 2011 #3
    I found the answer on my question at http://en.wikipedia.org/wiki/Angular_velocity" [Broken] :
    But [itex]W[/itex] is not a tensor, [itex]W[/itex] is a pseudotensor: [itex]W_{ij} = e_{iwj} \omega_{w}[/itex].
    Last edited by a moderator: May 5, 2017
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