Euler's Equations for Rotations in the Body-Fixed Frame
Suppose now that the body-fixed frame is rotating in the space-fixed
frame with angular velocity
. Then the total torque on the rigid
body becomes [270]
Similarly, the total external forces on the center of mass become
![$\displaystyle \underline{f}\eqsp \dot{\underline{p}} + \underline{\omega}\times\underline{p}.
$](http://www.dsprelated.com/josimages_new/pasp/img2966.png)
![$ \mathbf{I}=$](http://www.dsprelated.com/josimages_new/pasp/img2967.png)
![$ (I_1,I_2,I_3)$](http://www.dsprelated.com/josimages_new/pasp/img2968.png)
![$\displaystyle \underline{L}\eqsp \left[\begin{array}{c} I_1\omega_1 \\ [2pt] I_2\omega_2 \\ [2pt] I_3\omega_3\end{array}\right]
$](http://www.dsprelated.com/josimages_new/pasp/img2969.png)
![$ \underline{\omega}\times\underline{L}$](http://www.dsprelated.com/josimages_new/pasp/img2970.png)
![$ \,$](http://www.dsprelated.com/josimages_new/pasp/img196.png)
![\begin{eqnarray*}
\underline{\omega}\times\underline{L}&=&
\left\vert \begin{arr...
...1\,\underline{e}_2 +
(I_2-I_1)\omega_1\omega_2\,\underline{e}_3.
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/pasp/img2971.png)
Substituting this result into Eq.(B.30), we obtain the following
equations of angular motion for an object rotating in the body-fixed
frame defined by its three principal axes of rotation:
![\begin{eqnarray*}
\tau_1 &=& I_1 \dot{\omega}_1 + (I_3-I_2)\omega_2\omega_3\\
\...
...a_1\\
\tau_3 &=& I_3 \dot{\omega}_3 + (I_2-I_1)\omega_1\omega_2 \end{eqnarray*}](http://www.dsprelated.com/josimages_new/pasp/img2972.png)
These are call Euler's
equations:B.29Since these equations are in the body-fixed frame, is the mass
moment of inertia about principal axis
, and
is the
angular velocity about principal axis
.
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Examples
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Angular Motion in the Space-Fixed Frame