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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 $ \underline{\omega}$ . Then the total torque on the rigid body becomes [272]

$\displaystyle \underline{\tau}\eqsp \dot{\underline{L}} + \underline{\omega}\times \underline{L}. \protect$ (B.30)

Similarly, the total external forces on the center of mass become

$\displaystyle \underline{f}\eqsp \dot{\underline{p}} + \underline{\omega}\times\underline{p}.
$

If the body-fixed frame is aligned with the principal axes of rotation (§B.4.16), then the mass moment of inertia tensor is diagonal, say $ \mathbf{I}=$diag$ (I_1,I_2,I_3)$ . In this frame, the angular momentum is simply

$\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]
$

so that the term $ \underline{\omega}\times\underline{L}$ becomes (cf. Eq.$ \,$ (B.15))

\begin{eqnarray*}
\underline{\omega}\times\underline{L}&\!=\!&
\left\vert \begin{array}{ccc}
\underline{e}_1 & \underline{e}_2 & \underline{e}_3\\ [2pt]
\underline{\omega}_1 & \underline{\omega}_2 & \underline{\omega}_3\\ [2pt]
I_1\omega_1 & I_2\omega_2 & I_3\omega_3
\end{array}\right\vert\\ [5pt]
&\!=\!& \mbox{\small$(\underline{\omega}_2 I_3\omega_3 - I_2\omega_2 \underline{\omega}_3)\underline{e}_1 + (\underline{\omega}_3I_1\omega_1 - I_3\omega_3\underline{\omega}_1) \underline{e}_2 + (\underline{\omega}_1I_2\omega_2- I_1\omega_1\underline{\omega}_2) \underline{e}_3$}\\ [5pt]
&\!=\!&
(I_3-I_2)\omega_2\omega_3\,\underline{e}_1 +
(I_1-I_3)\omega_3\omega_1\,\underline{e}_2 +
(I_2-I_1)\omega_1\omega_2\,\underline{e}_3.
\end{eqnarray*}

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\\
\tau_2 &=& I_2 \dot{\omega}_2 + (I_1-I_3)\omega_3\omega_1\\
\tau_3 &=& I_3 \dot{\omega}_3 + (I_2-I_1)\omega_1\omega_2 \end{eqnarray*}

These are call Euler's equations:B.29Since these equations are in the body-fixed frame, $ I_i$ is the mass moment of inertia about principal axis $ i$ , and $ \omega_i$ is the angular velocity about principal axis $ i$ .


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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