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Embedded SystemsFPGAElectronics

Physical Audio Signal Processing
    Elementary Physics, Mechanics, and Acoustics
       Rigid-Body Dynamics
          Equations of Motion for Rigid Bodies
             Euler's Equations for Rotations in the Body-Fixed Frame

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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 [270]

$\displaystyle \underline{\tau}\eqsp \dot{\underline{L}} + \underline{\omega}\times \underline{L}. \protect$ (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{arr... ...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\\ \... ...a_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$.

Previous: Angular Motion in the Space-Fixed Frame
Next: Examples

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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