Euler's Equations of Motion for Rotation of Rigid Body

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Theorem

Let a rigid body $B$ rotate about an axis $\AA$ which is fixed in relation to $B$ and parallel to the principal axis of inertia of $B$.

Then the rotation of $B$ about $\AA$ is described by:

$\mathbf I \cdot \dot {\boldsymbol \omega} + \boldsymbol \omega \times \paren {\mathbf I \cdot \boldsymbol\omega} = \mathbf M$

where:

$\mathbf M$ is the torque applied to $B$ about $\AA$
$\mathbf I$ is the moment of inertia of $B$ with respect to $\AA$
$\boldsymbol \omega$ is the angular velocity about $\AA$.


Proof




Also presented as

Euler's Equations of Motion for Rotation of Rigid Body can also be seen presented in the form:

\(\ds I_1 \dfrac {\partial \omega_1} {\partial t} - \paren {I_2 - I_3} \omega_2 \omega_3\) \(=\) \(\ds M_1\)
\(\ds I_2 \dfrac {\partial \omega_2} {\partial t} - \paren {I_3 - I_1} \omega_3 \omega_1\) \(=\) \(\ds M_2\)
\(\ds I_3 \dfrac {\partial \omega_3} {\partial t} - \paren {I_1 - I_2} \omega_1 \omega_2\) \(=\) \(\ds M_3\)

where:

$I_1$, $I_2$ and $I_1$ are the components of the torque applied about the principal axes
$I_1$, $I_2$ and $I_1$ are the moments of inertia at fixed point $O$
$\omega_1$, $\omega_2$ and $\omega_3$ are the components of angular velocity along the principal axis.


Source of Name

This entry was named for Leonhard Paul Euler.


Sources