Euler's Equations in a Rotating Frame

Nd787 C3 L4 A06 Eulers Equations In A Rotating Frame V1

Euler's Rotation Equations

The general form of Euler's equations is:

\mathbf{M = I \dot{\omega} + \omega \times (I\omega)}

and note that the \mathbf{\omega} that shows up in this equation is actually a length 3 vector. In fact it's the vector of body rates you saw earlier:

\mathbf{\omega} = \begin{bmatrix} p \\ q \\ r \end{bmatrix}

and this means that

\dot{\mathbf{\omega}} = \begin{bmatrix} \dot{p} \\ \dot{q} \\ \dot{r} \end{bmatrix}

If you want to know where this equation comes from you might want to start with this wikipedia article but be warned you'll probably have to dig yourself into a pretty deep wikipedia before you'll get to a satisfactory answer.

The "cross product"

The cross product shows up in Euler's equations. It's a way of multiplying two vectors.

If you have two vectors \mathbf{a} and \mathbf{b} with an angle of \theta between them, then the cross product is given by \mathbf{a \times b = c}, where \mathbf{c} is a vector that's perpendicular to both \mathbf{a} and \mathbf{b}.

The direction of this \mathbf{c} vector is given by the right hand rule and the magnitude (size) is given by the following equation:

|\mathbf{c}| = |\mathbf{a}|\ |\mathbf{b}| \sin(\theta)

If you want to learn more, you can check out the wikipedia article on the cross product. It's very good!

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