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Angular Velocity Vector

When working with rotations, it is convenient to define the angular-velocity vector as a vector $ \underline{\omega}\in\mathbb{R}^3$ pointing along the axis of rotation. There are two directions we could choose from, so we pick the one corresponding to the right-hand rule, i.e., when the fingers of the right hand curl in the direction of the rotation, the thumb points in the direction of the angular velocity vector.B.18 The length $ \vert\vert\,\underline{\omega}\,\vert\vert $ should obviously equal the angular velocity $ \omega $ . It is convenient also to work with a unit-length variant $ \underline{\tilde{\omega}}\isdeftext \underline{\omega}/ \vert\vert\,\underline{\omega}\,\vert\vert $ .

As introduced in Eq.(B.8) above, the mass moment of inertia is given by $ I=mR^2$ where $ R$ is the distance from the (instantaneous) axis of rotation to the mass $ m$ located at $ \underline{x}\in\mathbb{R}^3$ . In terms of the angular-velocity vector $ \underline {\omega }$ , we can write this as (see Fig.B.6)

$\displaystyle I$ $\displaystyle =$ $\displaystyle mR^2
\eqsp m\cdot \left\Vert\,\underline{x}-{\cal P}_{\underline{\omega}}(\underline{x})\,\right\Vert^2$  
  $\displaystyle =$ $\displaystyle m\cdot \left\Vert\,\underline{x}-(\underline{\tilde{\omega}}^T\underline{x})\underline{\tilde{\omega}}\,\right\Vert^2
\protect$ (B.14)


$\displaystyle {\cal P}_{\underline{\omega}}(\underline{x}) \isdefs \frac{\underline{\omega}^T\underline{x}}{\underline{\omega}^T\underline{\omega}}\underline{\omega}\eqsp (\underline{\tilde{\omega}}^T\underline{x})\underline{\tilde{\omega}}

denotes the orthogonal projection of $ \underline {x}$ onto $ \underline {\omega }$ (or $ \underline {\tilde {\omega }}$ ) [454]. Thus, we can project the mass position $ \underline {x}$ onto the angular-velocity vector $ \underline {\omega }$ and subtract to get the component of $ \underline {x}$ that is orthogonal to $ \underline {\omega }$ , and the length of that difference vector is the distance to the rotation axis $ R$ , as shown in Fig.B.6.

Figure B.6: Mass position vector $ \underline {x}$ and its orthogonal projection $ {\cal P}_{\underline{\omega}}(\underline {x})$ onto the angular velocity vector $ \underline {\omega }$ for purposes of finding the distance $ R$ of the mass $ m$ from the axis of rotation $ \underline {\tilde {\omega }}$ .

Using the vector cross product (defined in the next section), we will show (in §B.4.17) that $ R$ can be written more succinctly as

$\displaystyle R \eqsp \left\Vert\,\underline{x}-{\cal P}_{\underline{\omega}}(\underline{x})\,\right\Vert \eqsp \left\Vert\,\underline{\tilde{\omega}}\times \underline{x}\,\right\Vert.

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2023-08-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University