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Embedded SystemsFPGAElectronics

Physical Audio Signal Processing
    Elementary Physics, Mechanics, and Acoustics
       Rigid-Body Dynamics
          Angular Velocity Vector

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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{\bf 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{\bf 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)

where

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

denotes the orthogonal projection of $ \underline{x}$ onto $ \underline{\omega}$ (or $ \underline{\tilde{\omega}}$) [451]. 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: Mass position vector $ \underline{x}$ and its orthogonal projection $ {\cal P}_{\protect\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}}$.
\includegraphics[width=1.5in]{eps/pxov}

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}}(\... ...\eqsp \left\Vert\,\underline{\tilde{\omega}}\times \underline{x}\,\right\Vert. $

Previous: Striking One of the Masses
Next: Vector Cross Product

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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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