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Physical Audio Signal Processing
    Elementary Physics, Mechanics, and Acoustics
       Rigid-Body Dynamics
          Mass Moment of Inertia Tensor
             Example with Coupled Rotations

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Example with Coupled Rotations

Now let the mass $ m$ be located at $ \underline{x}=[1,1,0]^T$ so that

\begin{eqnarray*} \mathbf{I}&=& m \left(\left\Vert\,\underline{x}\,\right\Vert^2... ... & 0\\ [2pt] -1 & 1 & 0\\ [2pt] 0 & 0 & 2 \end{array}\right]. \end{eqnarray*}

We expect $ \underline{\omega}=[1,1,0]$ to yield zero for the moment of inertia, and sure enough $ \underline{\tilde{\omega}}^T\mathbf{I}\,\underline{\tilde{\omega}}=0$. Similarly, the vector angular momentum is zero, since $ \mathbf{I}\,\underline{\omega}=\underline{0}$.

For $ \underline{\omega}=[1,0,0]^T$, the result is

\begin{displaymath} \mathbf{I}\eqsp \begin{array}{r}\left[\begin{array}{ccc} 1 ... ...egin{array}{c} 1 \\ [2pt] 0 \\ [2pt] 0\end{array}\right]m = m, \end{displaymath}

which makes sense because the distance from the axis $ \underline{e}_1$ to $ \underline{x}$ is one. The same result is obtained for rotation about $ \underline{\omega}=\underline{e}_2$. For $ \underline{\omega}=\underline{e}_3$, however, the result is $ I=2m = m \vert\vert\,\underline{x}\,\vert\vert ^2$, as expect.

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Next: Off-Diagonal Terms in Moment of Inertia Tensor

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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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