Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

Off-Diagonal Terms in Moment of Inertia Tensor

This all makes sense, but what about those $ -1$ off-diagonal terms in $ \mathbf{I}$ ? Consider the vector angular momentumB.4.14):

$\displaystyle \underline{L}\eqsp \mathbf{I}\,\underline{\omega}\eqsp
m\left[\begin{array}{ccc}
I_{11} & I_{12} & I_{13}\\ [2pt]
I_{21} & I_{22} & I_{23}\\ [2pt]
I_{31} & I_{32} & I_{33}
\end{array}\right]
\left[\begin{array}{c} \omega_1 \\ [2pt] \omega_2 \\ [2pt] \omega_3\end{array}\right]
$

We see that the off-diagonal terms $ I_{ij}$ correspond to a coupling of rotation about $ \underline{e}_i$ with rotation about $ \underline{e}_j$ . That is, there is a component of moment-of-inertia $ I_{ij}$ that is contributed (or subtracted, as we saw above for $ \underline{\omega}=[1,1,0]^T$ ) when both $ \omega_i$ and $ \omega_j$ are nonzero. These cross-terms can be eliminated by diagonalizing the matrix [452],B.25as discussed further in the next section.


Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

[How to cite this work]  [Order a printed hardcopy]  [Comment on this page via email]

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA