Now the system is in state space form, suppose we change to diagonal coordinates:

where

The
complex-conjugate-pair eigenvalues
represent the *modal frequencies* and *dampings*

- The main (desired) effect of the third order partial derivative
with respect to time is to make the damping factors
be smaller
for higher-frequency modes.
- A secondary effect is that the modal frequencies
are
slightly shifted.
- Another secondary effect of the third-order time derivative is
the introduction of
real rapidly decaying modes, characterized by
the
. These eigenvalues are typically
*negligible*.

We can reduce the size of the system to , and further change coordinates so that the matrix is made up of 2 by 2 real ``modal'' blocks as follows:

Download PianoString.pdf

Download PianoString_2up.pdf

Download PianoString_4up.pdf

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

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