Coupling the Modes

To “couple” many modes of vibration, we can create a bank of parallel digital resonators and provide a feedback path from their summed output back to each of their inputs that is controlled by a “coupling filter”.

The filter structure of Fig. 13, however, is problematic in this context because of its direct feedforward path. If the “coupling filter” also has a direct feedforward path, the coupling structure will have a delay-free loop and not be computable.

Van Duyne thus makes use of a modified filter structure as shown in Fig. 14.

**Figure 14:** The modified CMS resonator filter structure.
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The filter of Fig. 14 has a transfer function given by

$\displaystyle \frac{z^{-1} H_{k}(z)}{1 + z^{-1} H_{k}(z)},$ (4)

In terms of a magnitude response, the extra allpass filter $H_{k}$ and unit delay $z^{-1}$ in the numerator will have no affect. They will contribute an additional phase component but this will only modify the initial phase of the oscillations.

With this new resonator structure, the complete coupled mode synthesis block diagram is as shown in Fig. 15.

**Figure 15:** The coupled mode filter structure.
$\begin{figure}\begin{center} \begin{picture}(4.85,3.9) \put(0.27,0){\epsfig{fi... ....81){Coupling} \put(0.62,1.65){Filter} \end{picture} \end{center} \end{figure}$

Because the individual modal resonators have no attenuation, the “coupling filter” completely controls the modal decay rates.

The “coupling filter” can be considered akin to a lumped bridge impedance. If the “bridge” is fairly rigid, only a small amount of energy from the modes will be fed back to their inputs (the “coupling filter” will typically have a magnitude response gain on the order of 0.001 or less).

The “coupling filter” also introduces attenuation because its output is out of phase with the loop signal (the loop signal is subtracted at the loop input summer, while the input is added).

The cms.m Matlab script provides an example implementation of the CMS filter structure.

The CMS structure has a physical representation as illustrated in Fig. 16, whereby a group of mass-spring resonators are coupled via a common base which itself loses energy via a dashpot.

**Figure 16:** The CMS physical representation.
$\begin{figure}\begin{center} \epsfig{file = figures/cms-physical.eps,width=3.0in} \end{center} \vspace{-0.25in} \end{figure}$