Though the mass-spring-damper approximation may appear overly simplistic, more complex vibrational systems can often be reasonably analyzed in terms of nearly independent sets of decaying modes or resonances.
Modal synthesis involves the use of second-order digital resonators to synthesize sounds with a relatively small number of decaying resonant modes.
A second-order IIR filter can be designed to produce a single frequency magnitude peak at a specified frequency by setting its feedback coefficients as:
a_{2} = r^{2}
where r is a value between 0 - 1.0 that sets the pole radius and T_{s} is the sample period.
This filter can also be designed to have unity magnitude gain at the resonance peak with:
b_{0} = (1 - r^{2}) / 2
b_{1} = 0
b_{2} = -b_{0}
A digital resonator can be ``impulsed'' to produce a decaying sinusoid, as shown in Fig. 7 below.
Figure 7:
A digital resonance filter: impulse response (top) and frequency magnitude response (bottom).
A ``bank'' of resonance filters can be combined in parallel to simulate all the resonant modes of a system. Each resonator should have its own amplitude, center frequency, and rate of decay.
Any object that exhibits just a few decaying strong modes and is excited by striking or plucking is a good candidate for model synthesis.
A simple modal synthesis example (using arbitrary mode parameters) is provided in the Matlab script modal.m.