Numerical Experiments with Non-linear Double Membrane Drums

Drums with two membranes are very common; snare drums, tom-toms and bass drums can be found in Western music, but there are examples in Eastern music, as well (the Indian mridangam, the Japanese taiko, etc.) These instruments can have considerable physical dimensions; bass drum heads can sometimes reach a radius of half a meter. Given the size, the low tension at which the membranes are generally tuned and the amplitude of vibrations, it is unlikely that a linear model could capture the most salient features of the sound of these instruments.
Pitch glide effects and an increase of high-frequency energy has been observed at high excitation amplitudes for the bass drum [1] and more recently for tom-toms [2]. Similar phenomena have been observed, for example, in strings [3] and plates [4], and are often related to the presence of non-linearities in the system.
In this paper we present a finite difference time domain model of double membrane drums (i.e., tom-toms and bass drums) with air coupling and with non-linear terms (due to von Karm´ an) in the equations of motion for the two membranes. Some of the computational difficulties stemming from this particular choice will be discussed. Simulation results and sound examples will be presented.