Time-domain modeling and numerical simulation of a kettledrum

A kettledrum is made of a circular elastic membrane stretched over an enclosed air cavity. It is set into vibration by the impact of the mallet. The motion of the membrane is coupled with both the external and internal sound field. A time-domain modeling of this instrument is presented which describes the motion of the mallet and its nonlinear interaction with the membrane, the transverse displacement of the membrane, and the sound pressure inside and outside the cavity. Based on a variational formulation of the problem, which uses the pressure jump over the boundaries of the instrument as a new variable, a numerical scheme is derived by means of three- dimensional finite elements. Higher-order absorbing conditions are used to simulate the free space. The validity of the model is illustrated by successive snapshots showing the pressure fields and the displacement of the membrane. In addition, time histories of energetic quantities help in explaining how the energy is balanced between mallet, membrane, and acoustic field in real instruments. Simulated external pressures show particularly good agreement with the sound field radiated by real instruments in both time and frequency domains.