Time-domain modeling and numerical simulation of timpani
Leïla Rhaouti, Patrick Joly, Antoine J. Chaigne · The Journal of the Acoustical Society of America · 1998
Timpani are made of a circular elastic membrane stretched over a enclosed air cavity and set into vibrations by the impact of a mallet. The motion of the membrane is coupled with both the external and internal sound pressure field. A time-domain model of this instrument has been developed in order to investigate the influence of the main geometrical and physical quantities on the resulting sound. This model consists of a set of partial differential equations which govern the displacement of the membrane and the acoustic pressure inside and outside the cavity, respectively. These equations are coupled with a nonlinear differential equation which governs the excitation by the mallet. A numerical scheme has been derived from this model using three-dimensional finite element methods. Absorbing conditions have been implemented to simulate the free space. The validity of the model is illustrated by successive snapshots showing both the pressure field and membrane displacement. In addition, time histories of energetic quantities are presented for a better understanding of energy balance between membrane, cavity, and external space in real instruments. Sound examples are obtained by simulating the sound pressure at specific positions corresponding to the player’s ears.