Modelling the Wall Vibrations of Brass Wind Instruments

The vibration of the walls of brass wind instruments has been a subject of study in the field of musical acoustics throughout the last decades. The amplitude of such vibrations, stimulated by the oscillating air pressure inside the instrument bore, is very small compared to the dimensions of the instrument. However, it has been recently shown that at the flaring regions of the bell of brass instruments the internal air pressure can excite axi-symmetric vibrations of significantly large amplitude, which can affect the sound quality [1]. Using both twoand three-dimensional finite element modelling, this paper presents a structural analysis of brass wind instrument bells and compares the results of the simulations to those of experimental measurements and of simplified numerical methods. Finally, a multi-physics approach using an acoustic-structure interaction model studies the effect of the wall vibrations to the acoustic response of the instrument. Firstly, the problem is simulated in COMSOL using a two-dimensional axi-symmetric geometry and considering only the structural mechanics of the system. The air pressure inside a trumpet is applied as a boundary load to the walls of the instrument and their vibration is observed. This simulation is limited to identifying the axial resonances of the structure, so a three-dimensional geometry is used to study the elliptical modes of vibration. Finally, using the acoustic-structure interaction module, it is possible to compare the acoustic response of the instrument in the case of vibrating and rigid walls. As such the effect of the wall vibration on the function of the instrument can be discussed. Using the two-dimensional axi-symmetric model it is possible to calculate the amplitude of the total displacement at the rim of the bell over a frequency range (Figure 1), whence the first two axial resonances can be identified. The second axial resonance is of particular interest, since a node lies very close to the rim of the bell, as depicted in Figure 2. Furthermore, a rotational motion of the rim is observed, which can not be captured by a more simplified mass-spring model [2]. The radius of the rim was chosen such that the (2,1) mode of the bell (corresponding to two nodal diameters and one nodal circle, as shown in Figure 3) appears at the same frequency as that measured on a trumpet with the same bore profile using electronic speckle pattern interferometry [3]. Finally, the acousticstructure interaction simulation reveals the difference between the transfer functions of an instrument with fixed and free walls (Figure 4). The results for the amplitude of the wall vibration of brass instrument bells are of the same order of magnitude, as predicted by previous publications. However, finite-element modelling allowed the used of a more detailed analysis of the whole structure, that resulted in a deeper understanding of the vibration mechanism. Furthermore, it is expected that the latest version of the acoustics module, which includes new tools regarding thermoviscous acoustics, will enhance the acoustic simulations, since the viscous boundary layer affects the instrument response. Reference 1. W. Kausel et al. Influence of wall vibrations on the sound of brass wind instruments. J. Acoust. Soc. Am. 128(5):3161-3174, 2010 2. W. Kausel et al. More on the Structural Mechanics of Brass Wind Instrument Bells. In Forum Acusticum 2011, Aalborg, Denmark 3. T.R. Moore et al. The effect of bell vibrations on the sound of the modern trumpet. Acta Acust. Acust. 91, 578-589, 2005 Figures used in the abstract Figure 1: Rim displacement over frequency caused by a unity pressure at the bell. Figure 2: Second axial resonance deformation, scaled with a factor of 3000 to show the existence of a node very close to the rim. Figure 3: The (2,1) mode of vibration of the trumpet bell at 468 Hz. Figure 4: Difference in the transfer of a trumpet with fixed and free walls. The triangles indicate the frequencies of the impedance maxima of the instrument (playing frequencies).