Kausel, Wilfried; Chatziioannou, Vasileios, and Moore, Thomas R. (2011) "More on the structural mechanics of brass wind instrument bells," in Forum Acusticum Aalborg 2011 (European Acoustics Association, Aalborg, Denmark) p. 527-532.
Abstract: Previous work by the authors demonstrated that it is the flaring bell region of brass wind instruments where axi-symmetric vibrations of potentially considerable amplitude can be stimulated by the oscillating air pressure, often acting back on the air column to produce audible timbre changes. In work reported in JASA 128(5), the effect of vibrating walls on the input impedance and transfer function of the instrument was modelled and compared with experimental data. However, in the above mentioned article the structural behaviour was only roughly estimated and not rigorously analysed. Actual work on the structural mechanics of real trumpet bells now reveals an unexpected dependence on some parameters, such as the construction of rim and braces, which strongly affects these axial vibrations and their interaction with the air column. Here, results of structural simulations using the Finite Element Method in two and three dimensions will be presented. Alternatively a Finite Difference scheme for the distribution of mass and stiffness across a 2-D cross-section of the bell has been developed and compares well with these FEM results. It should be noted that this more rigorous treatment of the structural mechanics does deepen the understanding of how bells vibrate. However, it does not change the expected order of magnitude of the effect compared with previously published estimations.