The Bar-Resonator Interaction in Mallet Percussion Instruments: A Multi-Modal Model and Experimental Validation

In some mallet percussion instruments, like vibraphones and marimbas, acoustical resonators are placed under the tuned bars to enhance acoustic radiation. Despite its wide use in commercial instruments, the vibro-acoustic interaction between the tuned bars and their acoustic resonators has not been studied extensively, and previous modeling attempts often neglect important aspects of the coupling dynamics. This work presents a physical model describing the two-way vibro- acoustic interaction between a vibrating beam and an acoustic resonator. The proposed formu- lation leads to a system composed of a set of mechanical oscillators coupled to a set of acoustical oscillators, representing bar and resonator modes, respectively. The simplicity of the proposed model provides an intuitive understanding of the physics occurring in real instruments and show how specific design parameters will affect their behavior. The dynamics of the system are analyzed through energy balance, time-domain simulations and eigenvalue analysis, revealing a number of interesting features and highlighting three dimensionless parameters: the ratio of frequencies and quality factors, as well as a coupling "strength" coefficient, dependent on the bar- resonator distance, the placement of the resonator along the bar length, bar and resonator mode shapes, amongst others. The proposed model is also shown to be a useful tool for the design of instruments optimized for multi-modal coupling, where several resonator modes are tuned to bar frequencies. Finally, experimental results are presented to validate various aspects of the pro- posed model and demonstrate its capacity to emulate real instruments, both qualitatively and quantitatively.