A modal method for the simulation of nonlinear dynamical systems with application to bowed musical intruments

Abstract

Bowed instruments are among the most exciting sound sources in the musical world, mostly because of the expressivity they allow to a musician or the variety of sounds they can generate. From the physical point of view, the complex nature of the nonlinear sound generating mechanism – the friction between two surfaces – is no less stimulating. In this thesis, a physical modelling computational method based on a modal approach is developed to perform simulations of nonlinear dynamical systems with particular application to friction-excited musical instruments. This computational method is applied here to three types of systems: bowed strings as the violin or cello, bowed bars, such as the vibraphone or marimba, and bowed shells as the Tibetan bowl or the glass harmonica. The successful implementation of the method in these instruments is shown by comparison with measured results and with other simulation methods. This approach is extended from systems with simple modal basis to more complex structures consisting of different sub-structures, which can also be described by their own modal set. The extensive nonlinear numerical simulations described in this thesis, enabled some important contributions concerning the dynamics of these instruments: for the bowed string an effective simulation of a realistic wolf-note on a cello was obtained, using complex identified body modal data, showing the beating dependence of the wolfnote with bowing velocity and applied bow force, with good qualitative agreement with experimental results; for bowed bars the simulated vibratory regimes emerging from different playing conditions is mapped; for bowed Tibetan bowls, the essential introduction of orthogonal mode pairs of the same family with radial and tangential components characteristic of axi-symmetrical structures is performed, enabling an important clarification on the beating phenomena arising from the rotating behaviour of oscillating modes. Furthermore, a linearized approach to the nonlinear problem is implemented and the results compared with the nonlinear numerical simulations. Animations and sounds have been produced which enable a good interpretation of the results obtained and understanding of the physical phenomena occurring in these system