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

摘要

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 soundsthey can generate. From the physical point of view, the complex nature of thenonlinear sound generating mechanism – the friction between two surfaces – is no lessstimulating.In this thesis, a physical modelling computational method based on a modalapproach is developed to perform simulations of nonlinear dynamical systems withparticular application to friction-excited musical instruments. This computationalmethod 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 bowlor the glass harmonica. The successful implementation of the method in theseinstruments is shown by comparison with measured results and with other simulationmethods. This approach is extended from systems with simple modal basis to morecomplex structures consisting of different sub-structures, which can also be describedby their own modal set.The extensive nonlinear numerical simulations described in this thesis, enabled someimportant contributions concerning the dynamics of these instruments: for the bowedstring an effective simulation of a realistic wolf-note on a cello was obtained, usingcomplex identified body modal data, showing the beating dependence of the wolfnotewith bowing velocity and applied bow force, with good qualitative agreementwith experimental results; for bowed bars the simulated vibratory regimes emergingfrom different playing conditions is mapped; for bowed Tibetan bowls, the essentialintroduction of orthogonal mode pairs of the same family with radial and tangentialcomponents characteristic of axi-symmetrical structures is performed, enabling animportant clarification on the beating phenomena arising from the rotating behaviourof oscillating modes. Furthermore, a linearized approach to the nonlinear problem isimplemented and the results compared with the nonlinear numerical simulations.Animations and sounds have been produced which enable a good interpretation ofthe results obtained and understanding of the physical phenomena occurring in thesesystem.