Sound synthesis of a nonlinear string using Volterra series

• Published in: Journal of sound and vibration 2008-07, Vol.314 (1), p.275-306
• Main Authors: Hélie, Thomas, Roze, David
• Format: Article
• Language: English
• Subjects: Acoustics; Computer Science; Dirichlet problem; Dynamical systems; Engineering Sciences; Exact sciences and technology; Fundamental areas of phenomenology (including applications); Kernels; Mechanics; Nonlinear dynamics; Nonlinearity; Physics; Signal and Image processing; Solid mechanics; Sound; Strings; Structural mechanics; Transduction; acoustical devices for the generation and reproduction of sound; Vibrations
• ISSN: 0022-460X; 1095-8568
• DOI: 10.1016/j.jsv.2008.01.038

This paper proposes to solve and simulate various Kirchhoff models of nonlinear strings using Volterra series. Two nonlinearities are studied: the string tension is supposed to depend either on the global elongation of the string (first model), or on the local strain located at x (second, and more precise, model). The boundary conditions are simple Dirichlet homogeneous ones or general dynamic conditions (allowing the string to be connected to any system; typically a bridge). For each model, a Volterra series is used to represent the displacement as a functional of excitation forces. The Volterra kernels are solved using a modal decomposition: the first kernel of the series yields the standard modes of the linearized problem while the next kernels introduce the nonlinear dynamics. As a last step, systematic identification of the kernels lead to a structure composed of linear filters, sums, and products which are well-suited to the sound synthesis, using standard signal processing techniques. The nonlinear dynamic introduced through this simulation is significant and perceptible in sound results for sufficiently large excitations.