Modeling the dynamics of a vibrating string with a finite distributed unilateral constraint: application to the sitar

The free vibration response of an ideal string impacting a distributed parabolic obstacle located at its boundary has been analyzed, the goal being to understand and simulate a sitar string. The portion of the string in contact with the obstacle is governed by a different partial differential equation (PDE) from the free portion represented by the classical string equation. These two PDEs and corresponding boundary conditions, along with the transversality condition that governs the dynamics of the moving boundary, are obtained using Hamilton's principle. A Galerkin approximation is used to convert them into a system of nonlinear ordinary differential equations, with lower mode-shapes parametrized with respect to the location of the moving boundary as basis functions. This system is solved numerically and the behavior of the string studied from simulations. The advantages and disadvantages of the proposed method are discussed in comparison to the penalty approach for simulating wrapping contacts. Simulations with bridge-string parameters consistent with the configuration of a real sitar show that any degree of obstacle wrapping may occur during normal playing. Finally, the model is used to investigate the mechanism behind the generation of the buzzing tone in a sitar.