Introduction on bowed strings
Bowed string instruments include the violin and viol families, wheel-bowed instruments, as well as numerous lyras and traditional Chinese instruments. Playing a note on such instruments relies on the same underlying principle: a bow (or wheel) pushes down against a string, and slides transversally in order to bring about periodic self-oscillations, and thus, musical sound. The player also makes use of their finger(s) (or other stopping objects) to capture the string against the neck of the instrument, allowing them to play different notes on the same string.
The periodic motion of the string is brought about by the alternance of “stick” and “slip” phases, one cycle constituting a period . In ideal playing conditions, the so-called Helmholtz motion is achieved, with a corner travelling back and forth along the string, following an lens-shaped path.
Types of Bowed string instruments
A first step towards a physical model of a bowed string instrument is a model of the string, with player interaction. The string is linear, simply supported at both ends, stiff and lossy (with frequency dependent and independent energy losses). The string displacement is calculated in two polarisations: vertical (in a plane orthogonal to the top plate), and horizontal (in a plane parallel to the bow/top plate). A practical (and sufficient for most of our purposes) approximation consists in assuming that contact interactions occur in the vertical polarisation, and that the resulting normal forces give rise to friction phenomena in the horizontal polarisation.
The contact interactions use a nonlinear collision model, and a relative velocity dependent friction curve is used in the other direction. This allows for reproduction of the main bowing and finger-stopping gestures found in bowed string performance.
A numerical power balance is derived from the finite difference scheme, with a conserved energy term in the lossless case. Power is supplied through the external forces exerted by the player (downwards, sliding across and sliding along the string), and dissipated through internal string damping, nonlinear contact damping and friction between the string and the objects. Several bowed string gestures can be reproduced thanks to the dynamic variation of parameters such as bow normal force, transverse velocity, position, and finger position along time.
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