Friction and application to real-time physical modeling of a violin
Résumé
One key point in physical modeling of bowed string instruments is to
find a suitable equation that describes the friction
force between the bow and the string. This force is a highly non-linear
and eventually discontinuous function of bow parameters and dynamics
of the model.
The friction mechanism gives rise to the well known
stick-slip phase alternation, characteristic of the so-called
Helmholtz motion.
In this paper we propose different models of bow string interaction.
Evaluations of the models' influence on the quality of the synthesis
is done with a complete physical model of a violin.
We have chosen an hyperbolic friction function that fits measurements
carried out in real instruments. Solving the coupling of the bow and
the string usually requires to cope with a system of two equations.
This is often done by numerical methods or by table-lookup
techniques. However, we have shown that an hyperbolic friction
function allows us to obtain an analytical solution.
When the solution is not unique but triple, we choose the one that is
given by a physically based hysteresis rule: the system follows the
current state (stick or slip) continuously as long as it can and the
middle solution is not chosen.
We have furthermore refined the classical basic model of violin-like
instruments.
Starting from a single bow hair, we then consider a bow with many hairs.
The strings are represented by fractional delay lines, and losses are
lumped into low-pass filters. All the filter coefficients are estimated
according to impulse response of violin's strings.
Losses are adapted to the length of the vibrating part, and refined in
order to take into account the influence of terminations and
fingers. Finally, the body resonances are estimated on a measured
transfer function.
All our simulations are performed using the real-time environments
jMax or Max/MSP interchangeably. We built a graphical interface which
allows us to see the coupling between the non linear bow string
interaction and the linear wave propagation in the string.
In fact, we display the friction curve, (the shape of which depends
on the force exerted by the player on the bow), and the previously
mentioned analytical solution at each time, which
moves on the friction curve according to the coupling with the string.
>From a sound and musical point of view, the simulation results are very
satisfactory. Sound synthesis is even more realistic when the
model with many bow hairs is included. The model 's musical
capabilities will be highlighted at the conference using a real time
implementation and playing interface.