## 1 Introduction
Their gestures can be described with a handful of parameters, which must be perfectly coordinated at all times to allow a tone to be created and sustained; indeed, [PERSON] [21], following the work of [PERSON] [22] some decades earlier, was the first to analytically show that, under simplifying assumptions and for a certain bow velocity, only a relatively narrow triangular area (in logarithmic scale) in the downwards bow force versus bow-bridge distance parameter space gave rise to the characteristic stable Helmholtz motion desired by most musicians (his work was revised by [PERSON] _et al._[23], introducing more refined elements of bowed string motion). This area is tied to the concept of playability, and the so-called Schelleng diagrams are widely used in bowed string playability studies [24; 25; 26; 27]. Transient quality also constitutes a major part of playability: [PERSON] [28] investigated the relation between bow downwards force and bow acceleration regarding the quality of initial transients, producing triangular diagrams resembling those of [PERSON], again under simplifying assumptions; [PERSON] _et al._[29] produced Guettler diagrams with more refined numerical models as well as experimental data, showing the predicted wedge-shaped region. A detailed review of the published literature on bowed string mechanics (and, indeed, violin acoustics in general) was recently written by [PERSON] [30]. After the studies of [PERSON], an experimental study by [PERSON] [31; 32] yielded measured values for these control parameters, on a violin; he was the first to develop a measuring rig able to record them all simultaneously during performance. More recently, the variation of these parameters along various bowed string gestures was observed and analysed in detail [15; 33; 34]. The obtained signals can be mathematically reconstructed [35], or directly used, to be fed into a physical modelling algorithm such as the one presented in this work. The aforementioned studies are mainly interested in bowing gestures; to handle those of the left hand fingers (vibrato, legato, glissando, etc. ), one can include a finger model, along with a fingerboard. This work is concerned with a detailed model of bowed string vibration, emphasising the interactions of the string with the player. A linear bowed string is simulated in two polarisations. The model includes the following features:* in one polarisation, distributed nonlinear contact interactions between the string and the dynamic left hand fingers, dynamic bow, and fingerboard. A stable finite difference scheme for modelling distributed contact/collisions has recently been established [36; 37], that can be used in this stopped string-fingerboard setup [38; 39];
* in the other polarisation, orthogonally to the first, distributed nonlinear friction forces between the string and the same three objects. The friction force nonlinearity is modelled with a force/velocity friction curve for the bow [40]. Tangential Coulomb friction also keeps the string captured between the fingers and fingerboard during note production;
* full control over the physical parameters of the system, as well as dynamic variations of the playing parameters. This time domain model is therefore able to reproduce most bowed string gestures;
* introduction of an inertial term in the bow model, in the tangential direction. This leads to a horizontal force bow control, instead of the bow velocity input signal used in most of the aforementioned models;
* in the same direction, introduction of an inertial term and a restoring term in the finger model. The fingertip is massive, absorbs some of the tangential string vibrations, and oscillates about a fixed horizontal finger position. The main challenges associated with this model are computational. On one hand, the resulting numerical method must be stable; energy methods are used to derive a stability condition from the model system, and establish a power balance to keep track of the energy exchanges between the various parts of the system.