9.6.2 Varieties of thermal friction model

It is useful to understand exactly how a temperature-based friction model can be incorporated into the simulation approach described in section 9.2. This sheds some light on the different behaviour of alternative choices of model. First, a reminder of how the original friction-curve model is dealt with. At each time step, the new values of friction force and string velocity are found by the geometric construction illustrated in Fig. 1, which is a repeat of Fig. 7 from section 9.2.

Figure 1. Repeat of Fig. 7 for section 9.2, showing the graphical construction involving the intersection of a straight line and the friction force-velocity curve. Three examples of possible positions of the straight line are shown in blue, and two versions of the force-velocity curve (corresponding to different bow forces) are shown in red.

The history of the string motion allows us to compute the current value of the variable we called $v_h$, and this determines the position of a sloping straight line. We then have to find the intersection of that straight line with the curve representing friction force as a function of velocity, including the vertical segment which represents the range of possible values of force during sticking. Three possible positions of the straight line are shown, and two versions of the friction-velocity curve corresponding to different values of the bow force. For the solid red friction curve, the blue line labelled c indicates an ambiguity, which is resolved by a hysteresis loop involving jumps of force and velocity, as illustrated in Fig. 2 (which is another repeat of an earlier figure).

Figure 2. Repeat of Fig. 8 of section 9.2, illustrating the hysteresis loop and jump behaviour that arises if the maximum slop of the force-velocity curve is steeper than the straight line.

The original thermal friction model is based on the assumption that the the friction force is proportional to the normal bow force, and that the coefficient of friction is a function $\mu(T)$ of temperature $T$ alone. In order to ensure that the friction force always opposes the direction of sliding motion, the friction force $f$ must then take the form

$$f=-F_b \mu(T) \mathrm{sgn}(v) \tag{1}$$

where $F_b$ is the bow force, and $\mathrm{sgn}(v)$ denotes the sign function, equal to $\pm1$ depending on whether $v$ is positive or negative. What this means for the simulation is that at each time step we first calculate the temperature, and hence obtain $\mu$, then to deal with the $\mathrm{sgn}(v)$ term we solve a version of the graphical construction illustrated in Fig. 3. This function has a vertical portion representing the state of sticking, but during sliding the function is simply constant, so there can be no jumps and no hysteresis loop.

Figure 3. Version of the graphical construction shown in Fig. 1, appropriate to the original thermal friction model. Again, three examples of possible positions of the sloping straight line are shown in blue.

There is another possible temperature-based friction model we could have explored. The measurements on the bulk properties of rosin shown in Fig. 8 of section 9.6 suggest that over most of the relevant temperature range rosin behaves predominantly as a viscous liquid. That would suggest a friction law taking the general form

$$f=-F_b \zeta(T) v . \tag{2}$$

The function $\mathrm{sgn}(v)$ has been replaced by simple proportionality to $v$, because a viscous shear force is proportional to shear rate. This factor $v$ achieves the desired effect of reversing direction according to the sign of the sliding velocity, but it has a very drastic effect on the predicted behaviour. Figure 4 shows the graphical representation equivalent to Figs. 1 and 3. This time, there is no barrier whatever representing “sticking” behaviour. Simulations based on such a model will regularly show “forward slipping”, something that is never (or virtually never) seen in measurements.

Figure 4. Plot corresponding to Fig. 4, for a thermal model which assumed a temperature-dependent viscosity to represent the behaviour of the rosin layer.

A viscous-based model like this is clearly not, therefore, a good candidate for the true friction law. The explanation for this apparent conflict with the measured properties of rosin lies in the particular conditions of those measurements. The rheometer tests used extremely small strains and strain rates, in order to keep the behaviour in the linear range. But the bowed string motion we are interested in involves intervals of gross sliding. This will take the material well outside the range of the rheometer tests.

For an extension of the original thermal model to allow the possibility of an initial jump in velocity (and hence in measured bridge force) we need to look for something different. There is a very simple, if entirely ad hoc, approach to that question which is motivated by Figs. 1 and 3. We can replace the function $\mathrm{sgn}(v)$ in equation (1) by something like the function plotted in Fig. 5. The middle one of the three blue lines shows that this function can exhibit multiple intersections, and therefore sometimes predict jumps and hysteresis. But note that the range of velocity plotted here is much wider than in Fig. 1: the red curve here has a falling shape which is much gentler than the original friction curve.

Figure 5. Plot corresponding to Fig. 3, for the modified version of the thermal model as described in the text. The function $\mathrm{sgn}(v)$ has been replaced by a hyperbolic curve tending asymptotically to the values $\pm1$ at large sliding speeds.

For this example I have used a hyperbolic shape to replace the flat portion of the function $\mathrm{sgn}(v)$. It has a horizontal asymptote at the value 1, and it is then fully defined by two parameters: a vertical asymptote at a speed $v_{as}$, and a value $k_{lim}$ as the sliding speed tends to zero. The equation of the curve then takes the form

$$(f-1)(v-v_{as})=-v_{as} (k_{lim}-1) . \tag{3}$$

The particular function plotted here has $v_{as}=0.8$ m/s and $k_{lim}=2$, which are the parameter values used for the simulation examples shown in section 9.6.

With this modified model, we can still calculate a form for the required temperature-dependent friction coefficient by requiring that the simulated results for steady sliding agree with the original measurements. The result, when the term $\mathrm{sgn}(v)$ in equation (1) is replaced by the particular function plotted in Fig. 5, is shown in Fig. 6: it is still broadly in line with expectations based on the rheometer tests and glass transition range of violin rosin.

Figure 6. Coefficient of friction as a function of temperature, deduced from the modified thermal friction model shown in Fig. 5 after requiring that it should agree with the steady-sliding friction measurements.