Up until now, we have been using a particular model to describe the frictional behaviour between the bow and the string. This model is based on two assumptions: that the friction force is proportional to the normal force (i.e. the bow force), and that during sliding the friction force is a nonlinear function of the sliding speed. The time has come to look carefully at both these assumptions. What is the empirical evidence and/or theoretical understanding on which they are based?
This turns out to be an interesting but tricky question, and it will lead us right up to a research frontier. Most things we have discussed so far are governed by physical principles like Newton’s laws, which are well established and uncontroversial. But friction is different: there is no single underlying theory that has similar status. The details can be quite different, depending on the nature of the materials in contact, and in many cases we still don’t know the full story. Most of the time engineering calculations involving friction make use of grossly simplified approximations, but these are not good enough if we want to understand bowed-string transients in detail.
The first assumption, that friction force is proportional to bow force, we will defer for the moment. The question has an interesting history, but a discussion of it will fit more neatly into the next section when we talk about how a real bow might give different behaviour from a rigid rod. For now, we will concentrate on the second assumption, the “friction curve model”. Our starting point is the measurement we saw earlier, repeated here as Fig. 1.
Looking at this plot, it seems obvious that the friction force varies, rather dramatically, with the sliding speed. But we have to be careful to avoid a trap here. This plot shows the results of a particular type of measurement: for each separate data point, steady sliding was imposed at a particular speed, and the friction force was measured. Sliding speed is the only variable, so of course we can plot the measured forces against the speeds and we are bound to get something looking like a friction curve.
But now suppose we do a different measurement, in which the sliding speed varies in time (as it does with the actual bowed string). Do the results from Fig. 1 prove that the friction force is determined, moment by moment, by the instantaneous value of the sliding speed? Absolutely not! Other factors might come in, to do with the history of the motion. Figure 1 only says that if the speed stays constant for long enough, the force should settle down to the value plotted there. To resolve this question, we need to do measurements of a different kind.
It is not easy to observe the friction force directly during stick-slip vibration of a bowed string, or indeed of any other stick-slip system like a squealing vehicle brake. It is very difficult to insert any kind of force-measuring sensor right in the contact region, without drastically changing the behaviour. Instead, ingenuity must be exercised to design an experiment in which the friction force can be reliably inferred from measurements of some kind elsewhere on the vibrating structure.
For the case we are interested in, when the friction is mediated by violinist’s rosin, two different experiments of this kind have been carried out. The first, by Jonathan Smith [1], made use of a cantilever-like mass-spring oscillator. Once the effective mass, stiffness and damping of the device have been found by calibration tests, the motion of the mass during a stick-slip vibration can be measured and the friction force can be reconstructed by substituting the displacement, velocity and acceleration into the equation of motion of the oscillator.
The second experiment used an actual bowed string. Bob Schumacher designed a rig in which a violin E string was bowed by a glass rod which had been coated in rosin [2, 3]. The rosin was dissolved in a solvent, then a rod was slowly drawn up out this solution, leaving a thin film of rosin after the solvent had evaporated. At both ends of the string, force-measuring sensors of the kind described in section 9.1.1 were inserted. With a bit of ingenious processing in the computer, these two force signals can be combined to give estimates of the string velocity and the friction force at the bowed point. The next link gives a few details about how this can be done.
The Schumacher experiment gives the most illuminating information for our purpose, so we will show some results from this rig. Figure 2 shows a few cycles of the inferred waveforms of velocity (upper plot) and friction force (lower plot). The string was executing Helmholtz motion in this example. The pulses of slipping, with negative velocities, are clear. In between, the episodes of sticking do not show a velocity that is exactly constant. Small ripples in this waveform arise because the string can roll on the “bow”, as described in sections 9.5 and 9.5.3.
Now if we take these waveforms of velocity and force and we plot one against the other, we get the result seen in Fig. 3. There is a more-or-less vertical stripe at the right-hand side, showing the intervals of sticking. But during slipping, instead of a single friction curve we see a loop. The path around this loop moves anti-clockwise. The friction force is highest at the start of slipping (or in fact just after the start of slipping), then it falls as the slipping speed increases. But on the way back towards the sticking line, the force remains much lower than the maximum value on the way out. The details of the loop shape can vary in different bow strokes, but the qualitative behaviour is always very similar to this plot. Furthermore, Jonathan’s Smith’s experiment also gave very similar results, despite using a very different vibrating system.
The conclusion is that the friction force is not simply determined by the sliding speed: you get a different answer depending on whether the speed is increasing or decreasing. Rosin does not follow the kind of friction-curve model we have been assuming up to now. So what is going on? We should step back a little, and ask what is special about rosin. Why do violinists use it in the first place? Indeed, it is not only violinists: there are many bowed-string instruments around the world, and there are also other musical instruments which make use of stick-slip vibration, for example friction drums. Players of all these instruments coat their bowhair, or their hurdy-gurdy wheel, or in the case of friction drums their fingers, with rosin.
Rosin is a natural material, obtained from various species of pine tree. The raw tree resin is separated using distillation or solvents into a liquid component called “spirit of turpentine” and a solid component, which is rosin. It is manufactured in industrial quantities, because it has many uses: it is used in making soaps, printing inks, paper, adhesives, and many other applications. It is also frequently used for its high friction: not only for violin bows, but also to give dancers, weightlifters or baseball players a firmer grip on things.
Violin rosin is a clear, brittle material at room temperature: if you drop a block on the floor it is likely to shatter. But if you hold it in your fingers for a minute or two, it soon becomes sticky. It is a type of material known technically as a glass. As any glass-blower knows, if you heat ordinary glass up it does not show a sharp melting point at which it suddenly becomes a liquid (like melting ice). Instead, there is a broad range of temperatures over which the glass softens progressively. By holding the temperature in this range, the glass-blower can shape and mould it into laboratory glassware or decorative drinking glasses.
Rosin behaves in a similar way, but the changes happen at a lower temperature than with glass. The softening range of temperatures, usually characterised by the glass transition temperature marking the middle of the range, is not very far above room temperature. What this means is that the mechanical properties of rosin change rather quickly with a relatively small rise in temperature. As the temperature rises, the rosin becomes softer, so that the force required to deform a layer of rosin goes down. This is probably the origin of the behaviour seen in Fig. 1. When a rosin-coated bow or rod is forced to slide across another object (such as a violin string), heat is generated. The faster the sliding, the hotter the rosin becomes, and the lower is the resulting friction force. So the friction force does vary with the sliding speed, but only indirectly. The real cause of the variation is a change of temperature.
The idea that the friction force is strongly influenced by temperature, rather than sliding speed as such, gives an immediate qualitative explanation for the loop seen in Fig. 3. Heat is generated during episodes of sliding, but very little is generated during episodes of sticking. So think what will happen during a cycle of Helmholtz motion. At the end of a slip, the rosin will be relatively warm and so the friction will be low. But during the relatively long sticking interval, the heat has a bit of time to diffuse away into the body of the string and bow. So by the time the Helmholtz corner arrives back to trigger the next slip, the rosin layer has cooled down a bit, and the friction force is higher. As it slips, the rosin heats up rapidly and the friction falls. The result is a loop in the velocity—force plane, just as we saw in the measured results.
What this description amounts to is a claim that the rosin melts a little and re-freezes during every cycle of the string’s vibration, several hundred times a second. Is that really credible? Remarkably, we can get some direct evidence for this from the Schumacher experiment. We can use one of his glass rods for just a few bow strokes in the apparatus, rotating it a bit each time so that a different part of the rosined surface is used. We can then take that rod and look at it in the scanning electron microscope, and we see visible tracks left by the string’s vibration.
Figure 4 shows an example. The featureless grey background is the very smooth surface of the rosin coating on the glass rod. Running across this background we can clearly see three tracks, made up of a fairly regular row of little vertical lines. Each of those lines is the “footprint” of a single sticking event in the string’s vibration. The orientation of the string was vertical in this image, and the rod has been moved in the horizontal direction when performing the bow stroke. The three tracks have been created by three separate bow strokes.
Figure 5 shows a zoomed view of a portion of one of these tracks, and Fig. 6 shows an even closer view. Each of the vertical scars in the track shows churned-up rosin. This is the result of the string rolling back and forth a little during that sticking interval. But then the string slips rapidly across the surface of the rod, and we can see very clear evidence that the rosin has been partially melted: look at the streaks on the left-hand side of Fig. 6. These are surely “threads” of hot rosin, drawn out by the string as it slipped across the surface.
Figure 7 shows a low-angle view, to give a different perspective on the terrain revealed by these images. The churned-up texture created by the rolling string during the sticking event which caused the scar in the middle of the image is particularly clear here.
There are standard laboratory measurements that can give information about the mechanical properties of rosin as a function of temperature. As a first step, the usual way to determine the glass transition temperature is with a device called a “differential scanning calorimeter”. This measures the amount of heat absorbed by a small sample of rosin (or whatever material is being tested), as the temperature is slowly raised. Whenever a material undergoes a phase transition from solid to liquid, or from liquid to gas, heat energy must be supplied in the critical temperature range. The resulting peak in the heat absorbtion identifies the temperature of the transition. In the case of a glass transition the peak is quite broad, but its highest point can still be used to put a number on the glass transition temperature.
But we are most interested in mechanical properties, rather than thermal properties as such. For this, we can use a device called a “rheometer” which measures the force needed to deform a layer of the material. The specific deformation we are interested in is shear, because a sliding frictional contact requires shearing between the two surfaces. Figure 8 shows some results of measurement of the shear viscosity of two different types of commercial rosin for bowed instruments, which represent the extremes of available behaviour. The temperature has been varied slowly during the test. The red points are for violin rosin, of the kind we have been talking about so far. The blue points are for a brand of rosin aimed at double bass players. This bass rosin is supplied in a pot: unlike violin rosin, which is supplied in a solid block, the bass rosin will flow, slowly, at ordinary room temperatures so it needs to be prevented from getting away.
The glass transition temperatures of the two types of rosin are shown by vertical dashed lines in corresponding colours: $16^\circ$C for the bass rosin, and $49^\circ$C for the violin rosin. The variation of viscosity with temperature reflects this difference: the two curves have very similar shapes, but they are separated along the temperature axis by approximately the difference of these two temperatures. These plots make it rather clear why the bass rosin needs to be kept in a pot: it is already above its glass transition temperature at normal room temperature.
There are two things to notice about this viscosity plot. First, there is a gap in the sequence of red points. This is simply because two different types of rheometer were needed to cover temperature ranges where the material was “solid” and “liquid”. But it is easy to see that the two would join up if the intermediate range of temperature could have been tested. The second thing to note is the vertical scale. A logarithmic scale is needed for the viscosity because the values have changed by almost 7 orders of magnitude over this range of temperature. When I said that the mechanical properties of rosin changed sensitively with small changes in temperature, I was not joking!
The next step is to try to extend the computer simulation model to incorporate temperature-dependent friction. First, we need to calculate the temperature in parallel with simulating the string motion. The simplest version of such a calculation is easy to describe. Figure 9 shows a schematic close-up of the contact region of rod, string and rosin. The rod (being used as a bow as in the Galluzzo or Schumacher experiments) carries a layer of rosin. It is moved across the string, seen in cross-section in the sketch. There will be a small “contact footprint” between rod and string, with a size that depends on the diameters of rod and string, and also on the normal force (as will be discussed in the next section). In that footprint region, the rosin will be warmer than ambient temperature, indicated by the red patch.
To work out the temperature in this contact patch, we can do a “heat balance” calculation. There are four effects to consider. Heat is being generated by friction, at a rate which is simply the product of the friction force and the speed of relative sliding between the two surfaces. Heat is being lost by two mechanisms. First, there is diffusion of heat by conduction into the bodies of the rod and the string. The second mechanism is associated with the moving rod: cold rosin is carried into the contact area, while warmer rosin moves out at the other end, taking some heat away with it. Finally, there is a term associated with changes in the heat stored in the red contact region: if the temperature is rising, then heat is being added; if it is falling, heat is being removed. Putting these four things together yields a governing equation that we can solve in the computer, alongside the simulation of the string’s motion. In the unlikely event that you are keen to see the gory details, they are explained in reference [1] (you can find a PDF at number 61 in the list here).
Now for the difficult part: we need to choose a specific model for exactly how the friction force is influenced by the temperature. So far, I have been talking as if the force will only depend on temperature, but this is rather misleading. One thing we definitely know about any friction force is that it always opposes the sliding motion. It must change sign if the sliding direction is reversed — but of course the temperature will be the same, regardless of the direction of sliding. So the friction “law” we are looking for must involve the sliding velocity as well as the temperature, in some way. We don’t really have enough experimental data to determine the correct answer, so we need to resort to a bit of guesswork and empirical exploration.
The simplest model, and the one that has been most extensively investigated, is to assume that the coefficient of friction is a function of temperature alone [4,5]. This coefficient would be multiplied by the sign of the sliding speed ($+$ or $-$) to give the necessary reversal of friction force when sliding reverses. We can use the measurement from Fig. 1 to infer what the variation with temperature must be. The same computer code can be used to simulate the steady sliding experiment, and thus obtain the contact temperature as a function of sliding speed. This can be used to convert the results of Fig. 1 into a function of temperature: the result is shown in Fig. 10. These results are based on violin rosin similar to the red points in Fig. 8, so it probably has a similar glass transition temperature, around $49^\circ$C. It is reassuring to see that this temperature falls near the middle of the downward slope in Fig. 10.
Armed with this function of temperature, we are ready to incorporate the thermal friction model into our bowed-string simulation. Figure 11 shows an example of a Guettler constant-acceleration transient, simulated in this way. Specifically, it is the case corresponding to the pixel (10,14) in the Guettler plots seen in section 9.5 (10th acceleration from the left, 14th force from the bottom, bow position $\beta=0.0899$). The red curve shows the new simulation, while the black curve in this figure shows the corresponding measured bridge force. In this case, both of them lead to the Helmholtz sawtooth, after a rather short transient.
Figure 12 shows the corresponding computed temperature variation. The mean temperature is predicted to rise to some $70^\circ$C, and in every cycle of the final Helmholtz motion there is a fluctuation by some $30^\circ$C. This certainly seems like a big enough change to account for the appearance of the tracks on the glass rod seen in Figs. 5—7: the rosin looked “melted” during slipping, but semi-solid during sticking.
However, we only learn a rather limited amount by looking at a single transient like this. The next step is to use the new model to scan a family of transients and construct a Guettler diagram. Figure 13 shows an example, chosen to match the cases we saw in Fig. 7 of section 9.5, which is reproduced here as Fig. 14 for easy comparison. It is immediately obvious that the new model behaves quite differently to the friction-curve model. It has produced a fairly solid patch of colour (marking cases that led to Helmholtz motion), in contrast to the “spotty” texture of the friction-curve case.
So far, so good: but the patch of colour in Fig. 13 is nowhere near as big as the corresponding patch in the measured Guettler diagram. A player of the friction-curve “cello” would find their job almost impossible: every pale pixel marking a good, fast transient is not far from some black pixels. This means that if the cellist tried to repeat a successful bow stroke, they are likely to be frustrated by sensitivity to small details of the gesture. The “thermal model cello” of Fig. 13 would be preferred: there is a region of the Guettler plane where every transient gives a successful outcome, with relatively small variations in transient length. However, the real cello would surely win hands down in this playability comparison: it offers a far larger range of transients that “work”. From the point of view of a beginner, this means that it is easier to perform a bow gesture that is not a disaster. From the perspective of an expert, the larger available area of the plane will open up a wider “sound palette” of musical possibilities.
So can we do any better than this? In terms of improved models that are firmly based in understanding of the underlying physics, there is no good answer to that question at present. But to give a hint that better models are out there waiting to be found, I will show a few examples from a slightly enhanced version of our thermal friction model. Comparing the two waveforms in Fig. 11, we can see one qualitative difference at the very start of the string’s vibration. Both waveforms begin with a similar rising curve. What is happening here is that the string is sticking to the bow, so it is being pulled to one side by the accelerating motion of the bow. The motion we are interested in can only begin after the first moment of release, when slipping starts. In the measured waveform (in black), that first release happens with a jump. The simulated waveform in red shows no such jump. Furthermore, as explained in the next link, this version of the thermal friction model can never show a jump.
There is a simple (but rather ad hoc) way to enhance our friction model to allow the possibility of jumps. The argument is explained in the previous link. An example of this enhancement in action is shown in Fig. 15. The two waveforms from Fig. 11 are now followed by a third, in blue. This new waveform does indeed show an initial jump that looks broadly similar to the measurement (in black).
Using this new model to scan a range of transients, we obtain the Guettler diagram shown in Fig. 16. Looking back at Figs. 13 and 14, is this better or worse than the original thermal model? Better, surely: coloured pixels are found over a larger area of the Guettler plane than in Fig. 13. The terrain in both plots is a bit spotty, with black pixels dotted around, but this is not necessarily a bad thing: we have seen in section 9.5 that the real bowed cello string showed clear evidence of “sensitive dependence” leading to spottiness rather like this.
It is worth digging into the details a bit more, to understand what is going on and get a sense of what a player might think about it. This particular presentation of the Guettler results certainly does not tell the whole story. It is based on an automated routine for detecting Helmholtz motion, and then establishing the length of each transient. The problem with that kind of routine is that it is drawing a hard line between “Helmholtz”and “not Helmholtz”. Sometimes, this distinction is clear, but not always: there are “nearly Helmholtz” waveforms, and placing the threshold is a matter of judgement on the part of the programmer of the automated detection routine. The result is that a gradual change from Helmholtz to nearly-Helmholtz, which might not bother a player very much, is turned into an abrupt change from a coloured pixel to a “failed” black one.
If we process the results in a different way, we can get a different view of the relative merits of the various simulation models we have described. One approach is to look at the final waveform, near the end of each simulation, and simply ask how similar this is to the corresponding measured waveform: no attempt is made to classify the waveforms. Specifically, we will look at two period-lengths near the end of each measured Guettler transient. A corresponding chunk can be extracted from the simulated results, adjusting the phase of the two-period chunk to give the best match. We can then calculate a numerical measure of the difference between the two: the particular measure chosen to be plotted here is the root-mean-square (RMS) difference, after each separate waveform has been scaled to have unit RMS value.
If we plot how this difference measure varies over the Guettler plane, we get the results shown in Figs. 17, 18 and 19. Figure 17 compares the measurements with the original thermal model, Fig. 18 with the modified thermal model, and Fig. 19 with the friction-curve model. The paler and “hotter” the colour, the more similar the waveforms are. Of the three figures, Fig. 18 has the hottest colours overall. Figure 17 shows a similar pattern but with less white and more yellow. Figure 19, for the friction curve model, performs the worst. Based on these plots, the modified thermal model looks the most promising.
To see what lies behind these plots, the next set of figures shows the detailed results for a single row of the Guettler diagram. Figure 20 shows the 20 waveform comparisons which give the difference results shown in Fig. 21, based on the original thermal model. In each case, the measured waveform is shown in black, and the phase-matched simulation is shown in red. The first two cases of Fig. 20 show rather irregular string motion for both measurement and simulation. The details are quite different, so it is no surprise that the difference measure is quite big. The next 11 cases show something looking like Helmholtz motion in both measurement and simulation, and the first few of these show quite a good match.
Cases 9–13 are interesting: the simulations appear to show Helmholtz motion, as does the measurement, but the shapes are distinctively different in the simulations. The bottom “points” of the sawtooth waveform have been rounded off, in a way that is not echoed in the measurement. These waveforms are sufficiently different from the ideal Helmholtz sawtooth wave that the automatic classification routine has rejected them: that is why the right-hand half of Fig. 13 was solid black. This is a good example of the difficulties of automatic classification — but the routine has correctly identified the fact that the simulation is not a good match to the measurements, which is the question we are most interested in at the moment. The remaining cases for Fig. 20 show versions of double-slipping or other Raman “higher types” in both measurement and simulation. The simulations still have the characteristic rounded points, but the qualitative comparison is otherwise reasonably good.
Figure 22 and 23 show corresponding results for the modified thermal model. A first thing to notice is that none of the simulated waveforms here show the rounded bottom points, as we noticed with the original thermal model. As in Fig. 20, the first two cases show irregular motion in both measurement and simulation. Cases 3, 4 and 5 show Helmholtz motion in the measurement, while the simulations either show Helmholtz motion with large Schelleng ripples, or perhaps they show S-motion. This is another example of the difficulty of automatic classification: it is far from clear how these waveforms should be labelled.
The middle two rows of Fig. 22 show Helmholtz motion in both measurement and simulation, in most cases. Case 11 is an exception: it has Helmholtz motion in the measurement but double slipping in the simulation. We need to be a bit careful in how we interpret this. We have already commented that the bowed string, both in reality and in simulation, shows some evidence of “sensitive dependence”. Back in section 9.5 we saw some results of repeat measurements of the Guettler plane, and although the qualitative picture remained the same, individual pixels could change between takes. What we are seeing in case 11 of Fig. 21 may be illustrating this sensitivity: perhaps the choice between Helmholtz motion and double-slipping is rather finely balanced for this case, and therefore subject to sensitivity. The disagreement between measurement and simulation is not necessarily pointing to incorrectness in the model.
The final row of Fig. 22 shows some variety of double-slipping in both measurement and simulation, in all cases. There are differences of detail, leading to variations in the RMS difference measure, but actually the qualitative agreement seems rather good throughout. In some cases, the difference simply reflects a small difference in the phasing of the two slips: I would suspect that such a difference has very little influence on the perceived sound. Overall, then, the modified thermal model performs rather well in this comparison: slightly better than the original thermal model. That is not to say, by any means, that this model is perfect, but it is very encouraging. Is the agreement good enough that this model could be used reliably to do systematic simulation studies on playability differences? The answer to that question is not yet clear: I said we would reach a research frontier in this section.
Finally, for completeness Figs. 24 and 25 show corresponding results for the friction-curve model. We need not go through and describe the waveforms individually. Some cases show reasonable agreement with the measurements, but it is pretty obvious that this model gives unsatisfactory results for many cases.
[1] J. H. Smith and J. Woodhouse, “The tribology of rosin”; Journal of the Mechanics and Physics of Solids 48 1633–1681 (2000).
[2] J. Woodhouse, R. T. Schumacher and S. Garoff, “Reconstruction of bowing point friction force in a bowed string”; Journal of the Acoustical Society of America 108 357–368 (2000).
[3] R.T. Schumacher, S. Garoff and J. Woodhouse, “Probing the physics of slip-stick friction using a bowed string”; Journal of Adhesion 81, 723–750 (2005).
[4] J. Woodhouse, “Bowed string simulation using a thermal friction model”; Acta Acustica united with Acustica 89 355–368 (2003).
[5] P. M. Galluzzo, J. Woodhouse and H. Mansour, “Assessing friction laws for simulating bowed-string motion”; Acta Acustica united with Acustica 103, 1080-1099, (2017). DOI 10.3813/AAA.919136