The discussion in this section is in part the result of a research workshop held in Cambridge in May 2024. It is pleasure to acknowledge the other participants, who all contributed ideas and data: Joseph Curtin, Andreas Hudelmayer, Luca Jost, Colin Gough, Sean Hardesty and Claudia Fritz.

The topics to be discussed fall under three main headings: approximations made in putting together the theoretical model of the bridge, issues associated with the measurements that are a necessary ingredient of the model calculations, and questions about whether it is possible to devise a reliable workshop-level method of measuring the parameter values for a particular bridge to help a maker control the adjustment process. We will not be able to give definitive answers to all these questions, but at least we can make explicit where there are known shortcomings, and explore the level of confidence we should have in the model predictions.

**A. Missing factors in the theoretical model**

One approximation in the theoretical model has already been mentioned. The idealisations of the two bridge modes sketched in Figs. 2 and 3 of section 7.5 showed the force on the body in the vertical direction. However, there may also be a sideways force component at the feet: this is most obvious for the rocking motion shown in Fig. 2, where a side force is needed to balance the lateral applied force. In the derivation of the extended model, described in section 7.5.1, the side force is taken into account to a certain extent. However, it was assumed in that model that there was no sideways *motion* at the bridge feet: in other words, that the admittance of the body in the tangential direction was zero.

Measuring the tangential admittance is not as easy as measuring the normal admittance, but with a bit of ingenuity an approximate measurement can be made — the method is described in the next link. The results, for the two bridge-foot positions of our standard test violin, are shown in Figs. 1 and 2. In both cases the blue curve shows the normal admittance, more or less in agreement with the version seen previously in Fig. 5 of section 7.5, while the red curve shows an estimate of the corresponding tangential admittance.

Both plots show that the tangential admittance is much lower than the normal admittance at lower frequencies, but it has a systematically rising trend, and by the highest frequencies the two become comparable. As will be explained shortly, there are good reasons to expect this measurement to give an over-estimate of the true tangential admittance. As a result, these plots give good justification for the original approximation, in which the tangential admittance was assumed to be negligible. This is surely true over most of the relevant frequency range, but we are warned that the model predictions at the highest frequencies may become unreliable because of the missing tangential effect.

As explained in the side link, the rising trend shown by the two red curves in Figs. 1 and 2 is the signature of excitation of bending vibration in the violin top via a *torque* (or *moment*) applied at the bridge foot. As illustrated by the sketch in Fig. 3, a tangential force on the surface of the plate does indeed create a torque about the centre-plane of the plate. This will make the plate twist, and so excite bending vibration. (The simplest theory predicts that the ratio of the tangential admittance to the normal admittance should rise proportional to frequency, and the side link shows that this gives a fairly reasonable match to the measured trend in both cases.)

The magnitude of the torque is given by the force multiplied by the distance to the centre of the plate, $Fh/2$ in the notation defined in Fig. 3. However, in order to measure the tangential admittances plotted in Figs. 1 and 2, a small block was stuck to the surface of the violin to give a target for the impulse hammer to strike and the laser beam to reflect from. This means that the actual tangential force was applied a small distance above the surface of the plate. This increases the torque for a given force, and means that the measured admittance will be over-estimated.

Ignoring effects of the sideways force at the feet is not the only approximation that has been used in the model development. The next one is a related effect: as well as the two components of force, each individual bridge foot could directly apply a torque to the violin body (in addition to the torque we have just talked about, from the sideways force). This would be associated with twisting of the bridge feet, or conversely with such twisting being resisted by the bridge. For a violin bridge such twisting would involve flexing the “ankles”, but for the long legs of a cello bridge the details would be significantly different. No attempt has yet been made to model this effect — it is a task for the future.

Another approximation is of a rather different kind. The rocking mode has been represented as a loaded bar, hinging about a fixed pivot point on the bridge base. This amounts to an assumption that the top part of the bridge, above the “heart”, rotates as a rigid body. This seems a very plausible first approximation, but it is far from clear how accurate it is, or whether there is really a well-defined “pivot point”. Probably the best way to probe this question would involve detailed Finite Element calculations — such calculations have been carried out for violin bridges, but they have not been analysed to probe this particular question in detail.

*B. Issues arising from measurement*

A rather different source of concern comes from errors and uncertainties associated with measurements. In order to make predictions of the effect of bridge adjustment, we have used two kinds of mechanical measurement: the rocking and vertical admittances on the bridge of the complete instrument, and the set of admittances at the bridge-foot positions on the body without the bridge. In order to make audio demonstrations, a third measurement was also used: the radiated sound from tapping at the bridge-foot positions. A number of issues are raised by these measurements.

Perhaps the first one to occur to a violin maker is that the bridge admittance measurements include the effects of the tensioned strings, but both measurements made at the bridge-foot positions inevitably require the strings to be removed. Surely the body behaviour will be different without the effect of the strings? It is worth discussing this issue rather carefully, because there are several different ways in which the strings might influence the body vibration.

The first aspect to consider is the static force exerted downwards through the bridge as a result of the combined tension of the strings together with the break angle of the strings over the bridge. This static force will slightly deform the body, and it is likely to affect the tightness of the soundpost. But, counter-intuitively to many people, it has no direct effect on the vibration behaviour of the body other than through these small geometric effects. Vibration is only affected by internal forces which *change sign* during a cycle of vibration — such forces contribute to the energy balance of the vibrating system, and hence to the stiffness which determines vibration resonance frequencies. But a static force does no net work through a cycle of vibration, and so it does not contribute to these things.

Far more important is the effect of the *stiffness* of the strings along their length. During a cycle of vibration, if the top of the bridge moves it must carry the strings along with it. Because of the geometrical arrangement of strings, nut, tailpiece and bridge, this will change the length of each string: it will be a little longer in one half of the cycle but a little shorter in the other half. This in turn will modulate the string tension by a small amount, and crucially this effect changes sign during the vibration cycle. The result will be an extra stiffness contribution, raising the frequencies of all the body modes slightly. We met a similar phenomenon when we discussed the behaviour of the banjo in section 5.3: any stringed instrument featuring a tailpiece, a “floating” bridge and a break angle of the strings over the bridge will behave in the same way.

Finally, we should consider whether the *vibrational properties* of the strings have an influence on the body: for example, it might be thought that the strings will add some effective mass to the top of the bridge. But this turns out not to be the case. Instead, the main effect is that the strings add some extra *damping* to the measurements of bridge admittance.

As the bridge vibrates in response to a tap from the impulse hammer, it will generate transverse waves in the strings which travel outwards in both directions, away from the bridge. Admittance measurements are normally made with the strings damped, and if that damping were to be 100% effective the waves that are sent out along the strings would be absorbed before any reflected waves could travel back to the bridge. The influence is thus as if the strings were *infinitely long*, stretching away into the distance. But the behaviour when a force is applied to an infinite string can be analysed very easily, and the result is that it behaves exactly like an ideal damper (or *dashpot*). Strictly, that is the mathematical answer if the bending stiffness of the string is ignored — but it gives a very good first approximation to the behaviour of real musical strings.

There are other effects associated with measurements, that we should take into consideration. We have rather blurred the distinction between bridge-foot forces that are horizontal and vertical, and forces that are normal and tangential to the arch of the top plate. The measurements are always normal to the surface, so strictly the forces in the bridge model should be at a small angle — but this is a small effect.

A more serious issue of the same general kind affects the measurements of bridge admittance with the strings in place. The vertical admittance at the bridge centre is straightforward: both hammer tap and laser beam can be oriented close to the vertical at that point. But the rocking admittance raises an issue, which is illustrated in Fig. 4. An important quantity that enters the bridge model is the perpendicular distance from the effective centre of rotation to the line of action of the applied force. Rocking admittance is usually measured by tapping with the impact hammer at a corner of the bridge. But the line of action of that force could vary a lot, depending on the details.

Figure 4 shows two examples: the blue arrow corresponds to a horizontal impact at the corner, while the green arrow corresponds to a tap in the bowing direction of the G string. The purple and red lines show the corresponding perpendicular distances from a plausible position of the centre of rotation of the rocking mode of the bridge with its feet clamped rigidly. It is immediately obvious that these two distances are different: the red line is about 50% longer than the purple line. This would translate into a difference of about 3.5 dB in the measured admittance if the bridge was held with its feet in a vice, a significant amount in the context of the subtleties of bridge adjustment. A similar argument would apply to the line of action of the measured response, whether by laser vibrometer or accelerometer: there could be a big difference between a horizontal measurement and a measurement tangential to the curve of the bridge top.

Uncertainty about the magnitude of the measured admittance leads directly to uncertainty in the parameters of the bridge model, deduced by the best-fitting process described in section 7.5. We will come back to this question shortly, after we have mentioned one final issue connected to measurements on the instrument body.

When making the body admittance measurements that are needed for the bridge model, it rapidly became apparent that the results were very sensitive to the exact position at which the measurements were made. This sensitivity also affects the measurement of sound radiation, and the effect is illustrated here by sound radiation results. Figure 5 shows, to scale, the footprints of the two bridge feet, which each have dimensions $12 \times 4$ mm. A series of measurements were done in which the impulse hammer tapped at 2 mm intervals along the mid-line of each footprint, and the response recorded at a fixed microphone position.

Coloured dots on each footprint in Fig. 5 mark a selection of positions from this scan, and the corresponding results are shown in Figs. 6 and 7. For the bass foot, positions at the centre and the extreme edges are shown. For the treble foot two additional positions are included, 2 mm on either side of the centre. At the bass foot (Fig. 6), the differences are small at low frequency, but above 1 kHz or so, differences of up to 10 dB can be seen in some frequency ranges. At the treble foot (Fig. 7) there are comparable differences at high frequency, but also large differences at low frequency, in the “signature modes” range. Even a shift of 2 mm away from the centre (the two dashed curves) results in changes of 3 dB or so. Presumably, this low-frequency sensitivity is associated with the proximity of the soundpost to this bridge foot. We will return to the interaction of bridge and soundpost in section 7.6.

The sensitivity revealed by Figs. 6 and 7 arises largely from the vibration behaviour of the violin body, rather than from differences in sound radiation as such. This is confirmed by the fact that comparable sensitivity was seen in measurements of the body admittance matrix. Differences in the admittance matrix give rise to differences in the process of tuning the bridge model to best reproduce the measured bridge admittances. Some examples of that sensitivity will be shown shortly, when we turn to questions of approaches to estimating the bridge parameters.

*C. Estimating bridge model parameters*

To use the modelling developed here and apply it to a particular bridge on a particular violin, we obviously need to establish the parameter values for that bridge. The most important of these are three parameters for the rocking motion (moment of inertia $I$, torsional spring stiffness $K$ and the effective length $L$ from the top of the bridge to the pivot point) and two parameters for the bouncing motion (mass $m$ and stiffness $k$). The discussion here will focus on the three parameters of the rocking mode, which has the most important effect on bridge adjustment. Three approaches have been tried, but all of them show sensitivity to experimental details and it requires some care to reconcile the values they give, at the level of accuracy relevant to realistic bridge adjustments.

The first approach is the one that has already been described: with the bridge* in situ* on the instrument, measure the rocking and vertical bridge admittances, then use a numerical optimisation procedure to find the set of 5 parameters that give the closest match with the admittances predicted by the model. This method has the advantages that the bridge is in its natural environment, and that we are using the same model for the fitting process that we then wish to use to investigate the effects of adjustments to that bridge. However, it has disadvantages too: the approach is rather indirect, it is computationally complex, and there is no clear route to justify the individual parameter fits by physical arguments.

The second approach swaps these advantages and disadvantages. As we have seen in section 7.5.1, the parameters we are trying to fit have an interpretation in terms of the resonance frequencies and mode shapes of the bridge when the feet are rigidly fixed. We can do measurements in precisely that way: the bridge feet can be clamped in a suitable vice, and we can do direct measurements of the rocking and bouncing resonances. The two frequencies immediately give us the ratios $K/I$ and $k/m$. This approach has the advantage that a violin maker could use it as a workshop procedure: it does not require complicated equipment.

However, we need to do more than this to determine all 5 parameter values. One possible approach is to attach small masses at carefully controlled places on the clamped bridge, and monitor how the resonance frequencies change. As explained in the next link, attaching a single known mass at the top of the bridge gives information about $m$ and $I$, while for the rocking configuration if we also attach the same mass at a second position, the distance $L$ can be estimated.

The third approach is a sort of combination of the first two: clamp the bridge feet as in the second approach, then measure the two bridge-top admittances in that condition. From calibrated measurements like this, it should be possible to deduce the frequency, damping factor and effective mass of each bridge resonance using the approach of experimental modal analysis (which will be explained in section 10.5).

The simplest way to compare these three approaches is through their predictions of the *effective mass* for the rocking mode of the bridge. But we need to be a bit careful, because “effective mass” could have more than one meaning. The starting point is something mentioned above: the frequency of the rocking mode is governed by the balance between the rotational stiffness $K$ and the moment of inertia $I$. But moment of inertia is not the same thing as mass: the units of moment of inertia mean that it is equal to a mass times the square of a length.

In other words, to convert the moment of inertia into an effective mass, we need to divide it by the square of a length. But which length? There is more than one possibility, implicit in the different approaches listed above. For a given value of the moment of inertia a shorter length will give a higher effective mass; a longer length will give a lower mass. To compare the masses obtained by the different approaches, we need to be clear about the relevant length for each.

There are two possibilities which seem physically sensible, but which do not in fact relate directly to the three measurement methods. An obvious question is: “can we use a length such that the deduced effective mass would be something we could in principle find by weighing?” Remember that the top part of the bridge is assumed to rotate as a rigid body in the rocking resonance. If the moment of inertia is to be expressed via the actual mass of that rigid body, the length we would need to use is something called the “radius of gyration”. But there is no simple measurement that would tell us this length.

Another natural possibility involves a parameter of the rotating rigid body which enters the detailed modelling (see section 7.5.1): this is the *centre of mass*. There is a very general result in mechanics (called the “parallel axis theorem”) which tells us that the distance to the centre of mass must be less than the radius of gyration, and so the effective mass we would deduce using that distance would be greater than the actual mass. But in any case, there is again no simple measurement procedure that would tell us the value of this length.

Turning to our three actual measurements, the second approach gives the most unambiguous answer to the question of which length it assumes. As explained in the previous link, the method to infer the effective mass involves sticking a small known mass at the top of the bridge, then tracking how the clamped-foot rocking frequency changes. The effective mass given by this method is at the position of the added mass, so the relevant length is $L$. Almost certainly, this is greater than the radius of gyration. The result is that the effective mass we deduce will be less than the actual mass.

Furthermore, the results will be sensitive to exactly where the mass is attached: depending on the method used, it might be slightly above or slightly below the top of the bridge curve, and those variations will influence the deduced effective mass. And there is another complication: the calculation described in the previous link makes an assumption that the added mass is sufficiently small that it does not affect the *mode shape* of the rocking resonance. So you need to use the smallest possible mass for the test, but of course this has the effect of reducing the frequency difference you are measuring, and thus of reducing the accuracy. A compromise is necessary.

A simple experiment with a particular bridge gives an idea of the magnitude of this effect: without changing anything else, frequencies were recorded after different numbers of 0.05 g masses were fixed to the top of a particular bridge. The calculated effective masses for the rocking mode were 0.60 g, 0.56 g and 0.52 g after adding 1, 2 and 3 masses respectively. A pattern of “diminishing returns” is seen when more mass is added, and this is what would be predicted from simple vibration theory. When a mass is added is added to any structure, the mode shape will tend to adjust itself to give a little less motion at that point, so that when a second mass is added at the same point, it has slightly less effect than the first one. So the result with a single mass is probably the best estimate. Possibly an even better estimate would be given by extrapolating the pattern to *zero* added mass, suggesting an effective mass for the rocking mode of 0.64 g. We take note of the fact that this approach will always tend to slightly underestimate the true effective mass.

In the first approach to measuring the effective mass, the model fitting process gives estimates of the values of the moment of inertia $I$ and the effective length $L$. To deduce an effective mass, the natural choice is the one we already used to generate Fig. 9 of section 7.5: we can use the length $L$ to obtain a mass $m_e=I/L^2$, which can be visualised as a point mass located at the very top of the bridge. This would be the same as in the second approach, but there is a complication. There is another aspect of this approach which involves an effective length, as sketched in Fig. 4. The measured rocking admittance plays an essential role in the process, and there is uncertainty about the line of action of the hammer force during that measurement, and thus in the perpendicular distance from the pivot point. The resulting variation in admittance will change the fitted values of the parameters $I$ and $L$, and hence the value of $m_e$: specifically, the method may tend to over-estimate the mass.

A very similar uncertainty affects the third approach, in which the bridge feet are clamped in a vice and then the rocking admittance is measured. For any calibrated driving-point admittance, the modal amplitude that we would deduce from experimental modal analysis (for example by “circle fitting”, see section 10.5.1) is the inverse of the effective mass at the position and along the line of action of the admittance measurement. So looking back at Fig. 4, a hammer force aligned along the green arrow would give a smaller effective mass than a hammer force aligned along the blue arrow.

Furthermore, the usual measurement of rocking admittance is not really a point measurement at all: the motion is measured at the treble corner of the bridge top. The orientation of the sensor used to make that measurement will introduce a similar uncertainty, and the actual deduced mass will involve a combination of both — as well as possible uncertainty because the measurement position is different from the excitation position.

There is an additional issue affecting this approach to measuring effective mass. The circle-fitting procedure actually gives the product of two factors: the inverse effective mass, and the Q-factor governing the damping of the bridge resonance. So to deduce the mass, we need to estimate the Q-factor by an independent means, then divide it out. But accurate measurement of damping of any vibrating system is notoriously tricky. You are doing well to measure a Q-factor to within 10% accuracy. Unfortunately, 10% error in the effective mass of a bridge would probably be regarded by a violin maker as unacceptably high for the purposes of controlling bridge adjustment.

Having described the three approaches and their issues it is time to see some results, for the same bridge and violin which featured extensively in section 7.5. This violin bridge has been repeatedly tested by all three methods, and the various resulting estimates of the effective mass are shown in Fig. 8. The left-hand column of symbols shows masses deduced from the first approach, by fitting the bridge model to measured admittances with the bridge in place on the violin. The four estimated masses are derived from independent measurements of the body admittance matrix and the two bridge admittances, on different occasions over a period of some two months.

The middle column shows masses deduced from the second approach. The feet of the bridge were clamped, and the rocking frequency was measured with and without the addition of a 0.1 g mass at the top centre. The four symbols correspond to variations in the details: three different vices, and for one of them, two different clamping pressures on the feet. (This pair is indicated by the diamond and dot symbols.) The right-hand column shows results by the third approach, using the unloaded admittances from the same four tests with variations of clamping conditions: the symbols indicate the corresponding cases to those in the middle column.

Figure 8 shows that all three methods give a considerable spread in the estimated masses. For the second and third columns, notice that the symbols are in the same order in both cases, but the masses are systematically higher for the third method. The most likely reason for this is the issue highlighted in Fig. 4: the third approach relies on the calibrated value of admittance, and this is subject to doubts about the line of action of the applied force. The result of this uncertainty would be a bigger spread in estimated effective mass, but there is a systematic tendency to give a higher value.

The first two approaches are, in principle, estimating the same effective mass (located at the top of the bridge), so it is reassuring that they give comparable results. The second approach, via frequency shifts when a mass is added, is the simplest and most robust method, so it is perhaps a little surprising that this gives such a large spread of estimated masses. Figure 9 shows some of the underlying measured admittances. For this plot the bridge has been held in the same vice throughout, and the three curves show the effect of increasing the clamping force in stages. It is immediately clear that changing the clamping force has affected both the frequency and the damping of the main peak.

We learn several things from Fig. 9. The somewhat unexpected sensitivity to clamping force must lie behind the spread of effective masses seen in Fig. 8. What is the “correct” mass? There is no good answer to that. But if a violin maker wants to use this approach to help them control bridge adjustment, they need to use a careful procedure to ensure that bridges are always held in exactly the same way so that consistent effective masses can be obtained. These measurements were made with a clamping arrangement routinely used by violin maker Andreas Hudelmayer, and he deals with this issue by using a torque wrench to control the clamping force to a standard level. The red curve in Fig. 9, corresponding to the diamond symbols in Fig. 8, uses his standard setting.

Another thing revealed by Fig. 9 is that there are other peaks in this frequency range, adding complication to the interpretation of the results. These are presumably out-of-plane bending modes of the bridge, which would be entirely different when the bridge was mounted on the violin in the normal way because of the constraining effect of the strings at the top of the bridge. A maker wanting to use this approach in the workshop would need to take care that these extra resonances do not interfere with the identification of the main rocking peak, both without and with the added mass.

Figure 10 shows, in a similar format to Fig. 8, estimates for the effective rocking length $L$ obtained by first and second approaches (the third approach doesn’t allow $L$ to be estimated). For the second approach, two different positions were tried for the second added mass, and the corresponding estimates of $L$ are shown with different colours of the same symbol. Both methods show a spread of values: a relatively narrow spread for the first approach, but a very wide spread for the second. The pairs of symbols in different colours are always quite close, but nevertheless one is left with the feeling that the second approach is not very useful.