A. Background of bellringing
Our final example of a musical instrument influenced by impacts and bouncing is the church bell. The sound of bells, when rung in the “full circle” style, depends on multiple bouncing in a very unexpected way. We gave a brief account of full-circle ringing back in section 3.4, when we were mainly interested in the vibration frequencies of bells, but now we need to look a bit more carefully at the details. This section will draw heavily on a published account .
Change-ringing on church bells, as practiced mainly in the U.K., involves the ringing of complex sequences of notes on bells that can weigh up to a tonne or more. A bell suitable for change-ringing is supported on a pivot so that, for each note struck, it can rotate through a full circle. The motion is controlled by the ringer’s rope, which passes round a wheel fixed to the bell. The bell rotates in opposite directions for alternate strikes, called “handstroke” and “backstroke”. The typical arrangements at the start of each are illustrated in Fig. 1. The bell starts from an inverted position, the ringer pulls the rope, and the bell rotates through about 360˚. At some time during the swing the clapper strikes the bell. By the time the swing is completed the clapper has normally come to rest against the side of the bell. The ringer pulls the rope again more or less immediately to start the reverse swing and the next strike. You can see a brief video of some bells in action in this way in Fig. 2.
By making subtle changes in the amplitude of the swing near the top of the circle the ringer can make the necessary timing adjustments for controlling the position of the bell in the ringing sequence. As we will see, the sound and the “ringability” of bells depend not only on the linear acoustics of the modes and natural frequencies of the bells, but also on the rotational dynamics of the bell and clapper, interacting through impact each time the clapper strikes the bell surface.
B. Clapper bouncing
To see what actually happens when a bell is rung, a small bell was mounted in the laboratory on a makeshift “bell tower”: you can see the arrangement in Fig. 3. Various sensors were fitted, to reveal what was going on when the bell was rung. Accelerometers were attached to both bell and clapper, and also a simple electrical circuit in which the bell-clapper contact acted as a switch allowed times of contact and non-contact to be detected directly.
Figure 4 gives an example of the result, for a single ring. The top trace shows the output of the contact detector: when the level is high, the clapper is out of contact with the bell, and when it is low it is in contact. What is revealed is very complicated. The clapper is initially out of contact, but very soon it makes a first strike which shows as a very short downward spike in the signal. Zooming in reveals that the length of this contact is about 0.3 ms: we will see the significance of that number shortly. This first strike is followed by a long series of further contacts, getting progressively closer together until they merge. The signal is then flat at the low level for a while, showing that the clapper is resting against the side of the bell. At the right-hand side of the plot, the bell has started to swing the other way, and the clapper lifts off the bell so that the signal returns to the high level. The clapper is then “in flight”, but it does not strike the bell again within the range of this plot.
The middle trace of Fig. 4, in blue, shows the output of the clapper accelerometer. The pattern follows the prediction of the top trace: there is a big pulse of clapper acceleration following the first strike, then a series of further pulses corresponding to the multiple impacts just described. You can listen to the result in Sound 1. For this, the signal has been processed in the computer to give velocity rather than acceleration: as we have found with previous sound demonstrations, velocity gives a better surrogate than acceleration for the radiated sound of a vibrating object. You can hear the sequence of multiple impacts very clearly. At the end, you may hear a rather faint high note: this is one of the vibration resonances of the bell, audible in the clapper signal because the clapper is now resting in contact. If you listen carefully, this high note ends a bit before the sound sample is over: the clapper has lifted off the surface of the bell.
The bottom trace of Fig. 4, in red, shows the output of the bell accelerometer. You can see that the bell starts vibrating at the moment of first impact, but the complicated pattern of multiple impacts is not really visible because the bell vibrates for a relatively long time following each impact, and these effects overlap.
Unfortunately, it was discovered on revisiting this data that the bell accelerometer was faulty, and the signal is distorted. The waveform plotted in Fig. 4 is qualitatively correct, but in order to generate a sound file a new recording was made on the same bell. Turning this new signal into velocity yields Sound 2, which does not exactly match the pattern of multiple clapper impacts as the original but is as similar as could be achieved after the lapse of time.
It is interesting to compare this sound with the corresponding response when the bell was “chimed” by pulling the rope so that bell and clapper swung just enough for contact to be made, but without turning the bell through the full circle. There is just a single contact, and the velocity waveform deduced from the bell accelerometer can be heard in Sound 3. Sounds 2 and 3 are somewhat different, even though both were recorded using the same accelerometer and processed in the same way to convert to velocity. In the “chimed” case the bell rings on longer, whereas the circle-rung Sound 2 is “quenched” more abruptly.
To understand this sound difference, we can look at spectrograms: Figs. 5 and 6 show spectrograms of Sound 2 and Sound 3, with the original velocity signals plotted alongside. Both show a set of vertical lines corresponding to the strongly-excited vibration resonances of the bell. Figure 5 also shows traces of horizontal rows of “blobs”, which are a direct manifestation of multiple clapper impacts: the bell vibration is given another kick by each impact. (As a side note, comparing these images with the spectrograms shown in Fig. 3 of reference  shows the accelerometer distortion in action: those earlier spectrograms are misleading.)
If we concentrate on the frequencies of the first few strong resonances of the bell, we can learn other things about the sound of the circle-rung bell. First, we should note that only a few resonances of the bell are strongly excited: Figs. 5 and 6 show five in the frequency range plotted here. This number fits very well with the analysis of bouncing from section 12.1 and its side links. In section 12.1.2 we gave a simple estimate of this number: the maximum possible number was predicted to be, roughly, 1/3 of the mass ratio of bell to clapper. This estimate was on the basis that all the kinetic energy of the clapper is turned into kinetic energy of vibration, with no rebound. For the bell tested here, the two masses were 45.8 kg and 1.65 kg, so the predicted maximum number is in the vicinity of 9. In reality there was a small rebound of the clapper after the first impact, so we would expect to get a slightly smaller number than the maximum, which indeed we do.
The analysis from section 12.1.2 also told us something about the time of contact: if there is to be a rebound, the contact has to be long enough that the frequency spectrum of the contact force covers the correct range of frequency to cover the predicted number of modes. We noted earlier that the measured contact time between clapper and bell for the first impact in Fig. 4 was 0.3 ms, which translates to a bandwidth of the order of 2—3 kHz. Figure 7 shows the frequency spectrum of Sound 3, plotted over a wider frequency range. Noting that the vertical axis in this plot shows a very wide decibel range, we can see that a bandwidth estimate around 3 kHz is about right: above that range, peaks heights are at least 30 dB below the highest level.
Returning to the spectrograms in Figs. 5 and 6, we can learn something interesting by pulling out the decay profiles of the first few bell modes. Figure 8 shows the comparison of these profiles from the columns of the spectrograms corresponding to the first three modes: the results from Fig. 5 are plotted in red, and those from Fig. 6 are in blue. The vertical scale is in decibels, so exponential decay would be indicated by a straight line in the plot. All three blue lines, for the chimed bell, show this exponential pattern, but the red lines do not.
The clearest pattern is shown in the left-hand plot. For the first half of the time range, the red curve falls a little faster than the blue line, but still with an approximately exponential decay. But then the red curve plunges rapidly downwards, to very low levels: this is the “quenching” in action. The other two plots in Fig. 8 show a similar pattern, albeit less dramatically. Both red curves begin by approximately tracking the corresponding blue curves, but by the end of the time range they have fallen to lower levels.
We can give a tentative explanation for the pattern shown in the left-hand plot of Fig. 8. The transition between the two parts of the decay curve comes at about 0.7 s, and we know from Fig. 4 that this is approximately the time when the clapper stops bouncing and comes to rest against the bell. While the clapper is bouncing, there will be some energy loss associated with each impact, including some transferred to other modes as a result of energy redistribution across the frequency spectrum. This energy loss is probably the reason that the red curve falls a little faster than the blue curve. But once the clapper has come to rest, things change. The strongest candidate for the observed dissipation after that time is friction. Bending vibration in the rather thick-walled bell will produce some tangential surface motion, and thus cause sliding against the resting clapper. Frictional damping is very effective, and the bell motion is rapidly quenched.
It seems likely that the relatively long time during which the clapper bounces plays an important, and somewhat counter-intuitive, role in the sound of the bell. The sound of a church bell rung full circle is significantly different from the sound of the same bell “chimed”. This sound difference is influenced by factors not relevant to this study, such as the Doppler effect of the moving bell, but the “quenching” behaviour surely plays a part. If the clapper “stuck” to the bell surface immediately on first impact, without the bouncing, the frictional damping effect would come into play immediately and the sound would be deadened. However, while the clapper is bouncing, the bell sound is able to ring on roughly as it would if chimed: it only switches to the faster decay when the clapper comes to rest. The frictional sliding effect then damps the sound out rather abruptly, in time for the next strike to be heard clearly without much residual vibration from the previous strike.
Of course, this hypothesis would only be plausible if all bells show the same kind of long-lasting clapper bouncing. In order to check whether there was anything odd about the small bell studied in the laboratory, tests with an electrical contact sensor were made on a wider range of church bells. Figure 9 shows two examples of the results: these are for two bells in Great St Mary’s church in Cambridge. It is immediately obvious that both show a similar pattern of bouncing to the laboratory bell, and all the bells tested gave comparable results.
C. A playability diagram for bells
Bellringers encounter playability problems, just like players of other musical instruments. One major issue concerns the fact that there is an ambiguity about the initial state of a bell. Look back at Fig. 1: these pictures show a bell ready to swing, in the two directions of bell motion. In both cases, when the rope is pulled the bell will start to move in the direction such that the clapper is initially resting against the trailing side. In order to strike, the clapper needs to swing a little faster than the bell so that it catches up and strikes the leading side. This state of affairs is called “ringing right”.
But it is obvious that the clapper could have been moved across so that its initial position was against the leading side of the bell. It would then need to swing a little slower than the bell, and strike the trailing side. This would be called “ringing wrong”. You can guess from these terms that ringers prefer the first one: “ringing right” makes the bell a little easier to handle when making the subtle timing adjustments that change-ringing relies on. This immediately raises some questions. Can all bells ring both right and wrong? What can the bell-hanger do in terms of the detailed configuration of the bell, clapper and wheel to encourage ringing right?
To address these questions, we need to analyse or simulate the motion of bell and clapper during a swing. Curiously, for this purpose we do not need to model the bell vibration in detail, we only need to take account of the energy loss to vibration when the clapper impacts the bell. The next link gives the resulting governing equations, and some details of their implications. It reveals something important: if we make some reasonable assumptions (the clapper much lighter than the bell, the influence of damping small), it turns out that the behaviour is governed, approximately, by just two dimensionless parameters. That immediately suggests that it might be useful to plot simulation results in the plane of these two parameters, to give a “playability diagram” of a similar kind to the ones we have used several times now for bowed strings, wind instruments and other things.
The two dimensionless parameters are constructed from ratios between three lengths, all of which can be determined from an in situ bell in a church tower. Two of these lengths relate to the swinging periods of bell and clapper, for small-amplitude motion with the bell hanging downwards rather than facing upwards as in Fig. 1. These periods are easy to measure by timing a few swings of bell and clapper, separately. We then calculate the lengths of simple mass-on-a-string pendulums that would swing with the same periods. A formula for this is given in the side link, equation (14) — or you could measure the length directly by adjusting the length of a piece of string with a weight on the end, so that it swings in synchrony with the bell or the clapper.
We can call these two lengths $L_b$ for the bell, and $L_c$ for the clapper. The third length we need is the distance $r$ between the swing axes of the bell and clapper. This one can be found directly with a tape measure, by measuring upwards from some convenient horizontal surface to the centre of the bell pivot and the centre of the clapper bearing, then subtracting one from the other. Finally, we find the ratios $r/L_c$ and $L_b/L_c$, which will be the two axes of the diagrams we are about to plot.
First, it is useful to see some typical simulated results. To describe the swinging bell and clapper we need two angles, illustrated in Fig. 10. The angle between the axis of the bell and the downward vertical is called $\theta$ (Greek letter “theta”), and the angle between the clapper and the bell’s axis is called $\phi$ (“phi”). The blue and red curves in Fig. 11 show how these two angles vary in time, for a simulated case matching the parameters of the laboratory bell.
At the left-hand edge of the plot, the bell is near the upward vertical. It then swings down, and until the angle $\theta$ reaches about $60^\circ$ the clapper angle $\phi$ is a flat line. Throughout that time, the clapper is “sticking to the bell” and being carried downwards by it. The clapper then lifts off the surface of the bell, and is in free flight for a while until approximately the time 1 s on the horizontal axis, at which moment the clapper strikes the bell. The angle $\phi$ has reached the negative of the original flat-line value, because the clapper has reached the limit of its travel across the inside of the bell.
The moment of striking is perhaps more clear in the black curve in Fig. 11. This shows the rate of change of $\phi$, and at the moment of striking that curve shows an abrupt upwards jump: the clapper velocity jumps, as it bounces off the bell surface. The clapper is then in flight for another short time, before it strikes the bell again, on the same side, giving another jump in the black curve. After a few more impacts, the clapper settles down to a new flat line at a negative angle: it is resting against the opposite side of the bell to where it started. After that, the whole sequence repeats — but all the curves are inverted because the bell swings back the other way, and the clapper motion goes through the same sequence of events in mirror image. In total, Fig. 11 shows what the ringer would describe as four strikes of the bell: two handstrokes and two backstrokes.
In Fig. 11 the bell is “ringing right”: Fig. 12 shows the contrasting case of the same bell “ringing wrong”. The sequence of events is essentially the same as just described, but the initial flat-line value of the clapper angle $\phi$ is negative, and during the first downward swing of the bell it switches to being positive. Notice that the waveform details are not identical to Fig. 11: the two black curves are not mirror images of each other. For example, the jump in the black curve at the first strike is bigger in Fig. 12. The bell will not sound the same when it is rung “right” and “wrong”.
Now we are ready to use the simulation program to map out behaviour in the playability diagram suggested earlier, parameterised by the two ratios $r/L_c$ and $L_b/L_c$. This diagram has been called the “clappering plane” , because it gives practical guidance for bell-hangers wishing to adjust the bell and clapper suspension details to deal with perceived “ringability” issues. We can take a grid of values of the two ratios, run the simulation program for each one, then analyse the results to show various aspects of the predicted behaviour.
The first issue to investigate is the one we have already mentioned: to map out regions of the plane in which a bell can ring “right”, or “wrong”, or both, or neither. The result is shown in Fig. 13, based on a $40 \times 40$ grid of simulations. Bells can ring “right” in the orange and red regions, and “wrong” in the red and yellow regions. Within the red region, both types of ringing are possible. In the white region, neither is possible. The white star indicates the configuration of the laboratory bell. It lies in the red region, so it is capable of ringing both right and wrong. We have already seen simulated examples in Figs. 11 and 12, and this prediction was confirmed by the behaviour of the real bell.
To understand what governs the boundaries of these regions, we can use the same set of simulations and colour-code them to bring out other aspects of the behaviour. In Fig. 14, all the bell/clapper configurations capable of ringing right are coloured to indicate the value of the bell angle $\theta$ at which the first strike occurs. Figure 15 shows the same information for ringing wrong. In both figures, we can see a curving boundary on the left-hand edge of the allowed region where the colour becomes very pale, showing that the first strike only happens when the bell has almost come back to the top of its swing. Beyond these boundary curves, the clapper does not manage to strike at all before the bell starts its reverse swing.
To see the physical origin of the other main boundary of these two regions, we can look at Figs. 16 and 17. This time, the same two regions are colour-shaded to show the striking speed at the first strike: for the case of the laboratory bell this is the magnitude of the first jump in the black curves in Figs. 11 and 12. This striking speed will correlate, roughly, with the loudness of the bell. In Fig. 16 we see that as the upper boundary line of the “ringing right” region is approached, the striking speed tends towards zero. Figure 17 shows the same thing on the lower boundary of the “ringing wrong” region. At these two boundaries, the clapper makes a “soft landing” on the bell, rather than giving it a definite strike.
D. “Double striking”
The simulations make use of a coefficient of restitution between clapper and bell, to represent energy transfer from the clapper into vibration of the bell as described in the previous link; more detail is given in the published account . The aspects of behaviour discussed so far do not depend on what happens after the first impact, so the value of the coefficient of restitution used in the simulations makes virtually no difference to the plots. The next aspect to be discussed, however, does depend on the bouncing of the clapper. This is the question of “double striking”. Some bells produce a clear audible impression of a double strike on each stroke. The first question to ask is why this is not the case for all bells, since the results of Figs. 4 and 9 show that there are always multiple impacts between the clapper and the bell during normal ringing.
The answer to this probably lies in a psychoacoustical phenomenon known in different manifestations by a variety of names, including “echo suppression”, “forward masking” and the “precedence effect”: the idea was introduced back in Section 6.2. The human hearing system has evolved to cope with sounds in the presence of echoes from environmental features like trees or walls. The result is that if we hear a sound followed quickly by a recognisable copy of the same sound, especially if it comes from a different direction, our brains identify the second sound as probably being an echo. We are then, ordinarily, not consciously aware of the echo as a separate event, although it contributes to our sense of the acoustical environment we are in.
The sound of a church bell excited by multiple clapper strikes may tap into this mechanism, so that the later impacts are perceptually discounted to a greater or lesser extent. In the case of the bell there is no directional difference between the sounds, so the echo suppression effect is less strong than in the case of, for example, wall reflections in an enclosed space. Nevertheless, it seems to be empirically the case that most bells are not perceived as producing multiple strikes.
A simple listening test was conducted in which experienced ringers were played computer-synthesised sounds and, for each one, asked to say whether they would describe it as a single strike or a double strike. The results showed a clear pattern, governed by two factors: the time delay between the first two strikes, and the ratio of the striking speeds. This pattern could be reproduced by a simple formula in which a “double striking propensity” could be calculated from the time delay and speed ratio.
These factors can both be deduced (at least approximately) from the simulation results, which allows us to plot “double strike propensity” in the clappering plane. Figures 18 and 19 show some results, for the cases of ringing right and ringing wrong respectively. The simulations underlying these plots used the value 0.1 for the coefficient of restitution, at the high end of the range of possibilities (see the previous link). The outcome of the listening test what that if the “double strike propensity” gets to 10 or higher, virtually all listeners agree that there is a double strike. The two figures suggest that, with this value of the coefficient of restitution, most ringable bells should be heard as double-striking.
This does not agree with common observation of the sound of church bells, so a second set of simulations was run using the value 0.05 for the coefficient of restitution. This value is near the lower end of the range of possibilities. The results, in the same format as Figs. 18 and 19, are shown in Figs. 20 and 21. All four of these plots are rather “speckly”, probably as a result of the tricky nature of consistently extracting the times and magnitudes of the first two strikes using an automated procedure. Nevertheless, the trends are clear. The smaller value of coefficient of restitution shrinks the region where double striking is predicted, leaving quite a lot of space to be occupied by bells that can ring right without producing the perception of double striking.
Amusingly, the laboratory bell falls in the worst possible place for ringing both right and wrong. We would predict that this bell should produce a very clear double strike under all circumstances, and this was exactly what happened when the bell was rung in the laboratory as seen in Fig. 3. Looking back at earlier plots, we can see that this is not the only thing wrong with the setup of this bell. Figures 16 and 17 show that, whether it is ringing right or wrong, the sound would be very quiet because it lies close to the “fade-out” boundary in both plots so that the striking speed should be very low. Again, this prediction was borne out. Basically, this small bell could serve as a case study in how not to set up a bell for satisfactory ringing!
 J. Woodhouse, J. C. Rene, C. S. Hall, L. T. W. Smith, F. H. King and J. W. McClenahan, “The dynamics of a ringing church bell”, Advances in Acoustics and Vibration 681787 (2012). The article is available here: http://www.hindawi.com/journals/aav/2012/681787/