The Sound of Bells

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Clappering and partial intensity

Analysis of many different bells has shown that the relative intensities of the various partials can differ greatly between different bells, and that this leads to a distinct difference in the sound of the bell - from the soft, dull sound of a bell with strong low partials and weak high partials, to the bright, steely sound of a bell with strong high partials. Bells of particular founders, dates and positions in a peal show consistent effects, but it is not clear whether the variation is due to bell shape / metal, clappering, or both. The experiment described here was devised to investigate the effect of clappering on partial intensity. As far as I am aware it is the first such investigation since some research in the 1960s described in Rossing.

As an example of the difference in partial intensity profiles, here is a plot and recordings of two bells - the second at Llandaff Cathedral (weak high partials) and the treble at Dorking (strong high partials). The variations can be more extreme than in these two bells.

Spectra of Llandaff and Dorking

This experiment consists of two parts, one investigating differences due to how hard the clapper hits the bell, and one the differences due to the point at which the clapper hits. In the first part, the major issue in experimental design was to find a way of delivering blows of known strength to the bell, at exactly the same point each time, using equipment which was portable allowing use on multiple bells. The original work in the 1960s was carried out on a single bell under laboratory conditions, using a special test rig.

The results of this experiment were conclusive, at least on this bell. Partial intensity distribution is substantially independent of the strength of the clapper blow. Partial intensity is more dependant on the point at which the bell is struck. The variations observed are not typical of the differences seen from one bell to another.

This experiment was carried out on 5/1/02 on the treble at Dorking, a Taylor bell cast in 1998. I am very grateful to David and Kate Cameron, steeple keeper and tower captain at Dorking, for permission to use their bell for this investigation.

Experimental arrangement

Bell, tilted sideways

Objectives in the experiment design included:

  • capable of repetition in different bell towers
  • using only portable equipment
  • no significant mechanical adaptation of the bell or its mountings.
It was decided to use the bell's own clapper to strike the bell for the experiments on strength of blow. The experimental set-up involved strapping the bell tilted over to one side, so that the clapper, hanging vertically, just touched the soundbow. Blows can then be delivered to the bell by pulling the clapper sideways a measured distance, releasing it, and catching it on the rebound so that it does not strike twice. The arrangement, with the bell tilted over using straps between the wheel spokes and the frame, can be seen in the photo. Four straps were used to hold the bell firmly in place.

To calibrate the energy of the blows, a spring balance was used to pull the clapper sideways through different displacements while measuring the displacement force. Although the position of the clapper's centre of gravity and its weight are not known, a simple mathematical model of the clapper dynamics shows that this information is not needed. The kinetic energy in the clapper as it hits the bell can be derived from the displacement forces and measurements with a tape measure. I had hoped to measure the position of the centre of gravity of the clapper by timing its swing, but unfortunately the clapper bearing in this bell was so stiff that the clapper did not swing for long enough to take accurate timings. In any case, a clapper is a compound pendulum and any measurements would have been of dubious value.

The photograph below shows the clapper being 'weighed'. The spring balance is just visible on the left, with a piece of scrap timber across the frameside of another bell used to provide a horizontal pull.

Weighing the clapper

When the clapper is pulled sideways with a horizontal force F, the angle A it moves is given by F = m * g * h * tan(A) / l where m is the mass of the clapper, g the acceleration due to gravity, h the distance from the pivot of the clapper's centre of gravity, and l the distance from the pivot to the attachment point of the spring balance. The angle A was calculated from the distance s from pivot to the strike point on the clapper ball, and the distance d this spot was pulled away from the bell: A = 2 * arcsin(d / (s * 2)). The clapper dimensions were: pivot to impact spot on ball (s) 46cm; pivot to attachment point of spring balance (l) 50cm. The graph below shows various measurements of displacement force against angle, and a straight-line fit to the data. From this graph I calculate that m * h for this clapper is 3.17 kg-m.

clapper displacement

When the clapper is pulled sideways through angle A, it gains potential energy given by E = m * g * h * (1 - cos(A)). All this energy appears as kinetic energy of the clapper if it hits the bell at bottom dead centre. In the experiment, the clapper was pulled successively greater distances from the bell, released, and caught before the second strike. The sound of this impact was recorded using a laptop and microphone. In an attempt to avoid overload (the initial sound of a bell is very loud) the microphone was placed in a room below the bells and a level check taken. All the sounds were then recorded in a single session to avoid intensity differences due to the recorder or microphone position.

A similar but much simpler arrangement was used to collect recordings for blows at different positions. The bell was returned to its normal vertical position, and marked on the outside every 2cm from lip to waist. The bell was then hit with a hammer at each point while recordings were taken. The previous part of the experiment had shown that the strength of the blow was not of great importance to the intensity profile.

Both sets of recordings were then analysed as described in the following sections.

Blow strength and intensity

Here is the waveform recorded as a result of eight different blows of increasing strength.

blow strength waveform

Unfortunately, as can be seen despite a level check beforehand the recorder has overloaded on the loudest blows. However, to my surprise and relief the results are still of considerable interest, not least because the louder blows allow investigation of the effect on relative intensity of recorder overload. For calibration, it is shown at the end of this page that the intensity of the sound when the bell is chimed normally with its clapper is roughly half the maximum in this experiment.

Some of the partial frequencies of the treble at Dorking are as follows:

PartialFrequency
Hum411.5
Prime823.5
Tierce974.5
Quint1286.5
Nominal1651
Eleventh2042.5
Superquint2450.5
Octave Nominal3341
-4292

Each of the clapper blows was analysed with wavanal, over a duration of half a second (as opposed to the normal duration of one second). This shorter period was chosen to give greater emphasis to the initial clapper impact. All eight blows plotted together appear as follows. Each is labelled with the distance the clapper fell against the bell. The blow gets stronger in the graphs further into the page.

3d plot of blow strength

This bell has high intensity high partials, which is one reason why it was chosen for the experiment. It is clear from this plot that all the spectrums have a very similar profile. However, it is quite difficult to see the detail in this plot. To make the intensity change of individual partials with blow strength clearer, the next graph plots this for some of the louder partials.

2d plot of blow strength

Over the four weakest blows all the intensities track in good proportion. For the stronger blows, this pattern is spoilt, presumably due to recorder or microphone overload. However, even in these extreme circumstances there is no gross change in the relative intensities of high and low partials - certainly, nothing like the changes seen in the next part of the experiment. The conclusion is that the partial intensity profile of this bell is substantially independent of the strength of the clapper blow, over the ranges of energy used in this experiment.

Clappering position and intensity

The conclusion of the first part of this experiment - that the partial intensity profile is largely independent of blow strength - makes the second part rather simpler. There is no need to supply a blow of exact strength, and for this part of the experiment the bell was hit, on the outside, with a hammer. The bell was marked at 2cm intervals up from the lip to the middle of the waist. The highest mark was 20cm up the bell, the total straight-line distance from rim to shoulder being 48cm. The clapper normally strikes 5cm up from the rim on the inside of the bell. The following plot shows all the spectra plotted together. Unlike the previous plot, the amplitudes of the sounds were normalised so that the loudest partial is always the same intensity.

3d plot of blow position

It is clear from this plot that there is considerable variation in partial intensity. However, again it is difficult to see the detail. The next graph shows how the intensity of the main partials varies with the position of the clapper blow.

2d plot of blow position

Note that the relative intensities of the partials are different from those arising from blows with the clapper. The bell certainly sounded different when struck with the hammer - and very different when struck in the waist. The differences when the bell is struck with a hammer on the soundbow are presumably due to the different material but deserve further investigation. In the waist, the amplitude of the usual partials is low and is replaced by strong partials not heard when the bell is struck normally.

Despite these intensity variations, I did not see the gross change in amplitude between high and low partials typically seen in different bells.

Energy in the radiated sound

The chain of events whereby the potential energy stored when the clapper is pulled aside, eventually ends up as amplitude in a digitised waveform, is quite convoluted, and involves

Is it possible to relate the recorded amplitude to the initial potential energy?

The energy in a sound wave is as the square of its amplitude. For each of the blow strengths in the first part of the experiment, I calculated the potential energy of the clapper, and summed the squares of the amplitudes (over 0.5s, from zero frequency to 6000 Hz) to get an indicator of the radiated energy. The results plot as follows:

radiated energy

The result is much better than I had anticipated. The four weakest blows show an almost linear relationship. The stronger blows have quite a lot of scatter, but even so the relationship is maintained, despite the recorder and microphone overload. The amplitude-squared measured under indentical circumstances when the bell is chimed via it's rope in the normal fashion is shown as a mauve line on the graph.

Conclusions

This was very much a preliminary experiment to establish the feasibility of the approach. It produced rather better results than anticipated for a first attempt, which gives me a considerable incentive to repeat it for a number of other bells with different intensity profiles, with some of the procedural points (especially the recorder overload) improved on. It is clear even from this attempt that clappering will not explain the gross differences in intensity profile seen in different bells. A further experiment is needed to investigate clapper hardness.


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Last updated January 20, 2002. Site created by Bill Hibbert, Great Bookham, Surrey