The Sound of Bells

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The work of Perrin, Charnley and DePont

The investigation in the early 1980s by Perrin, Charnley and DePont published in their paper 'Normal Modes of the Modern English Church Bell' must be the most extensive examination ever done of a single bell. The abstract of their paper reads:

Experimental measurements of the frequencies and nodal patterns of all the partials of a good quality 214kg English church bell up to about 9 kHz have been made. By matching these with the results of finite element calculations an understanding of the physical mechanisms generating the various partials has been achieved. This has made possible the production, for the first time, of a classification scheme for the partials with a firm physical basis, and has given considerable new insight into church bell design. In particular it is now clear just how crucial to the production of the bell's characteristic timbre is the thick ring near its rim.

The bell on which all the work was done was one of a peal cast by Taylors destined for St Margaret's Church, Leigh on Sea, Essex. However, it was never delivered, and it was one of the more exciting moments of this investigation to discover the bell, still hung in the iron framework described in the paper, in the museum at the Loughborough foundry. Taylors kindly allowed me to take a recording. The comparison between Perrin and Charnley's results and the actual sound made by the bell has been enlightening.

Perrin et al identified 134 modes of vibration, many of which were doubleted giving almost twice this many partials. Each mode of vibration was identified by comparing experimental results with a computer simulation of the bell structure. This allowed each mode of vibration to be characterised as described below. However, there were two areas not reported by Perrin et al in their paper - the variation in amplitude of each partial over time, and the actual partial strengths stimulated by a clapper hit when the bell is rung. Both are of great importance. A crude example of the latter is shown by striking a bell first on its soundbow, and then on the waist. The sound is quite different; dissimilar sets of partials are prominent in the two cases. Grützmacher et al give examples of the difference in sound due to different clapper materials.

Vibrational modes of a bell

The classification scheme devised by Perrin et al is comprehensive - bells vibrate in many different ways. The modes were classified using the number of nodal circles (i.e. stationary rings in the same plane as the soundbow), nodal meridians (i.e. stationary lines running from bow to crown) and whether the bell was breathing (metal moving away from and towards the clapper stem) or twisting (metal at a constant distance from the clapper stem). The basic types of vibration and the notation for them used in the paper are as follows:

Perrin also used a parameter 'm' in classifying partials where 2m is the number of axial nodes. All the 134 modes of vibration were placed into this classification scheme.

Comparison with analysis of a recording

The table below lists the Perrin et al partials up to 4 kHz. Against them are listed actual results from analysis of a recording, taken by striking the bell with a clapper suspended inside hitting the soundbow, and then permitting the bell to freely vibrate. Partials above 4 kHz are omitted because their intensity in the recording is low. Amp (1s) is the relative amplitude measured over 1s.

Perrin quotes the frequencies for a doublet by giving the higher frequency of the pair, together with the splitting. This convention has been followed in the results from Wavanal. Closely spaced doublets are hard to distinguish, and only the wide splits are given in the measured results. This is particularly true for the nominal; the higher frequency of the pair is so weak compared with its lower partner that it's frequency could not be measured. Note that the measured nominal frequency (1171.39 Hz) corresponds exactly with the lower of the two Perrin frequencies (1172 - 0.6 = 1171.4 Hz) - so no correction for recorder speed was made.

Positions of the nodal circles (i.e. the positions measured from the soundbow upwards where the bell is at rest for that mode of vibration) measured by Perrin et al. are given. The figures are given in cm. - the bell is 70.2cm high.

Perrin et al results Measurements on recording Partial
P&C no. Freq. Split Class Nodal circles Cents Freq. Split Amp (1s) Name
1 292.72 0 RIR none -2402 292.77   20.5 hum
2 585.92 0 R=1,2m=4 16 -1200 586.13   48.4 prime
3 692.94 0 RIR 29 -910 693.02   32.5 tierce
4 882.53 0.32 R=1,2m=6 10 -491 882.53 0.5 10.1 quint
5 1172.0 0.6 RIR 29 0 1171.39 see above 69.8 nominal
6 1199.7 0 R=1,2m=2 13, 41, 66 40        
            1278.2   1.87  
7 1393.8 - other 16, 38, 54 300        
8 1470.1 - other 21 392        
9 1472.8 0.3 R=1,2m=8 10 396 1472.55   1.81  
10 1525.6 0 R=2,2m=6 11, 26 456 1526.9   3.24  
11 1559.8 0.6 R=2,2m=4 10, 28 495 1560.44 0.7 6.75  
12 1619.8 13.1 RA 11 560        
13 1764.3 3.2 RIR 29, 57 708 1764.16 3.1 10.5 superquint
14 1776.2 - R=1,2m=0 11, 54 720 1775.6   0.62  
15 1948.9 0 R=2,2m=8 11, 29 880 1949.58   4.11  
16 2040.3 0 R=2,2m=2 9, 26, 47, 58, 64 960        
17 2146.8 1.2 R=1,2m=10 9, 60 1048        
18 2155.9 - R=2,2m=0 3, 41, 59 1055        
          1200 2342.80   4.14  
19 2441.2 4.3 RIR 15, 53 1270 2441.22 4.2 11.4 octave nom.
20 2443.0 4 other 52 1272        
21 2485.7 0 R=3,2m=4 7, 21, 38, 66 1302        
22 2540.3 2.7 RA 8, 39, 53, 64 1339        
23 2550.4 4.6 R=3,2m=6 8, 21, 37 1346        
24 2617.8 1.5 R=2,2m=10 9, 32, 63 1391 2617.84 1.3 0.71  
25 2718.1 1.5 other 37, 55 1456        
26 2804.5 - R=3,2m=0 8, 20, 50, 55 1511        
27 2832.5 3.9 R=3,2m=8 9, 22, 37, 66, 72 1528        
28 2858.5 0.3 R=3,2m=2 4, 20, 36, 51, 56
63, 66, 68
1544        
29 2921.6 0.9 R=1,2m=12 9 1581        
30 3028.2 - R=4,2m=0 8, 20, 35, 42, 63 1643        
31 3172.2 - other 5, 18, 32, 54, 63 1724        
32 3183.1 7.1 RIR 31, 54 1730 3184.60   3.88  
33 3233.5 8.3 R=4,2m=4 7, 20, 30, 44 1757        
34 3369.6 4.6 R=4,2m=2 6, 19, 30, 42, 59 1828        
35 3376.8 0 R=3,2m=10 9, 24, 38, 62 1832        
36 3403.5 0.5 RA 8, 29, 39, 57, 62 1846        
37 3433.4 2.1 R=2,2m=12 9, 32 1861        
38 3514.7 2.6 R=5,2m=4 8, 12, 30, 40, 54 1901 3514.40   0.94  
39 3607.1 3.5 R=4,2m=6 8, 18, 30, 40, 56 1946        
40 3678.5 - other 13, 46 1980        
41 3698.1 - R=5,2m=0 6, 17, 28, 40, 55, 64 1989        
42 3797.6 2 R=1,2m=14 10, 59 2035        
43 3867.0 3.5 R=4,2m=8 8, 18, 30, 41 2067        
44 3934.8 19.4 R=5,2m=2 4, 17, 29, 39, 49, 59 2097        
45 3982.0 3.6 RIR 27 2117 3982.30 4.1 0.49  

Here is a plot of the partial intensity against frequency, taken over 1 second, up to 4 kHz.

Partials of the Perrin bell

For an interpretation of these results, go to the next page.


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Last updated August 20, 2000. Site created by Bill Hibbert, Great Bookham, Surrey