Acoustics of bell plates

This page is an appendix for the scientific paper:
Lavan, D., Hogg, S. and Wolfe, J. (2003) "Why do bell plates ring?" Acoustics Australia, 31, 55-58,
and vice versa.

3 bell plates 3 hammers Bell plates (sold commercially as Belleplates) are metal plates that, when struck, ring with a strong initial transient, followed by a nearly pure decaying sinusoidal tone. They are played like handbells but are rather cheaper and less loud.

The photo shows three plates: the first two are in the standard shape used for bell plates, the third is an equilateral triangle with a tang at one corner. The beaters used to make the sound files below are shown: a soft fabric beater, a soft rubber mallet and a hard rubber mallet.

These three shapes all play, that is to say they all ring for tens of seconds when struck. However, as we shall see below, only a limited class of shapes works for bell plates: in general, polygons with a handle at one corner just go 'clunk'.

 

 

larger bell plate smaller bell plate equilateral bell plate

Larger bell plate, standard shape

wav mp3 soft mallet
wav mp3 fabric beater
wav mp3 hard mallet

Smaller bell plate, standard shape

wav mp3 Smaller bell plate

 

Equilateral bell plate

wav mp3 with second partial
wav mp3 without second partial

In the photographs above, the shape of the largest plate is shown in comparison with those of the others. As expected, the smaller plate is higher in pitch than the larger plate of the same shape. The equilateral plate is interesting because, as well as the principal mode, there is another sounding mode at about three times the frequency (a twelfth higher in pitch), which fades more rapidly than the principal mode. This higher mode is the ring mode of the equilateral triangle. One can, however, strike the plate at a node of the higher mode and thus obtain just the lower tone. Perhaps this complication is one reason why the equilateral plate is not used in commercial instruments.

The shape of bell plates is important to the sound. We demonstrate this by progressively slicing strips of the end of a plate, recording the sound for each shape. We then slice vertical strips off the corners until we regain the initial shape. The soft rubber mallet is used in all sound files.

untrimmed bell plate 5 mm trim 10 mm trim 15 mm trim
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The orginal bell plate shape: it rings well

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5 mm removed: the pitch has risen, but it now rings for a shorter time

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10 mm removed: higher pitch and shorter ring

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15 mm removed

20 mm trim 25 mm trim 30 mm trim 35 mm trim
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20 mm removed

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25 mm removed: the ring is so short that it is hard to identify the pitch

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30 mm removed

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35 mm removed: the ring is not only short but very soft.

 

We now start taking slices off the corners, to return to the original proportions.

corners trim 1 corners trim 2 corners trim 3
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A slice off both corners:
a slight increase in pitch

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Second slice from corners

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Third slice from corners

corners trim 4 corners trim 5 corners trim 6  
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Here we're back to
the original proportions

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Fifth slice from corners

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Sixth slice from corners

 

 

So why is the shape critical?

In the paper on bell plate acoustics, we explain that, for the conventional bell plate shape, the two nodes of the (0,2) mode converge at the tang. This is not true for general shapes. We give a summary of the argument here. (For an introduction to modes of vibration of plates, see Chladni patterns.)

Consider the two simplest modes of vibration of a rectangular plate. These are named according to the numbers of nodes parallel to the short and long axes respectively: (0,2) and (2,0). Their Chladni patterns are shown below. The pale lines are nodes: lines of no displacement. areas on either side of a nodal line move in opposite directions. Because it is easier to bend a regtangle about its short axis than about its long, the (0,2) mode has the lower frequency. (What happens for a square, when the (0,2) and (2,0) mode have the same frequency? If you add these two vibrations together in phase, you get the mode shown in the third photograph.)

photo of (0,2) mode of rectangular plate     photo of (2,0) mode of rectangular plate     photo of the ring mode of square plate

These (2.0) and (0,2) modes are sketched below. Cross sections along the lines a-a' and b-b' are sketched to show the shape at two instants separated by half a cycle.

sketch of (0,2) and (2,0) modes of rectangles and bell plates

The lower half of the figure shows the analogous modes of a bell plate. The (2,0) mode (left) is the more important. If we imagine cutting away the corners of a rectangular plate, we move the nodal lines closer towards the centre. Eventually they intersect. This is the shape used for bell plates.

Now at a nodal line, the position of the plate isn't changing. So a node is the place to hold a plate if you want it to ring. However, on either side of the node, the plate is moving in opposite directions, so the plate is rotating locally. (See the cross sections in the sketches above.) If you hold a plate with your fingers at a simple nodal line, as we do in an example below, you damp the sound a little.

At the intersection of two nodal lines, there is no rotation: both displacement and slope are zero. This is the place to put the handle, which becomes an extended node itself. Let's see how this works:

bellplate held away from node A. Here I am holding it at the normal position, which is an extended region formed from the fusion of the two nodes of the (2,0) mode.

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B. Here I'm holding it at a point on one of the nodes of the (2,0) mode. Still the same mode, so it has the same pitch. But because it is a simple node, rather than the intersection of two nodes, the plate is rotating locally about this node. This rotation transfers energy to my fingers, so it rings for a shorter time.

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held at node C. This point is an antinode of the (2,0) mode but a node of the (0,2), so of course it is the latter mode that sounds: about one tone lower than the (2,0).

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held at node D. This point is at the intersection of a node of the (0,2) and a node of the (0,2). So, by striking at a point that is not a node of either, it is possible to sound the two modes together. (In B above, I also held it at this point, but struck at a node of the (0,2), so only the (2,0) sounded).

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The finger and thumb damp both modes, so the interference between the two modes is not very clear. So...

 
suspended bell plate E. Here the plate is suspended by a thread through a hole drilled at the intersection of the two nodes. The local rotation of each mode transfers rather little energy to the thread, so the plate rings in both modes. The transients are clearer too. Notice that the interaction of several different modes that are not harmonically related makes the sound more like that of a bell.

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How does the sound depend on direction?

bell plate suspended 2 Here I strike the plate so that, at first, the microphone is on a line normal to the plane of the plate. Then I rotate it so that the microphone lies in the plane of the plate. In the latter situation, the plate radiates approximately as a dipole, so the sound is weaker.

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It's worth noting that there is not just one 'special' shape for bell plates, but rather a family of shapes. If you return to the top of this page, you'll see that an equilateral triangle with a tang at one corner rings well. Again, this shape has two nodes that meet in the tang. If we make the angle at the tang larger, we must cut successively more off the other two corners so that their momentum doesn't bring the node back into the main part of the plate.

Why are bell plates made the shape they are? The equilateral triangle has two sharp corners which could annoy the player's neighbours, and there is the complication of the non-harmonic overtone from the ring mode that we saw/heard at the top of this page. But that still leaves a range of shapes. Perhaps the current shape was found by accident and has been conserved ever since.


  • More detail is given in the paper Why do bell plates ring?, by Lavan, Hogg and Wolfe.
  • Daniel Lavan did a MSc thesis on bell plates at the University of Technology, Sydney, where he was supervised by Suzanne Hogg. He did the experimental work at the music acoustics lab at UNSW with Joe Wolfe.
  • Australian Bell has an excellent web site about (three dimensional) bells that covers history, physical principles and pitch perception.
  • Here is a report of another type of bell plate. These are simply square plates and are suspended by cords that (presumably) attach to points on the node of the desired mode. The (1,1) mode would be the obvious one, but might cause balance problems in suspension.
This page was written by Joe Wolfe, with sound recording and photographs by John Tann.

 

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