Bell Intensity: CR4 Challenge (07/29/08)

It's a poorly worded question.
There will be a standing wave around the rim, so some points will be relatively stationary and thus theoretically not producting much sound intensity, however this standing wave at the rim isn't the only part of the bell contributing to the wavefront.

One would need to clarify exactly what you mean by 'sound intensity in a point at the rim'

But my short answer is.

No.... do I get some Tuna now?
Del

There will be a standing wave around the rim (?)

How so, considering that, unlike an insturment sting, all waves must return continuously to and over the point of initial displacement; and all other points of amplitude excursion? And remember, the challenge ask about measuring (within the limits and sampling time constraints) of the measuring instrument.

What is a ring clapper - I've never heard the term? Is it a ring that is rotationally uniform and that is pulled into a rotationally uniform bell so as to provide uniform excitation? If so, and in an ideal world, the vibration will be entirely axial or radial in direction, and will be uniform around the radius.

Similarly, even if the clapper is the usual sort of hammer, the vibration in the main modes of the bell will also be uniform along the rim.
However, many other modes will be excited, and these will not be uniform around the circumference. Nevertheless, these will die away more rapidly than the main bell mode, so over time the vibration will tend to become uniform around the rim of the bell.

Of course, that is for an ideal bell. Real bells need to be mounted or suspended, cast bells inevitably have some built-in strains, unless the bell is in a rotationally uniform universe other objects nearby will interfere with the vibration, and...

Fyz

P.S. I vote tuna for the cat.
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This post may contain elements of clappertrap

I think the clapper is the dangling part in the middle. Part 9 below. Picture from Wikipedia.

That is indeed a clapper - but what is a "ring clapper"?

In my entry below, it is the clapper that strikes a ring. Maybe they meant something like this:

When the bell is struck by the clapper, the various natural modes of vibration of the bell will be excited. If the bell were a ring magically suspended in the air, it would vibrate in the modes where the impact point was at an anti-node for each of the modes. The lowest frequency mode would have a mode shape resembling an oval with four locations on the ring that are nodes and do not have radial motion. In the figure below, the thicker blue line represents the ring and the red line represents the mode shape of the lowest frequency mode. The other colors represent the mode shapes for the next three natural frequencies. In the figure, the ring would have been struck by the clapper on the right side.

For a bell, the mode shapes and frequencies would be much more complex since it is a three dimensional object and waves could be moving up and down the bell surface as well as around the circumference. Designing a good sounding bell must be fine art indeed.

At some distance from the bell, I don't think that you would notice a detectable difference in the intensity of the sound from the bell for different angular positions relative to the clapper impact point. By angular, I mean angle as measured about the axis of symmetry. Very close to the bell rim, the frequency content of the sound would be different because some points are at nodes and some at anti-nodes for some modes. The intensity of the sound at these frequencies would be different for different locations. Since the majority of the sound energy if probably associated with the few lowest natural frequencies, I'd guess the sound intensity does vary as a function of rim location.

I should have said (but wasn't thinking clearly) that purely radial/axial vibrational modes (of the types that could be uniform at the rim) would not be the lowest frequency modes, and I imagine the lowest of them would be well damped by the suspension of the 'standard' church-bell design, because its (circular) node is some way down the side of the bell. The radial/axial modes are readily seen (as purely axial movements) on a suitably suspended circular plate - as seen in the second moving illustration in this article. Cupping the plate increases the stiffness of these modes, and suspension at the centre will damp at least the lowest of these circular (or ring) modes.

the intensity will be the same, but the intesity will vary widely depending on the modulus of the local fog index

I am considering the validity of the question in view of the misspelling in the title. Hmph...

Mike

This would also depend on the actuation of the clapper on the bell too.

if the impact is square to the side of the bell (clapper/striker/solenoid) hits a single point then returns down the same trajectory, then the standing wave would be different than if the striker/clapper/solenoid were to hit on a angle creating a rotating wave around the bell rim...

Another way of saying this, if the bell is struk directly on the side, the bell will swing in the direction of impact, so a let/right swing (depending on your perspective) if the bell is struck on a "glancing blow" the bell will swing in a rotating orbit.

if the impact is square to the side of the bell (clapper/striker/solenoid) hits a single point then returns down the same trajectory, then the standing wave would be different than if the striker/clapper/solenoid were to hit on a angle creating a rotating wave around the bell rim...

Good point, only that the rotating wave will die soon, unless the wave length of sound, when traveling around the bell, happens to be exact multiple of the rim's circumference. This must be impossible for any bell you might find on earth, I guess.

Bells are generally cast and in particular the big ones (the smaller ones can be spun on a lathe or even beaten into shape although the quality is of sound is very different). Their shape gives rise to many natural frequencies and the bell designer needs to know (from experience more than anything else) how the many different aspects of bells affect the harmonics:

• shape (lets call it bell shape but the geometry of the bell shape varies from one to another)
• material (bronze is used for very large bells as casting is easier but cast steel is also used)
• casting process: thickness of material, flow of molten metal (how to stop two flows from meeting and creating an interface that could become a crack or at least a barrier to the vibrations), internal stresses (vary over the surface of the bell), one of the major difficulties for the large bells is the avoidance of cracks (some of the most famous bells in the world have cracks such as Big Ben however this still chimes well and could even be called distinctive )

At any point on the surface, the interference patterns of the many natural frequencies will make the overall amplitude at any moment in time vary. The clapper is basically an impact signal and so will give rise to all the natural frequencies of the bell.

Perhaps it is worth adding that the rim can be considered a special case of the surface vibrations. The first natural frequency of the rim will show up as an in and out oscillation at 1/4 points almost like squashing the rim between your fingers. This can be done on straight sided beer glasses (slight curvature will give too much strength to the rim; obviously precautions should be taken such as strong gloves).

This seems to evade the point. The question specified measuring only at the rim. We must intuit that a uniform bell shape was impliied, such that (in addition to downward) any vibration propagated vertically/radially, up and over the bell and returning to the rim should do so equally along either side of a midline extending vertically from the point of clapper strike.

I would expect the bell to have higher and lower intensities at certain points around the circumference. The bell may have a point or points where it is more rigid than at other points. A rigid part would act as a node and produce a lower intensity sound, the more flexible parts would act as antinodes and vibrate more (larger amplitude) producing a louder sound. Here's how a trumpet does this.. http://la.trompette.free.fr/Smith/IOA/figure3b.jpg And check out the salt on the nodes on this marimba... http://www.lafavre.us/tuning-marimba.htm There is the added complication of the sound waves reinforcing or canceling out when they have been produced, peaks and troughs in the sound intensity could coincide at certain points in the air around the bell producing quiet areas. Two peaks coinciding would produce a loud area. Sound refections from nearby objects / walls could do this. Consequently you can move the sound meter around the bell and find high and low values for sound intensity, you can also find the sound intensity varying up and down as you move the sound meter closer to the bell. One last thing! the phenomenon of "beats" results from interference from overtones with frequencies that are higher or lower interacting to make the whole thing louder and quieter, a kind of wah wah effect. http://dev.physicslab.org/Document.aspx?doctype=3&filename=WavesSound_Beats.xml gives you an excel spread sheet that you can change showing you beat frequencies of two simultaneous sounds and the intensity variations in the 'combined' and 'subtracted' sounds. Hope this helps Jim Tarver

Everyone learn from

"Here's how a trumpet does this.. http://la.trompette.free.fr/Smith/IOA/figure3b.jpg And check out the salt on the nodes on this marimba... http://www.lafavre.us/tuning-marimba.htm"

preeze to not use the "Insert/Edit Hyperlink" tool that CR4 provides ... for to do so would make reading SOOooo much easier! (Just as breaking-into multiple paragraphs would do, like):

"Here's how a trumpet does this... And check out the salt on the nodes on this marimba..."

I expect that the vibration is principally at the point the clapper hits and directly opposite. The clapper striking the bell produces a minute deformation which propagates in both directions, reinforcing at the opposite side. This could be tested by touching the bell at various points and seeing where the sound is damped. I believe you will find the nodes (where vibration is not damped) at 90 degrees from the point the clapper hits.

Although the sound is instantaneous, the intensity is a function of the distance away from the point of contact and attenuation of the vibration in the bell metal itself giving the distinct long bell wave sound. The diferential intensity at all points may be negligible however to as the traveling wave goes around the rim of bell and dampens out.

Scott Taper

NERAC

I would hazard a guess and say that the intensity would be the same. This is because the intensity is proportional to square of the amplitude as far as i remember. with all the waves traveling along the rim and assuming each wave returns to its original position and then attenuates, the superposition of all these waves would be equivalent to a single wave of say amplitude A. This amplitude would keep reducing due to attenuation.

In case my assumption si hihgly suspect and not applicable then nope, they wont be the same.

Where in the question does it state "Assume Ideal World Conditions" ?

That's just explanation. The (correct) implication is that in a non-ideal world there are still differences.

The bigger issue is the use of the term "ring clapper" in the challenge. This could imply that the bell shape is such that it could be uniformly excited by a ring, and thus give uniform intensity along its periphery (and, yes bells do exist where that is a close approximation - although strictly nothing in our universe can be truly "uniform", except possibly ignorance)

I must disagree with the "And the Answer is...." The bell has many different natural frequencies and their associated modes shapes. The given solution assumes that only the oval mode shape is exicted. In fact, all mode shapes with an anti-node at the impact point will be excited. I do agree that the intensity will not be uniform along the rim but the sound of the bell with have a rich frequency content and not just one frequency.

You don't much like oversimplifications either...

hows this for simplifying it..

strike bell, lower vertically into water so the lip enters slightly, watch for surface disturbances, this should show you answer ;o)