"Bell" modes? for a circular tube [Son of Chime] 2

[Dinosaur]

Guys,

I'm back to discuss the design of a chime, a circular tube to make a sort of bell like sound during a church service. For you old timers, we talked about this about four years ago. Back then, my time ran out and I decided to order a single chime tube from an orchestral chime producer. Makes me feel like an engineering wimp that I couldn't do the calculations, but a wise man once said (e.g. Clint Eastwood) "A man's got to know his limitations." But seriously folks ...

It is my belief that to make a proper chime, you cut a tube to the length corresponding to the acoustic resonance length for an open-open tube resonating at the requisite frequency. For a concert A of 220 Hz for example,

344 m/s / (2 x 220 Hz) = 0.782 m -or- 30.78 inches

The exact frequency is not important, but let us assume we will be shooting for the concert "A" for the purpose of this discussion. (220 Hz = 1382.3 rad / s)

So the tube will be 30.78 inches long. However, my experiments led me to conclude that the mode of vibration I am seeking in the design is not the vibration of a free-free rod in its second mode, which I originally believed, but rather a "bell" mode as suggested in this forum years ago. I interpret that to mean a shell mode of vibration of the tube wall through the thickness.

If this is the case, then I guess that if the tube were considered narrow relative to its length, the only variables we are working with are the material properties of the tube (the mass density and Young’s modulus) as well as the diameter and the wall thickness.

I still have some aluminum tube from the original experiment so I would like to stick with the aluminum for now. (E=10,000,000 psi & p=165 pcf) The diameter of the tube is around one inch. Using these values, what would the wall thickness have to be to get a good "Bell" mode? Is it possible to describe the arithmetic in a post? Thanks for everyone's help. - Dinosaur

 

[MikeyP]

electricpete,

In an attempt to clear up your confusion about the ring frequency (although I fear it may have the opposite effect), I offer the following.

The "ring frequency" (yes it is the correct term) is the frequency where one longitudinal wavelengh fits exactly around the circumference of the ring or cylinder. It is easier though to think of just the ring for now.

The ring frequency is often called the "breathing mode" becuse the ring simply expands and contracts whilst retaining its circular shape. Imagine a thin ring divided into equal sized elements about the circumference. Now stretch each element slightly along its length. The ring remains a circle but it now has a larger radius. Now shorten each element and the reverse happens. It is still a circle but the radius is smaller than when we started.

For a ring, this is indeed a mode of vibration but a rather peculiar one (for a start it either has no nodal points or every point is a nodal point, depending on how you look at it!). The strain on each element is entirely in the plane of the element (you are just making it longer or shorter). It is a longitudinal motion. However the effect on the displacement of ring as a whole is ENTIRELY OUT-OF-PLANE (i.e. the ring expands and contracts).

So now you can see why the formula for the ring frequency involves the longitudinal wave speed (sqrt(E/rho)). It is becasue the motion is caused by longitudinal strain in the cylinder material.

This is an extreme example of what cylinder dynamics is all about. Out-of-plane motion of the cylinder surface be caused by in-plane strain of the cylinder material and vice versa. As I said, the ring frequency is an extreme example. Most modes of vibration of a cyliner are a complex mixture of out-of-plane and in-plane behvaiour.

For a cylinder, the ring frequency is just that - A frequency. It is not necessarily a mode of vibration. It serves as a useful marker. Below the ring frequency we have the complex mixture described above. Above the ring frequency this behaviour starts to sort itself out into something we are more familiar with. By the time the frequency reaches about 1.5 times the ring frequency, out-of plane motion is caused purely by bending strain and in-plane motion is caused purely by longitudinal strain. As a result the whole thing starts to behave in the same way as a flat plate making the maths a whole lot easier (thank God).

I think I had better stop now.

M

--
Dr Michael F Platten

 

[Dinosaur]

Chime Update,

After getting my piece of 0.94" dia conduit, I set up a spreadsheet to compute the frequency of the first three beam modes along with the first seven acoustic modes. Then I plotted them on a graph with the length of the rod on the horizontal axis and the frequency on the vertical. This provided an interesting result in that the second and third beam modes are very nearly double and can therefore be tuned to two acoustic frequencies at the same time.

At the point where an acoustic frequency intersects a beam frequency, I expect to hear a clear resonant frequency. It is my belief that quality chimes are manufactured so that you obtain a nice blend of notes all resonating at the same length, but at different acoustic resonant frequencies. When I get a little time to cut my conduit, I hope to verify this.

The first step will be to place a hole @ 22% L and hang the tube. Then I will observe the behavior when I strike it. 63 inches is not a resonant length so I should get a short lived "harsh" sound when I strike it.

Then I will trim the tube above and below the hanging point so this point stays 22% L from the top. When I get near L = 45 inches, I should begin to hear the second and third beam modes resonating. For the musically minded, these two notes are an interval of an octave apart in the scale. So if I have everything right I will get a 300 Hz and 590 Hz note when I strike the tube.

After I hear this, I will decide if I am going to proceed to the next note of interest which occurs at a length of 33.25 inches. At this length, my spreadsheet shows the first beam mode intersects the first acoustic mode at around 200 Hz, a little below the concert A = 220 Hz I would like to get. However, in order to get this frequency, I would have to mill off a tiny bit of the wall thickness, which I am not interested in doing right now.

I should be able to play with my tube sometime in the next week. I'll keep you posted.

 

[GregLocock]

Dinosaur, this is a quick FEA problem, if you give me the dimensions of your tube (OD ID length) I'll run a model and let's see if we get any correlation.



Cheers

Greg Locock

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[Dinosaur]

Off the top of my head, I am working with two steel tubes. One is a galvanized fence post with an O.D. of 1.515 inches and a wall thickness of 0.068 inches. It is currently about five feet nine inches long, but I plan to cut it down to a resonating length where a beam mode matches an acoustic mode.

The other is a galvanized piece of electrical conduit (also steel) with an O.D. of 0.94 inches and a wall thickness of 0.05 inches. It is currently 63 inches long and I will cut it down to a resonating frequency.

I plan to hang each tube at 22% L from one end and hit it with a wooden hammer made from oak just for this purpose.

I gave you the resonant frequencies I think will work so try those lengths and see if you get the same answer.

I couldn't work on it today because I had to take my son to his ballgame. Tomorrow I have choir practice, so maybe I will work on it Thursday and have something to report on Friday. Thx - Dinosaur

 

[electricpete]

I looked at Tom's articles and I see the variation of frequency vs length follows the free free beam very closely.
Chime___L____f1___f2
1_____34.7__244__663
2_____32.6__278__753
3_____30.7__312__850

Should follow f~1/L^2
chime three fundemantal should be 244*(34.7/30.7)^2 = 311
pretty darned close.

But the series of tones for a given chime seems to vary somewhat more from the ideal. The ratio I see for the first 4 frequencies from Den Hartog
a1=22, a2=61.7, a3=121, a4=200

For chime 1 the measured frequencies were
f1=24, f2 = 663, f3 = 1272, f4 = 2050

If the measured frequencies follow Den Hartog's ratios, then fi/ai would be a constant.

Using f1, f2, f3, f4 for the first chime we have:
f1/a1 ~ 11
f2/a2 ~ 10.75
f3/a3 ~ 10.5
f4/a4 ~ 10.25
Close enough to fit the pattern but makes me wonder why it varies this way. Any ideas?

A new question - what determines, which bending mode is the one that sustains (and presumably the one that we hear as the amin musical note).

I think you have touched on two hypotheses.

First would be that the bending mode with same frequency as the an air-column frequency might be the one that is amplified or perhaps resonantes. But the direction of vibration is transverse for tube bending and longitudinal for air column vibration, right? Doesn't seem like there should be any coupling. Maybe if the resonant frequencies are very ver close to each other, even very small cross coupling between longitudinal and lateral would become important?

Second you alluded to 22% as the point to suspend the chime. Presumably this arises from the fact that the first nodal point for free free beam is 22% of length from the end. It makes sense to me that the mode that has a node at the point of suspension would last the longest. In Tom's examples, the 2nd mode always lasted the longest. Where were those suspended? Applying the same logic as above it would have to be 13% from the end to correspond to the node of the 2nd mode.

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[electricpete]

Mike - thanks for your comments. I'm going to follow up with some questions of my own on that subject in another thread.

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[GregLocock]

Back to the FEA - by inspection I'd expect the shell modes of those tubes to be in the 2-4 kHz area, that is, everything below 1 kHz will be adequately explained by beam bending. For reasons that I am yet to determine my FEA model isn't solving. Grr.

Electricpete - you've raised a good point, I doubt that there will be any strong coupling between the axial air mode and the radiation from the tube bending modes. The radiation efficiency of each bending mode is strongly affected by the ratio of the bending speed of the tube compared with the speed of sound in air. But, as the nodes and antinodes get closer together (as frequency increases) the radiation efficiency plateaus as the contributions start to cancel.

Possible reasons why the harmonic frequencies don't follow theory - shear stiffness of the tube (small), second order effects from the suspension system (small), fluid loading of the structure (very small). Might be worth asking an instrument maker.











Cheers

Greg Locock

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[Dinosaur]

Greg

Did you check the mass of your tube? Often people don't convert the weight of steel into mass by dividing by gravity, or the other way around, they input the value of the mass incorrectly. I had that problem myself when I was originally working on the beam modes by hand. Fortunately, with Tom's worked example, I could verify I had the right units on my mass term.

The FE programs I used made it difficult to set up the boundary conditions. I was never able to get a free-free mode. As a result, I had to either work on the PDE or look up solutions. I don't like looking up solutions because there is a good chance you miss something in the assumptions. What I prefer to do is derive the expression and then look it up to see if I got it right.

But since you have access to a nice FE program ...

My next question will be, how can I tune the chime to a precise frequency?

What I want to do is be able to take a specific diameter tubing, like the fence post, and make a whole scale from the same stuff. Well, at first some may say, no problem. Just cut the tube to the different lengths and you will get the notes you want. Since I support the acoustic mode (resonating air column) theory right now, the tube will only match the acoustic mode at discrete frequencies.

Anyway, my theory is that you can place a mass like a large hex nut on the end (top) and that will change the frequency of the beam mode. If the upper six inches of the tube were threaded, you could move the nut up or down to bring the tube into tune.

Let me know what you think of tuning a chime in this way.

 

[electricpete]

I was mistaken in stating that the 2nd mode was the longest sustaining in Tom's data... it was the first mode that was the longest sustained.


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[electricpete]

For chimes 3 - 5 the 2nd mode dominated for the 1st 5 seconds, then died away while 1st mode lasted longer.

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[MikeyP]

I suspect that the reasons for the frequency ratios not following the theory precisely are due to the shear deformation effects that Greg mentioned and also the effects of rotatory inertia. Normally these effects can be ignored for long thin beams with a wavelength to thickness ratio of >10:1. Tom's beam has a wavelength to diameter ratio of about 12:1 for the 4th mode.

However, we have a thin walled tube with a correspondingly lower shear stiffness and larger radius of gyration than for a same-sized solid beam which means that these secondary effects could be noticeable.

Using a Timoshenko beam model instead of a Euler-Bernoulli model should take into account these differences. I suspect that Greg's FE model will show a similar deviation from the Den Hartog formula.

M

--
Dr Michael F Platten

 

[electricpete]

I recently got S. Rao Vibration 3rd ed vibration textbook used for $20 to supplement my low-budget library of mechanical and vibration books. I saw where he talked about Tomshenko beam theory. I was looking for the punchlines (simple formula to plug and chug) but couldn’t find it. A little over my head I think.

I have a critical speed analysis program which uses the transfer matrix method. It allows to specify a hollow shaft with no bearings and free/free end conditions. It also allows to exclude gyroscopic effects. Critical speed of a hollow shaft excluding gyroscopic effects would be the same as the bending modes that we’re talking about, wouldn’t it?

Note the program also allows to include or exclude shear effects.

Would be interesting to try it out when I get time... probably not until the weekend.

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[btrueblood]

As far as acoustic coupling to the tube is concerned, it is possible to excite combined acoustic modes, such as the 1T-1L mode (1st tangential and 1st longitudinal mode combined), etc. This should result in possible acoustic resonances at frequencies between the 1T and 1L frequencies IIRC. It seems likely to me that a tangential acoustic(think "slosh") mode or combined slosh+longitudinal acoustic mode could couple to beam bending motions more so than coupling between pure longitudinal acoustics modes and beam bending modes. Damn, that was a hard sentence to write, and it still doesn't sound good... hopefully you get the idea.

To find the acoustic frequency for combined modes, use the formula

L = sqrt( L1^2 + L2^2)

where L = the combined mode wavelength, L1 = the longitudinal wavelength (is it 2*tube length?), and L2= the tangential mode wavelength, given for the 1st order slosh mode by:

L2 = pi * D / 1.841,

where D is the cylinder diameter.

The combined mode frequency is then given by the equation
f = c/Lt, c is the speed of sound, and L is the combined mode wavelength.

 

[GregLocock]

Yes, but look at the proportions of the tube. L/D is 40:1, so roughly speaking the duct will be one dimensional for the first 10 modes at least.

electricpete - yes the critical frequency of a shaft is the bending mode.




Cheers

Greg Locock

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[Dinosaur]

Greg,

I'm glad you understood that cause it all went way over my head. I guess I have never delt with a media that could have such coupling.

Dinosaur

 

[btrueblood]

Greg,

I agree, the modes wouldn't couple very efficiently, and the tangential mode contribution would be pretty highly damped within a few wavelengths down the tube. Just was looking at the prior post that was questioning the discrepancy between apparent resonant frequency and the beam bending frequency, and suggested that some combined mode coupling could explain some of the apparent frequency shift. Also, was trying to picture how a coupling mechanism between beam bending and a longitudinal mode might occur, so suggested the above. Wouldn't trust what I wrote without some acoustic pickup data to back it up.

Dinosaur, forgive and ignore my earlier post, it's just a comment from the peanut gallery. This thread is quite interesting, and it will be pretty neat to see if acoustic/structural coupling (i.e. tuning the tubes to produce a beam bending mode that is very close to the longitudinal acoustic mode) will produce an audible/noticeable change to the human ear. Good luck and keep us posted, please.

 

[Dinosaur]

Guys,

I drilled a hole in the 0.94 inch tube yesterday and cut it down to 58.5 inches long. I hung the tube on the wall in my shed (sure wish I had a shop) and struck it. At 58.5 inches, it was supposed to be tuned to both the third beam mode and the third acoustic mode. I couldn't hear any substantial difference in the tone quality from the 63 inch tube before I cut it down.

However, I did note that the hanging mechanism played a significant role in the chime behavior. When the chime tube was resting on only one side of the two holes, it "rang" better (clearer and more sustained). But when I balanced it better so it rested on both holes, the sound dampened out much more quickly and did not have as clear a "ring".

I am not ready to declare anything concerning the merits of coupling the longitudinal acoustic mode with the transverse beam mode. The problem of the chime support and the fact that I started with a third mode (which probably doesn't contain much energy) while hanging the tube from the first mode node (say that five time fast) indicates more testing is required. Maybe I will get a chance to try a couple more this weekend.

 

[electricpete]

Interesting results. Just one question. Did you try striking in a direction perpendicular to the line formed by the two support holes. That would allow movement where the support acts like a true pivot, whereas if you struck 90 degrees from there you could get a reaction moment from the support.

Also just thinking out loud, I'm wondering whether the type of string used to support the chime makes a difference. My intuition would tell me a rubber band would be a bad support material and fine metal bailing wire would be good. String might be in between. It seems like a lot of the cheap small chimes I've seen used fishing line for some reason.

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[Dinosaur]

Well, I did some tests last night with the 0.94 inch dia conduit.

But first ... My suspension is a galvanized nail through a hole drilled through both sides. The nail is supported on each side by cotton string because that is what I had laying around the shed. I struck the tube from the "front", perpendicular to the axis of the nail, and from the "side", parallel to the axis of the nail, as well as at points near the top and the middle and the bottom. Striking the tube from the front was noticably better than striking the tube from the side and I agree it is for the reasons electricpete stated. Another important observation is that striking the chime near a node of the first mode shape produced poor results. I ended up striking the tube at about 10% L from the bottom with an oak dowel. I also noticed that when I struck the tube with the end of the dowel I had better results than when I struck it with the side of the dowel.

Now for the results of my experiments last night ...

The first night I had cut the tube to about 54.5 inches which was computed to match the third beam mode and the third acoustic mode. The ring was not great but it was not bad. Last night, I cut the tube to 45 inches which was predicted to match the second beam mode and nearly the third beam mode also with the second acoustic and fourth acoustic modes respectively. The ring was about the same quality as the first test and it sounded either about the same pitch or lower.

I know several guys will raise their eyebrows at that but remember the first length was computed to resonate at the third beam mode, a higher frequency than the second test at a second beam mode. However, because I didn't record the sounds, I am not sure what frequencies I was hearing.

Next I cut the tube to about 33 inches which was computed to be about one-quarter inch longer than the matching first beam mode with the first acoustic mode. This chime had a noticably poorer sound than the first two. It had less sustain and a harsher ring sound. But, the first beam mode at this length has a lower frequency than the second beam mode or third beam mode of the earlier experiments.

Now I am beginning to have doubts about the importance of acoustic modes. I am also beginning to think that there is something about the longer tubes that makes them more suitable. As a matter of fact, I am beginning to think it may have to do with the amount of striking energy that is stored in the tube as vibration energy.

When I hit the tube, it vibrates. But, it also swings. So I have vibration energy, swinging pendulum energy and I suppose some energy of deformation lost in the oak striker. The pendulum energy is lost for the purposes of hearing any sound. The shorter chime has a comparatively small mass so it swings more as you would expect, but it also has an even greater reduction in polar moment of inertia. Now instead of just swinging, it also kinda bobs up and down. I think this is part of the reason my third chime was so poor.

I did a final test after cutting one-quarter inch off the tube to get it to match the first mode frequency, but the results were much the same.

My theory at this point is that I want to work with longer chimes for the energy reasons mentioned previous. Also, if possible, I hope to get a chime that resonates at the combined second and third modes. In a few days, I will have time to experiment with the galvanized fence posts with an O.D. of about 1.52 inches and a wall thickness of 0.068 inches. The tube is 72 inches long right now. I am also considering testing this tube with a hang point at the second beam mode node which I think is around 13% L. If anyone knows the location of this node, please let me know.

If anyone is interseted in doing their own experiments, it turns out that it is not that difficult to cut the tube with a hacksaw. After I cut it to length, I take a file and round off the edge a bit for safety. My son likes to hang around me when I am working and I don't want him to cut himself in his curiosity.

 

[electricpete]

It seems reasonable for whatever reason it is not always the 1st or 2nd or 3rd mode that is dominant... changes when we change the pipe size. That can be seen on Tom's data - in Chime 1 the first mode is dominant, while in chimes 3,4,5 the 2nd mode is dominant (at least for the first 5 seconds).

The reasons why the dominant mode may change?
- One may be air column resonance.
- Another may be the support location which favors certain modes (when it falls on a node) and suppreses others.
One more way might be the ability/inability of the striker to excite high frequencies. The curves I see for hammers used in bump tests have flat response for low frequencies, and when frequency exceeds a certain cutoff they begin to tail off. A harder tip generally has a higher cuttoff. So if you decrease the lenght which moves one of the modes above the range which you can excite, the absence of that mode could make a new lower mode may become dominant. I think the tip of your dowel acts harder than the center of the dowel so you are more successful at exciting high frequencies with the tip of yor dowel.

There was nothing new in my discussion (all of these factors you've discussed before). Just wanted to throw in some comments.

If you want to drag the chime into your study, you can record the sound into your pc microphone (or buy one for $2 if you don't have one) to create a wave file which can be analysed for pitch using a wide variety of free tools from excel to cooledit. I have just started messing around with scilab which looks like a pretty powerful free version of matlab. Or if you'd like to email me any files I'd be glad to send you back a spectrum.

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