Designing A Chime, resonance of a metal tube? 3

[Dinosaur]

Hello,

I participate in other forums here and I would like some help with a problem I have accepted for my church. I am trying to make an inexpensive chime to be used during the service. I have looked in physics texts and my finite elements books but I can not figure this one out. What I want is a metal tube of a standard size pipe stock that when struck lightly with a wooden hammer will make a nice, low, pleasing tone. Can anyone help me get a formula to predict the frequency of a metal tube in free vibration supported by a string at the top? Thanks in advance for any help. - Ed

 

[GregLocock]

End correction for hemholtz resonators is 0.6*diameter, offhand I can't think why it would be different for a quinkie (quarter wave) or half wave tube.

Cheers

Greg Locock

 

[magicme]

hello there
i am new to this forum, and hope i don't make too big a fool of myself, but this question jumped out at me.

i think the important modes of the tube, in terms of resonating with the air column, are not the lateral (beam) bending modes....they are the bell modes (or diametral modes). i have not run any numbers on this, but isn't the chime/tube more a bell than a beam?

Dave Leo

 

[EnglishMuffin]

Well, reading the literature seems to confirm that the fundamental is primarily a free-free bending mode, but some of the the higher harmonics are probably more bell-like, depending on how and where it is struck. I agree with you that those would be the only ones that could have much effect on the air column. Personally, I don't believe the air column resonance is significant - even the added mass effect on the bending modes is almost negligable. But you never know for sure till you experiment a bit.

 

[vanstoja]

To daveleo
I've also wondered why lobar shell response modes are not mentioned in the musical chimes discussions by Lee Hite (see vanstoja & mikeyp citations above) and Chuck (cited by Hite). Lobar responses (particularly 2,3 and 4-lobed) are definitely a factor in large diameter thin shells and bell-like structures such as high specific speed impeller front shrouds. Perhaps the critical factors causing changeover from shell lobar to beam bending modal responses is the shell L/D and t/R ratios. For Hite's brass orchestral chime the L/D and t/R ratios of 43.6 and 0.087 may be high enough to preclude lobar responses and bring up only beam bending responses when subjected to well-spaced single impacts on the outer diameter. This may be generally true of chimes geometries. Possibly, continuous excitation in the form of random vibrations (perhaps even induced by turbulent flow around a shell) might engender lobar shell responses in chime configurations. For bells, I understand that the responses are actually plate modes rather than beam or shell modes. There is an intriguing article about ancient Chinese bells (see Shen,S. "Acoustics of Ancient Chinese Bells", Scientific American, Apr.1987, pp. 104-110) that discusses plate modes in oblate bells with two different striking positions that excite two arrays of response harmonics both starting around 700-800 Hz. Cast-in nipples on the outer diameter act to suppress or moderate some of the harmonic responses presumably providing a variety of sounds. This is a 5th century BC acoustic art form that was apparently lost and and only partially recovered in the 12th century AD.
My favorite shell frequency equation (though only approximate) gives 2,3 and 4 lobe response frequencies for Hite's brass orchestral chime of 452-689Hz, 1009-1429Hz and 1968-2647Hz, respectively for the m=1 axial halfwave mode which matches the open-open organ pipe fundamental waveform. However, Hites orchestral chime had a cap with hole on one end which may make it a quarter wave resonator which would be better excited by the m=2 shell mode which for the long L/D of 43.6 gives almost the same frequencies as the m=1 mode. The calculated shell lobar mode frequencies are apparently within the "musical" range.

 

[EnglishMuffin]

vanstoja : So you appear to be saying that orchestral chimes are indeed designed such that the air column resonates at the same frequency as one of the shell type resonances - in which case Dinosaur is sort of on the right track. There would not seem to be much point in putting a cap on one end if the air column was not involved in some way, but what is the reason for the hole in the middle? Do I understand your post correctly ? I too recall that Scientific American article quite vividly - probably because it's one of the few that you don't have to be a theoretical physicist to understand.

 

[vanstoja]

EnglishMuffin
Yes. It appears that some coincidence or near coincidence of beam structural bending and organ pipe aeroacoustic harmonics is involved in chime tones. Chuck, in discussing Hite's orchestral chime in

http://www1.iwvisp.com/cllsj/windchimes/

shows a plot of the 4th beam and the 5th air harmonics intersecting at a length of about 72 inches whereas the orchestral chime length is reported by Hite to be 62.625 inches. My calculations for a 1.5 in. OD, 1.375 in. ID brass chime of that length finds the 4th free-free beam harmonic at 585 Hz and the 5th open-open pipe air harmonic at 536 Hz both of which are close to Chuck's curves at a length of 62.625 in. I presume that the failure of these two calculated frequencies to match exactly, is because the end cap with a hole screws up the end conditions of the beam and/or the organ pipe thus deviating from the ideal theoretical model. I suspect that the primary note being excited is C5 at 523 Hz on the musical scale. Apparently, the hole in the end cap is big enough to keep the organ pipe near the open-open modeshape though I don't know what it's there for.
In my prior post, I erred in stating that a quarter-wave organ pipe would match the modeshape of an m=2 shell axial mode. Actually there are 4 quarterwaves in an m=2 shell mode (ie, a full sinewave) which clearly will not match a quarterwave in the fluid within the shell.

 

[EnglishMuffin]

vanstoja :
Well, I've read Chuck's link, but I'm still skeptical. He says "Others have written that you can't predict the frequency of a tube and that the column of air is unimportant. They are wrong." This suggests that I am not the only skeptic, and that the issue is somewhat controversial, as with a lot of things having to do with musical instruments, Hi-Fi etc. What does he mean by "cut much shorter than the ideal length" ? He does not say precisely. If there was anything to this, I would have thought that you should see the effect without cutting the tube "much shorter". What happens if the tubes are, say, 5% shorter for instance? Very short tubes will not ring as long in any case - and if the frequency of those has been matched by increasing the mass, other effects might be responsible. And how was the striking force controlled ? I see there is a patent involved, but people have patented perpetual motion machines in the past, so that may not necessarily mean anything. However, I'm still ready to be convinced!

 

[Dinosaur]

In my reading, I have also run across the end cap mentioned in professionally manufactured chimes. It would seem to have two effects on the performance of the tube. First, by dramatically changing the mass, the frequency related to the sqrt (K/m) would be effected. Second, with a significant mass at one end, the mode shape would be effected so the relation of mass distribution is being put to use. Also, I wouldn't be surprised if the hole at one end is made useful as a tuning device for the final step. If the tube is a little out of tune, maybe the hole can be increased in diameter to get it just right. At any rate, the professional chime makers use a fair size metal plug at the top and this is where they are struck to make them sound correctly. I thought this was going to be a fairly simple study with three variables, but it is quite a bit more involved than that. Dinosaur

 

[EnglishMuffin]

If you strike the tube at a point coincident with the plug, that would also perhaps have the effect of not exciting the shell modes that were being discussed. My guess would be that the plug would be there partially for that reason - ie to achieve a purer tone. Using FEA you should be able to study the resonance of all these more complex cases quite easily, if you have access to a program.

 

[Dinosaur]

EnglishMuffin,

Well I don't have access to a FE program with the power to handle that one. I also don't know how to get the boundary conditions set up for a problem like that. A chime tube hanging from a cable is not strictly a stable structure and would return an error message saying the matrix is singular. That is what I would expect. So I don't know how to set that one up. My PDEs are not very good but given enough time I could probably get a few modes doing it that way. Let's face it . . . I can get a tube and do some trial and error in less time. I do think it is an interesting problem and I hope to get a handle on the mathmatics of it someday. - Dinosaur

 

[EnglishMuffin]

Dinosaur:
I don't know why you would say that a chime hanging from a cable is not stable - it is stable both dynamically and statically - (assuming of course, in the latter case, that it is suspended from a point above its center of gravity). Like you, I do not at present have access to an FEA program, but I don't know which matrix you are referring to that would be singular. If, for example, you write the matrix equation for two point masses suspended in space, totally unsupported and connected only by a spring, with a sinusoidal force applied to one of them, you don't get any singular matrices, so why would you (in general) for a beam suspended in space with a sinusoidal force applied at the proposed striking point ? FEA programs should give you free body modes without any problem, unless they are based on some aspect of Newtonian mechanics of which I am not aware. As a matter of fact, the very last problem in the current edition of "Vibration Problems in Engineering - Timoshenko, Young & Weaver" is to set up the FE matrix equations for a two element prismatic beam with no restraints and calculate the frequencies and mode shapes!

 

[EnglishMuffin]

By the way - I've just realized what the hole in the end cap is probably for - IT'S FOR THE SUSPENSION CABLE TO PASS THROUGH (dummy!) Everyone seems to support at the node which is .224 from the end.

 

[ewesson]

Another purpose of the cap is there for a striking surface. The directions are adamant about only striking the cap; the body of the tube will dent. I guess they really whack them in an orchestra setting.

They are also adamant about striking the cap at a ninety degree angle, or the chime will sound "out of tune". I don't know what sort of resonances this would induce.

I got fascinated by the subject of wind chimes a while back, and in doing research came across Chuck's page. He was kind enough to share his spreadsheet model for the resonant frequencies of pipes of various sizes and materials.

I share the skepticism about the importance of the acoustic resonances. Not only are the waves inside the tube orthogonal to the tube's vibration, they are open ended tube resonances, which are weaker than closed tube resonances. How much weaker? I couldn't find a formula for this; if anyone can help me, I'd greatly appreciate it.

However, just to be safe, I modified Chuck's model to find the closest overall fit between a set of chime resonances and their acoustic resonances. If you are interested, I would be happy to email you this model.

Designing chimes for their primary resonance mode is interesting, but designing them for their 4th, 5th, and 6th modes is more rewarding: These modes resonate at close to a 2:3:4 ratio, meaning they are much more musical than the other modes.

In part because these frequencies are harmonically related, they also encourage the ear to hear additional lower frequencies -- the "missing fundamental" Lee Hite talks about.

Not only does the chime not natively produce these frequencies, it would be less able to produce them as the output rolls off at 6dB per octave. This rolloff also helps hide the inharmonic lower order resonances.

Ed, if your church is not very big, or if you've got amplification available, you can probably make a pretty good set of chimes by tuning for F4, F5, F6 and hanging the chimes by a point near these modes' nodes: They're at 0.051, 0.06, and 0.073. Then strike the chimes either gently at the end or more vigorously at a point about 15% of the way down -- this is the antinode for F5.

I built my large chimes from 1-1/2" type L copper pipe, available at Home Depot. These pipes are quite similar to the brass pipes used in orchestra chimes, though the brass may have some characteristics (such as less damping?) that I don't know about.

- Eric

 

[cllsj]

I see my website has been discussed here and felt I should join the discussion. First of all because a patent has been applied for doesn't make this a perpetual motion machine. If you can find where I violate any law of physics please tell me. I firmly believe that I can only try to work within those laws or I'm wasting my time.

How short, or for that matter how long, is too short. This is really a gray area. Sort of like how old is old. I refer to the length where the acoustic and transverse frequencies meet as the "ideal" length. As one moves away from the ideal length the next lower or higher mode of the acoustic length starts to affect the tube. At shorter lengths it appears to stop the vibration of the tube. At longer lengths it tends to support the second mode of the tube which is not an even harmonic of the first mode and therefore they don't sound good. In my length calculator I set some limits but the limits are just my best guess.

Does the acoustic mode even couple with the tube? I wondered my self about that. I purchased a book on acoustic and when I couldn't find the answer I wrote the author. He suggest another book and told me what I had on my webpage was "good enough" for what I was doing. He also told me how to proceed but my knowledge of acoustic is rather limited and I'm have a difficult time. If anyone would like to see his respond I will post it.

I also attend a two day class on FEA of cavity acoustic related to aircraft. I asked the instructor about this problem. He said that at resonant I would have to use the more complex (heavy) forumula to correctly model the problem. So I'm convince that here is coupling between tube and air column but I would have trouble proving it with an equation.

I've recently been looking at bell modes or circular modes. I've not included this on my website yet. I do believe these modes make the type M copper tubing used my many for chimes sound terrible. I suspect if larger diameter (3 inch?) steel conduit was used it may also sound terrible for the same reason. I've been looking at trying to tune the tubes such as this first circular mode is an even harmonic of the first mode. I have used a digital filter to remove it and the tubes do sound better.

Tubular bells (chimes tuned to the four natural frequency) are capped at one end. I made some from steel conduit and have the plans on my website. I could not hear any difference between capping the ends and suspending them at the first node. Then my hearing may not be as good as it should be. I've also read that some sets of tubular chimes use different diameter at the upper and lower frequencies. This would appear to support my "ideal" length. Does the weight of the cap change the response of the tube? It is going to depend on how heavy the cap is. In my case I used a flat washer which is very small fraction of the total weight. Since I don't have a tubular bell I just don't know.

To the question that started this thread I've considered this building a set of tubular bells for my church. Based on what I know at this point I would use rigid aluminum conduit suspended at the first node point (which is not 22% of the length for the 4th mode). I would tune them to the 4th natural frequency. I would use the same diameter tubing for as many of the tubes as I could as there appears to be a difference in tones produced when the diameter changes.

chuck

 

[EnglishMuffin]

Hi cllsj:
1. I didn't intend to hurt your feelings when I commented that just because something is patented does not necessarily mean that it will work - it all depends on the patent examiners involved. Some of them are better lawyers than scientists. If you happened to get Albert Einstein, for example, you'd obviously get a very rigorous analysis. At about the time that he was working as a patent examiner in Switzerland, there were people successfully patenting perpetual motion machines in the USA, something it's hard to imagine you could have got past him at the time, but on the other hand in later life he believed in stuff like pole shifts in the earths crust, which is definitely fringe science today - nobody's infallible after all, and that certainly includes me!
2. It's not clear (at least to me) what your FEA instructor meant : he may have been referring to the fact that for a really precise computation of the natural frequency, you need to consider the effect of the added mass of the air inside and outside the tube, although the effect is small, unlike the case with liquids. But this has nothing to do with axial resonance of the internal air column. However, maybe he was talking about something else which I don't understand. But in any case you shouldn't need to go to FEA to calculate simple longitudinal resonances of air columns. If your effect is real, the simple basic physics formulae discussed on this thread should suffice, with appropriate end corrections of course.
3. If your theory about the air column resonance is correct, it should be possible to prove it with experiments, such as, for example, carrying out tests at different temperatures. Since the speed of sound in air is proportional only to the square root of the absolute air temperature, then temperature should have a marked effect on the resonant frequency of the air column. A change from, say, -25deg C to 25 deg C would change the air column frequency by about 10 percent, whereas the beam resonance would hardly change at all, even counting added mass effects, so if your effect were significant you should notice something. I have no suggestions of how to achieve such a temperature change, other than using mother nature, but with the right lab facilities it would be a simple matter.
4. Are you the same person as Dinosaur, or are there two of you guys building chimes for your churches ?

 

[cllsj]

Nope, I'm not Dinosaur. I'm the Chuck that wrote the website Chuck's Chimes or perhaps better known as

http://www1.iwvisp.com/cllsj/windchimes/

I'd have to dig out my notes from the class but off the top of head NASTRAN (?) can use two different formulations for a fluid. For air, other than at resonance with the structure, a light (less complex) formulation can be used. I would assume this formulation would require less computer time to solve.

I had not thought about temperature. If I could do this at work this would be rather easy to do; however, I don't think my employer would be willing to support my hobby.

I would point out again that author of the acoustic book I contacted didn't appear to have a problem with coupling of the axial air column and transverse vibration. Here is what he wrote.

I'm afraid my book was the wrong place to go! It deals with the production of sound by flow, i.e. where the energy is extracted from the flow.

>From my brief look at the your website (and another linked on your site)
I gather that the excitation mechanism is a striker. In that case I
should think sound generation is certainly via the resonant excitation
of a resonant acoustic mode in the tube by a structural mode of the
tube. The resistance of the air is so small that it seems to me that the
method of calculation you appear to have been using is quite adequate to
determine the frequency.

If you want to express the acoustic amplitude in terms of the tube
motion, however, you will have to calculate the tube mode shape and use
it to determine the normal velocity distribution on the inner surface of
the tube as a function of time (the amplitude will decay, predominantly
because of structural damping). You would then have to solve the
acoustic wave equation within the tube (say with zero pressure open end
conditions, in a first approximation) for the sound generated by a point
source. Then imagine the inner surface divided into infinitesimal
surface elements, the normal velocity of each of which determines its
effective source strength, and calculate the sound produced by each of
these sources using your point source solution. Then integrate over the
inner area to get the net effect of all the sources. The result (i.e.
the solution of the wave equation due to surface forcing) will have a
large peak at the matching resonant frequency of the air in the tube,
the magnitude being dependent on the decay rate of the tube vibration.
Next use the solution to calculate the fluctuating mean axial volume
velocity of the air at the open ends, and use this as the effective
source strength of the two monopole sources (one at each end) of the
sound radiating into the ambient free pace.

chuck

 

[EnglishMuffin]

I don't know anything about FEA as applied to acoustics, so its not clear to me whether he is talking about writing a program to do this from scratch, or whether you can do all this with built-in NASTRAN capability. Sounds like quite a project! And from the tone of his answer, it does not sound as though this analysis has ever been done in detail. However, the bottom line is - how big is the effect? Did he express an opinion? If so, I won't deny that it should obviously be given some weight, since he is an acoustics expert, something I certainly cannot claim to be.

 

[EnglishMuffin]

By the way, one way you could crudely investigate the temperature effect would be to heat the air inside the tube by using a propane torch near the lower opening. If you were investigating a capped tube, the air inside would get fairly hot, since continuous convection effects inside would be minimal. I expect you could also heat the tube itself, but I'm not sure what other effects that might produce under such uncontrolled conditions.

 

[Dinosaur]

Thanks again everyone for your comments. cllsj (chuck), I will run by the home improvement store and see if they have the copper pipe you mentioned.

Let me bring y'all up to date. The rector wanted this in place by Advent I, today. I ordered a 36 inch length of aluminum tube from a wind chime supplier and hung it from a chord through holes drilled eight inches from one end. This is at the proper location of the node according to the website cited earlier. I built a stand for it and a small wood mallet for striking. The head was cut to an edge to provide a crisper stricking device rather than a flat round head. When you strike it in a "sweet spot" it rings a little but if you don't it makes a sort of thud. The inside of the nave will seat about two hundred (I'd like to see it full someday, but hasen't happened yet). It is finished in wood and tile so a small amount of sound goes a long way. I had hoped the tube would ring better so I am still looking for a good tube. For the purpose of the service, we need only one note so precise tuning is not necessary, but I appreciate the additional discussion on this subject as well.

So, given where I am and where I'd like to be, I expect to do two things. First, I am going to buy a 36 inch length of the copper tube and see how that sounds. Second, I am going to experiment with adding an end cap to the top for striking and to see how this effects the note and the resonance. I don't plan to give up on this before I have a nice ring in the sound. My worry is that I will have to either purchase a tube of some variety of exotic material or that it will have to be machined to produce a satisfactory ring. Wish me luck and thanks again. - Ed

 

[Dinosaur]

Whoops! Glad I checked my facts. I guess I won't be going to the hardware store after all. So that leaves me with fixing my aluminum tube.

Well the bad news appears to be that I have hung the tube in the wrong spot, 8 inches from one end. The good news is that when I ordered the tube, I asked for three sections. Now I have two more tries to get it right, before I go back and try and change the length of the original tube.

O.K., it appears I should try again and support the tube at a node 7.35% from the top. Do you have a recommendation where I should strike it?

I am glad to hear that my sense that the "resonant air column length" is not misplaced. I truely believe this will play a factor in getting the tubes to sound correct.

I'll let you know how it comes out. - Ed

P.S. Chuck, what will your patent cover? Have you considered manufacturing chime tubes commercially?