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

 

[EnglishMuffin]

I answered this question once before, and for some reason the thread disappeared. Its a genuine design question, and presumably not a student post, so I can't imagine why it happened. In the previous post there was something about an air resonator, which didn't make a lot of sense, (at least to me). This was my reply as far as I can remember it :

The lowest natural frequency in Hz of a free-free single span beam is given approximately by :

fn = 4.73^2/(2*pi*L^2)*(E*I*gc/m)^0.5 .. (eg : Blevins)

where L = length : in
E = Youngs modulus : lbf/in^2
I = Second moment of area : in^4
m = mass per unit length : lbm/in
gc = 386.4
pi = 3.14159

The ideal suspension point would be at a node, of which there are two, situated approximately 0.22*L from each end.

You will find that tubular wind chimes are often suspended at such a location.

 

[vanstoja]

As I found out from a very recent web search of the subject "musical chimes frequencies", the design of chimes is a rather complicated engineering problem. The #1 hit on the list is the following

http://home.fuse.net/engineering/Chimes.htm

This discusses, at considerable length, the best range of musical tones, the overtone frequency ratios and provides length formulas. It also cites a more recent review by a kindred spirit that can be called up. This later treatise complements the first one and includes a discussion of best material for chimes (steel and aluminum much better than copper) and talks about the design of "strikers" and the locations of tube bending mode nodal points for best response. Your hoped for "pipe whacked with a wooden mallet" looks like a forlorn hope after reading the above stuff. Good luck and hope you make it in time for Christmas services!

 

[Dinosaur]

Yes, I suspect it was my post in another area of this forum that was deleted. I was disappointed because this is a real design question. I thought this would be a great place to try my luck, and so far it has generated the only quality response. Also, as you recall, I asked about seeking a length that would correspond to both the fundamental frequency of vibration of the tube and the resonant frequency of an air column. To explain further what I mean there I will describe a simple physics experiment we did in high school (about twenty years ago). We had a beaker of water, a glass tube and a tuning fork. We held the glass tube in the beaker so one end was immersed in the water and the other end at the top was open to the air. Next we struck the tuning fork and held it over the "air" end of the glass tube. Then by moving the glass tube up and down we were able to change the length of the resonating chamber. When we had the length matched to the frequency of the tuning fork, it became quite loud because the air in the tube resonated at the same note.

Getting back to my problem, it seems to me the best effect will be acheived when the note of the tube matches the resonant length of a column of air in the tube. That is why I am trying to work the problem from two variables. Essentially, the length of the chime is known because that is a direct correllation between the desired frequency and the resonant length. I don't remember what the number was excatly, but it was about sixty inches. So now I have to find a material, a wall thickness and a diameter that resonates at about the right frequency.

Since I first posted, my dad purchased a stock piece of steel tube. I may just solve the problem the brute force way. I will strike the tube and measure the frequency using a program I downloaded this past weekend. Then I will plot the frequency against the length. Then I will cut an inch off and do the same thing. I should be able to develop a curve that will give me a reasonable chance of solving the problem.

If you have any more theoretical help, I will keep checking back here. Thanks for those who have answered. My appologies for rambling and/or any severe spelling errors. In this you will know I am a stereotypical engineer. - Ed

 

[EnglishMuffin]

vanstoja : very interesting - at least your reference seems to confirm the 0.22L node support point - so, Dinosaur, I would use that as a hanging location. I think your column of air idea is somewhat questionable, since even if the longitudinal air column resonant frequency can be arranged to match the bending natural frequency of the tube, its not certain that the one resonance will sympathetically excite the other very much, at least to an audible level. But maybe it will. Good luck.

 

[EnglishMuffin]

Dinosaur:
Further note : I have a copy of Lord Rayleigh's classic "Theory of Sound", which you can probably still get from Dover. After wading through his somewhat circumlocuitous Victorian prose, I think I am correct in saying that, for the axial vibration of an air column inside a tube, the frequency in Hz of the first mode is given by V/(2*L) for a tube open at both ends, and V/(4*L) for a tube with one end blocked, where L is the length of the tube and V is the speed of sound in air in length units per second. The diameter does not affect the frequency. However, I'm still of the opinion that bending vibration of the tube is not likely to impart much axial momentum into the air column, and hence you would not notice much sound amplification if the mechanical bending and air column frequencies were arranged to coincide. But I've been wrong before! There are of course going to be many complex harmonics above the fundamental, which will give the chime a distinct character. As with many musical instruments, the nature of the striking device will have a considerable effect on these harmonics. A soft striker will damp out the higher frequencies, and a hard one will permit them.

 

[GregLocock]

Yes those are the right relationships for wavelength to tube length. An organ pipe is a quarter wave long, and an open tube is like two quarter waves back to back, ie half a wave long. A sticky toffee to the first person with the correct explanation why they don't have to be one wavelength long.

I agree that I'd be pessimistic about the chances of getting an axial pressure wave excited by a transverse bending mode of the pipe, but it can't hurt. Oh, here's how to test it - find a pipe with the right length diameter and thickness to give matching bending mode and acosustic mode, then compare the noise made with and without one end blocked off.

Incidentally there is another effect that may be important, the coincident frequency. That's the frequency at which the wave speed of air matches wave speed of the bending mode of the pipe - which can give radiation efficiencies >1. I suggest you look at Bernek for that, there's too much handwaving for a post.

Cheers

Greg Locock

 

[GregLocock]

That's Beranek.

Cheers

Greg Locock

 

[Dinosaur]

Thanks everyone for your assistance. I had already looked up the length of the open and closed tube and recognised it did not depend on the diameter. My motivation for getting the transverse vibration length to match the resonant air column length is an effort to enhance the final sound. First, by getting the lengths to match, a softer hammer may be used thus eliminating upper harmonics. Also, because the fundamental will be enhanced, the upper harmonics will hopefully be even further dominated by the fundamental.

My back up plan is to build a resonating chamber at the mouth of the tube that is tuned to the proper frequency of the chime. One benefit of this is that if made of the proper wood, it will give the chime a softer tone.

The reason I suspect to derive a noticable benefit from the resonant air column is because the air inside the tube will have much of the sound energy and it has to escape at the end of the tube. So I suspect most of the sound radiating from the chime emanates from the end of the tube. I do understand the transverse mode vs. the longitudinal mode decreases the effect but I am doing this as much for fun as anything else. In the end I do hope to get a nice low ringing sound though. Oh, and one more thing, I am trying to get this ready for Advent, not Christmas, so I have one month less to fool around. Thanks again. Dinosaur

 

[EnglishMuffin]

GregLocock : You seem to be implying that you know the correct answer to your question - you just want to see if anyone else does. I suppose I could reproduce Lord Rayleigh's extremely involved explanation, which I'm sure nobody would dare to argue with, but I'm not sure whether I want one of your sticky toffees, even if it didn't have to make the trip from Australia. Actually, I wouldn't be surprised if your toffees come from the UK, in which case it would have travelled an even greater overall distance! (Can you still get Sharps by the way ?)

 

[GregLocock]

Dunno about Sharps. OK, no bites, no toffees. The velocity wave undergoes a 180 degree phase shift at the reflection plane at the end of the tube, so the characteristic length is twice what you'd expect. The reason I asked is that I hoped someone had an intuitive explanation - it certainly surprises me that the quarter wavelength is the important property of a closed ended tube.





Cheers

Greg Locock

 

[MikeyP]

This looks pretty interesting:

http://home.fuse.net/engineering/Chimes.htm

Note especially the section on the PERCEIVED pitch of a chime in relation to its natural frequencies!

Also: When discussing acoustic modes in pipes don't forget that there is a radiation loading on the moving mass of air at the open ends of the tube. This depends on the radius of the tube. Dimensionally, this load impedance seen by the tube is a mass. If you add mass to a dynamic system you decrease its natural frequencies. It effectively makes the natural frequency of the tube lower than it would be without the radiation loading.

Often you will see this described as an "end correction" to the length of the tube (L + delta) because adding length to the tube also decreases the natural frequency.

M

PS Greg: I tend to think of it in terms of boundary conditions. The particle displacement at the closed end must be zero and the particle displacement at the open end must be maximum. So the simplest sinusoidal profile along the tube which satisfies these boundary conditions is...?

 

[MJKing]

I have found the windchime discussion to be very interesting. I am not sure that I understand what the confusion is about the quarter wavelength tubes. But if you go to my website, I have some theory on quarter wavelength tubes as they are applied to stereo speaker design.

Hope that helps,

Martin

Quarter Wavelength Loudspeaker Design

http://www.quarter-wave.com

 

[MikeyP]

Oh yes, and sorry to be picky, Greg, but most organ pipes are more like 1/2 wavelength resonators. The end nearest the wind chest (throat) has a boundary condition which approximates that of an open end. This type of pipe forms the majority of the organ (diapaisons, flutes, reed and trumpets). An 8 foot pipe sounds about middle C. The exceptions are stopped ("gedeckt&quot pipes which are usually square section wooden flutes which have a bung in the top. These sound 1 octave higher. Just to confuse the issue some open pipes can be "over blown" with higher pressure air. These also sound 1 octave higher than normal because the 2nd mode is excited to a greater degree. Try blowing gently into a tin whistle or recorder and then blow full pelt and you will see what I mean.

M

 

[GregLocock]

I wondered about that. It seemed to me that exciting a system at, or near, a node is a rather odd way of doing things.



Cheers

Greg Locock

 

[MikeyP]

Clarification of my last post

An 8ft organ pipe does not sound at middle C. An "8 ft" stop in organ terms means that the largest pipe in the rank is 8 ft long (approx.). On an 8 ft stop you press middle C on the keyboard and the sound you get is at middle C. On a 4 ft stop you press the middle C key and get a sound at C above middle C.

M

 

[EnglishMuffin]

Since the generally accepted value for middle C is 256 Hz, I fail to see how you could get a fundamental resonance anywhere close to that with a 4 ft or 8 ft pipe, either closed at one end or open. See the following for a calculator.

http://hyperphysics.phy-astr.gsu.edu/hbase/waves/opecol.html#c1

 

[MikeyP]

EnglishMuffin: See my post above yours.
M

 

[EnglishMuffin]

MickeyP : I've seen it, and I'm afraid I don't understand it. Presumably, most of us are not organists, so we don't know what an 8ft "stop" or a "rank" are supposed to be. All I know is that for an open-open pipe, you would need a 26.43 in long pipe to get middle C, assuming air at 20 deg C. For a closed-open pipe, it would be 13.21 in. (Of course, this ignores all the slight corrections etc).

 

[Dinosaur]

Thanks again. I checked the link provided by Vanstoja and it was very helpful.

Regarding the length of an organ pipe for resonance, if it is a half wave long with open-open end conditions it will resonate at the fundamental frequency. Except I have learned in a physics book dealing directly with sound that there is an end correction on the overall length of something near one diameter of the pipe. The pipe acts as a pipe a little longer than it measures according to observations. I don't recall exactly what the correction factor was reported to be in the book.

If the velocity of sound in air is 344 m/s and you wish to resonate a 256 Hz sound, the half wavelength is:
344/(2*256) = 0.672 m -or- 2.21 ft. It appears you can get a "C" from either an 8 ft stop or a 4 ft stop. Also remember, if you overblow the pipe you should be able to excite it to its first overtone and get a pitch twice as high. How this effects the sound produced is something I would like to hear more about.

Looking forward to more insight. - Ed