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Welcome! On this page I present what I think is the best information one will find on tubular wind chimes. Many years ago I started searching the web for information to build my own chimes. After finding a little bit of information I tried to build a set of chimes using copper tubing. Mostly they made noise but they sure didn't compare to expensive commercial chimes. A few years later I decided to give it another try. This time I found an equation to help me determine the length of a chime. I had not seen this equation before and being an engineer I set out on my current quest for more knowledge. On this page I present the information needed to make chimes that I think will be as good as or better than any of the expensive commercial chimes. The best part is the satisfaction that you personally built them, and at a fraction of the cost of purchase.

I've not done this totally on my own. I owe Jack Maegli a big thanks for his help particularly in regards to the patent, even though the patent was denied. Lee Hite and his web page has also provided some additional insight and I don't want to forget the members of the Yahoo Windchimes group.

Plans

In a hurry and just want to make some chimes, here are some plans to allow you to do just that.

A Minor       D Major       F Major

Tube Frequency / Length Calculator

Having read the comments on the Yahoo Windchimes group over the last few years, it appears my methods works best if one stays with the first mode chimes like my plans for the A minor chimes. Even then some people have problems and are not successful. People seem to have more problems with tuning if they attempt chimes tuned to the fourth natural frequency like my plans for the F major chimes.

If you are using a microphone to check the frequency of your chimes, you are measuring the frequency of the air at the microphone. You cannot automatically assume that this measurement is in any way related to the frequency of the metal chime itself. There has been more than one time when something was not working the way I expected only to find my method of measurement was at fault.

On the web page length calculator used javascript to calculate the frequencies, length, and suspension point based on your inputs. The first input is the type of material. Your choices are aluminum, steel, and copper. If your tubes are brass use copper since brass is largely copper. Steel will work for all types of stainless and regular steels. Next pick the units you want to use inches or millimeters. 

Length Calculator

The spreadsheet has similar inputs. Cell E5 for the material is a drop down list to pick a material. If you want to add a material, you can do so my moving to the material sheet and adding your own values for modulus of elasticity and density for mat1, mat2, or mat3. Even more materials could be added, but you will have to modify the drop down list.

Spreadsheet - Open Document Format


Definitions

I want to give some definitions first.  When a tube, such as a wind chimes, is struck it will start to vibrate.  The tube can and will bend in many different shapes at many different frequencies all at the same time.  The tubes can also vibrate in the axial direction and the radial direction.  All these different vibrating shapes are called modes.  The mode with the lowest frequency is the first mode.  The frequencies the tube vibrates will sometimes be referred to as natural frequencies.

The lowest frequency can also be called the first natural frequency,  fundamental, primary frequency, or first harmonic.  Higher modes are called overtones.  Harmonics are frequencies that are even multiples of the fundamental.

The quality factor or Q refers to the width of the frequency response.  A high Q has a narrow frequency response.

Technical Stuff

Frequency versus Length

The bending natural frequency of a tube is quite easy to predict using Euler's equation.

w = (B*l)2 * SQRT (E*I/(rho*l4))

w - frequency radian per second
E - modulus of elasticity
I - area moment of inertia = PI*d3*t/8 for a thin wall round tube
d - mean diameter
t - wall thickness
rho - mass per unit length = Area * mass per unit volume = PI*d*t*density
l - length of tube

w= (B*l)2 * (d/l2) * SQRT (1/8) *SQRT (E/density)

(B*l)2 - Constants based on the boundary conditions.
For a wind chime (Free-Free Beam)
(B*l)2= 22.373 for the first natural frequency.
(B*l)2= 61.7 for the second natural frequency.
(B*l)2= 121 for the third natural frequency.
(B*l)2= 199.859 for the fourth natural frequency.

To get the units correct you must multiply the values inside the square root by gravity (g).

g = 386.4 in/sec2 for the units I'm using.

for frequency in cycles per second (Hz)
f = w/(2*PI)

So for a given material then the frequency of a thin wall tube reduces to

f = constant * d / l2

for steel and the first natural frequency
SQRT(E/density) = SQRT (30e6*386.4/.283) = 202388.56 in/sec
constant = 22.373 * 202388.65 * SQRT(1/8) / (2*PI) = 254792

for aluminum and the first natural frequency
SQRT(E/density) = SQRT (10e6*386.4/.100) = 196570.60 in/sec
constant = 22.373 * 196570.60 * SQRT(1/8) / (2*PI) =  247467

Note the natural frequency does not depend on the hardness, temper, surface finish, or any other material property of the tube beside the modulus of elasticity and density.  A coating such as zinc may have some  very minimal effect on the natural frequency of the tube by changing the overall density.

mode shapes

Here are the first three mode shapes.  The first mode is at the left.  As I said before all of the mode shapes and frequencies will be excited when the tube is struck sharply.  Remember the tube is in motion bending back and forth.  Also note that these are only the transverse modes.  I'll discuss the other mode shapes and the roles they play further down the page.

So I given a formula for determining the frequency of a steel or aluminum tube.  I can head down to the hardware store buy some tubing, plug in some numbers, and I've created a great sounding set of chimes.  As you probably guessed it isn't so simple.  





air column
Just using the equation I've provide above ignores the interaction between the air in the tube and the tube or fluid structure interaction (FSI).  While the subject is quite complex, for wind chimes one just needs to get the frequency of the air column to match the frequency of the tube.

For a column of air the natural frequency

Frequency =  n * speed of sound of air / 2 * length of the tube

where

speed of sound of air = 13 527 inches/second 
n = 1,2,3 .....   mode number.

The mode shapes are shown at the right.  On the page (http://hyperphysics.phy-astr.gsu.edu/)where I got this formula and picture from there was a note that 0.6 times the tube radius should be added to each end of the tube to get the acoustic length of the tube.  Inside the tube the air is forced move axially along the tube.  At the ends the air is free to move in a spherical pattern.  This 0.6 factor accounts for this change at the end of the tube.

The red lines represent the velocity of the air in the axial direction.  For n=2, the second mode, the maximum air velocity occurs at the center and ends of the tube.  At the intersection of the lines the velocity of the air is zero and the alternating air pressure is at it maximum.  The result is an alternating flow of air inside the tube which at least in part produces the sound that we hear.  Of course the pressure at the ends of the tube is same as the surrounding air.

Now we put these two equations into a spreadsheet and plot frequency versus length/diameter.  The plot to the right is for a 1 inch diameter steel tube but as I've shown previously aluminum would not be significantly different.  At a length/diameter ratio the 2nd natural frequency of the air column crosses the the 1st natural frequency for the tube.  I call this the "ideal" length.  At this length the natural frequencies reinforce each other and damp out all other natural frequencies of the tube.  If you are wondering, there are other places where a natural frequency of the tube crosses a natural frequency of the air column.  I will discuss this further further down the page.  However, for standard chimes this combination seems to work the best. 






Jack Maegli created the next three frequency response graphs for use within the patent application.  The first graph is for a set of chime tubes cut at or near the ideal length.  Lines one through six are the response as the primary frequency of the tubes.  Lines 7 through 11 are the response of the same tubes at the second natural frequency.  Notice the length of the response at the primary frequency compared to that at the second natural frequency.  This is what you want a strong response at the primary frequency and a short, leak response for all other modes.






In this graph 1 and 4 are the response of a tube cut to the ideal length.  For 2 and 5 or 3 and 6 the tube was cut much shorter than the ideal length.  The response at the primary frequency for these two short tube is much less than for the ideal tube.





Again in this graph 1 and 4 are the response of a tube cut to the ideal length.  For 2 and 5 or 3 and 6 the tube was cut much longer than the ideal length.  Once again the response at the primary frequency is diminished compared to the ideal length.  Also notice now strong the response is at the second natural frequency for these two long tubes.  Since the second natural frequency  is not an even  multiple (actually about 2.75) of the first natural frequency  it is not harmonic.

Others have written that you can't predict the frequency of a tube and that the column of air is unimportant.  They are wrong.  In fact I believe because they have ignored the column of air that they say that the frequency of the tube can't be predicted.  Look at the last graph once again.  If I cut the tube at a length longer than ideal I will hear a blending of the two tone which will not be pleasant.  Clearly, if I cut the tube near the ideal length the only frequency I will hear is the primary.


As I mentioned way back at the top of the page Lee Hite's web page An Engineering Approach to Wind Chime Design or What Makes Toast, Toast? has provide some useful insights.  Lee wanted to create a more bell like sounding chime similar to a tubular bell.  He was able to gain access to a set of orchestra chimes and includes some dimensions and the frequency for one of the chimes.  The length of the tube is much longer than my ideal length but I could not imagine that it didn't work.   Based on my theory  I started looking at higher natural frequencies.  What I found is shown in the graph at the left.  The tube was tuned to where the fourth natural frequency of the tube crosses the fifth natural frequency of the air column. After some search on the internet I learned that the fourth, fifth, and sixth natural frequency of a tube form a very close ratio of 2:3:4.  From this we hear the fundamental.  Now this is not the fundamental of the beam but the fundamental of the progression of notes.  In the case of the tube Lee mentions, 523 Hz-"C", the fundamental would be the 261 Hz-"C", the sixth natural frequency would be the 1024 Hz-"C", and the fifth natural frequency would be 784 Hz-"G".  This "G" would be the 5th note in a "C" chord.

Be warned that trying to make tubes tuned to the fourth natural frequency are much more difficult than tuning them to the first natural frequency. There is no tuner that I know of will help with this perceived fundamental note. Brent Hamilton wrote that this perceived note is the brain's fuzzy logic trying to produce analog "addition" of the many, musically unharmonic frequencies being fed to it by inputs from the ear.

 Circular Modes and Copper Tubing

Of course I have another equation

f = (t/(2*d^2))*SQRT(E/density)

f - frequency
E - modulus of elasticity
d - mean diameter
t - wall thickness

The circular mode is not a function of the length of the tube.  The chart to the left, again from Jack Maegli, shows the response of 4 - 1" nominal diameter type M copper tube. (1.125 OD 1.055 ID).    The type of copper has to do with its intended use.  Type M is for lower pressure / strength applications and is thinner than type L or K copper tubing.  Type M is also the most commonly available at the hardware stores.  Notice the response at 2100 Hz.  This is the approximately the predicted frequency from the equation above. 

I don't like the sound of copper chimes and I believe the circular modes are the reason.   I've taken the audio output as a wave file and used a digital filter to remove the frequencies above 2000 Hz  (ie remove the circular mode).  I feel the sound is much improved.  To do this of course for a set of chimes is impossible.    Increasing the wall thickness will force this frequency higher making it less responsive.  Jack has made some chimes from type L copper tubing and found they sound better than the type M copper tubing.

Another way to handle this would be to tune the first natural frequency to be a harmonic of the circular frequency.  That would mean, however, that each tube could only be tuned to one or two frequencies.

Recently I was reminded that orchestra chimes are struck with a downward blow at the top.  When I tried this with a copper chime tuned to the first natural frequency to my surprise the circular mode didn't appear.  Orchestra chimes are also capped at the top.  This cap would reinforce the top and help prevent the activation of the circular modes.  I'm not sure how to accomplish either or both of these with chimes, but it does appear to reduce or elimanate the circular modes.


Suspension Point

So after a very long discourse on frequency and tube length determining the point at which to hang the tube is relatively easy.  As the tube bends back and forth there are points along the tube that don't move.  For the first mode there are two points, the second has three, and so on.  Again Euler's equation makes these easy to predict.  For the primary mode there is a node 22.4% of the length from either end.  For the fourth mode there is a node 7.35% of the length from either end.  However, I noticed that Lee mention that tubular bells were suspended at the end of the tube.  When I tried this I could tell no difference between suspending the tube at the end or the node.  Warning,  don't suspend the an ideal length tube at end.  Use the 22.4% location for the node for tubes tuned to the primary frequency.  I used an internal axle to suspend the tube from mostly which I think is a cleaner look.  I've also used the more conventational method of running a string through the holes.  This puts the chime at the bottom of a "V" and helps prevent the tubes from swinging into each other.

Striker / Sail

Most books on structural dynamic have a dynamic amplification factor chart.  The chart shows that as the impact time constant approaches the time constant for structure more energy is transferred into the structure.  So that means for lower frequency chimes a soft, heavy striker and for higher frequency chimes a harder, lighter striker.  Beyond that I can't shed much light on the subject but I'm still working on it.  Currently I use a plastic disk laminated between layers of wood.  The plastic disk is larger in diameter and strikes the tube while the wood adds weight.  Jack Maegli suggested that a striker to tubes distance of 0.75-1.00 works well under moderate wind conditions.  He also suggest the sail be about 32 square inches and have a weight equal to 25% of the striker.

Note Selection and Tones

I have a limited knowledge of music but I will try to give some explanation of a chord because I've been asked several times. The western scale, and there are many scales, is formed by 12 semi tones progressing from one note to it's octave.  The notes of C major scale are C - D - E - F - G - A - B - C.  There is a semi tone between E - F and B-C all the other the other notes are separated by whole tones. A chord is formed by playing two or more of these notes.  A major chord based on the C major scale would consist of the 1st, 3rd, and 5th note or C, E, and G.  The  Jaz Class web site gives a large selection of chords.  Also note that pentatonic chords are popular for chimes.  Lee Hite's page has more information on this subject.

I've been told many times that larger tubes are more mellow.  If two tubes are the same length but one is larger in diameter, the larger tube will have a higher natural frequencies.  Putting different diameters into the length calculator should quickly make this clear.  Of course, the "ideal" length of the larger tube will be longer and have a lower frequency.

Tube Arrangement

The motion of the striker is much too random too attempt to arrange the tubes to play a tune if you will.  Mostly I attempt to arrange the tube to balance the chimes.  That is so the top or the part the tubes are suspended from is level.  I don't always succeed. 

If you feel I've not answered your questions, you may write me at cllsjd @  google mail.com   (gmail.com)






Subpages (2): Euler Length Calculator