Radial Breathing Mode for hollow cylinder - formula needed

[GenSoft3d]

Hi, I'm trying to locate a formula for determining the radial breathing mode of a hollow cylinder.

The dimensions of the cylinder of interest are approximately .5" OD x .4" ID x .5" L

Any help would be greatly appreciated. Thanks!

 

[GenSoft3d]

Researching further I came across this formula that someone posted for determining the "ring frequency" for a hollow cylinder:

Ring frequency = sqrt(E / mu) / (Pi * d)

In the post he suggested that "The ring frequency is often called the breathing mode because the ring simply expands and contracts whilst retaining its circular shape."

This definitely sounds like the radial breathing mode vibration to me however he further describes the ring frequency as follows, "The ring frequency is the frequency where one longitudinal wavelength fits exactly around the circumference of the ring or cylinder."

Can someone please confirm this for me? Is the formula mentioned for this ring freq the same one for determining the RBM? I would very much appreciate it. Thanks!

 

[AlephZero]

Ring frequency = sqrt(E / mu) / (Pi * d)

That looks right for the "breathing mode" frequency in Hz (assuming you mean rho for density, not mu)

In the post he suggested that "The ring frequency is often called the breathing mode because the ring simply expands and contracts whilst retaining its circular shape."

This definitely sounds like the radial breathing mode vibration to me

Agreed.

however he further describes the ring frequency as follows, "The ring frequency is the frequency where one longitudinal wavelength fits exactly around the circumference of the ring or cylinder."

That doesn't make much sense to me. The "breathing mode" is different from the other vibration modes, because it doesn't involve any bending of the ring. A thin ring is approximately a state of uniform hoop stress in this vibration mode, so I don't see what "wavelength" is meant.

IMO, either the author is confused (or forgot that the breathing mode is different from the other vibration modes), or else it needs some more context to understand exactly what it's talking about.

 

[GenSoft3d]

Thanks so much for your help AlephZero... I really appreciate that.

So, using rho for mu means that I would actually be entering the mass per unit length value I assume. If so then I should be able to determine the rho by multiplying the Area by the mass per unit volume (i.e., PI*d*t*density)... can you confirm this for me?

 

[AlephZero]

In your formula "mu" is the mass density (in kg/m^3). Not the mass per unit length, or whatever. And at risk of stating the obvious, E is Young's modulus.

It must be the mass density to make the units correct. sqrt(E/rho) is a speed (the speed of sound in the material) and d is a length. So speed / length = 1 / time = frequency.

The symbol mu is used for line density (and several other things!) but that doesn't make sense here.

FWIW I checked your formula from the Rayleigh quotient Strain energy / Kinetic energy (which is easy, since you know the exact vibrating shape and the the only stress is the hoop stress in the ring). That comes out as the mass density (as I expected it would).

 

[bbbeard]

Researching further I came across this formula that someone posted for determining the "ring frequency" for a hollow cylinder:

Ring frequency = sqrt(E / mu) / (Pi * d)

In the post he suggested that "The ring frequency is often called the breathing mode because the ring simply expands and contracts whilst retaining its circular shape."

This definitely sounds like the radial breathing mode vibration to me however he further describes the ring frequency as follows, "The ring frequency is the frequency where one longitudinal wavelength fits exactly around the circumference of the ring or cylinder."

Can someone please confirm this for me? Is the formula mentioned for this ring freq the same one for determining the RBM? I would very much appreciate it. Thanks!

This is the formula in Timoshenko's book "Vibration Problems in Engineering", if μ is the mass density (kg/m³) as AlephZero pointed out.

As an interesting side note, the higher modes are multiples of this breathing mode frequency. For n waves around the circumference, multiply the breathing mode frequency by sqrt(1+n²). Note for the breathing mode, n=0, not 1.

BBB

 

[GenSoft3d]

Thanks for the responses... this is a ton of help.

I really appreciate the detailed explanation AlephZero... that helps a lot. And I am definitely interested in calculating the higher modes so thanks bbeard for providing the info on that and saving me time.

So, the d is the length then? I would have suspected it to be the diameter since it would seem that the radial nature of the breathing mode would be independent of the length. How excactly does the length of the cylinder effect the mode I wonder?

 

[bbbeard]

Thanks for the responses... this is a ton of help.

I really appreciate the detailed explanation AlephZero... that helps a lot. And I am definitely interested in calculating the higher modes so thanks bbeard for providing the info on that and saving me time.

So, the d is the length then? I would have suspected it to be the diameter since it would seem that the radial nature of the breathing mode would be independent of the length. How excactly does the length of the cylinder effect the mode I wonder?

"d" is the diameter. The breathing frequency is independent of the length of the ring. Of course, a finite length tube will have longitudinal modes as well as radial modes.

BBB

 

[AlephZero]

So, the d is the length then? I would have suspected it to be the diameter since it would seem that the radial nature of the breathing mode would be independent of the length. How excactly does the length of the cylinder effect the mode I wonder?

Yes d is the diameter. I said it was a length not the length - i.e. it had the dimensions of length. Sorry if that confused you.

The length of the cylinder does slightly affect the frequency, because of Poisson's ratio. As the ring "breathes" in and out its length changes slightly. The kinetic energy from the axial motion is ignored in your formula.

If you want to go that level of detail, you probably need a 3-D finite element model rather than looking for a "simple" formula.

 

[GenSoft3d]

Ok, I understand... that makes sense. Thanks for the clarification on that. But should I be using the O.D. for the diameter or the I.D.? And to get the diameter correct I would need to use metric units (e.g., .5 in diam = 0.0127m) for SI compatibility in this case correct? Seems obvious but just want to be certain.

Also, when you say that I would need a 3-D finite element model to determine the actual ring (RBM) freq are you referring to an autocad model?

Thank you for all the help!

 

[AlephZero]

Since all this assumes the cylinder is thin, use the mean diameter.

Yes, you have to use consistent units. You can work pounds and inches if your density is in slugs / cu in rather than lb / cu in.

By an FE model I meant using a program like Ansys, Abaqus, Nastran, etc. Some CAD programs have a built in FE analysis option, or you can use the CAD geometry to create the FE mesh, though if the part is literally just a hollow cylinder, it's arguable whether that would save any time compared with creating the geometry directly in the mesh generator program.

 

[GenSoft3d]

Ok, that's what I thought. I ran the model through a CAD simulation and here is what I ended up with:

There appears to be two modes that have the breathing effect (F1, F5). When viewing F1 axially it appears to expand out from it's original position. F5 appears to contract in the central region while the outer edges seem to keep their original shape. Looking at this image can you tell which one is the actual breathing mode?

Using the simple formula and going with the OD value (.49"), I came up with 92521.89 Hz which is much closer to F5 so I am leaning toward that mode, however it's off by 14197 Hz (92521 - 78324) which seems to be a considerable amount. Am I missing something or is that to be expected given the kinetic energy from the axial motion you mentioned? Perhaps I need to locate a more accurate possion's ratio value... I wasn't able to find much info on that.

FYI, I'm working with c101 copper and am using the following properties:

density = 8940 kg/m^3
young's = 117000000000 Pa
poisson's = .345

Thanks again!

 

[AlephZero]

Using the mean diameter I get 102 KHz.

Has your model got any restraints imposed on it? If it doesn't, you should have got 6 zero frequency modes (rigid body translations and rotations). Compared with 100 KHz, 50 Hz could be a zero frequency. I suspect it is actually rigid body rotation of the ring about its axis. The motion of any point on the ring is along a tangent, and when you scale up the animation to a finite size it looks as if the diameter is increasing, even though it isn't really. But if there should be 6 rigid body modes, why is mode 5 obviously not a rigid body mode?

Mode 5 could be an axial vibration mode. That would have frequency sqrt(E/rho) / (2L) which is the same order of magnitude as the breathing mode.

My suspicion is that your model is restrained somehow, which it shouldn't be. Plctures of all the first 7 or 8 modes (and the frequencies) might help understand what's going on, if you can't figure it out.

What do the colors represent?

 

[GenSoft3d]

Ok, you were correct... I was restraining it. After removing the restraint I'm now getting the first 6 rigid body modes which are showing a 0 value for their freq. Now I am seeing what appears to be a breathing mode at F19 (78.3KHz) and F30 (102.46KHz). They are both doing different things though and I'm still not sure which one can be considered the actual breathing mode. I created some animations of them though if you would like to have a look at them:

http://www.yantra3d.com/cyl/F19_78312.avi

http://www.yantra3d.com/cyl/F30_102460.avi

Also, this 6th 0 freq mode appears to be "breathing" to me... can you verify that it's indeed a rigid body mode?

http://www.yantra3d.com/cyl/F6_0.avi

Thanks again for all your generous and invaluable help btw... I really appreciate it!

 

[AlephZero]

Mode 6 has got to be a rigid body mode, from the frequency.

Your 78 kHz mode looks the same as when you had some restraints. It looks like longitudual vibration of a rod to me. Presumably the restraints in your first model didn't affect that vibration mode.

I don't know what to make of the 102 kHz. I wonder if your FE models aren't refined enough to get a "clean" picture. If you are generating a "random" mesh automatically, that will tend to break the perfect symmetry.

One way to check that is to look at the table of frequencies. For any modes that have "wiggles" round the circumference, (or that look like bending of a short beam) you should be getting pairs of modes with identical frequencies, but the two mode shapes clocked round by "half a wiggle" relative to each other (similar to graphs of sine and cosine functions). The single frequences left over will be the radially symmetrical modes. If most of the frequencies aren't coming out in pairs (equal to at least 3 or 4 figures) the finite element model has lost the symmetry of the structure somewhere.

If you can create a regular mesh (e.g. solid brick elements with say 10 along the length, 30 round the circumference, and 2 or 3 through the thickness) that should give a model that is "perfectly" symmetrical, and might be easier to understand.

 

[bbbeard]

Using the mean diameter I get 102 KHz.

I get

sqrt(117E9 Pa / 8.94E3 kg/m3)/(3.1416*0.01143 m) = 100.75 kHz.

This seems really high to me. I suppose the diameter is pretty small. But how big would a copper ring have to be to produce, say, a 100 Hz (not kHz) tone? This calculation suggests you get there only when the diameter reaches 11.4 meters. Food for thought.

 

[AlephZero]

I get

sqrt(117E9 Pa / 8.94E3 kg/m3)/(3.1416*0.01143 m) = 100.75 kHz.

This seems really high to me. I suppose the diameter is pretty small. But how big would a copper ring have to be to produce, say, a 100 Hz (not kHz) tone?

Usually, this is not the lowest vibration frequency of the ring or cylinder. That is the case for the OP's model, since it seems to be about mode number 30.

For a thin cylinder, there will be several modes with lower frequences, where the radial displacement is similar to \cos n \theta around the circumference, for n = 2, 3, etc.. Those frequencies depend on the thickness, since the ring is bending similar to a curved beam. The "breathing mode" frequency doesn't depend on the thickness.

I don't have a problem with the order of magnitude of the frequency here. I wouldn't be surprised to measure a this at about 1 kHz in a 1 meter diameter ring. but the lowest (nonzero) frequency for that size of ring might only be 5 or 10 Hz.

 

[GenSoft3d]

I'm a little confused I guess... I was under the impression that the breathing mode (n=0) was the very first mode possible (referred to as monopole) followed by the dipole or oscillating mode (n=1) which I believe is actually a "bending mode". Next is the "bell mode" or quadrupole (n=2) where the cylinder squashes and stretches alternately along the horizontal and vertical axis. N=3 would be considered the sextapole, n=4 would be then be the octupole (n=5 is the 5-star shape but not included in the image below) and n=6 would be the interesting 6-star shape where no farfield appears to exist. These modes are shown below in their respective order:

http://www.hasdeu.bz.edu.ro/softuri/fizica/mariana/Mecanica/Waves_2/cylmodes.gif

What confuses me is that the n=0 breathing mode does not seem to be properly represented in either the equation or in our modeled simulations because it suggests that the breathing mode (n=0) is a much higher mode than the other supposed higher modes (i.e., n=1, n=2...). This is what is throwing me off. How could n=0 be at 100 KHz (like bbbeard mentions) when the other higher modes are well below that frequency in the model simulations? Why is it not showing up first - before the other modes (like the dipole for instance)?

From my perspective, the breathing mode derived from the equation is actually not the first breathing mode vibration (n=0) and so the equation appears to me to be misleading. But how to calculate this first mode of vibration (what is the equation?) and why is it not also showing up in the modeled simulations? There must be another formula for finding the actual first mode of vibration right?

 

[AlephZero]

I didn't really notice this earlier post till now.

As an interesting side note, the higher modes are multiples of this breathing mode frequency. For n waves around the circumference, multiply the breathing mode frequency by sqrt(1+n²). Note for the breathing mode, n=0, not 1.
BBB

I haven't got Timoshenko's book available right now but I think there is some misunderstanding there about exactly what modes Timoshenko is talking about, and/or whether that general formula applies when n=0.

I'm a little confused I guess... I was under the impression that the breathing mode (n=0) was the very first mode possible (referred to as monopole) followed by the dipole or oscillating mode (n=1) which I believe is actually a "bending mode". Next is the "bell mode" or quadrupole (n=2) where the cylinder squashes and stretches alternately along the horizontal and vertical axis. N=3 would be considered the sextapole, n=4 would be then be the octupole (n=5 is the 5-star shape but not included in the image below) and n=6 would be the interesting 6-star shape where no farfield appears to exist. These modes are shown below in their respective order:

http://www.hasdeu.bz.edu.ro/softuri/fizica/mariana/Mecanica/Waves_2/cylmodes.gif

What confuses me is that the n=0 breathing mode does not seem to be properly represented in either the equation or in our modeled simulations because it suggests that the breathing mode (n=0) is a much higher mode than the other supposed higher modes (i.e., n=1, n=2...). This is what is throwing me off. How could n=0 be at 100 KHz (like bbbeard mentions) when the other higher modes are well below that frequency in the model simulations?

The animations look right. Actually the "n=1" mode that you drew is a rigid body translation of a ring. For a cylinder, there are other "n=1" modes that do look like beam bending modes, as you said.

The reason n = 0 doesn't fit the pattern is because the stress and strain in the ring is fundamentally different. It isn't obvious from looking at the animations, but for n > 0 the length around the circumference remains constant (to first order). At the points where there appears to be no motion (i.e. no radial motion) the ring is actually moving tangentially, because the parts of the curve that are outside the undeformed circle are longer than the circle but the parts inside are shorter. The n=0 mode does not have constant length when the whole circle expands and contracts.

The kinetic energy of all the modes is similar order of magnitude, but the stiffness for a thin ring is much higher for n = 0 than for n > 0. The difference in stiffness is between squashing the ring out of round (flexible) compared with expanding it with uniform internal pressure (stiff). The frequencies for n > 0 depend on the thickness of the ring (the bending stiffness is roughly proportional to t³ but the mass is proportional to t) but the frequency for n = 0 is independent of the thickness.

 

[GenSoft3d]

Well, I think I discovered the source of my problem w/ not seeing the breathing mode in my simulations. I did some tests and with the length of the cylinder at < .3 I can definitely see the breathing motion as predicted by the formula. However, anything > .3L the breathing motion becomes distorted by interference from the longitudinal modes and so the motion was not obviously perceivable to my untrained eye. (At least that is what I believe is happening)

At about .3L the breathing shape begins to favor one end over the other causing it to contract more severely at one end compared to the other. At > .5 the ends appear to stay stationary while the inner section of the cylinder breath in and out (expand/contract) as expected.

Your help has been most valuable so thanks for walking me through all of this... I will play with this some more and will continue to post my findings or questions here.

 

[AlephZero]

At about .3L the breathing shape begins to favor one end over the other causing it to contract more severely at one end compared to the other. At > .5 the ends appear to stay stationary while the inner section of the cylinder breath in and out (expand/contract) as expected.

Hm... that sounds like your FE model, or the solver, isn't good enough to handle it properly. The mode shape should (obviously) be symmetrical end-to-end.

I'm guessing you are using the FE package that is bundled with your CAD program and you are pretty much going with the defaults. If there is an easy way to tell it to generate a finer mesh, that might help.

 

[GenSoft3d]

I took your advice earlier and increased the subdivisions of the mesh but am still getting the same effect. It's actually triangulating the mesh so subdividing may not be the correct term here.

It may be the way I have it set up. The options I have available are single point and parametric dimension although both are producing the same results. Within those options I have settings for "contacts" and by default it's set to bonded (which is what I have been using). The other available options for contacts are; separation, sliding / no separation, separation / no sliding, shrink-fit / sliding, shrink-fit / no sliding, and spring.

Do you think it may have to do with the "contacts" settings?

 

[AlephZero]

Contacts of what with what? I would hope there aren't any, so it doesn't matter which option you pick. But of that list of options, "bonded" seems most plausible, followed by "separation" (if "bonded" is bonding to something that it shouldn't).

This is probably heading outside of my experience. We use a CAD package plus separate FE programs (more than one, for different applications) rather than its built-in FE analysis option. In fact we've had some bad experiences with people going off and doing their own thing with the built-in software, and either not fully understanding what they were doing, or not understanding the limitations of the software (or both!). (I won't name the package because I don't have any reason to think it is fundamentally broken - but using it isn't as "automagical" as the CAD sales people might want you to believe).

Beyond the general comment that I think your FE model has somehow broken the symmetry of the physical structure, you probably need somebody with expertise in your particular software.

 

[GenSoft3d]

Ok, thank you for all your help AZ... It 's most likely what you are saying I would think because I am using an CAD prgm and not a specialized FE prgrm and so that is likely the case here. But, I'll spend some more time working with it and see what I can find out. If I can track down the problem then I'll post what I find back here.

Thanks again... you've been extremely helpful!