Resonance modes in cylindrical open steel tubes

[Pipedreamers]

Hi Fellows,

I like to ask for some help here. I am trying to calculate
the resonant frequencies for a sample of an open stainless steel
tube, free and mounted rigid on one end.
I have done some measurements with and spectrum analyzer
and got a couple resonance peaks.
My goal with the calculations is to predict the resonant
frequencies (or modes) on stainless steel tubing with
different dimensions.
I have encountered some difficulty to find the equations
to solve such problem.
Is there some software available I could use to solve this ?

Thanks

 

[electricpete]

I think sreid used an equation based on radius rather than diameter.

Attached is a spreadhseet which calculates the 5 modes using the equations I provided above. I have cross-checked it with another program (the beam program from Tom Irvine) and the results agree:

http://home.comcast.net/~electricpete/eng-tips/PipeDreamersBeamCalculatorMetric.xls

The calculated lateral resonant frequencies for this pipe in fixed/free (cantilevered) configuration using Euler Bernoulli method are:
389 (hz)
2430
6814
13363
22087

Since this is a fairly long thin beam, I think the Euler Bernoulli method should certainly be very close. I don't think something like a method of shells is required. If I am mistaken, I am always willing to learn.

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[electricpete]

My apologies sreid. I didn't read your last post before my last post.

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[mtnengr]

Shell behavior is when the cross section deforms with angular variations at one station. The general pattern can be expressed of the form An(Sin pi*k*L)*(Sin pi*n*theta) + (Cos pi*n*theta) + Bn(Cos pi*k*L)*(Sin pi*n*theta) + (Cos pi*n*theta)). An and Bn are functions of the natural frequency.

When n=0, the cross section simply moves in and out (breathing mode) with no circumferental variation (usually at a very high in frequency). When n=1, the motion is a rigid beam motion (generally very low). In both cases, the longitudial variation (sin series) provides multiple modes and are based on the boundary conditions. Many of the calcs. earlier presented by others are for this latter case.

If the shell has an R/h ratio < 10, the behavior is usually a rigid body (beam) motion. For higher ratios, in the order of R/h > 30 to 5000, the cross section begins to respond with a Fourier variation. So high rigid body normal mode frequencies may be higher than those associated with the Fourier normal modes. The higher the excitation frequencies, for cylindrical shells, the closer the response is to a diamond shape on the surface.

You may also subject the shell to a radial and uniform impulse. If it is large enough, the object may first start out with a breathing mode (n=0) and then respond in the lowest Fourier mode. The wave form will have a beating appearance over time. This is called Mathieu instability.

The extent of the "ping" behavior observed in the experiment has to do more with the damping of the material and the non-linear effects of a so called "fixed" BC.

 

[electricpete]

Thanks. So if I understand correctly, the shell theory is required to study modes other than simple lateral beam vibration type. The Euler Bernoulli should be good enough for that particular set of modes, right>?

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[Pipedreamers]

Great stuff, many thanks for posting suggestions.

Many Thanks electricpete for the Excel file. This is most helpful.

But I am still a bit confused, as the Excel calculations
indicate Frequencies that do not match my measurements.
I wonder if I am wrong with my measurement ?

Anyway, Thanks everyone for trying to help me.

Cheers

 

[electricpete]

You're welcome. A small labeling error: w1 through w5 should be labeled f1-f5 since they are regular hz frequencies, not radian frequnencies.

Why doesn't it match:
A - it could be something other than simple lateral cantilever beam motion. Acoustic, longitudinal, and those complicated modes described by mtnengr

B - There could be a simple error in the material properities that you reported.

I noticed the coincidences that the ratio of your frequencies is 10514/3789=2.8. That also happens to be the ratio of the 2nd and 3rd lateral frequencies above 6813/2430~2.8. That may point toward explanation B. i.e. you may be looking at the 2nd and 3rd simple lateral modes, but some error in the constants creates a multiplicative error (the same error for each frequeency).

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[electricpete]

To confirm the hypothesis B, you might look very carefully to see if you can find signs of what would be the first mode down around 597 hz (that's the same ratio above my first computed frequency as your 2nd and 3rd are above my 2nd and 3rd computed frequencies). Perhaps try bumping right at the end, or pulling the end and then suddenly releasing.

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[GregLocock]

Can you post the frequencies you measure, and those you'v calculated?

Cheers

Greg Locock

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

 

[mtnengr]

Ah, the real world. I would look very closely to the true bounday condition. If you have just clamped one end and not buried the end into a very stiff material, the BCs are not fixed (but close). The "clamp" may be over only a portion of the circumference something like a shop vise).

Now, the response is non-linear, coupling between Fourier modes associated with the clamp and the rest of the cylinder.

 

[mtnengr]

Oops, I forgot the other real world component--the stainless material being used. Some SS are really non-linear. Their stress-strain curve follows what is called the Ramberg-Osgood Law and bends so you can not determine a true Elastic modulus for any strain level. A tangent modulus may be used for an approximation.

 

[Pipedreamers]

Hi,

well the material data came from a website, a company that makes ultrasonic transducers. I felt it was quite safe to use data they had posted for SS136.
I had also found material data elsewhere which varied in some parameters but I am not knowledgeable in this field so I don't know if it makes a significant difference. Once I have the right equation, then it should be easy to predict.

GregLocock,
The frequencies I had measured had two very strong spectrum peaks, one at 3729Hz and the other at 10514Hz. I have not been able to calculate any of these as I am not sure what is the correct equation for the to me unknown resonance mode.

Different Material Data for SS316:
Young's Modulus = 2.1*10^11N/m^2
Density = 7800 kg/m^3
Poisson's Ratio = 0.29

Anyway, here are a few pictures of what I have done. I have been using SpectraLab for analysis. For those that are interested, here are the wav. files as well.

Again, many thanks for all the help.

http://files.engineering.com/getfil...add7-4a64-9be8-6dd8ef42f427&file=DSC08830.JPG

http://files.engineering.com/getfil...de9c-4ae9-8212-ef61c19d94ef&file=DSC08855.JPG

http://files.engineering.com/getfil...0958-480d-84a6-bd8d80a0dbb2&file=DSC08883.JPG

http://files.engineering.com/getfil...84f1682&file=short_larger_tube_in_air___2.wav

http://files.engineering.com/getfil...arger_tube_in_air_held_by_vise_on_one_end.wav

 

[sreid]

How rigid is your fixed end support? If it's not entirely rigid, your frequencies may be somewhere between Fixed-Free and Pinned-Free.

As a sanity check, pull down and release the free end of the tube and see it it vibrates near the 389 Hz electricpete has given.

 

[electricpete]

Looking at the pictures made me remember that you got the same frequencies whether it was fixed/free or free/free. I forgot about that during my recent comments about theory B. Seems to shoot theory B down.

That would seem to point away from the simple lateral beam modes and perhaps towards acoustical resonance.

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[electricpete]

Here is another spreadsheet to analyse acoustical resonances.

It is set up to identify the length of pipe that will cause a given frequency.

http://home.comcast.net/~electricpete/eng-tips/PipeDreamerAcousticPipeResonance.xls

To apply it to this particular problem, proceed as follows:

Go to Calculations tab
* put f=3729hz into the frequency (cell C3)
* Observe the length 217 appears in cell N28 (5th mode for open/open pipe)

* put f=10514 into the frequency (cell C3)
* Observe the legnth 217 appears in cell N35 (14th mode for open/open pipe corresponding to L=Lambda)

The above MIGHT lead you to conclude these are some higher order acoustic resonances. BUT, why would it only respond to just a few of the higher order resonances and completely skip the others? Sounds a little strange to me. Some other caustions:
* I took a guess at air density - depends on humidity and barometric pressure
* The higher order modes are so close together (as a fraction of the frequency) that odds are one of them will be close to a given frequency


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[electricpete]

Correction - I should have included the following items identified in bold:

"(5th mode for open/open pipe corresponding to L=2.5*Lambda)"

(14th mode for open/open pipe corresponding to L=7*Lambda)


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[electricpete]

Another correction if you are using my spreadsheet for something else... in the boundaries tab it says that an elbow acts like a closed boundary... not true... elbow acts not much different than a straight pipe. An open-ended L shaped pipe with 3' on one side of elbow and 5' on other side of the elbow acts the same as an open-ended 8' straight pipe .

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[GregLocock]

Well, if it is a tructural resnace why not measure he mode shape, instead of trying to guess wht mode is which?

Now there is a problem there, the beam modes of axisymmetric beams are notriously hard to measure with accelerometers. I suggest using two at the same location at 90 degrees.





Cheers

Greg Locock

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

 

[Pipedreamers]

sreid,

have a look at the picture I had posted. Considering I had used a small vise and considering as you stated fixed does
not equal fixed I am absolutely not sure how fixed it is.

electricpete,

awesome, many thanks for taking the time to help me and to come up with the Excel sheet.

Well there are other peaks in the spectrum as well, but they are not as dominating as the two I had mentioned. I thought it made more sense to try to analyze the dominant modes.

When you speak of acoustic modes, you are hinting that air plays a role. That is really throwing me off, as I thought this would only be applicable if air travels through the pipe or if flowing air under pressure is used to excite the pipe.

Also, in one attempt to tune it, I had milled a 15mm slot 1/8inch wide into one end of the pipe, but it seemed to make
little to no difference where the two dominant peaks occur.
This lead me to believe that the peaks originated in a 'bell' mode and not over the length of the tube.

Thanks

 

[GregLocock]

Although that vice will modify the boundary conditions, I doubt it will really provide a fixed end.

especially not at 10000 Hz.

I really recommend that you measure the mode shapes, not just the frequencies.

I agree that ring modes of the tube are a possibility. You can measure those, although they are a bit tricky.





Cheers

Greg Locock

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

 

[Pipedreamers]

GregLocock,

how should I have to setup and measure the ring modes only ?

Thank