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]

Den Hartog page 432 (based on Euler Bernoulli):
w = an * sqrt{E*I/(MU*L^4)}
a1 = 3.52
a2=22.0
a3=61.7
a4=121
a5=200
I=pi*(OD^4-ID^4)/64


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[electricpete]

Forgot to say tha MU is mass per length. Of course E is Young's modulus, L is lenght, w = radian frequency = 2*Pi*f

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[40818]

If your also looking to get higher frequency modes, then it would only take a very short time to do a normal modes analysis in FE.

 

[Pipedreamers]

electricpete
Thanks for your help, but what is

an
a1=3.52
a2=22.0
a3=61.7
a4=121
a5=200

some fetch factors ?

40818
Also, forgive me but what does FE stand for ? Is this some
simulations software ? If so, can you post a link to it ?

Thanks so much

 

[electricpete]

Not sure I understand what your asking.

You plug a1, a2, a3 etc in for an to obtain the different possible resonant frequencies.

When you solve the Euler Bernoulli problem by separation of variables and apply these boundary conditions, you find that an must satisfy:
COS(SQRT(an))*COSH(SQRT(an))+1=0

The possible values of an that solve the above equation are 3.52, 22.0 etc. We gave them names a1, a2 etc.

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[Pipedreamers]

electricpete,

thanks for clearing this up. I did not understand where
these values came from. I am new to this matter.

So if I understand you right I will have to resolve this equation to f.

f = an * sqrt{E*(pi*(OD^4-ID^4)/64)/(MU*L^4)}/2*pi

I am trying to confirm my Experimental data for the 316 Stainless Steel tube.

Material Data:
Sound velocity (axial) = 4912m/s
Sound velocity (radial) = 5087m/s
Mechanical Impedance = 39290000kg/m^2/s
Young's Modulus = 1.93*10^11N/m^2
Density = 8000 kg/m^3
Poisson's Ratio = 0.26

Tube Data:
OD = 19.16mm
ID = 18.11mm
L = 216mm

I am getting two strong spectrum peaks, one at
3729Hz and the other at 10514Hz.

As I understand other modes may be present yet not as
dominant so they may not be seen in the spectrum.

Also, I was wondering in this equation, the speed of sound
of the material is not been considered.

Thanks again for your help.

 

[electricpete]

The frequencies that I calculated above were for lateral vibrations.

One could also compute resonant frequencies for torsional vibrations (unlikely wihtout torsional excitations), longitudinal vibrations, and air-column acoustic vibrations.

a few questions:
1 - how is the vibration being excited? Bump test? Bump at which location in which direction? Flow? Something else?
2 - Is the base truly acting rigid? i.e. zero vibration of the base?

To find the acoustic resonance, you would need to know speed of sound in air (1100 ft/sec). It is not necessary to know any speed of sound to calculate the lateral, longitudinal and torsional resonances. In some solution methods if you consider the vibration to be a wave or standing wave, there is an associated wavespeed in the material, but it's not a material property (depends on geometry).

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[Pipedreamers]

electricpete,

1. I am using a piece of hard plastic and strike the tube
on it side, somewhere half way in the middle of the tube length.

I experimented with different 'impact' locations along the
length of the tube but it seem to make not a big difference
in terms of spectrum peaks. Once I strike it, I get a quite
long lasting ping, I'd estimate about 1 second or even longer.

I was wondering, if this is a longitudinal resonance or a
breathing resonance over the diameter of the tube.

Or perhaps both ?

2. Well I tried different options.

a) The tube hung with a rubber band.

b) The tube clamped in a small vise at one end.

c) The tube held in the hand at one end.

Also, the spectrum peaks seem not to have moved much, perhaps
a couple hertz but I am not sure as I believe my
measurement setup may not be that precise. I am using
a sound card.
At most it seems more dampened as the ping does not last as
long compared with a 'free' tube.

I am not looking for the acoustical resonance as I believe this is not applicable.


Thanks

 

[ccw]

Another source: There is a ton of wind-chime do-it-yourself design aid sources on the internet. Just do a google search.

 

[sreid]

Wouldn't wind chimes have Free-Free vibration modes?

 

[sreid]

The Bible for Natural Frequencies

http://www.amazon.com/Formulas-Natural-Frequency-Mode-Shape/dp/1575241846

 

[ccw]

@sreid "...Wouldn't wind chimes have Free-Free vibration modes?"

Yes it would, but he said "free and one end rigid". I took that to mean that one of the cases he was attempting to analyze was free-free.

 

[electricpete]

Ah good catch. My initial post where I provided the lateral frequencies were fixed free. Maybe the O.P. can clarify the configuration tested.

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[mtnengr]

The vibration of cylindrical shells has been exhaustively studied. One of the better collection of data for shells of various shapes and boundary conditions is by A.W. Leissa, "Vibration of Shells", NASA SP-288, 1973. And its free. Leissa provides summaries of shell responses, particularly cylindrical shells.

There a number of formulae for the determination of natural frequencies presented by Leissa using different shell theories under different assumptions of type of responses (in-extensional, extensional and bending, etc.), boundary conditions and then compares them to one another. You can program these formulae and arrive at a solution to your problem.

There are a number of qualified shell programs (based on finite difference, finite element and numerical integration techniques) that also give one the ability to analyze any shell configuration.(see

http://www.volcano.net/~d.citerley)

The first eight entries under the software tag will give you what you need.

If you are using modal analysis gear to determine the experimental values, you need to very careful about the boundary conditions and where you are impacitng the shell. You can exite many modes with a single strike of a hammer. It is not like you are exciting one dominant mode like a shaker system can. Also, the magnitude of the impact impulse could be so extreme that you could excite a non-linear behaviour--especially with 301 stainless steel.

One vibration component often forgotten are the acoustic modes being excited.

 

[Pipedreamers]

Thanks everybody for trying to help me with this.

I am in particular interested in the fixed-free vibration mode.

Is this equation as I had posted above the correct one to
calculate this mode ?

f = an * sqrt{E*(pi*(OD^4-ID^4)/64)/(MU*L^4)}/2*pi

Thanks

 

[GregLocock]

Do you mean the cantilever mode of a beam?

Cheers

Greg Locock

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

 

[sreid]

For a cantilever, Blevins gives

fi = (ani^2/2piL^2)(sqrt(EI/M))

M= Mass per unit length

I for a tube = pi(OD^4-ID^4)/4

an1=1.875
an2=4.694
an3=7.854
an4=10.995

 

[electricpete]

Your ani^2 corresponds to my an
Your L^2 outside the radical corresponds to my L^4 inside

Our equations are the same except for the I. You have 4 in the denominator and I have 64

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[sreid]

My Bad; instead of OD and ID, the equation shuld have used radii (OD/2 and ID/2). Then our equations match.