vibration in hollow cylinder

[geovibe]

Hi there,

I'm a petroleum engineer in France and am looking for a good analytical model for vibrations in hollow cylinder simply supported.

First I tried to consider the cylinder as a beam and use the blevins formulas to compute natural frequencies:

f = [A(2*pi*L^2)]*sqrt(E*I/m)

where:

A= 9.87 for first mode
L= length of beam (m)
E= modulus of elasticity (N/m^2 = kg/(m-s^2)
I= area moment of inertia (m^4)
m= mass per unit length of beam (kg/m)

But the results are not close to the Finite Element Method (30% less).

So I tried a model taking in account the circumferential modes valid in the case of very thin shell. Results were closer to the FEM but still 10% less.

Questions :

Does somebody know a good analytical model of free vibrations in hollow cylinder ?

Same question when it is full of homogenous fluid ?

A formula to compute the area moment of inertia when the cylinder is full of water ?


Thanking in advance.

 

[GregLocock]

I doubt that Blevins is wrong. Does your system fit the assumptions for a beam theory solution? ie, is it relatively slender.

To get the frequency in that formula with the tube full of water, just bump up the m to suit. More generally, if you are just changing the mass in a simple system the frequencies will just vary by 1/sqrt(m)

If you give the relevant properties I could check it out- typically FEA models do come out too stiff, 60% is a bit much.



Cheers

Greg Locock

 

[geovibe]

Thanks Greg,

Indeed, I think that the cylinder is not enough slender to be considered as a beam (L=880mm, Diam ext.=116mm, Thickness=17).

With a E= 2E11 N/m^2, I= 6.87 E-4(m^4)
m= 43.22 (kg/m), I found 3260Hz for the third mode of natural frequency altough the FEA gives 4560Hz (which I didn't check).

Questions :

Does somebody know a good analytical model of free vibrations in hollow cylinder ?

Same question when it is full of homogenous fluid ?

Is there a change in the formula of the area moment of inertia when the cylinder is full of water ?


Thanking in advance.

 

[GregLocock]

Beam theory should be fine for that.

I get 3595 Hz using your figures in the corrected version of your formula in the first post, it should be A/(2*pi*L^2)*sqrt(E*I/m) I think.

However, your value for I is way wrong, it is pi/64*(d^4-di^4)

so the correct value is 354 Hz for the first mode

Are you looking at first mode or third mode? You need to get the first flexural mode (half wave bending) right before looking at the high frequency stuff.





Cheers

Greg Locock

 

[geovibe]

Sorry for the mistakes, indeed:
fn=A/(2*pi*L^2)*sqrt(E*I/m)
I= 6.87 E-6(m^4)
thickness= 18 mm
In this case, f1=361Hz, f2=1448Hz and f3=3259Hz;

The only result I know from FEA is f3=4560Hz.

I doubt that the beam vibration theory is reliable in this case and that is why I'm looking for other ones.

Cheers

 

[Bbird]

Geovibe,

I think you are looking for a theoretical solution finer with your simplified physical model.

All theoretical solutions should be regarded as approximations only because we seldom manage to replicate the theoretical behaviour in practice.

The simply supported condition will be difficult to achieve as you must have cylinder length beyond the supporting point.

I would suggest the 1st mode is the most important one to bear in mind as it is the easiest to be excited and set in motion. The frequency would go down immediately once the pipe is full of liquid. GreyLolock's suggestion of simply adding the mass in the calculation is correct as the fluid is not expected to contribute stiffness.

The beam theory should be used as a reference point. When the cylinder diameter is large relative to the length then the circular shape will suffer deformation during the vibration, resulting a reduction in frequency magnitudes which you can interpret as a loss of the original stiffness. The cylinder I-value in such case is different between the at-rest position and during movement. To overcome this short coming you will need to model the cylinder as a 3D shell elements. The 3D model should return the frequencies no higher than those from the beam theory. The two should not be radically different unless your cylinder is really flexible and cannot hold its original shape when vibrating, in which case the introduction of the fluid should probably cause a collapse.

So don't throw away the solution from the beam theory yet.

 

[GregLocock]

If you ask me, if you are trying to predict 4 kHz modes for a steel system a metre long then neither Blevins nor FEA is likely to be right.

Your FEA result may be able to get close, but you will need something like 1-5mm elements. Luckily by then your boundary conditions won't matter too much.



Cheers

Greg Locock

 

[geovibe]

Greg and Bbird,

First I want to thank you for all your explanations.

I am exactly trying to predict the third natural mode of a one meter long steel cylinder.
The idea is to build an analytical model to understand the impact on the frequential responses of each parameter:boundary conditions, axial load, temperature, fluid pressure, fluide flow...

But if you tell me that neither Blevins nor FEA can approach the experimental result for high frequencies, it is interesting. Maybe as Bbird says, I should focus on the first mode.

Thanks again.

 

[MikeyP]

I've messed around quite a bit with various models of cylinders in the past. The best analytical solution for your case would probably be Flugge shell theory. This is covered in Blevins I think (It certainly is covered in Flugge's book "Stresses in Shells&quot. Your shell is quite "thick" compared with the curvature so a simpler theory such as Sanders-Koiter or Donnel will not be sufficient.

The dimensions you quote mean that you are on the cusp between shell and beam theory. An easy way to tell is to do an eigenvalue analysis on the FE model (say 40 nodes long by 40m nodes around the circumference). If all the modes in your frequency range are "beam like" then us a beam theory. If not then use a shell theory. Make sure you use "thick" shell elements in your FE model.

If you find that beam theory is OK and there are no circumferential modes in your frequency range, then be sure to use Timoshenko beam theory rather than Euler-Bernouilli. A cylinder cross section has its mass concentrated at the edges and so the rotatory inertia in bending is high and must be included. Also your beam is quite "short" compared with its cross-sectional dimension and may well deform as much in shear as in bending so shear deformation effects should also be included. Most FE packages should have Timoshenko beam elements available which will deal with both the shear deformation and rotatory inertia issues. Both of these effects serve to bring the natural frequency of the first few modes down.

Finally, If you end up using a shell theory and enter the murky world of cylinder dynamics, be prepared for some very strange looking results. Counterintuitively, the modeshapes which "look" the simplest may not be the lowest in frequency! eg a mode with 5 circumferential wavelengths may be lower in frequency than one with 2! Another complication: for each individual pattern of axial and circumferential wavelengths there are three different modes of different freqencies!

Have fun.

M

 

[NicolasJobert]

Hello,

here is my two cents on the subject, namely how to take into account the fluid contained into the cylinder.

Pure mass addition should yield correct results as long as dilatationnal (acoustic) waves in the fluid have a wavelength much longer than the structural wavelength: This is for the "beam" modes, and should indeed work nicely.

If you are interested in having the net effect of the fluide on "circumferential" modes, you may experience very various (and sometimes deceiving) phenomenons, depending on the wavenumbers: for even wavenumbers, you will see only mass effects. For odd wavenumbers, you will also experience extra-stiffness because of fluid incompressibility. The criteria here is the following 'does the vibration pattern result in a net variation of the cylinder section ?'.

Beware, damping will increase (dramatically !).

Cheers.

Nicolas

BTW, there is an ANSYS Tutorial (the one on Fluid Structure Analysis) that deals with the subject and compares numerical and analytical solutions. I believe it is exactly what you are searching for. As far as I remener, it was a rev 5.0 tutorial so it may be as old as 1992 !

 

[dylan1450]

Hi, just some further pointers; the Engineering Sciences Data Unit (ESDU) produces a series of data sheets dealing with Vibration and Acoustic Fatigue - there are data items on the natural frequencies of shell structures. Also Bob Blevins's book "Flow Induced Vibration" has a chapter on "Vibrations of a Pipe Containing a Fluid Flow," which may be of some use to you.