Frequency analysis for cylindrical shell

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chlai
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Joined: Tue Jan 11, 2022 9:29 pm

Frequency analysis for cylindrical shell

Post by chlai »

Hi,

I am new in FreeCAD FEM. And I am doing a simple frequency analysis for the cylindrical shell as attached.
Cylindrical_Shell_Frequency_Analysis.FCStd
(254.58 KiB) Downloaded 86 times
The frequency result runs well with thickness t1 = 10 mm. I have some results from Ansys as a reference. The difference between the two is about 2~4% which is acceptable.
result_compare_t10.png
result_compare_t10.png (12.9 KiB) Viewed 4328 times
But I face a problem when the thickness t2 = 1 mm (t1/10). The result from Ansys shows the frequency is about 1/10 from t1 which is consistent with the theory. But in FreeCAD, the result is quite different.
result_compare_t1.png
result_compare_t1.png (14.13 KiB) Viewed 4328 times
I have tried to do a finer mesh and mesh with 1st order elements but the difference is still quite large.

Is there anything I could do to improve the results? Or am I missing anything? Any suggestion is welcome

Thanks in advance for your help

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Last edited by chlai on Mon Jan 17, 2022 9:48 am, edited 1 time in total.
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NewJoker
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Re: Frequency analysis for cylindrical shell

Post by NewJoker »

You said that you analyze two cases - with a thickness of 0.01 mm and 0.001 mm. But the file that you attached includes two analyses with a thickness of 10 mm and 1 mm. Maybe that’s the source of the problem.

What are these results actually - eigenvalues ? Wouldn’t it be easier to compare the frequencies?

Your model has only one constraint - the bottom edge is fixed in the Z direction. Also, the thicknesses are very low. Why did you choose such unusual settings for tests ? Does this model correspond to a real-life structure ?
chlai
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Re: Frequency analysis for cylindrical shell

Post by chlai »

Hi @NewJoker,

thanks for your reply :)
NewJoker wrote: Mon Jan 17, 2022 8:50 am You said that you analyze two cases - with a thickness of 0.01 mm and 0.001 mm. But the file that you attached includes two analyses with a thickness of 10 mm and 1 mm. Maybe that’s the source of the problem.
Oops... I wrote the wrong unit in my previous post :oops: both analysis use thickness 10 mm and 1 mm, so probably that's not the source of the problem
NewJoker wrote: Mon Jan 17, 2022 8:50 am What are these results actually - eigenvalues ? Wouldn’t it be easier to compare the frequencies?
Yes, it's frequencies. To be clear, I adjust the figures to better understand
NewJoker wrote: Mon Jan 17, 2022 8:50 am Your model has only one constraint - the bottom edge is fixed in the Z direction. Also, the thicknesses are very low. Why did you choose such unusual settings for tests ? Does this model correspond to a real-life structure ?
Hmmm... good question! I tend to use this simple structure to verify the frequency analysis of cylindrical shell model in FreeCAD. This structure settings is based on the Ansys file I have now, so that's not a real-life structure.

I tried a thicker one and got the result consistent with the theory in FreeCAD.
I guess there are some locking effect within the thin shell and am wondering if there is any settings or methods that can solve the problem in FreeCAD?
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NewJoker
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Re: Frequency analysis for cylindrical shell

Post by NewJoker »

I would definitely change this boundary condition to more realistic one. Preferably such that the results can be easily compared with analytical solution as well (references below).

Be careful when using triangular meshes generated by Gmsh or Netgen - their inaccuracies may sometimes lead to spurious mode shapes. This is even more problematic in the case of linear buckling analyses. Thus, you should also display each mode shape and make sure that it's correct.

Here are some books in which you will find analytical solutions:
- "Roark's Formulas for Stress and Strain" Budynas
- "Formulas for Natural Frequency and Mode Shape" Blevins
- "Formulas for Dynamics, Acoustics and Vibration" Blevins
thschrader
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Re: Frequency analysis for cylindrical shell

Post by thschrader »

NewJoker wrote: Mon Jan 17, 2022 12:53 pm I would definitely change this boundary condition to more realistic one.
+1
I ran the test-example from @chlai.
When using only z-fixings below, the first 3 Eigenmodes are missing.
I guess these are the rigid-body motions. The cylinder can xy-transverse
an rotate among the z-axis.
mac_the_bike
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Re: Frequency analysis for cylindrical shell

Post by mac_the_bike »

  • The first thing I would do is a "free-free" vibration analysis. "free-free" means no constraints on the model. This type of analysis can be performed even if the final analysis is going to be static as it will confirm that the model has:
    1. no constraints that absorb energy
    2. no mechanisms
  • I would then expect to see 6 Rigid Body Modes, RBM, these are modes that have "small" frequencies, or eigenvalues. It is possible that, due to rounding errors in the calculations, the first few modes could have negative eigenvalues. With version 18 of FreeCAD it is not possible to specify that the lowest eigenvalue is negative and as a consequence the first mode will always be non-negative. It is possible to tell that some modes have not been found because the output will have its first "Mode No" greater than 1, Note 1. (This is seen in both Pictures.)
  • To get the other RBMs the input file will have to be modified. After writing the input file, which will be something like FEMMeshGmsh.inp, look for the FREQUENCY card in the this file and change in to something like:
    *FREQUENCY
    10,-10.0,1000000.0

    where -10 is an arbitrary lower bound of the frequencies.
    and then run the analysis.
  • It is difficult to be exact about "small" but one criterion is that the ratio of the frequencies of the first non RBM to the last RBM mode should be above 1000 or something of this order.
  • One feature of a vibration analysis of a cylinder, which is completely axisymmetric, is that the non RBM modes come in pairs, i.e 2 modes will have the same frequency, see Note 3. The RBM modes will not be in pairs.
    In the figures there appear to be modes that are almost in pairs. This discrepancy can be assigned to at least the following:
    1. the mesh is not the same over the surface. If you look at the TDC of the mesh you will a line of elements - this pattern doesn't appear anywhere else.
    2. the nodes don't lie strictly on a cylinder, see Note 4.
  • Looking at the two analyses, because of the constraint that has been applied, it appears that the 4th mode is the first non RBM. This is what thschrader said. The modes are in pairs.
    It appears that the two sets of results are sensibly the same, if you take into account Note 2 and potential differences of element formulation.
  • Note 1: Calculation of the Sturm Sequence based on a shifted mass matrix can be used to calculate the number of modes below the shifted eigenvalue.
  • Note 2: I am also questioning the FreeCAD frequency values, of the second analysis, because if you divide the FreeCAD frequencies by PI the new numbers are close to the Ansys values.
  • Note 3: The so-called Fourier N=0 frequencies will be single. Examples of these are the axial extension and torsional modes.
  • Note 4: There are certain occasions that require the coordinates of the nodes to be specified to a high precision; this is one of them.
    Suggestion for bernd: there should be an option for the output of the coordinates in "double precision".
mac
davidosterberg
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Re: Frequency analysis for cylindrical shell

Post by davidosterberg »

chlai wrote: Mon Jan 17, 2022 10:37 am I tried a thicker one and got the result consistent with the theory in FreeCAD.
I guess there are some locking effect within the thin shell and am wondering if there is any settings or methods that can solve the problem in FreeCAD?
I think you are right. If I remember correctly Calculix has an unusual way that it implements shells. Instead of being "true" shell elements, the shells are actually expanded to one layer of 2nd order solid elements. Guido showed that this works reasonably well for his test cases. But for extreme aspect ratio I could expect the accuracy to go down.

Source:
The expansion to solid bricks is mentioned here. It is also mentioned that there are issues with extreme aspect ratio.
https://web.mit.edu/calculix_v2.7/Calcu ... ode39.html
thschrader
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Re: Frequency analysis for cylindrical shell

Post by thschrader »

Beam Eigenfrequencies, comparison volume elements 3D vs shells 2D vs theory
beam_eigenfrequencies_shell.FCStd
(10.57 KiB) Downloaded 58 times
beam_eigenfrequencies.FCStd
(8.71 KiB) Downloaded 59 times
beam_frequency.JPG
beam_frequency.JPG (140.53 KiB) Viewed 3952 times
chlai
Posts: 8
Joined: Tue Jan 11, 2022 9:29 pm

Re: Frequency analysis for cylindrical shell

Post by chlai »

NewJoker wrote: Mon Jan 17, 2022 12:53 pm I would definitely change this boundary condition to more realistic one. Preferably such that the results can be easily compared with analytical solution as well (references below).

Be careful when using triangular meshes generated by Gmsh or Netgen - their inaccuracies may sometimes lead to spurious mode shapes. This is even more problematic in the case of linear buckling analyses. Thus, you should also display each mode shape and make sure that it's correct.

Here are some books in which you will find analytical solutions:
- "Roark's Formulas for Stress and Strain" Budynas
- "Formulas for Natural Frequency and Mode Shape" Blevins
- "Formulas for Dynamics, Acoustics and Vibration" Blevins
Hi @NewJoker

Sorry for the late reply. It took me some time to read the references you gave. Many thanks for your advises and references. They help me a lot and I also find some interesting examples within them!

As for my initial problem, I solve it with quad mesh and the analysis settings as attached. I think the problem might come from the triangular meshes as you mentioned. I also recommend using "Packing Parallelograms" for algorithm2D when meshing, because it produces parallelogram elements which is similar to the shape used to calculate the analytical solution.
Cylindrical_Shell_Frequency_Analysis_correct.FCStd
(17.58 KiB) Downloaded 63 times
I also calculate the analytical solution (Rayleigh's method, see ref. below) to verify the results. The error is around 1% (except mode 2 of t=1mm, still finding the reason...)
Rayleigh_natural _frequencies_of_thin_cylindrical_shells.png
Rayleigh_natural _frequencies_of_thin_cylindrical_shells.png (36.68 KiB) Viewed 3923 times
result_compare_t10.png
result_compare_t10.png (16.22 KiB) Viewed 3923 times
result_compare_t1.png
result_compare_t1.png (16.15 KiB) Viewed 3923 times
The corresponding mode shapes are also quite similar to the theory.
mode_shape_compare.png
mode_shape_compare.png (281.08 KiB) Viewed 3923 times
Reference:
Rayleigh, J. W. S. B. (1894). The theory of sound (Chapter X)
Arnold, R. N., & Warburton, G. B. (1949). Flexural vibrations of the walls of thin cylindrical shells having freely supported ends.

If there is any question or suggestion, please feel free to contact me

Best regards
thschrader
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Re: Frequency analysis for cylindrical shell

Post by thschrader »

chlai wrote: Sat Jan 22, 2022 6:26 pm ...
Reference:
Rayleigh, J. W. S. B. (1894). The theory of sound (Chapter X)
Arnold, R. N., & Warburton, G. B. (1949). Flexural vibrations of the walls of thin cylindrical shells having freely supported ends.

If there is any question or suggestion, please feel free to contact me

Best regards
I like the Reference.
Our ancestors were not stupid.
Thomas
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