Dear all,
I am trying to find a way to find dispersion curves of plates using Rayleigh-Lamb frequency equations. I used a repetitive procedure by sweeping through the frequencies and inside it increasing the phase velocity incrementally to find the roots of the equation.
In each frequency, there is some roots of phase velocity. As symmetric and asymmetric equations are already separated it is easy to separate data points of the two different waves. However, at the end of the process we have just (W frequency , Cp phase velocity) datapoints and after plotting the result we can see the behavior of the curves to manually define the mode number.
Main question: One way of clustering the data to different modes is to sweep through the data and define the first Cp as the mode zero, the second one as 1st mode and so on.
1. What if we had a crossing between curves?
2. I got some identical roots derived from both symmetric and asymmetric equations. How can one distinguish these datapoints correctly for the type of wave they belong?
Many thanks in advance for your help.
Mohsen
Dear Dr Mohsen : lease take a look at the following useful links:
Article An alternative method for plotting dispersion curves
https://www.science.gov/topicpages/c/calculated+dispersion+curves
Chapter The Determination of Dispersion Curves from Measurements by ...
https://academic.oup.com/gji/article/173/2/519/728414
Regards--
Dear Mohsen,
You are using the Rayleigh-Lamb equations to obtain Lamb wave dispersion curves in isotropic plates. In this situation, the crossing of dispersion curves belonging to the same mode family, i.e., symmetric and antisymmetric modes, will never occur. This is possible only in anisotropic media, where the coupling of sagittal motion and shear horizontal motion occurs for wave propagation along other than the principal directions. Therefore, your routine for assigning the solutions to the different modes is sufficient in this case. I did it in the same way years ago. However, once you go to anisotropic media (and using the stiffness matrix method, for instance), you will face the crossing of modes belonging to the same mode family in the coupled case, and the simple mode-assignment method fails.
Your questions:
1. Basically, I see two methods. Firstly, you could check the mode shape for every found solution, namely the through-thickness displacement component along the propagation direction u1. As the mode order increases, more nodes (crossings through zero) will appear in u1. So you would calculate u1 at about 50 through-thickness positions x3, and count the number of nodes. This information would be used to assign the solutions to the corresponding mode. The second way is the one I applied in the Dispersion Calculator (DC) (download at: https://www.dlr.de/zlp/en/desktopdefault.aspx/tabid-14332/24874_read-61142/#/gallery/33485), namely dispersion curve tracing. Basically, you are tracing all dispersion curves one by one. The difference to sweeping through the full phase velocity range is that, based on the already found solutions on a dispersion curve, you make a guess in which range the next solution should be, and search only there. This is much faster also. It is possible though that the tracing algorithm jumps to a crossing mode (jumping mode problem). This problem can be reduced by excluding the already traced dispersion curves from the phase velocity sweep range. Another strategy is to apply mode-family specific boundary conditions, but this goes too far for now. My dispersion curve tracing algorithms are described in the DC user’s manual (to be accessed via the “Help”-button). Mike Lowe describes in his doctoral thesis the dispersion curve tracing algorithms, which he uses in the DISPERSE software (https://spiral.imperial.ac.uk/handle/10044/1/8011).
2. I suppose by finding identical roots from the symmetric and antisymmetric Rayleigh-Lamb equations you mean that the dispersion curves cross. The crossing of symmetric and antisymmetric mode dispersion curves is normal, and since you got your solutions from the respective symmetric and antisymmetric equation, the mode family is already determined. Of course, you are storing the solutions found from the two equations in different containers, so I don’t see your problem quite well. You might want to use the DC to validate your results (the DC in turn is validated by DISPERSE). You can also plot the through-thickness profiles mentioned above.
Best wishes,
Armin
Dear Mohsen,
The answers provided here may be helpful:
https://www.researchgate.net/post/How-to-calculate-Lamb-wave-dispersion-curves-with-a-complex-wavenumber-k
As you already mentioned, if the plate is perfectly symmetric, you can compute symmetric and antisymmetric modes separately and distinguish the modes at the crossing points by their symmetry properties. In the general case of non-symmetric (or anisotropic) plates, it can be arbitrarily difficult (if not impossible) to determine whether two dispersion curves cross or get very close and veer (also called 'osculation').
Generally, finding the roots of the dispersion equations is very tedious. It's much easier and faster to employ one of the methods that lead to an eigenvalue problem for the wavenumbers and mode shapes.
Dear Mohsen, you can have my thesis upon this matter. First of all, existence of both symmetric and asymmetric modes is observable in dispersion curves for higher frequencies depending on isotropic plate's thickness and material ( Sound velocity in material). Just in low or very low frequencies you have neat signals containing zero modes. Using phase velocity curves and dispersion curves simultaneously you can distinguish and track a specified mode. Using both wave number or stiffness matrix methods results in same curves to some extent. However, just by suppressing some sym or asym modes you can have more neat curves. I suggest symmetric actuation for forced suppressing asymmetric modes in low frequencies for experiments.
As usual, two neighbouring dispersion curves in anisotropic elastic cases veer, while in viscoelstic case cross may happen.
Article Accurate characterization of 3D dispersion curves and mode s...
Article Waves in a generally anisotropic viscoelastic composite lami...
These different cases can be ascertained by calculating dispersion curves in complex wavenumber and real frequency accurately .
Article A numerical algorithm to solve multivariate transcendental e...
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