How to calculate theoretically Rayleigh damping constants alpha and beta for a Cantilever Beam( length = 100m) with 2 elements ( each element length 50m) each having different material properties.?
Dear Reddy:
You may calculate in this way:
1. Build the FEA model for the Beam, obtain mass matrix M and stiffness matrix K.
2. Perform modal analysis with M and K. You'll have the natural frequency of the beam w1,w2,w3...
3. Select two frequency points (You may select those points with refering to the natural frequency of the beam) wa and wb(Eg. wa=w1 and wb=w3). Assign two damping ratio values da and db to the two points.
4. The constants alpha and beta can be obtained by using attached formulas.
Damping mechanism is still not very clear until now. This is just a crude approximation.
Best regards,
Bowei
In most cases you cannot calculate them theoretically. They are phenomelogical constants that can be used to match experimentally seen damping behavior. In that case you might use the approach described by the previous responder.
Maybe you have an andvanced numerical method to determine damping (e.g. cfd for viscous flow damping) in that case you can use numerical experiments in stead of real experiments to determine damping in a cantilever beam. And use the simpler model with Rayleigh damping for the bulk of the simulations.
First, you must specify some subcritical modal damping ratios. This is the part of the process with the greatest uncertainty. Ideally, these modal damping ratios should come from experimental data measured on the actual structure. But in the absence of such data, you might just have to guess some values based on experience with similar structures, and/or data from the literature, and/or intuition, etc.
If you have only two such modal damping ratios and the associated modal natural frequencies, then you can apply the method described by Bowei Li to calculate the Rayleigh constants. However, if you have more than two sets of data, there are also more general calculation procedures available, some of which are described in the book "Dynamics of Structures" by Ray W. Clough and Joseph Penzien, 1975, pages 194-199.
Answers above are very good.
I would like to add one more thing for higher damping materials, like rubber, in anti-vibration applications, I would suggest to use NFR (Natural Frequency Region) method based on Rayleigh damping to obtain accurate dynamic response.
Please see my paper “Article title: An energy dissipation approach on complete loading-unloading and dynamic impact predictions with experimental verification for rubber anti-vibration component”.
The following link can get the article free until 21 October 2017 from Polymer Testing.
https://authors.elsevier.com/ a/1VeviWW4yFehN
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