Rayleigh Damping Coefficients of a plate

[ghazwansaud]

Hello All;

I am working in the vibration analysis field , both theoretical and practical fields.

while working in the theoretical side I needed the Rayleigh Damping Coefficients of a vibrating plate.... I need to calculate (Alpha) and (Beta) of the following equation:

[C]=Beta*[K]+Alpha*[M]

I found in the literatures a method depending on the first natural frequencies from the equation :

Zeta=1/2(Alpha/Omega+Beta*Omega)

the problem is I do not have the (Zeta) to solve the above equtaion simultaneously using two Omega's.

how to calculate the Alpha and Beta ??

Thanks all in advance

 

[MikeyP]

What are the boundary conditions of the plate?
What is the material of the plate?
Is it just a single flat piece of material or is it made up of several components bonded together?

These might get you a little closer to an answer.

M

--
Dr Michael F Platten

 

[ghazwansaud]

Hello Dr Michael ,

Thank you for your response.

I have three boundary conditions (4-edge clamped, 3-edge clamped, 2-edge clamped).

The material is basically Steel (Poison's ratio=0.3,E=200Gpa).

The plate is simple thin plate only.

Is it any way to calculate the constants using the formula above? of course the natural frequincies are known.

Thanks

 

[MikeyP]

Damping is largely generated at the boundary for those type of structures so the clamping mechanism is very important. There will be some inherent damping of the plate (due to radiation and heating) but it will be less than 0.1% critical.
Depending on the clamping arrangement, the damping would probably be in the range 1% to 5% critical. You can use your equation to find a rayleigh model that gives you say 2.5% damping across the frequency range of interest.

This will always be an approximation. Real damping is pretty much impossible to predict and for accurate models measured values are usually used.

M

--
Dr Michael F Platten

 

[ghazwansaud]

Thank you So much prof.

I do understand now.

one more question please. Is there any reference talking about this issue ??? maybe I can find some tables or figures concerning this problem.

Best
Ghazwan

 

[hacksaw]

When it is used, the relative damping zeta is specified (1-5%) then alpha and beta are calculated using the modal frequencies.the text by C. Spyrakos provides an excellent description.

 

[ghazwansaud]

Hello Hacksaw,

Thank you for the information.

Do you mean the text "Finite Element Modeling: In Engineering Practice" For Constantine C. Spyrakos ???

because i did not find any thing else for him in books.google.com

Thank you

 

[hacksaw]

that's the book,

page 265 covers a sample calculation although there is a more comprehensive discussion of the coefficients on pages 151-152

 

[ghazwansaud]

Thank you,

To be honest I could not get the book !!

Is there any other reference?

 

[Vibac]

Hi ghazwansaud

Also, try googling the paper "Computation of Rayleigh Damping Coefficients for Large Systems" by Chowdhury and Dasgupta. I havn't had to use it myself, but I guess it might be helpful to you.

 

[hacksaw]

basically you have the natural frequencies and two unknowns.

pick two criticals, for example the lowest two and calculate alpha and beta for a given zeta.

 

[ghazwansaud]

@ Vibac

I already downloaded the paper , it is really useful.

thank you

 

[ghazwansaud]

@ Hacksaw

Thank you again

your tips reflects your background of knowledge.

I wil do it as you said.