I want to do a free vibration analysis for a plate. In this research, I want to use the Ritz method. For Ritz-shape functions, which are the best polynomial functions that give the most accurate answer?
If you have data for the vertical deformations of the plate under vibration you can get the best fitting polynomial using EXCEL regression tool, but it's accuracy will be limited because EXCEL tool can deal with polynomials up to X^6
For more accurate ones, you may use more advanced software such as MATLAB
dear Ahmed Ebid
I have no data. In the Ritz method for solving equations, we need shape functions that only need to satisfy the boundary conditions to obtain the displacement and natural frequencies. For example, polynomials of degree n, Chebyshev polynomials or trigonometric functions.
usually 1st mode shape is similar to static deflection shape, accordingly , for simple supported rectangle plate, you may use one of the following polynomials as shape function
Thank you dear Ahmed Ebid
Do you have an article about this polynomial?
you can find one of them in the link below
fenix.tecnico.ulisboa.pt/.../Shape_functions.pdf
and the other one in my derivation for static deflection of simple (beam/plate)under uniform load
Load = w
Shear = w (x - a/2)
Moment = w/2 (x^2 - ax), Moment = 0 at x=0, so integration constant =0
slope = w/2 (x^3/3 -ax^2/2 + a^3/12) slope = 0 at x = a/2
deflection = w/2 (x^4/12 - ax^3/6 + a^3 x/12 ) , def. = 0 at x=0, constant =0
deflection = w/24 ( x^4 -2 a x^3 + a^3 x)
shape function = ( x^4 -2 a x^3 + a^3 x) x ^m
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