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

Thank you dear Masoud Mohammadi .