# a question about vibration of plates having variable thickness

Hi everybody!

for plates with constant thickness and two opposite sides simply-supported B.Cs, we can write w(x,y)=f(y)*sin(m.Pi.x) and solving the problem from this viewpoin.

I wonder whether this approach is true for plates with (linearly) variable thickness?

many thanks

### Re: Vibration of Plates having Variable Thickness

Hi boreyri,

In Finite Elements the Displacement w(x,y) is chosen as a Polynomial function by using Pascal's Triangle.

Thus in your case we can use the Interpolation functions to define

the variation of your Linearly (or other variation) Variable Thickness before calculating the

Stiffness Matrix of the Plate Bending Element.

Is your approximation similar to this analysis ?

Sincerely

Mohammed lamine

### Unfortunately, my knowledge

Unfortunately, my knowledge on FEM is very shallow. functions we use to approximate the function w(x,y) must satisfy boundary conditions. therefore we choose for example sin(Pi.x) for simply-supported boundary conditions along x-axis. It is seems these approaches (FEM & semi-analytical) are a little different. I found two old papers "Forced vibrations of a non-uniform thickness rectangular plate with two free sides 1979" and "A semi-analytic solution for free vibration of rectangular plates 1978" in which w(x,y)=f(y)sin(Pi.x) for plates with variable thicknesses. It seems we can use this approach based on these two papers. many thanks Mohammed

### Why you don't use

Why you don't use a combination of Sin(m.Pi.x).Sin(n.Pi.y) which is a logical Choice in Vibration Analysis. Don't neglect the effect of Symmetric, Anti-symmetric modes,.....in all directions.

Your analysis is similar to Weighted Residuals which are Approximation Methods. In this case you need to Know the Differential Equation of Motion of your Plate, Choose the Trial (or approximation) Function and then Minimize the Integral of the Weighted Residual in the Analysed Domain. The Solution may Converge to the Exact in some Applications. It Depends on the Important Choice of the Trial Function.

mohammed lamine Moussaoui 