Solving rectangular plates vibrations wave equation

Question

I am a new Mathematica user (about 6 month) and I am trying to solve the wave equation for a rectangulare plate (not an elastic mebrane). I read some papers about this subject and my code structure is inspired by the 1D wave equation code found in the Wolfram's help sheets. Unfortunately, I have got the following message : Equation or list of equations expected instead of True in the first argument. Here is the theory I used to write my code :

ClearAll["Global`*"];
Ey = 2.078 10^11 ;(* N/m^2 *)
\[Nu] = 0.317756 ;(* unitless *)
\[CurlyRho]d = 8166; (* kg/m^3 *)
h = 10^-3;
Df = (Ey h^3)/(12 (1 - \[Nu]^2));
eqn = {D[w[x, y, t], {x, 4}] + 2 D[w[x, y, t], {x, 2}, {y, 2}] + 
    D[w[x, y, t], {y, 4}] + (\[CurlyRho]d h)/
     Df D[w[x, y, t], {t, 2}] == 0}
(* Initial conditions *)
ic = {w[x, y, 0] == Sin[m x ] Sin[n y], 
   D[w[x, y, 0], {t, 1}] == 
    0}; (* Sin[mx]Sin[n y] plate's initial vibration is sinusoidal \
waves in x and y *)
(* m and n are integers *)
(* Boundary conditions *)
bc1 = {Derivative[2, 0, 0][w][0, y, t] == 0, 
   Derivative[3, 0, 0][w][0, y, t] == 0, 
   Derivative[2, 0, 0][w][a, y, t] == 0, 
   Derivative[3, 0, 0][w][a, y, t] == 0};
bc2 = {Derivative[0, 2, 0][w][x, 0, t] == 0, 
   Derivative[x, 3, 0][w][0, y, t] == 0, 
   Derivative[0, 2, 0][w][x, b, t] == 0, 
   Derivative[0, 3, 0][w][x, b, t] == 0};
symbsol = 
 DSolve[{eqn, bc1, bc2, ic}, w, {x, y, t}, 
  Assumptions -> Element[{m, n}, Integers]]

Answer 1

In ic, the expression D[w[x,y,0],{t,1}]==0 is simply True because w[x,y,0] is not a function of t. You need to take the derivative w.r.t. t first, and then evaluate at 0, like so

(D[w[x,y,t],{t,1}]/.t->0)==0

In bc2, you're taking the $x$th derivative w.r.t. 0, presumably that should be the 0th derivative w.r.t. $x$.

Alas, Mathematica doesn't find a solution when I try with these changes. Trying an NDSolve with particular values for a, b might be helpful. I'm not the best at pde solving, but I might take some time to coax Mathematica into a solution (I thought rectangular plates had a solution in Fourier series).

Answer 2

Try this. Used Derivative instead. Also Rationlizing numbers helps a little. But DSolve still can't solve it.

ClearAll["Global`*"];
Ey = SetPrecision[2.078*10^11, Infinity];(*N/m^2*)
ν = Rationalize[0.317756];(*unitless*)
ϱd = 8166;(*kg/m^3*)
h =1/1000;
Df = (Ey h^3)/(12 (1 - ν^2));
eqn = {D[w[x, y, t], {x, 4}] + 2 D[w[x, y, t], {x, 2}, {y, 2}] + 
     D[w[x, y, t], {y, 4}] + (ϱd h)/Df D[w[x, y, t], {t, 2}] == 0};
ic = {w[x, y, 0] == Sin[m x] Sin[n y], Derivative[0, 0, 1][w][x, y, 0] == 0};
(Sin[m x]Sin[n y] plate's initial vibration is sinusoidal waves in x 

and y)(m and n are integers)(BC=plate is fully simply supported)
bc1 = {w[0, y, t] == 0, 
       Derivative[2, 0, 0][w][0, y, t] == 0, 
       w[a, y, t] == 0, 
       Derivative[2, 0, 0][w][a, y, t] == 0};
bc2 = {w[x, 0, t] == 0, 
       Derivative[0, 2, 0][w][x, 0, t] == 0, 
       w[x, b, t] == 0, 
       Derivative[0, 2, 0][w][x, b, t] == 0};
symbsol = DSolve[{eqn, ic, bc1, bc2}, w, {x, y, t}]

But DSolve can't solve it.

May be because you still have inconsistent B.C./I.C. ? or may because DSolve does not have template for it.

NDSolve solves it, but gives the above warning.

a = 1; b = 1; m = 1; n = 2;
numsol = NDSolve[{eqn, ic, bc1, bc2}, w, {x, 0, a}, {y, 0, b}, {t, 0, 1}, 
  Method -> {"StiffnessSwitching"}, MaxSteps -> 10^6]

 Plot3D[Evaluate[w[x, y, 0] /. numsol], {x, 0, a}, {y, 0, b}]

May be more refined NDSolve options are needed.