I've been trying to find the boundary conditions for a fourth order DE equation for a bending circular plate. I'm assuming angular symmetry and an even pressure distribution.

Displacement Equation

$$ D \nabla^2\nabla^2w(x,y,t) + h\rho \frac{\partial^2{w}}{\partial{t}^2}(x,y,t) = P(t)$$

D: Flexural Rigidity (Constant)

h: Thickness

$\rho$: Density

P(t): Pressure (Evenly applied across diaphragm)

Conversion to Polar Coordinates

$$ \nabla^2\nabla^2w(x,y,t) = \frac{1}{r^3}\frac{\partial{w}}{\partial{r}} - \frac{1}{r^2}\frac{\partial^2{w}}{\partial{r}^2} + \frac{2}{r}\frac{\partial^3{w}}{\partial{r}^3} + \frac{\partial^4{w}}{\partial{r}^4}$$

Now I need to solve

$$D(\frac{1}{r^3}\frac{\partial{w}}{\partial{r}} - \frac{1}{r^2}\frac{\partial^2{w}}{\partial{r}^2} + \frac{2}{r}\frac{\partial^3{w}}{\partial{r}^3} + \frac{\partial^4{w}}{\partial{r}^4}) + h\rho\frac{\partial^2{w}}{\partial{t}^2}(r,t) = P(t)$$

The boundary conditions I currently have

$\frac{\partial{w}}{\partial{t}}(r=a,t) = 0$

$w(r=a,t) = 0$

$\frac{\partial{w}}{\partial{t}}(r=0,t) = 0$

My goal is to discretize it with N divisions and transform the system into State Space so that I can use P(t) as a controller as well as create a Bode Plot and look at the system's resonance frequencies. I believe that I need one additional boundary condition due to the spacial derivative being of order 4.

I also plan on using central and forward/backward finite differencing of order 4 to replace the spacial differences. Possibly order 2 because that would simplify the problem.


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