I would like to model the vibration of a thin, square sheet using this equation: $$D\nabla^4w(x,y,t)+\rho h\frac{\partial^2{w}}{\partial{t}^2}=f(x,y,t)$$ where $\nabla^4$ is the biharmonic operator and $f$ is a given driving function. However, I would like to model it so that the boundaries are not free nor fixed but somewhere in between. The square sheet attaches to the encasing material that doesn't fully clamp vibrations, but damps and limits the amplitude of them.
How would I go about expressing this kind of boundary condition?
What you can do to get around this problem is model your plate using free-free boundary conditions, and then if (when) you get to testing, you can support the plate using soft springs, such as rubber bands.
You can model the plate as a seismic system. If the excitation at the attachment points is harmonic and has an amplitude of $Y$ we can express the transmissibility using the following transfer function. (I can derive the transfer function if necessary)
$\left|\frac{X}{Y}\right|=\left(\frac{1+(2\zeta\Omega)^2)}{(1-\Omega)^2+(2\zeta\Omega)^2}\right)^{1/2}$
Where $\Omega$ is non-dimensionalized forcing frequency that is normalized with respect to natural frequency.
We can see that $\lim\limits_{\Omega\to \infty}\left|\frac{Y}{X}\right|=0$, so to minimize transmissibility we want to maximize $\Omega$. We can do this by minimizing $\omega_{n}$, and since
$\omega_{n} \equiv \sqrt{\frac{k}{m}}$,
we can see that reducing k will reduce our transmissibility.
This system can also be termed a "vibration isolator". Note that this is significantly different from a tuned mass damper.
It is nearly impossible to get perfect clamped boundary conditions in vibrating systems, so this method of vibration isolation and using free boundary conditions is a common work around.