The von Karman Plate Model

One form of the so-called dynamic analogue of the system of von Karman [213], [151] is given by the system

$$\begin{eqnarray} \rho H \ddot{u} &=& -D\Delta\Delta u+{\mathca... ...i,u]\\ \Delta\Delta \phi &=& -\frac{EH}{2}{\mathcal L}[u,u] \end{eqnarray}$$ (11.1a)

Here $u(x,y,t)$ is the transverse plate deflection and $\phi(x,y,t)$ is the Airy stress function, both defined over region $(x,y)\in {\mathcal A}\subset {\mathbb{R}}^2$, and for time $t\geq 0$. $\Delta$, as before, is the Laplacian, and $\Delta\Delta$ the biharmonic operator. $E$, $H$, and $\rho$ are as defined for the Berger system in §11.1. The nonlinear operator ${\mathcal L}[\cdot,\cdot]$ is defined in Section 11.2.1. The system is initialized using the values $u(x,y,0)$ and $\dot{u}(x,y,0)$.

It is worth noting that there are several variants of the von Karman system; that given above is simplified from the so-called ``full" or ``complete" system, in which in-plane displacements appear explicitly [128], [151]; all such systems may themselves be derived from even more general forms [247]. Equation (11.1a) reduces, in the absence of the term involving the bracket operator, to the linear Kirchhoff model of plate vibration [100]; the Berger model also may be arrived at, under somewhat more subtle assumptions. All the analysis presented in this article extends, with ease, to the case in which a linear damping term is included--conserved quantities become dissipated, but all the resulting stability analysis remains virtually unchanged. This has been discussed earlier by this author, in the case of nonlinear strings [30], and is covered briefly in Section .