I am working on the biharmonic eigenvalue problem
∇4Φ=λΦ, λ∈R
on a rectangular and circular domain. The eigenfunctions Φ
correspond to the standing waves on a thin plate submitted to vibration and λ is related to the frequency of the vibrating source. I developed a code that finds the eigenfunction as well as the eigenvalues of the biharmonic operator in rectangular and circular domain. I would like to use this code for other applications. That means, I would like to find other physical problems where the biharmonic operator or its eigenvalue problem (or a modified version) appears.
Hello, Alejandro.
I can call the next cases:
The biharmonic equation is used in bending theory of linear elastic plates of Kirchhoff.
Also biharmonic equation determines Airy stress function in the plane problem of linear elasticity theory. But I did not see the usage of it in dynamics, so the equation is always static without time derivatives, only biharmonic operator.
Dear Alejandro, a second situation is the stationary Stokes equation, in which the stream function satisfies a biharmonic equation. I have no idea of what an eigenvalue problem would mean in this case, however.
Also solution of any 3-d linear elastostatic problem in the absence of body forces is a biharmonic vectorial function (Galerkin representation).
Dear Alejandro Daniel Paredes Cabrel , here are some models that biharmonic operator appears:
- Stommel model and the Stommel–Munk model from wind-driven ocean circulation
- Extended Fisher–Kolmogorov (EFK) equation in population genetics, traveling waves in reaction–diffusion systems, etc.
- Berger equation for the deflection of the thin elastic plate
Dear Alejandro Daniel Paredes Cabrel
The biharmonic equation is used in bending theory of linear elastic plates of Kirchhoff.
Also biharmonic equation determines Airy stress function in the plane problem of linear elasticity theory. But I did not see the usage of it in dynamics, so the equation is always static without time derivatives, only biharmonic operator.
Hello. Who knows how to calculate analytically eigenmodes of a biharmonic operator in Cartesian coordinates for the free edge rectangular plate?
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