Characterization of Vibrating Plates by Bi-Laplacian Eigenvalue Problems

TL;DR

This paper develops boundary integral identities for bi-Laplacian eigenvalue problems, establishing uniqueness, eigenvalue monotonicity with Poisson's ratio, and integral representations of strain energies for vibrating plates under various boundary conditions.

Contribution

It introduces new boundary integral identities and proves eigenvalue monotonicity with Poisson's ratio for simply-supported vibrating plates.

Findings

Eigenvalues increase strictly with Poisson's ratio for simply-supported plates.

Boundary integral identities are derived for different boundary conditions.

Strain energy representations are obtained for vibrating plates.

Abstract

In this paper we derive boundary integral identities for the bi-Laplacian eigenvalue problems under Dirichlet, Navier and simply-supported boundary conditions. By using these identities, we first obtain the uniqueness criteria for the solutions of the bi-Laplacian eigenvalue problems, and then prove that each eigenvalue of the problem with simply-supported boundary condition increases strictly with Poisson's ratio, thereby showing that each natural frequency of a simply-supported vibrating plate increases strictly with Poisson's ratio. In addition, we obtain boundary integral representations for the strain energies of the vibrating plates under the three boundary conditions.