Vibrational modes of circular free plates under tension

TL;DR

This paper analyzes the vibrational modes of circular free plates under tension by solving the eigenvalue problem for the bi-Laplace equation, identifying fundamental modes with simple angular dependence.

Contribution

It provides explicit eigenfunctions and eigenvalues for free circular plates under tension, highlighting the structure of fundamental vibrational modes.

Findings

Eigenfunctions and eigenvalues explicitly determined

Fundamental modes have simple angular dependence

Provides insights into vibrational behavior of tensioned plates

Abstract

The vibrational frequencies of a plate under tension are given by the eigenvalues $\omega$ of the equation $\Delta^2u-\tau\Delta u=\omega u$. This paper determines the eigenfunctions and eigenvalues of this bi-Laplace problem on the ball under natural (free) boundary conditions. In particular, the fundamental modes --- the eigenfunctions of the lowest nonzero eigenvalue --- are identified and found to have simple angular dependence.