The (small) vibrations of thin plates

TL;DR

This paper develops a numerical scheme to analyze small vibrations of thin elastic bodies using discretized vector fields, successfully approximating resonance modes consistent with experimental data.

Contribution

It introduces a novel discretization method leveraging algebraic topology for modeling small vibrations of thin plates, improving numerical approximation accuracy.

Findings

Numerical approximations of resonance modes match experimental results.

The scheme effectively models thin bodies with one dimension significantly smaller.

The approach exploits topological properties of the body for better solutions.

Abstract

We describe the equations of motion of elastodynamic bounded bodies in 3-space, and their linearizations at a stationary point. Using the latter as an approximation to model small motions, we develop a scheme to find numerical solutions of these equations. We discretize the solution in the space of PL vector fields associated to the oriented faces of the first barycentric subdivision of a given smooth initial triangulation of the body, in order to exploit the algebraic topology properties of the body that these vector fields encode into the sought after solution, and solve a weak version of the linearized equations in that context. We apply our scheme to a couple of relevant examples of thin bodies, bodies where one of the dimensions is at least one order of magnitude in size less than the other two, and determine numerical approximations to some of their resonance modes of vibration. The results obtained are consistent with known vibration patterns for these bodies derived experimentally.