# A boundary integral equation method for mode elimination and vibration confinement in thin plates with clamped points

## TL;DR

This paper introduces a high-order boundary integral equation method to analyze and manipulate vibrational modes in thin elastic plates with clamped points, enabling eigenvalue elimination and mode confinement for improved control.

## Contribution

A novel boundary integral equation approach for efficiently computing and controlling vibrational modes in plates with discrete clamped points, including eigenvalue elimination and mode confinement strategies.

## Key findings

- Careful placement of clamped points can eliminate specific eigenvalues.
- Clamping can partition the domain, confining vibrational modes.
- The method is effective for various geometries and defects.

## Abstract

We consider the bi-Laplacian eigenvalue problem for the modes of vibration of a thin elastic plate with a discrete set of clamped points. A high-order boundary integral equation method is developed for efficient numerical determination of these modes in the presence of multiple localized defects for a wide range of two-dimensional geometries. The defects result in eigenfunctions with a weak singularity that is resolved by decomposing the solution as a superposition of Green's functions plus a smooth regular part. This method is applied to a variety of regular and irregular domains and two key phenomena are observed. First, careful placement of clamping points can entirely eliminate particular eigenvalues and suggests a strategy for manipulating the vibrational characteristics of rigid bodies so that undesirable frequencies are removed. Second, clamping of the plate can result in partitioning of the domain so that vibrational modes are largely confined to certain spatial regions. This numerical method gives a precision tool for tuning the vibrational characteristics of thin elastic plates.