On the variation of longitudinal and torsional frequencies in a partially hinged rectangular plate

TL;DR

This paper investigates how the eigenfrequencies of a partially hinged rectangular plate change with shape deformations, revealing the non-existence of a perfect shape functional for torsional instability prediction.

Contribution

It demonstrates the non-existence of an ideal shape functional for torsional instability and critiques existing functionals in the literature.

Findings

No simple shape functional can quantify torsional instability accurately.

Existing literature functionals are unreliable.

Eigenvalues vary with domain deformations in complex ways.

Abstract

We consider a partially hinged rectangular plate and its normal modes. There are two families of modes, longitudinal and torsional. We study the variation of the corresponding eigenvalues under domain deformations. We investigate the possibility of finding a shape functional able to quantify the torsional instability of the plate, namely how prone is the plate to transform longitudinal oscillations into torsional ones. This functional should obey several rules coming from both theoretical and practical evidences. We show that a simple functional obeying all the required rules does not exist and that the functionals available in literature are not reliable.