# On the first frequency of reinforced partially hinged plates

## TL;DR

This paper investigates how to optimize the placement of stiffening material in a partially hinged rectangular plate to improve its stability by analyzing the first eigenvalue of a weighted eigenvalue problem.

## Contribution

It establishes the existence of optimal weight distributions for minimizing the first eigenvalue in specific classes and characterizes the associated eigenfunctions.

## Key findings

- Existence of minimizing weights within certain classes.
- Characterization of eigenfunctions corresponding to optimal weights.
- Insights into stabilizing stiffening strategies for plates.

## Abstract

We consider a partially hinged rectangular plate and its normal modes. The dynamical properties of the plate are influenced by the spectrum of the associated eingenvalue problem. In order to improve the stability of the plate, it seems reasonable to place a certain amount of stiffening material in appropriate regions. If we look at the partial differential equation appearing in the model, this corresponds to insert a suitable weight coefficient inside the equation. A possible way to locate such regions is to study the eigenvalue problem associated to the aforementioned weighted equation. In the present paper we focus our attention essentially on the first eigenvalue and on its minimization in terms of the weight. We prove the existence of minimizing weights inside special classes and we try to describe them together with the corresponding eigenfunctions.