# In-Plane Vibration Analysis of Rectangular Plates with Elastically Restrained Boundaries Using Differential Quadrature Method of Variational Weak Form

## TL;DR

This paper presents a numerical method to analyze in-plane vibrations of rectangular plates with flexible boundaries using a variational approach and differential quadrature.

## Contribution

The study introduces a weak-form differential quadrature method for in-plane vibration analysis of elastically restrained FGM plates.

## Key findings

- The method avoids higher-order derivative transformations by using a weak formulation.
- The impact of gradient parameter, aspect ratio, and elastic constraints on frequency is analyzed.
- Modal properties of FGM plates are investigated using the proposed approach.

## Abstract

An efficient numerical approach utilizing a variational weak form, grounded in 2D elastic theory and variational principles, is proposed for analyzing the in-plane vibrational behavior of rectangular plates resting on elastically restrained boundaries. The differential and integral operators can be discretized into matrix representations employing the differential quadrature method (DQM) and Taylor series expansion techniques. The discretization of dynamics equations stems directly from a weak formulation that circumvents the need for any transformation or discretization of higher-order derivatives encountered in the corresponding strong equations. Utilizing the matrix elementary transformation technique, the displacements of boundary and internal nodes are segregated, subsequently leading to the derivation of the generalized eigenvalue problem pertaining to the free vibration analysis of the Functionally Graded Material (FGM) rectangular plate. Furthermore, the study examines the impact of the gradient parameter, aspect ratio, and elastic constraints on the dimensionless frequency characteristics of the FGM rectangular plate. Ultimately, the modal properties of an in-plane FGM rectangular plate are investigated.