# A discrete model for analyzing the free vibrations of a non-uniform 2D-FGM beam under elastic foundations and different support conditions

## TL;DR

This paper introduces a discrete model to analyze the vibrations of non-uniform 2D-FGM beams resting on elastic foundations with various support conditions.

## Contribution

The novelty lies in the development of a discrete physical model for 2D-FGM beams with spatially variable elastic foundations and general support conditions.

## Key findings

- The model accurately predicts natural frequencies and mode shapes of non-uniform 2D-FGM beams.
- The model is computationally efficient and suitable for parametric studies involving geometric and material parameters.
- The approach is validated against published results, confirming its accuracy and reliability.

## Abstract

This paper proposes a discrete physical model (DPM) for the transverse free vibrations of non-uniform bi-directional functionally graded (2D-FGM) beams. The material properties vary in both the axial and thickness directions according to exponential laws, and the beam rest on a spatially variable elastic foundation and satisfy general support conditions. In the proposed formulation, the continuous beam is replaced by a multi-degree-of-freedom chain of lumped masses connected by bars and linear rotational and vertical springs. An adaptive discretization strategy is employed to construct consistent mass, stiffness, and foundation matrices. By applying Hamilton’s principle, the governing equations are reduced to an algebraic eigenvalue problem, from which nondimensional natural frequencies and associated mode shapes are obtained. Comparison with published results confirms the accuracy and reliability of the DPM. Owing to its simplicity and low computational cost, the model is well suited for extensive parametric studies and design-oriented analyses, including frequency tuning through geometric parameters, the material gradation exponents \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(n_x, n_y)$$\end{document}, the taper ratio \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\gamma$$\end{document}, and foundation characteristics (position \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\chi$$\end{document}, intensity \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\eta$$\end{document}) under various boundary conditions. The proposed approach provides a practical and efficient tool for analyzing and optimizing complex FGM beam structures.