Undamped Free Vibration Analysis of Functionally Graded Beams: A Dynamic Finite Element Approach

Abstract

A Dynamic Finite Element (DFE) method for coupled axial–flexural undamped free vibration analysis of functionally graded beams is developed and subsequently used to investigate the system’s natural frequencies and mode shapes. The formulation is based on the Euler–Bernoulli beam theory and material grading is assumed to follow a power law variation through the thickness direction. Using the closed-form solutions to the uncoupled segments of the system’s governing differential equations as the basis functions of approximation space, the dynamic, frequency-dependent, trigonometric interpolation functions are developed. The interpolation functions are used with the weighted residual method to develop the DFE of the system. The resulting nonlinear eigenvalue problem is then solved to determine the coupled natural frequencies. Example elements using DFE, Finite Element Method (FEM) and the Dynamic Stiffness Method (DSM) are implemented in MATLAB for testing, verification, and validation. Good agreement was observed and the DFE formulation exhibited superior convergence performance compared to the FEM.

Advanced Dynamic Thermal Vibration of Thick FGM Cylindrical Shells

“…In 2021, Saeedi et al [5] applied differential quadrature (DQ) method to study the stresses distributions of thick FGM cylindrical shell with aluminum and silicon carbide under internal pressure and thermal loadings. In 2022, Gee and Hashemi [6] used the dynamic finite element (DFE) method to investigate the numerical results of natural frequencies for the FGM beams. In 2019, Khoshgoftar [7] used the virtual work principle and second-order shear Disclaimer/Publisher's Note: The statements, opinions, and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s).…”