A Discrete Physical Model for Analyzing the Vibrations of a Non-Uniform Bi-Directional FGM Beam under Elastic Foundations and Different Support Conditions

This study employs a discrete physical model (DPM) to investigate the small-amplitude transverse free vibrations of non-uniform beams made from bidirectional functionally graded materials (2D-FGM), the material properties of the beam are varied continuously in both the axial and transverse directions by exponential gradation laws. The beam is analyzed on variable elastic foundations and under various boundary conditions. The model represents the continuous beam as a multi-degree-of-freedom (MDOF) system that incorporates masses, bars, and spiral linear springs. After calculating the system's mass and linear stiffness matrices, as well as the foundation's rigidity matrix, which are computed using an appropriate discretization that aligns with the properties of the examined beam, Hamilton's principle is applied to derive a set of algebraic equations predicting the non-dimensional natural frequencies and mode shapes of the beams. The numerical results obtained from this discrete model were compared with those from previous studies to validate the proposed approach. The flexibility of this model allows for precise adjustment of linear vibration frequencies by modifying the geometrical properties of the beams, the gradation exponents in the $x$ and $y$ directions $(n_x, n_y)$, the taper ratio $\gamma$ of the non-uniform beam, and the foundation's position $\chi$ and intensity $\eta$, as well as the boundary conditions. This highlights its utility in analyzing and designing complex beam structures for diverse engineering applications.