Modal analyses of nonuniform axially functionally graded Euler-Bernoulli beams

Abstract

The dynamic response of functionally graded beams is of immense importance in modern engineering applications where structural elements often exhibit material inhomogeneity and geometric nonuniformity. This study investigates flexural free vibrations of axially functionally graded Euler-Bernoulli beams with nonuniform cross-sections, where both geometric dimensions and material properties vary along the beam axis. The governing equations of motion were discretized and solved using the Haar wavelet method, which provides an efficient numerical scheme. Four classical boundary conditions: clamped-free, pinned-pinned, clamped-pinned, and clamped-clamped were analyzed to demonstrate the versatility of the approach. The accuracy of the method was verified by comparison with benchmark solutions available in the literature. Extended case studies were then performed for tapered and cone-shaped beams with linearly varying depth or width, considering axially functionally graded material. The results demonstrate that variations in axial cross-sectional geometry have a greater impact on natural frequencies and mode shapes than material gradation. This study revisits the Haar wavelet method and extends its application to conditions that were previously unstudied, such as different functionally graded material models in tapered and cone configurations. The validated results are in good agreement with existing literature. Finally, detailed graphs and tables present the results obtained for previously uninvestigated cases