Free vibration analysis of functionally graded carbon nanotube-reinforced beams with variable cross-section using the differential transform method

The paper studies the free vibration of carbon nanotube-reinforced composite (CNTRC) beams with variable cross-sections. The carbon nanotubes embedded in a polymeric matrix are assumed to be functionally graded (FG) across the beam's thickness, with their material properties determined using the rule of mixtures. Various CNT distribution patterns and cross-sectional variation profiles are considered. The study employs Timoshenko beam theory, deriving the governing equations via Hamilton's principle. These differential equations with variable coefficients are solved using the differential transform method (DTM), which is formulated as a unified eigenvalue problem applicable to various boundary conditions. The computed results are validated against available literature to ensure accuracy and reliability. Subsequently, a comprehensive parametric study examines the influence of geometrical and material parameters on the vibration behavior of FG-CNTRC beams. The findings reveal that natural frequencies are significantly affected by taper parameters, CNT content, and nanotube distribution patterns. Finally, the study identifies the CNT distributions that offer the most favorable vibration characteristics.