Power Series Solution to the Natural Frequency of a Rotating Non-Uniform FG-CNTRC Beam Considering Boundary Relaxation

This paper delves into the free vibration analysis of a rotating non-uniform functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beam with symmetric material distribution, taking into account boundary relaxation. Three common carbon nanotube (CNT) distributions, namely FG-X, UD, and FG-O, are considered. The governing equations of a rotating FG-CNTRC beam with variable cross-section and boundary relaxation are formulated via Hamilton’s principle. Some factors, including the centrifugal force induced by rotation, boundary relaxation, cross-section gradient, and others, substantially complicate the boundary conditions, making it challenging to directly obtain an analytical solution with variable coefficients. To address this, a novel power series solution based on the differential transformation method (DTM) is introduced to discretize the vibration equation and obtain the natural frequency of the rotating FG-CNTRC beam, which forms the core novelty of this study. Comprehensive numerical calculations are carried out, and the reliability of the DTM results is fully verified via comparisons with finite element (FEM) outputs and published reference data.