Free vibration exploration of axially functionally graded carbon nanotube-reinforced beams through an innovative perturbative methodology

Axially functionally graded (AFG) carbon nanotube-reinforced composite (CNTRC) emerges as a promising solution for tailoring the stiffness and strength of various beam-like structures. Applications range from optimizing the weight-to-strength ratio of airplane wings to fine-tuning the frequency of micro-beam sensors, and even alleviating stresses induced by centrifugal forces or temperature gradients in turbine blades. However, the challenge of solving variable coefficient partial differential equations (VCPDEs) arising from the axial material gradient limits conventional methods to numerical and simulation techniques. In this study, a novel perturbation approach (PA) customized for AFG-CNTRC is highlighted for solving the problem analytically, simply, and precisely. First, the vibration model of an AFG-CNTRC beam is established according to the Euler-Bernoulli theory (EBT). Second, the vibration governing equations are solved by the PA and the state space method (SSM) analytically. Finally, the adaptivity and accuracy of the PA are validated and the effects of carbon nanotubes (CNTs) distribution patterns are analyzed. It is found that increasing CNT content at either end will increase the natural frequency for symmetric boundary conditions (BCs). The frequency decreases with the degree of nonlinearity distribution of CNTs. The frequency under the C-F BC exhibits low sensitivity to parameter effects.