Nonlinear Free Vibration Analysis of Functionally Graded Carbon Nanotubes Reinforced Composite Beams on the Winkler Elastic Foundation

PurposeThis study based on the Euler-Bernoulli theory and Von-Karman nonlinear theory investigates the nonlinear free vibration characteristics of functionally graded carbon nanotubes reinforced composite (FG-CNTRC) beam on the Winkler foundation under thermal environments.MethodsThe nonlinear vibration governing differential equation of FG-CNTRC beam is derived using the generalized Hamilton principle. The Galerkin method is employed to discretize the nonlinear partial differential equation into nonlinear ordinary differential equations, which are then solved using the multiple scale method to obtain the system's nonlinear natural frequency. The results are compared with dimensionless natural frequencies from existing literature to validate the accuracy and effectiveness of the proposed method. Based on carbon nanotubes (CNTs) distribution types, including UD type, FG-X type, FG-O type, and FG-A type. The effects of distribution type, CNTs mass fraction, temperature gradient, and foundation stiffness on the nonlinear natural frequency are analyzed under simply-simply (SS) supported and clamped-clamped (CC) boundary conditions.ConclusionThe results indicate that the FG-O type significantly influences the nonlinear natural frequency. The CNTs mass fraction and foundation stiffness parameters are positively correlated with the nonlinear natural frequency, and the temperature gradient is negatively correlated with the nonlinear natural frequency.