Nonlinear free vibration of sandwich beams with CNT–reinforced composite core and elastically restrained ends resting on Kerr foundations

This paper aims to investigate the nonlinear free vibration of sandwich beams with carbon nanotube (CNT) reinforced composite core and homogeneous face sheets. Unlike previous studies, the combined influences of elasticity of tangential constraints of ends, interaction from Kerr foundations and elevated temperatures on the nonlinear vibrations of sandwich beams with CNT core are examined. CNTs are reinforced into polymer matrix through uniform distribution and functionally graded distributions. The properties of constituents are assumed to be temperature–dependent and effective properties of nanocomposite core are estimated by means of an extended rule of mixture. Motion equations are established within the framework of Timoshenko beam theory taking into account von Kármán nonlinearity, geometric imperfection and interaction from Kerr elastic foundations. These equations are solved by using analytical solutions and Galerkin method to derive a time–dependent nonlinear ordinary differential equation. Fourth–order Runge–Kutta scheme is employed to determine the nonlinear frequencies of sandwich beams. Parametric studies detect that the support of Kerr foundations increases natural frequencies and decreases the nonlinear to linear frequency ratio. Meanwhile, the frequency nonlinearity is considerably stronger when the ends are constrained more severely and temperature is more elevated. Furthermore, the study also reveal that initial geometric imperfection enhances the natural frequencies and weakens the frequency nonlinearity in the deep region of deflection amplitude of sandwich beams.