Nonlinear vibration of single-walled carbon nanotubes with nonlinear damping and random material properties under magnetic field

This paper copes with the statistical dynamic behaviors of nonlinear vibration of the single-walled carbon nanotubes (SWCNTs) under longitudinal magnetic field by considering the effects of the geometric nonlinearity and nonlinear damping. Both the Young's modulus of elasticity and mass density of the SWCNTs are considered as stochastic with respect to the position to actually characterize the random material properties of the SWCNTs. Based on the theory of nonlocal elasticity, the small scale effects of the nonlinear vibration of the SWCNTs are investigated. By using the Hamilton's principle, the nonlinear governing equations of the single-walled carbon nanotubes subjected to longitudinal magnetic field are derived. The Monte Carlo Simulation, Galerkin's method and the multiple scale method are adopted to solve the nonlinear governing equation and to calculate the statistical response of the SWCNTs. Some statistical dynamic responses of the SWCNTs such as the mean values and standard deviations of the midpoint deflections are computed, the effects of the small scale coefficients, magnetic field, nonlinear damping and the elastic stiffness of matrix on the statistical dynamic responses of the SWCNTs are investigated and discussed.