Uncertainty and sensitivity analysis for nonlinear torsional vibration of carbon nanotubes embedded in a nonlinear viscoelastic medium in a thermo-magnetic environment

In this article, the nonlinear torsional vibration of single-wall carbon nanotubes (SWCNT's) imbedded in a nonlinear viscoelastic medium under the effect of thermo-magnetic environment is investigated. Nonlocal elasticity theory is applied to capture the effect of small-scale effect for the SWNCTs. The presence of magnetic field results in a fourth order differential equation with only two boundary conditions which makes impossible to obtain the analytical mode functions. Rayleigh–Ritz method is used to obtain the mode functions from a fourth order linear equation of motion. The nonlinear torsional damped natural frequencies are obtained by solving the nonlinear equation of motion by multiple time scale method. Uncertainty analysis is carried out to depict the effect of the uncertain parameters on torsional frequencies. Moreover, Sensitivity analysis is conducted to demonstrate the sensitivity of uncertain parameters on nonlinear torsional frequencies. Magnetic field and temperature effects on the nonlinear torsional vibration of SWCNTs are examined. The magnetic field contributes to a rise in the nonlinear damped frequencies of SWCNTs, regardless of thermal conditions. Sensitivity analysis shows that the frequencies significantly depend on the shear modulus of SWCNTs and damping constant of the viscoelastic medium at different temperatures. The lower and upper bounds of frequencies during uncertainty analysis strongly depends on nanotube temperature.