Stochastic Analysis for the Embedded Single-Walled Carbon Nanotube Under Random Vibrations

It is inevitable that carbon nanotubes (CNTs) are subjected to random vibrations due to environmental noise, which may impair their mechanical properties. The nonlinear dynamics associated with single-walled carbon nanotubes (SWCNTs) under random noise has been investigated in this work. First, based on the nonlocal theory and the Euler–Bernoulli beam model with fixed supports at both ends, the governed equations and boundary conditions are established according to Hamilton’s principle. Second, the Galerkin method is used to approximate the differential equations of motion and the stochastic averaging method (SAM) is applied to predict the approximate response of the original system. Subsequently, a detailed parametric study is conducted to analyze the nonlinear random vibration response of the SWCNT with nonlocal effects, and the effectiveness of the proposed method is verified by comparing the analytical results with those from the Runge–Kutta method. Finally, by solving the backward Kolmogorov (BK) equation and the generalized Pontryagin (GP) equation simultaneously, the conditional reliability function (CRF) and mean first-passage time (MFPT) for determining the reliability of SWCNT systems are obtained.