Pull-in analysis of an electrostatically actuated nano-cantilever beam with nonlinearity in curvature and inertia

In this paper, the nonlinear behavior of electrostatically actuated carbon nanotubes (CNTs) is investigated based on a comprehensive model with nonlinearity in curvature, inertia and electrostatic force. The aim of this study is to show when the nonlinear formulation needs to be taken into account and when the linear formulation can simulate the system behavior accurately. The model comprises a cantilevered CNT suspended over a fixed electrode plate from which a DC potential difference is imposed. A relatively large gap between the CNT and the ground plate is considered. The versatile Galerkin method is employed to reduce the nonlinear integro-differential equations of motion to a set of nonlinear ordinary differential equations in time, and then, the reduced equations are solved by direct numerical integration. Dynamic response of the system before and beyond the pull-in voltages and effect of gap to length ratio of the CNT are studied. It is shown that in a large gap to length ratio, when the applied voltage is close to the corresponding pull-in voltage the nonlinear terms have a profound role in the dynamic behavior of the system. Eventually, the contribution of nonlinear terms are examined and it is found that the nonlinear inertia and curvature terms have softening and hardening effects, respectively, whereas the hardening effect of the nonlinear curvature has a major contribution.