Dynamic and static pull-in instability analysis of partially affected nano-cantilevers using modified couple stress theory

In this paper, the modified couple stress theory is used to study static and dynamic pull-in instability of a general model of a nano-cantilever under a sudden applied DC voltage in the presence of the surface effects. A partial part of the nano-cantilever is subject to the electrostatic and capillary forces. Euler-Bernoulli theory is used to model the beam and the equation of motion is derived by using Hamilton’s principle. The governing equations are transformed into a non-dimensional form and then solved using finite element method (FEM). The results, obtained using FEM are compared with the data available in the literature and found in good agreement. Basic parameters for engineering design at the nanoscale, such as deflection and pull-in voltage have been calculated for both of the dynamic and static modes. The results of dynamic analysis of the beam show that as the voltage increases, the beam goes into an oscillating mode with large amplitudes just before pull-in phenomenon occurs and the beam collapses into the substrate (fixed electrode). Moreover, it is found that a decrease in the length of the fixed electrode (increase of the partially affecting parameter), and the increase of the fringing field effect, the size effect and the surface effect increases the pull-in voltage of the nano-cantilever beam.