Dynamic Analysis of Electrostatically Actuated Nanobeam Based on Strain Gradient Theory

In this study, dynamic response of a micro- and nanobeams under electrostatic actuation is investigated using strain gradient theory. To solve the governing sixth-order partial differential equation, mode shapes and natural frequencies of beam using Euler–Bernoulli and strain gradient theories are derived and then compared with classical theory. Galerkin projection is utilized to convert the partial differential equation to ordinary differential equations representing the system mode shapes. Accuracy of proposed one degree of freedom model is verified by comparing the dynamic response of the electrostatically actuated micro-beam with analogue equation and differential quadrature methods. Moreover, the static pull-in voltages of micro-beams found by one DOF model are compared with the reported data in literature. The main advantage of proposed method based on the Galerkin method is its simplicity and also its low computational cost in analyzing the dynamic and static responses of micro- and nanobeams. Additionally, effect of axial force, beam thickness and applied voltage are analyzed. The results obtained based on strain gradient theory, are compared with classical and modified couple stress theories which are the special cases of the strain gradient theory. It is shown that strain gradient theory leads to higher frequency and lower amplitude in comparison with two other theories.