Vibration Analysis of Circular Nanoplates under Nonlinear Electrostatic Excitation Considering the Surface Energy and Size Effects

This article investigates the primary resonant behavior and static pull-in instability of a circular nanoplate under nonlinear electrostatic actuation. The consistent couple stress theory, Gurtin-Murdoch surface elasticity theory and Hamilton principle were utilized to derive the governing differential equation of transverse vibration Kirchhoff nanoplate by considering the fluid damping and Casimir forces. The governing equation were solved for small amplitude vibrations. To this end, it is assumed that the elastic nanoplate is deflected using a DC bias voltage and then driven to vibrate around its deflected position by a harmonic AC load. The weighted residual method of Galerkin was used to obtain a reduced order model. The method of multiple scales is used to solve the nonlinear equation of motion and, the primary resonance mode frequency response equation is derived. The obtained numerical results were compared to those of previous research works, and a good agreement observed between them. The numerical results revealed that electrostatic actuation and Casmier force have softening effects; but the surface energy can has hardening or softening effect depending on the surface mechanical properties, dimensions and boundary condtions of the nanoplate.