Nonlinear vibration analysis of an electrostatic functionally graded nano-resonator with surface effects based on nonlocal strain gradient theory

In this paper, a comprehensive analysis of the nonlinear vibration of an electrostatic nanobeam resonator is presented based on the nonlocal strain gradient theory (NSGT) and by incorporating the Gurtin-Murdoch surface elasticity theory. The Von-Karman geometrical nonlinearity and inter-molecular dispersion forces, i.e., van der Waals and Casimir forces are included in the equation of motion. Both DC and AC components of the electrostatic actuation are regarded as the excitation terms. The nanobeam is considered to be composed of a power-law functionally graded (FG) material. Utilizing Hamilton's principle, the size-dependent nonlinear equation of motion of the system is derived. Multiple scales technique in conjunction with the differential quadrature method (DQM) is adopted to analytically obtain the solution. Results obtained are shown to be in good agreement against available literature. Static deflection and fundamental natural frequency are obtained for different size-dependent and volume fraction index parameters. Meanwhile, the variation of the oscillation amplitude by the quality factor, excitation magnitude, and frequency is determined near the primary resonance. The acquired results revealed that the nonlocal and strain gradient parameters can significantly displace the multi-valued portions and instability thresholds of the dynamical response diagrams. It is shown that the increment of the volume fraction index reduces the pull-in voltage while increasing any of the size-dependent parameters enlarge the instability voltage. Moreover, the surface effects induce the stiffness hardening behavior, whereas the inter-molecular forces impose the stiffness softening effect.