Size-dependent dynamic pull-in instability of vibrating electrically actuated microbeams based on the strain gradient elasticity theory

• The governing equation of microbeams based on the strain gradient elasticity theory is considered.
• Size-dependent pull-in instability of microbeams is investigated.
• The impact of vibrational amplitude on microbeams pull-in instability is studied.
• The second-order frequency–amplitude relationship of microbeams vibration is presented.
• The influences of significant parameters on the dynamic pull-in voltage are investigated.

This paper presents the impact of vibrational amplitude on the dynamic pull-in instability and fundamental frequency of actuated microbeams by introducing the second order frequency–amplitude relationship. The nonlinear governing equation of microbeam predeformed by an electric force including the fringing field effect, based on the strain gradient elasticity theory is considered. The predicted results of the strain gradient elasticity theory are compared with the outcomes that arise from the classical and modified couple stress theory. The influences of basic nondimensional parameters on the pull-in instability as well as the natural frequency are investigated by a powerful asymptotic approach namely the Parameter Expansion Method (PEM). It is demonstrated that two terms in series expansions are sufficient to produce an acceptable solution of the microstructure. The phase portrait of the microstructure shows that by increasing the actuation voltage parameter, the stable center point loses its stability and coalesces with unstable saddle node.