Comprehensive numerical modeling of the nonlinear structural behavior of MEMS/NEMS electrostatic actuators under the effect of the van der Waals forces

The mechanical behavior of MEMS/NEMS based-actuators has been studied in this work while considering the effect of both electrostatic and van der Waals forces. The assumed structural model was constructed based on an Euler–Bernoulli beam continuous nonlinear model, where both the mid-plane stretching (geometric nonlinearity) and the electric fringing-fields effects have been taken into consideration. The nonlinear equation of motion of the actuator governing its static behavior has been derived. An original Galerkin expansion based reduced-order model has been developed to avoid problem arising from all the possible nonlinearities in the nonlinear differential equation. The obtained reduced-order model equations have been resolved mathematically using the so-called Newton-Raphson technique. The developed numerical method was shown to be powerful in examining the basic pull-in design parameters such as the electric load amplitude and the beam static deflection at the pull-in instability state. The model also proved its capability in capturing the effect of the actuator parameter on its detachment length due to the attractive van der Waals forces. Some of the obtained numerical outcomes have been compared with other numerical approaches such as finite-elements and finite-difference methods. The assessment revealed acceptable agreement among all assumed numerical techniques.