A comparison between two approaches for solving the governing nonlinear equation of vibrations of electrostatic micro-sensors

Electrostatic micro-sensors as a part of microelectromechanical systems (MEMS) play an important role in modern technology. So, precise modeling and suitable solutions for solving the governing mechanical and vibrational equations of them are of great importance. Due to the nonlinear nature of the electrostatic excitation, numerical methods are used to solve the governing equations. This paper presents a comparison between two Galerkin-based approaches to solve them. In the first approach, as used by many researchers in the literature, both sides of the equations are multiplied with the denominator of the electrical force term and then the Galerkin method is applied, whereas in the second approach, we apply direct Galerkin method to solve the equation. As a case study the nonlocal elasticity theory has been used to obtain the governing equation. The results show that for a given beam, although the both approaches predict same pull-in voltage in most cases, but the first approach cannot predict the pull-in instability in some cases and also misses some fixed points. So, the bifurcation diagrams and phase portraits have different quality in the two approaches. Also, the results show that the singular point which is the position of the substrate plate, acts as a strong attractor in the phase diagrams which the first approach is unable to predict it.