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Size dependent behavior of materials appears for a structure when the characteristic size such as thickness or diameter is close to its internal length-scale parameter. In these cases, ignoring this behavior in modeling may leads to incorrect results. In this paper, strong effects of the size dependence on the static and dynamic behavior of the electrostatically actuated micro-beams have been studied. The equilibrium positions or fixed points of the gold and nickel micro-beams have been determined and shown that for a given DC voltage, there is a considerable difference between the fixed points gained using the classic beam theory and the modified couple stress theory. In addition, it has been shown that the static and dynamic pull-in voltages gained using the couple stress theory are several times higher than those gained using the classic beam theory. Some previous studies have applied the classic beam theory in their models and introduced a considerable hypothetical value of residual stress to match their experimental and incorrect theoretical results. It has been shown that using the modified couple stress theory decreases considerably the difference with the experimental results.

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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.

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In this paper, the fluid- solid interaction in an electrostatic microbeam by using three- dimensional aerodynamic theory has been studied. Modified couple stress theory is used to model the elasticity depends on the size of the microbeam. The proposed model can be used as a mass micro- sensor. To analyze the dynamic behavior of the microbeam a DC voltage applied to the system and then by applying an AC voltage dynamic characteristics of the system around static deformed condition is analyzed. Because of non-linear nature of the governing equations to solve them reduced order model based on Galerkin is used. Results have shown that considering the couple stress and also increase the size of the length characteristic parameter reduces the size of the fluid pressure differential created between the two sides of the microbeam. However, according to the three- dimensional aerodynamic theory for fluid-solid interaction, change of the pressure difference created does not lead to creation difference in predicting the size of the added mass between the classical and modified couple stress theories. In another part of the results has been shown that the presence of added mass to what extent can makes changes in the frequency response curves drawn for the system. Also applied the couple stress theory and increase the size of the length characteristic parameter makes the system more rigid and consequently reduce the amplitude of the vibration and frequency response curves shift to the right.