Nonlinear mathematical modeling of microbeam MEMS

Microbeams and microcantilevers are the main part of many MEMS. There are several body and contact forces affecting a vibrating microbeam. Among them, there are some forces appearing to be more significant in micro and nanosize scales. Accepting an analytical approach, we present the mathematical modeling of microresonators dynamics and develop effective equations to be utilized to study the electrically actuated microresonators. The presented nonlinear model includes the initial deflection due to polarization voltage, mid-plane stretching, and axial loads as well as the nonlinear displacement coupling of electric force. It also includes the thermal and squeeze-film phenomena. The equations are nondimensionalized and the design parameters are developed. In order to have a set of equations, depending on depth of accuracy and difficulty, we present equations of motion for linearized and different level of nonlinearity. The simulation method makes it easy for investigators to pick the appropriate equation depending on their design and application. It is shown that the equation of motion for microresonators is highly nonlinear, parametric, and externally excited. The most important phenomena affecting the motion of microbeam-based and microcantilever-based microresonators are reviewed in this chapter and the corresponding forces are introduced. The mechanical and electrical forces are the primary forces that cause the microresonators work. There are also two specific phenomena: squeeze-film and thermal damping, that their effects on MEMS dynamics are considered secondary compared to mechanical and electrical forces. Some tertiary phenomena such as van der Waals, Casimir, and fringing field effects are also introduced. There are a few reported investigations on secondary phenomena, and their effects are defined.