Voltage Response of Superharmonic Resonance of Electrostatically Actuated MEMS Resonator Cantilevers Using the Method of Multiple Scales

This work investigates the voltage response of superharmonic resonance of second order of electrostatically actuated Micro-Electro-Mechanical Systems (MEMS) resonator cantilevers. The results of this work can be used for mass sensors design. The MEMS device consists of MEMS resonator cantilever over a parallel ground plate (electrode) under Alternating Current (AC) voltage. The AC voltage is of frequency near one fourth of the natural frequency of the resonator which leads to the superharmonic resonance of second order. The AC voltage produces an electrostatic force in the category of hard excitations, i.e. for small voltages the resonance is not present while for large voltages resonance occurs and bifurcation points are born. The forces acting on the resonator are electrostatic and damping. The damping force is assumed linear. The Casimir effect and van der Waals effect are negligible for a gap, i.e. the distance between the undeformed resonator and the ground plate, greater than one micrometer and 50 nanometers, respectively, which is the case in this research. The dimensional equation of motion is nondimensionalized by choosing the gap as reference length for deflections, the length of the resonator for the axial coordinate, and reference time based on the characteristics of the structure. The resulting dimensionless equation includes dimensionless parameters (coefficients) such as voltage parameter and damping parameter very important in characterizing the voltage-amplitude response of the structure. The Method of Multiple Scales (MMS) is used to find a solution of the differential equation of motion. MMS transforms the nonlinear partial differential equation of motion into two simpler problems, namely zero-order and first-order. In this work, since the structure is under hard excitations the electrostatic force must be in the zero-order problem. The assumption made in this investigation is that the dimensionless amplitudes are under 0.4 of the gap, and therefore all the terms in the Taylor expansion of the electrostatic force proportional to the deflection or its powers are small enough to be in the first-order problem. This way the zero-order problem solution includes the mode of vibration of the structure, i.e. natural frequency and mode shape, resulting from the homogeneous differential equation, as well as particular solutions due to the nonhomogeneous terms. This solution is then used in the first-order problem to find the voltage-amplitude response of the structure. The influences of frequency and damping on the response are investigated. This work opens the door of using smaller AC frequencies for MEMS resonator sensors.