On the response of MEMS resonators under generic electrostatic loadings: theoretical analysis

Abstract

We present an investigation of the dynamic behavior of an electrostatically actuated resonant structure, resonator, under the simultaneous excitation of primary and subharmonic resonances. A comprehensive analytical solution is obtained via the method of Multiple Time Scales (MTS), which is applicable for generic electrostatic loading cases. Results using different MTS scaling methods in the equations of motion and loading conditions are compared. These results are further verified against results obtained using direct time integration of the equation of motion. It is observed that for a generic parallel plate electrostatic loading case, the direct forcing component at the excitation frequency, and the direct and parametric excitation components at double the excitation frequency must be considered for accurate prediction of the structure's response. Further, the case of simultaneous excitations of primary and subharmonic resonance, where both excitations are of comparable strength, is examined under various electrostatic loading conditions. We show mixed behaviours of the resonator transiting from a subharmonic dominated response, characterized by the sudden jumps in amplitude and smaller monostable regime, to primary dominated response 2 exhibiting gradual amplitude increase and larger monostable regimes. This transition behaviour can be potentially used for applications, such as electrometers.

On the Application of the Multiple Scales Method on Electrostatically Actuated Resonators

“…Next, we substitute Eqs. (17), (22), and (23) into Eq. (19), and obtain 2 22 2 2 2 2 2 2 2 2 i 10 1 11 3 3 2i 3 3 3 1 2i e 2 0…”

“…The higher order parametric terms are retained here in order to improve accuracy of the solution [17,18].…”

“…In order to tackle this challenge significant focus has been dedicated to develop techniques for investigating the dynamics of MEMS structures that are efficient and can provide results with satisfactory level of accuracy [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]. Perturbation techniques such as the Method of Multiple Time Scales [11,12] have been effectively used to reveal the dynamic characteristics of oscillators under primary [13], secondary [14][15][16][17][18], parametric [19,20], and mixed source excitations [21,22]. Furthermore, MTS has also been used to study the selfsustained MEMS oscillators with linear feedback systems [23], nonlinearities in MEMS comb drive actuators [24], and dynamics of MEMS arch resonators [25].…”

“…Several investigations have been conducted to explore the dynamic behavior of MEMS/NEMS resonators under electrostatic excitations. These are based on the method of Multiple Time Scales (MTS) [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25] and other techniques, such as reduced order models [26,27], and shooting techniques [28]. These studies have focused on revealing limit cycles, understanding the nonlinear behavior, investigating the stability, and studying the dynamic pullin phenomenon.…”

“…The second half of the manuscript presents a comprehensive analytical solution based on MTS, for a simultaneous superharmonic and primary excitation of a MEMS/NEMS resonator. When a MEMS resonator is electrostatically actuated with its AC harmonic component around a certain excitation frequency, due to the quadratic form of the electrostatic voltage, the actuation is transformed into two-source excitation with one frequency around the excitation frequency and the other at twice of that [17,18]. In the case of excitation around half of the fundamental resonance frequency (superharmonic excitation), the contribution from the double frequency component is significant as it directly excites the resonator at its primary resonance frequency.…”