Nonlinear Response of Parametrically-excited Mems

Due to the position-dependent nature of electrostatic forces, many microelectromechanical (MEM) oscillators inherently feature parametric excitation. This work considers the nonlinear response of one such oscillator, which is electrostatically actuated via non-interdigitated comb drives. Unlike other parametricallyexcited systems, which feature only linear parametric excitation in their equation of motion, the oscillator in question here exhibits parametric excitation in both its linear and nonlinear terms. This complication proves to significantly enrich the system’s dynamics. Amongst the interesting consequences is the fact that the system’s nonlinear response proves to be qualitatively dependent on the system’s excitation amplitude. This paper includes an introduction to the equation of motion of interest, a brief, yet systematic, analysis of the equation’s nonlinear response, and experimental evidence of the predicted behavior as measured from an actual MEM oscillator. ∗Please address all correspondence to this author. †Currently at Hewlett-Packard Research Labs, Palo Alto, CA INTRODUCTION The emergence of practical uses for electrostaticallyactuated microelectromechanical (MEM) oscillators coupled with the inherent existence of parametric excitation (due to the position-dependent nature of electrostatic forces) in many such devices, has led the authors, amongst others, to consider both the modeling and response of systems involving generalized forms of parametric excitation and its associated resonances [1–7]. In this paper the response of a simple model of a representative parametrically-excited MEM oscillator is considered. Unlike most parametrically-excited systems, which feature only linear parametric excitation, the system of interest features parametric excitation in both the linear and nonlinear terms of its equation of motion. This simple, yet fundamental, difference proves to have a dramatic effect on the system’s dynamics [1, 8]. In particular, it can be shown that such systems fail to feature a single effective nonlinearity which characterizes the nonlinear behavior of the system. Rather, such systems exhibit branchspecific nonlinearities which, when collectively analyzed, yield the system characteristics. One result of this complication is that 1 Copyright c © 2005 by ASME such systems can exhibit not only typical softening or hardening behavior, but also mixed behavior which corresponds to the nontrivial response branches bending toward or away from one another near resonance [1, 8]. In addition, it can be shown that the qualitative nature of their nonlinear frequency response depends on the amplitude of excitation. This paper begins with a brief introduction to the equation of motion of interest and a summary of the analytical procedure used to reach the results summarized in this work. The relevant nonlinear behavior is discussed in the context of the motivating example, a parametrically-excited MEM oscillator, and experimental evidence of the predicted behavior is presented. The paper then concludes with some closing remarks and an outline of ongoing and future work. It should be noted that, where relevant, issues pertaining to the practical design of MEM oscillators are included. The full details of this work can be found in [8]. THE EQUATION OF MOTION AND PERTURBATION ANALYSIS As mentioned above, the equation of motion examined in this work was originally formulated by the authors to model a parametrically-excited MEM oscillator. It is worth noting, however, that such equations naturally arise in other problems as well, including the analysis of parametrically-excited columns fabricated from nonlinear elastic materials [9, 10] and Paul trap mass spectrometers [11]. Accordingly, the principal motivation for this study is treated only as an example here. The equation of motion of interest in this work is of the form z′′+2εζz′+ z(1+ εν1 + ελ1 cosΩτ)+ εz3 (γ3 +λ3 cosΩτ) = 0, (1) where ε represents a ‘small’ scaling parameter introduced solely for the sake of analysis and prime designates the derivative with respect to τ [1, 5, 7, 8]. In order to simplify the analysis of Eq. (1), it is convenient to employ standard perturbation techniques, in this case, the method of averaging. To assist with this approach, a standard coordinate transformation is first introduced which transforms the equation into amplitude and phase coordinates: z(τ) = a(τ)cos ( Ωτ 2 +ψ(τ) ) , (2) z′(τ) =−a(τ) 2 sin ( Ωτ 2 +ψ(τ) ) . (3) In addition, since near resonant behavior is of primary interest, a detuning parameter σ is introduced, which is defined with respect to the principal parametric resonance condition, namely,