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MEMS Cantilever Resonators Under Soft AC Microelectromechanical Systems Cantilever Resonators Dumitru I. Caruntu1 Mem. ASME Under Soft Alternating Current Mechanical Engineering Department, University of Texas Pan American, 1201 W University Drive, Voltage of Frequency Near Edinburg, TX 78539 e-mail: [email protected] Natural Frequency Martin W. Knecht This paper deals with nonlinear-parametric frequency response of alternating current Engineering Department, (AC) near natural frequency electrostatically actuated microelectromechanical systems South Texas College, (MEMS) cantilever resonators. The model includes fringe and Casimir effects, and damp- McAllen, TX 78501 ing. Method of multiple scales (MMS) and reduced order model (ROM) method are used e-mail: [email protected] to investigate the case of weak nonlinearities. It is reported for uniform resonators: (1) an excellent agreement between the two methods for amplitudes less than half of the gap, (2) a significant influence of fringe effect and damping on bifurcation frequencies and phase–frequency response, respectively, (3) an increase of nonzero amplitudes’ fre- quency range with voltage increase and damping decrease, and (4) a negligible Casimir effect at microscale. [DOI: 10.1115/1.4028887] Introduction linear Mathieu’s equation as the governing equation of motion. Yet, the parametric coefficient was only in the linear terms. Non- Microelectromechanicals systems (MEMS) find their use in linear behavior of electrostatically actuated cantilever beam micro biomedical, automotive, and aerospace. MEMS beams are com- resonators, including fringe effect, has been investigated [11] mon and successfully used as chemical sensors [1], biosensors [2], using MMS and ROM. Yet, the nonlinear behavior was due to AC pressure sensors [3], switches [4], and energy harvesters [5]. voltage of frequency near a system’s half natural frequency of the MEMS systems are nonlinear due to actuation forces, damping resonator, which resulted in primary resonance. However, effects, large structural deformation, and intermolecular surface resonances occur at various excitation frequencies due to the para- forces, such as Casimir and/or van der Waals, which are signifi- metric excitation [11–13,23]. cant for gaps less than 200 nm [6]. Structural nonlinearities are Partial differential equations of motion of such MEMS continu- significant for short beams and/or large deflections. Nonlinearities ous systems can be solved using analytical methods such as aver- due to damping effects, such as squeeze film damping, can be aging, harmonic balance, or MMS [11,24], and numerical neglected within certain pressure regimes [7,8] or when consider- methods such as finite element (FEM), boundary-element (BEM), ing steady state behavior [9]. Conversely nonlinearities from elec- finite difference (FDM), and ROM method [24]. FEM, BEM, and trostatic forces cannot be neglected [10]. FDM methods are relatively accurate. Yet, they are only adequate Parametrically excited MEMS resonators via electrostatic for static systems, which require less computational time and cost actuation are highly dependent on parameters [9,11–13]. MEMS than dynamic systems. cantilevers with periodic coefficients in both linear and nonlinear In this paper, the nonlinear parametric behavior of MEMS can- terms of the equation of motion [11] have been reported. Clam- tilever resonators due to AC of frequency near a system’s natural ped–clamped resonators using the method of multiple scales frequency electrostatic actuation is investigated. This leads to (MMS) and reduced order model (ROM) method have been inves- parametric resonance as shown afterwards. An Euler–Bernoulli tigated [14]. Yet, fringe correction and Casimir effect have not MEMS cantilever including electrostatic force, fringe effect, been considered, and the second harmonic has been neglected due damping, and Casimir effect is investigated using MMS and ROM to the direct current (DC) voltage much larger than the AC voltage [11,24–26]. MMS is applied directly to the dimensionless partial in Ref. [14]. Electrostatically actuated MEMS resonators modeled differential equation of motion of the system in order to obtain as mass–spring–damper system have been investigated using the analytically the amplitude–frequency response. It is showed that method of averaging [9] and their frequency response and stability at micro scale the Casimir effect is negligible. found. Yet, it lacked numerical validation, continuous system To the best of our knowledge, this is for the first time when for modeling, and fringe correction effect. Frequency response of electrostatically actuated MEMS cantilever resonators: (1) para- cantilever resonators has been reported [15–21]. Forces driving metric resonance due to only soft AC voltage leading to a soft the resonator in these investigations were due to thermal (Brown- force of two components, constant and second harmonic [11], is ian) excitation and fluid hydrodynamics. Yet, such forces did not considered. Soft actuation can be used for microbalances due to produce parametric excitations or nonlinear behavior. Parametric very small power consumption. (2) An approach in which the resonance of a MEMS cantilever [22] has been created via a feed- model includes periodic coefficients in the nonlinear terms is con- back scheme on a piezoelectrically driven microcantilever using a sidered. The model includes linear parametric, nonlinear, and most importantly parametric-nonlinear terms. This is in contrast with other works on parametrically excited beams. For instance, 1University of Texas Pan American becomes University of Texas Rio Grande AC voltage was much smaller than the DC voltage in Ref. [27], Valley starting Fall 2015, e-mail: [email protected]; [email protected]. which led to very small nonlinear parametric terms that were Contributed by the Dynamic Systems Division of ASME for publication in the neglected in the MMS. (3) The convergence of ROM frequency JOURNAL OF DYNAMIC SYSTEMS,MEASUREMENT, AND CONTROL. Manuscript received November 6, 2012; final manuscript received August 16, 2014; published online response using two, three, four, and five terms is reported as well. January 9, 2015. Assoc. Editor: Qingze Zou. (4) A direct comparison between MMS and ROM is reported. Journal of Dynamic Systems, Measurement, and Control APRIL 2015, Vol. 137 / 041016-1 Copyright VC 2015 by ASME Downloaded From: http://dynamicsystems.asmedigitalcollection.asme.org/ on 01/17/2015 Terms of Use: http://asme.org/terms sffiffiffiffiffiffiffiffi This work shows that: (1) the approach in which the MMS model 1 EI includes the periodic coefficients in the nonlinear gives an excellent 0 u ¼ w=g; z ¼ x=‘; s ¼ 2 Á t (3) agreement between MMS and ROM of any number of terms for ‘ qA0 amplitudes less than 0.5 of the gap. (2) The pull-in from large Constant q is the beam density, A0 is the cross-sectional area, w is amplitudes is predicted only by four or more terms in the ROM, (3) the dimensional beam deflection, t is the dimensional time, E is while two and three terms ROM and MMS fail in this prediction. the Young modulus, I0 is the cross-sectional moment of inertia, x (4) Casimir effect is negligible at microscale. This effect is a point is the dimensional longitudinal coordinate, ‘ is the cantilever of future investigation at nanoscale levels. Also, (5) the influences length, and g is the gap (Fig. 1). The dimensionless parameters of of damping, voltage, and fringe effect on the frequency response of Eq. (2), Casimir effect a, electrostatic excitation (voltage) d, the MEMS cantilever resonator are reported. fringe effect f , damping bÃ, and dimensionless frequency xÃ, are [11,25,26] given by Equation of Motion 2 4 4 p hcW ‘ e0W‘ 2 0:65g Electrostatically actuated system to be investigated consists of a ¼ 5 ; d ¼ 3 V0 ; f ¼ ; 240g EI0 2g EI0 W a conducting elastic MEMS cantilever over a rigid stationary con- sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffi (4) 4 ducting plate (ground plate) (Fig. 1). Between the beam and plate ‘ qA0 is a dielectric fluid such as air. Motion equation is based on bà ¼ b ; xà ¼ x‘2 qA EI EI Euler–Bernoulli theory (structural nonlinearities are neglected) 0 0 0 and includes electrostatic, viscous damping, and nanoscale surface where h ¼ 1.055  10À34 J s is Planck’s constant divided by 2p, forces such as Casimir. The electrostatic force includes a first- c ¼ 2.998  108 msÀ1 is the speed of light, e ¼ 8.854  10À12 C2 order fringe field correction based on Palmer’s formula [10]. The 0 NÀ1 mÀ2 is the permittivity of free space, W is the beam width, electrostatic force is produced by applying a potential difference b is the damping coefficient, and V is the amplitude of the between the flexible MEMS cantilever and the rigid plate, Fig. 1, 0 AC voltage. From this point forward, dimensionless natural fre- VðtÞ¼Vp þ V0 cosðXtÞ, where Vp is the polarizing DC voltage, quencies of the MEMS cantilever are denoted simply by xk. and V0 and X are the AC voltage and excitation frequency. Since the electrostatic force is proportional to the square of the voltage, à 2 2 AC Near Natural Frequency X xk, Parametric there are three components: constant Vp þ 0:5V0 , first harmonic 2 Resonance 2VpV0 cosðXtÞ, and second harmonic 0:5V0 cosð2XtÞ [11]. In this work, Vp ¼ 0, therefore, only the second harmonic is present The dimensionless AC voltage considered in this investigation is given by rffiffiffiffiffiffiffiffi 2 2 2 V0 V0 à à 2 qA0 VðtÞ ¼ þ cosð2XtÞ (1) VðsÞ¼cos X T0; X ¼ X‘ (5) 2 2 EI0 à However, V contributes to a static voltage, despite the lack of where X and X are
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