Mechanics Research Communications 52 (2013) 25–31 Contents lists available at SciVerse ScienceDirect Mechanics Research Communications j ournal homepage: www.elsevier.com/locate/mechrescom Voltage–amplitude response of alternating current near half natural frequency electrostatically actuated MEMS resonators ∗ Dumitru I. Caruntu , Israel Martinez, Kyle N. Taylor University of Texas-Pan American, Mechanical Engineering Department, Edinburg, TX 78541, USA a r t i c l e i n f o a b s t r a c t Article history: This paper uses the Reduced Order Model (ROM) method to investigate the nonlinear-parametric dynam- Received 7 November 2012 ics of electrostatically actuated MEMS cantilever resonators under soft Alternating Current (AC) voltage Received in revised form 10 April 2013 of frequency near half natural frequency of the resonator. The voltage is between the resonator and a Accepted 7 June 2013 ground plate, and provides a nonlinear parametric actuation for the resonator. Fringe effect and damping Available online xxx forces are included. The resonator is modeled as an Euler–Bernoulli cantilever. Two methods of inves- tigations are compared, Method of Multiple Scales (MMS), and Reduced Order Model. Moreover, the Keywords: instabilities (bifurcation points) are predicted for both cases, when the voltage is swept up, and when the MEMS resonators voltage is swept down. Although MMS and ROM are in good agreement for small amplitudes, MMS fails Nonlinear dynamics to accurately predict the behavior of the MEMS resonator for greater amplitudes. Only ROM captures the Soft AC excitation Primary resonance behavior of the system for large amplitudes. ROM convergence shows that five terms model accurately Voltage response predicts the steady-states of the resonator for both small and large amplitudes. © 2013 Elsevier Ltd. All rights reserved. 1. Introduction become predominant with continuous reduction in device size, usually occurring at the submicron scales, Ramezani et al. (2007). Due to their attractive features, Microelectromechanical Sys- Clamped–clamped micro-switches under electrical actuation have tems (MEMS) devices have gained tremendous interest over the been investigated, Jia et al. (2012). A perturbation-based method years. Excellent review papers have been reported in the liter- has been used to obtain the amplitude–frequency response, which ature, Bogue (2007), Chuang et al. (2010), Tanaka (2007), and then has been then compared with experimental data. Yet, volt- van Beek and Puers (2012). Microbeams are used extensively in age response has not been investigated. The dynamic pull-in applications such as switches, filters, resonators, accelerometers phenomenon for a clamped–clamped microbeam using Reduced and sensors, Dumas et al. (2011), Badri et al. (2010), Tang et al. Order Model (ROM) method has also been reported in the lit- (2011), Dai et al. (2011), and Zhang et al. (2011). Accelerome- erature, Nayfeh et al. (2007). Moreover, the amplitude–voltage ters are used in many fields, from automobile (air-bag sensor and response of MEMS resonators under super-harmonic excitation, rollover detection sensor) to mobile phone applications, Braghin Younis et al. (2004), has been obtained using a perturbation-based et al. (2007). Besides piezoelectric and electro-magnetic actua- method and validated using ROM. Al Saleem and Younis (2009) tion methods, electrostatic actuation is also used in driving and investigated the dynamic pull-in escape in electrostatic MEMS. sensing such systems, Stulemeijer et al. (2009) and Abdel-Rahman Amplitude–frequency responses have been generated using sim- et al. (2003). Electrostatic actuation offers the advantage of pro- ulation and experimental results. Yet, the system was modeled viding enough force that can be controlled through an electric as a nonlinear single-degree-of-freedom model (lumped system). supply requiring low energy consumption. Since these resonators However, the amplitude–voltage response of MEMS cantilever res- are small in size, they have to be driven close to or even into onators for AC actuation frequency near the system’s half natural nonlinear regimes to store enough energy for sufficiently good frequency has been reported in the literature, Caruntu and Martinez signal to noise ratio, Mestrom et al. (2008). Nonlinearity usu- (2012). Yet, results were numerically obtained using only a two ally arises from sources such as the electrostatic actuation itself, term ROM. A subsequent investigation used both the Method squeeze-film damping, geometric nonlinearities and intermolec- of Multiple Scales (MMS) and a five term ROM, Caruntu et al. ular forces, i.e. Casimir and van der Waals. Intermolecular forces (2013). Yet, only amplitude–frequency response has been inves- tigated. To the best of our knowledge, investigations reported in the literature are limited to either (1) the pull-in phenomenon for ∗ clamped–clamped microbeams, (2) lumped models, (3) influence Corresponding author. Tel.: +1 956 665 2079; fax: +1 956 665 3527. of parameters other than air damping and fringe effect, (4) MMS, E-mail addresses: [email protected], [email protected], [email protected] (D.I. Caruntu). or (5) ROM. 0093-6413/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mechrescom.2013.06.001 26 D.I. Caruntu et al. / Mechanics Research Communications 52 (2013) 25–31 where A and I are current cross-section area and moment of iner- tia. In the case of uniform beams, A* and I* are equal to one. A0, I0, and are the reference cross-section area, reference cross-section moment of inertia, and beam length, respectively. If uniform can- tilevers, then they are the cross-section area and moment of inertia of the cantilever. If the cantilevers are nonuniform, then they could h be considered where the cross-section area is maximum, Caruntu (2007, 2009a). The first term at the right hand side of Eq. (1) is the dimen- g sionless damping force, the second term is the dimensionless W electrostatic force, and the last term is the additional electrostatic force due to fringe effect. The dimensionless voltage V() consid- ered in this work is given by ∗ x V() = cos ˝ (4) The dimensionless parameters in Eq. (1) are b* the dimen- Fig. 1. Uniform MEMS resonator. sionless damping coefficient, ı the dimensionless amplitude of the electrostatic excitation force (also called dimensionless volt- age parameter), f the dimensionless fringe coefficient, and * the This paper deals with the amplitude–voltage response of MEMS dimensionless frequency of excitation, and they are as follows, resonator cantilevers under soft electrostatic actuation. The elec- Caruntu and Knecht (2011a), and Caruntu et al. (2013): trostatic actuation is of AC voltage of frequency near half natural frequency (resulting in primary resonance) of the system. This is 4 4 ∗ b ε0W 2 = = a nonlinear parametric excitation. This is for the first time when b , ı V , 3 0 g A0EI0 2g EI0 for the amplitude–voltage response of such MEMS resonator can- tilevers (1) a ROM (based on the Galerkin procedure) up to five 0.65g ∗ 2 A0 = = terms is employed to develop a set of non-explicit ordinary dif- f , ˝ ˝ (5) W EI0 ferential equations and AUTO 07P software is used to numerically obtain the amplitude–voltage response, (2) a direct comparison where V0 is the amplitude of the AC voltage, b the viscous damping between MMS and ROM method using two-, three-, four-, and per unit length coefficient, W the beam width and εo the permit- five-term is reported, (3) the influences of AC frequency of actu- tivity of free space. ation, air damping, and fringe correction on the response using both MMS and ROM are provided. This work shows that the insta- 3. Soft AC voltage frequency near half natural frequency bilities for high amplitudes of the resonator are not captured by the MMS. Only four or five term ROM would capture this The frequency of the AC voltage is near half the natural fre- behavior. ≈ quency, ˝* ωk/2 in this work. The nearness of the excitation frequency can be written as 2. Differential equation of motion ∗ = + 2˝ ωk (6) Fig. 1 shows an electrostatically actuated MEMS cantilever res- where is a detuning parameter. Substituting Eqs. (4) and (6) into onator that consists of a deformable electrode suspended over a Eq. (1), it results ground plate. The dimensionless boundary value problem is given by Caruntu et al. (2013) and Caruntu and Knecht (2011a) as follows: ⎧ 2 2 2 2 2 · · · ⎪ ∗ ∂ u (, z) ∂ ∗ ∂ u (, z) ∗ ∂u (, z) ı V () f ı V () ⎨ A (z) + I (z) = −b + + 2 2 2 2 ∂ 1 − u , z ∂ ∂z ∂z [1 − u (, z)] [ ( )] ⎪ (1) ⎩⎪ ∂u ∂2u ∂3u u(, 0) = (, 0) = (, 1) = (, 1) = 0 ∂z ∂z2 ∂z3 4 2 where u = u(z,t), z and are dimensionless beam transverse ∂ u (, z) ∂ u (, z) ∗ ∂u (, z) + + b 4 2 ∂ displacement, dimensionless longitudinal coordinate, and dimen- ∂z ∂ sionless time, respectively. These dimensionless variables are as ∗ ı f · ı (1 + cos 2˝ ) follows: = + (7) 2 1 − u , z 2 [1 − u (, z)] [ ( )] w x 1 EI0 u = ; z = ; = t · (2) 2 One can notice that although the frequency of the AC volt- g A0 age is near half natural frequency, the frequency of the excitation where w is the dimensional transverse displacement of the can- force is near natural frequency; notice the cosine in Eq. (7). This is tilever, x the dimensional longitudinal coordinate, t dimensional nonlinear-parametric primary resonance of the structure. time, is the beam length, g is the gap, the density, and E Young modulus. Dimensionless cross section area A* and moment of iner- 4. MMS and ROM of uniform MEMS resonators tia I* are ∗ A ∗ I The Method of Multiple Scales (MMS) has been used by Caruntu A = , I = (3) A0 I0 and Knecht (2011a) to investigate the frequency response of the resonator. The steady state response of the resonator has been D.I.