Dumitru I. Caruntu 1 Mem. ASME Department of Mechanical Engineering, University of Texas Rio Grande Valley, 1201 W University Drive, Edinburg, TX 78539 e-mails: dumitru.caruntu@utrgv.edu; caruntud2@asme.org Reynaldo Oyervides Department of Mechanical Engineering, University of Texas Rio Grande Valley, 1201 W University Drive, Edinburg, TX 78539 e-mail: reynaldo64@hotmail.com Voltage Response of Primary Resonance of Electrostatically Actuated MEMS Clamped Circular Plate Resonators This paper investigates the voltage–amplitude response of soft alternating current (AC) electrostatically actuated micro-electro-mechanical system (MEMS) clamped circular plates for sensing applications. The case of soft AC voltage of frequency near half natural frequency of the plate is considered. Soft AC produces small to very small amplitudes away from resonance zones. Nearness to half natural frequency results in primary reso- nance of the system, which is investigated using the method of multiple scales (MMS) and numerical simulations using reduced order model (ROM) of seven terms (modes of vibra- tion). The system is assumed to be weakly nonlinear. Pull-in instability of the voltage–amplitude response and the effects of detuning frequency and damping on the response are reported. [DOI: 10.1115/1.4033252] Introduction MEMS are used in automotive industry and medical field [1], as microswitches, transistors, sensors, micromirrors, microgrip- pers, microvalves, resonators [2], resonator sensors [3], and actua- tors for mechanical stimulation of living cells [4]. Electrostatic actuation is preferred due to low energy consumption of operation and high precision with which they can be controlled [5,6]. Other forms of actuation are piezoelectric and magneto-electric. Most MEMS device structures include elements such as cantilevers, bridges, and plates of different shapes and boundary conditions. A large class of electrostatically actuated MEMS structure con- sists of flexible cantilevers, bridges, or plates suspended above a parallel, rigid ground plate. A direct current (DC) voltage applied between the flexible beam/plate and the fixed ground plate pro- duces an attracting electrostatic force between them. The flexible plate deforms toward the ground plate into a new equilibrium posi- tion [7] where electrostatic and elastic forces balance each other. The elastic restoring force within the flexible plate opposes defor- mation. AC voltage is used on top of the DC voltage to cause the system to vibrate around this equilibrium position. Such systems are called resonators [6–8]. If the electrostatic force overcomes the elastic restoring force, the system becomes unstable and the flexi- ble plate collapses onto the ground plate in a phenomenon known as the pull-in instability. The voltage at which this instability phe- nomenon occurs is known as the pull-in voltage [7]. Electrostati- cally actuated MEMS response depends on voltage and frequency of actuation, and initial displacement and velocity of the plate. Due to the high cost and time consumption of experimentation, extensive research has been done to accurately model and simu- late these systems mathematically. These models can be used to predict pull-in instability and the pull-in voltage [9,10]. Investiga- tions have been conducted for MEMS devices such as clamped–clamped microbeams at primary, superharmonic, and subharmonic resonances [7,11]. These structures may experience hardening or softening effects depending on the excitation vol- tages and frequencies used [7]. Amplitude frequency response is widely used to investigate the behavior of MEMS resonators. This type of response explains how the steady-state amplitude of vibration of the system changes with the excitation frequency while the voltage is kept constant [8,10,12]. Bifurcation points in the frequency response give the frequencies at which the stability of the system changes from sta- ble to unstable and vice versa. Amplitude frequency responses are used to predict the changes in stability, pull-in, and hardening or softening effects. The behavior of circular plates under axisym- metric vibration was investigated [13] using the MMS to obtain the amplitude frequency response of the system and show that cir- cular plates may undergo internal resonance due to interactions between the mode of vibration at certain frequencies. Investiga- tions using MMS and ROM method to obtain the amplitude fre- quency response of other structures such as cantilever resonators were reported in the literature [6,8,14,15]. Voltage–amplitude response is important for MEMS resonators. This response predicts the change in stability due to change in voltage. A continuous model of the pull-in effect in electrostati- cally actuated MEMS circular plates, limited to DC voltage, has been reported [16]. The static pull-in, i.e., the DC voltage is grad- ually increased until the center of the plate deflection leads to pull-in, and dynamic pull-in, i.e., the DC voltage is applied to the undeformed plate producing a rapid deflection leading to pull-in, have been investigated. Another model [17] of electrostatically actuated circular microplates, limited to DC voltage actuation, based on von Karman’s nonlinear bending theory, and including Casimir force, was used to investigate pull-in instability and vibration of prestressed microplates. Specifically, using the shoot- ing method, the static deformation and the pull-in parameters, and the small amplitude free vibration about predeformed bending position, were reported. A simplified model of elasto-electrostatic analysis of thin circular microplates used as ultrasonic transducers [18] was used to predict the pull-in DC voltage. The electrostatic force was expanded in Taylor series and terms up to the squared term were retained, and then the Galerkin-weighted residual tech- nique was used for predictions. The results were in agreement with ANSYS simulations. A similar study [19] reported the pull-in voltage and free vibrations of functionally graded material circu- lar microplate subjected to DC voltage and mechanical shock. The effect of surface stress on the pull-in instability [20] of elec- trostatically actuated circular nanoplates subjected to DC voltage and hydrostatic pressure was reported. It has been found that the surface stress effect is more significant in the pull-in of lower thickness nanoplates. Size-dependent behavior of capacitive circu- lar microplates [21] was reported as well. Using Hamilton’s 1 Corresponding author. Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received January 31, 2015; final manuscript received March 22, 2016; published online May 13, 2016. Assoc. Editor: Daniel J. Segalman. The United States Government retains, and by accepting the article for publication, the publisher acknowledges that the United States Government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this work, or allow others to do so, for United States government purposes. 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