IOP PUBLISHING JOURNAL OF MICROMECHANICS AND MICROENGINEERING J. Micromech. Microeng. 19 (2009) 045013 (14pp) doi:10.1088/0960-1317/19/4/045013 On the nonlinear resonances and dynamic pull-in of electrostatically actuated resonators Fadi M Alsaleem, Mohammad I Younis and Hassen M Ouakad Department of Mechanical Engineering, State University of New York at Binghamton, Binghamton, NY 13902, USA E-mail: myounis@binghamton.edu Received 1 August 2008, in final form 11 February 2009 Published 17 March 2009 Online at stacks.iop.org/JMM/19/045013 Abstract We present modeling, analysis and experimental investigation for nonlinear resonances and the dynamic pull-in instability in electrostatically actuated resonators. These phenomena are induced by exciting a microstructure with nonlinear forcing composed of a dc parallel-plate electrostatic load superimposed on an ac harmonic load. Nonlinear phenomena are investigated experimentally and theoretically including primary resonance, superharmonic and subharmonic resonances, dynamic pull-in and the escape-from-potential-well phenomenon. As a case study, a capacitive sensor made up of two cantilever beams with a proof mass attached to their tips is studied. A nonlinear spring–mass–damper model is utilized accounting for squeeze-film damping and the parallel-plate electrostatic force. Long-time integration and a global dynamic analysis are conducted using a finite-difference method combined with the Floquet theory to capture periodic orbits and analyze their stability. The domains of attraction (basins of attraction) for data points on the frequency–response curve are calculated numerically. Dover cliff integrity curves are calculated and the erosion of the safe basin of attraction is investigated as the frequency of excitation is swept passing primary resonance and dynamic pull-in. Conclusions are presented regarding the safety and integrity of MEMS resonators based on the simulated basin of attraction and the observed experimental data. (Some figures in this article are in colour only in the electronic version) 1. Introduction There has been an increasing interest in the area of dynamics of MEMS in recent years [1–28]. This has been boosted by the demand to design new MEMS devices of lower cost and smarter functions as well as the need to improve the performance and reliability of existing MEMS devices. Many of these employ a microstructure or more that moves or vibrates. The inherent nonlinearities in many MEMS devices, such as those due to electrostatic forces and squeeze- film damping, have led to a growing interest in the field of nonlinear dynamics of MEMS. Recent researches have revealed a multitude of nonlinear phenomena in these systems. These include dynamic pull-in [1, 2], parametric resonance [3, 4], hardening-type and softening-type behavior [5], hysteresis [6, 7], multi-valued responses and jumps [2], secondary resonances [8–10], period-doubling bifurcations [2, 11] and chaos [11, 12]. Because of the common use of microbeams in MEMS, their dynamic behavior has been also under extensive research. Liu et al [13] simulated an electrostatically controlled cantilever microbeam. They showed period- doubling bifurcation, chaos, Hopf bifurcation and strange attractors using the Poincar´ e map method. They concluded that for some cases, the stable operation range is reduced because of a chaotic response. Lenci and Rega [14] used a Melnikov analysis to design a controller, which is based on the application of coupled harmonic and superharmonic voltage signals, to stabilize the motion of beam-based systems. The proposed controller is based on shifting the homoclinic bifurcation associated with dynamic pull-in to higher excitation amplitudes. Using a Duffing-like equation 0960-1317/09/045013+14$30.00 1 © 2009 IOP Publishing Ltd Printed in the UK