High-order subharmonic parametric resonance of nonlinearly coupled micromechanical oscillators

Abstract.This paper studies parametric resonance of coupled micromechanical oscillators under periodically varying nonlinear coupling forces. Different from most of previous related works in which the periodically varying coupling forces between adjacent oscillators are linearized, our work focuses on new physical phenomena caused by the periodically varying nonlinear coupling. Harmonic balance method (HBM) combined with Newton iteration method is employed to find steady-state periodic solutions. Similar to linearly coupled oscillators studied previously, the present model predicts superharmonic parametric resonance and the lower-order subharmonic parametric resonance. On the other hand, the present analysis shows that periodically varying nonlinear coupling considered in the present model does lead to the appearance of high-order subharmonic parametric resonance when the external excitation frequency is a multiple or nearly a multiple (≥3) of one of the natural frequencies of the oscillator system. This remarkable new phenomenon does not appear in the linearly coupled micromechanical oscillators studied previously, and makes the range of exciting resonance frequencies expanded to infinity. In addition, the effect of a linear damping on parametric resonance is studied in detail, and the conditions for the occurrence of the high-order subharmonics with a linear damping are discussed.

[1]  C. Broeck,et al.  Coupled parametric oscillators , 1999 .

[2]  A. Mioduchowski,et al.  Structural instability of a parallel array of mutually attracting identical microbeams , 2006 .

[3]  Steven W. Shaw,et al.  Institute of Physics Publishing Journal of Micromechanics and Microengineering the Nonlinear Response of Resonant Microbeam Systems with Purely-parametric Electrostatic Actuation , 2022 .

[4]  Jeff Moehlis,et al.  Generalized parametric resonance in electrostatically actuated microelectromechanical oscillators , 2006 .

[5]  C. Mastrangelo.  Adhesion-related failure mechanisms in micromechanical devices , 1997 .

[6]  R. Syms,et al.  Characteristic modes of electrostatic comb-drive X-Y microactuators , 2000 .

[7]  L Q English,et al.  Study of intrinsic localized vibrational modes in micromechanical oscillator arrays. , 2003, Chaos.

[8]  S. K. Korovin,et al.  Approximation Procedures in Nonlinear Oscillation Theory , 1994 .

[9]  Michael L. Roukes,et al.  Electrically tunable collective response in a coupled micromechanical array , 2002 .

[10]  W. K. Tso,et al.  Parametric Excitation of a Nonlinear System , 1965 .

[11]  C. Hierold.  From micro- to nanosystems: mechanical sensors go nano , 2004 .

[12]  T. Michalske,et al.  Frequency-dependent electrostatic actuation in microfluidic MEMS , 2003, Journal of Microelectromechanical Systems.

[13]  Yaron Bromberg,et al.  Response of discrete nonlinear systems with many degrees of freedom. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[14]  M. Roukes,et al.  Metastability and the Casimir effect in micromechanical systems , 2000, cond-mat/0008096.

[15]  Philip Dowd,et al.  Tilted folded-beam suspension for extending the stable travel range of comb-drive actuators , 2003 .

[16]  J. Pelesko,et al.  Modeling MEMS and NEMS , 2002 .

[17]  Collective modes in parametrically excited oscillator arrays , 2002 .

[18]  Charles M. Lieber,et al.  Nanobeam Mechanics: Elasticity, Strength, and Toughness of Nanorods and Nanotubes , 1997 .

[19]  N. Aluru,et al.  Calculation of pull-in voltages for carbon-nanotube-based nanoelectromechanical switches , 2002 .

[20]  A. J. Sievers,et al.  Optical manipulation of intrinsic localized vibrational energy in cantilever arrays , 2004, nlin/0403031.

[21]  Parametric resonance in coupled oscillators driven by colored noise , 2004 .

[22]  R. Legtenberg,et al.  Stiction in surface micromachining , 1996 .

[23]  T. Ebbesen,et al.  Exceptionally high Young's modulus observed for individual carbon nanotubes , 1996, Nature.

[24]  B. Hubbard,et al.  Observation of locked intrinsic localized vibrational modes in a micromechanical oscillator array. , 2003, Physical review letters.

[25]  Maxim Zalalutdinov,et al.  Frequency-tunable micromechanical oscillator , 2000 .

[26]  A. J. Sievers,et al.  Colloquium: Nonlinear energy localization and its manipulation in micromechanical oscillator arrays , 2006 .

[27]  Jian Zhu,et al.  Surface-forces-driven instability of comb-drive microcantilevers in MEMS , 2006 .

[28]  M. Mond,et al.  Stability Analysis Of The Non-Linear Mathieu Equation , 1993 .

[29]  C. R. Willis,et al.  Discrete Breathers , 1997 .

[30]  R. Howe,et al.  Critical Review: Adhesion in surface micromechanical structures , 1997 .

[31]  T. Kenny,et al.  Design of large deflection electrostatic actuators , 2003 .

[32]  M. Blencowe.  Nanoelectromechanical systems , 2005, cond-mat/0502566.

[33]  N. Mclachlan.  Theory and Application of Mathieu Functions , 1965 .

[34]  W. D. Heer,et al.  Electrostatic deflections and electromechanical resonances of carbon nanotubes , 1999, Science.

[35]  Ron Lifshitz,et al.  Response of parametrically driven nonlinear coupled oscillators with application to micromechanical and nanomechanical resonator arrays , 2003 .

[36]  M. Roukes.  Nanoelectromechanical systems face the future , 2001 .