Parametric Excitation of a Microbeam-String With Asymmetric Electrodes: Multimode Dynamics and the Effect of Nonlinear Damping

[+] Author and Article Information
Karin Mora

Department of Mathematics,
University of Paderborn,
Paderborn 33098, Germany
e-mail: kmora@math.upb.de

Oded Gottlieb

Faculty of Mechanical Engineering,
Technion—Israel Institute of Technology,
Haifa 32000, Israel
e-mail: oded@technion.ac.il

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received November 30, 2016; final manuscript received April 27, 2017; published online May 30, 2017. Assoc. Editor: Steven W. Shaw.

J. Vib. Acoust 139(4), 040903 (May 30, 2017) (9 pages) Paper No: VIB-16-1569; doi: 10.1115/1.4036632 History: Received November 30, 2016; Revised April 27, 2017

The dynamic motion of a parametrically excited microbeam-string affected by nonlinear damping is considered asymptotically and numerically. It is assumed that the geometrically nonlinear beam-string, subject to only modulated alternating current voltage, is closer to one of the electrodes, thus resulting in an asymmetric dual gap configuration. A consequence of these novel assumptions is a combined parametric and hard excitation in the derived continuum-based model that incorporates both linear viscous and nonlinear viscoelastic damping terms. To understand how these assumptions influence the beam's performance, the conditions that lead to both principal parametric resonance and a three-to-one internal resonance are investigated. Such conditions are derived analytically from a reduced-order nonlinear model for the first three modes of the microbeam-string using the asymptotic multiple-scales method which requires reconstitution of the slow-scale evolution equations to deduce an approximate spatio-temporal solution. The response is investigated analytically and numerically and reveals a bifurcation structure that includes coexisting in-phase and out-of-phase solutions, Hopf bifurcations, and conditions for the loss of orbital stability culminating with nonstationary quasi-periodic solutions and chaotic strange attractors.

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Hornstein, S. , and Gottlieb, O. , 2012, “ Nonlinear Multimode Dynamics and Internal Resonances of the Scan Process in Noncontacting Atomic Force Microscopy,” J. Appl. Phys., 112(7), p. 074314.
Hacker, E. , and Gottlieb, O. , 2012, “ Internal Resonance Based Sensing in Non-Contact Atomic Force Microscopy,” Appl. Phys. Lett., 101(5), p. 053106. [CrossRef]
Buks, E. , and Yurke, B. , 2006, “ Mass Detection With a Nonlinear Nanomechanical Resonator,” Phys. Rev. E, 74(4), p. 046619. [CrossRef]
Souayeh, S. , and Kacem, N. , 2014, “ Computational Models for Large Amplitude Nonlinear Vibrations of Electrostatically Actuated Carbon Nanotube-Based Mass Sensors,” Sens. Actuators A, 208, pp. 10–20. [CrossRef]
Ilic, B. , Yang, Y. , Aubin, K. , Reichenbach, R. , Krylov, S. , and Craighead, H. , 2005, “ Enumeration of DNA Molecules Bound to a Nanomechanical Oscillator,” Nano Lett., 5(5), pp. 925–929. [CrossRef] [PubMed]
Vidal-Álvarez, G. , Torres, F. , Barniol, N. , and Gottlieb, O. , 2015, “ The Influence of the Parasitic Current on the Nonlinear Electrical Response of Capacitively Sensed Cantilever Resonators,” J. Appl. Phys., 117(15), p. 154502. [CrossRef]
Vidal-Álvarez, G. , Agustí, J. , Torres, F. , Abadal, G. , Barniol, N. , Llobet, J. , Sansa, M. , Fernández-Regúlez, M. , Pérez-Murano, F. , San Paulo, Á. , and Gottlieb, O. , 2015, “ Top-Down Silicon Microcantilever With Coupled Bottom-Up Silicon Nanowire for Enhanced Mass Resolution,” Nanotechnology, 26(14), p. 145502. [CrossRef] [PubMed]
Hassanpour, P. A. , Nieva, P. M. , and Khajepour, A. , 2011, “ Stochastic Analysis of a Novel Force Sensor Based on Bifurcation of a Micro-Structure,” J. Sound Vib., 330(23), pp. 5753–5768. [CrossRef]
Rhoads, J. F. , Shaw, S. W. , and Turner, K. L. , 2006, “ The Nonlinear Response of Resonant Microbeam Systems With Purely-Parametric Electrostatic Actuation,” J. Micromech. Microeng., 16(5), pp. 890–899. [CrossRef]
Nayfeh, A. H. , Younis, M. I. , and Abdel-Rahman, E. M. , 2007, “ Dynamic Pull-In Phenomenon in MEMS Resonators,” Nonlinear Dyn., 48(1–2), pp. 153–163. [CrossRef]
Karabalin, R. B. , Cross, M. C. , and Roukes, M. L. , 2009, “ Nonlinear Dynamics and Chaos in Two Coupled Nanomechanical Resonators,” Phys. Rev. B, 79(16), p. 165309. [CrossRef]
Zaitsev, S. , Shtempluck, O. , Buks, E. , and Gottlieb, O. , 2012, “ Nonlinear Damping in a Micromechanical Oscillator,” Nonlinear Dyn., 67(1), pp. 859–883. [CrossRef]
Gutschmidt, S. , and Gottlieb, O. , 2012, “ Nonlinear Dynamic Behavior of a Microbeam Array Subject to Parametric Actuation at Low, Medium and Large DC-Voltages,” Nonlinear Dyn., 67(1), pp. 1–36. [CrossRef]
Shoshani, O. , and Shaw, S. W. , 2016, “ Generalized Parametric Resonance,” SIAM J. Appl. Dyn. Syst., 15(2), pp. 767–788. [CrossRef]
Wang, F. , and Bajaj, A. , 2010, “ Nonlinear Dynamics of a Three-Beam Structure With Attached Mass and Three-Mode Interactions,” Nonlinear Dyn., 62(1–2), pp. 461–484. [CrossRef]
Ruzziconi, L. , Younis, M. , and Lenci, S. , 2013, “ An Electrically Actuated Imperfect Microbeam: Dynamical Integrity for Interpreting and Predicting the Device Response,” Meccanica, 48(7), pp. 1761–1775. [CrossRef]
Nguyen, C. H. , Nguyen, D. S. , and Halvorsen, E. , 2014, “ Experimental Validation of Damping Model for a MEMS Bistable Electrostatic Energy Harvester,” J. Phys.: Conf. Ser., 557(1), p. 012114. [CrossRef]
Kacem, N. , Hentz, S. , Pinto, D. , Reig, B. , and Nguyen, V. , “ Nonlinear Dynamics of Nanomechanical Beam Resonators: Improving the Performance of NEMS-Based Sensors,” Nanotechnology, 20(27), p. 275501. [CrossRef] [PubMed]
Nayfeh, A. H. , and Mook, D. T. , 1995, Nonlinear Oscillations, Wiley Classics Library, Wiley, New York.
Kacem, N. , and Hentz, S. , 2009, “ Bifurcation Topology Tuning of a Mixed Behavior in Nonlinear Micromechanical Resonators,” Appl. Phys. Lett., 95(18), p. 183104. [CrossRef]
Sobreviela, G. , Vidal-Álvarez, G. , Riverola, M. , Uranga, A. , Torres, F. , and Barniol, N. , 2017, “ Suppression of the A-f-Mediated Noise at the Top Bifurcation Point in a MEMS Resonator With Both Hardening and Softening Hysteretic Cycles,” Sens. Actuators A, 256, pp. 59–65. [CrossRef]
Kacem, N. , Baguet, S. , Dufour, R. , and Hentz, S. , 2011, “ Stability Control of Nonlinear Micromechanical Resonators Under Simultaneous Primary and Superharmonic Resonances,” Appl. Phys. Lett., 98(19), p. 193507. [CrossRef]
Shabana, A. A. , 1991, Theory of Vibration, Vol. 2, Springer-Verlag, New York.
Meirovitch, L. , 1997, Principles and Techniques of Vibrations, Vol. 1, Prentice Hall, Upper Saddle River, NJ.
Pandey, A. K. , Gottlieb, O. , Shtempluck, O. , and Buks, E. , 2010, “ Performance of an AuPd Micromechanical Resonator as a Temperature Sensor,” Appl. Phys. Lett., 96(20), p. 203105. [CrossRef]
Nayfeh, A. H. , 2005, “ Resolving Controversies in the Application of the Method of Multiple Scales and the Generalized Method of Averaging,” Nonlinear Dyn., 40(1), pp. 61–102. [CrossRef]
Gutschmidt, S. , and Gottlieb, O. , 2010, “ Bifurcations and Loss of Orbital Stability in Nonlinear Viscoelastic Beam Arrays Subject to Parametric Actuation,” J. Sound Vib., 329(18), pp. 3835–3855. [CrossRef]
Govaerts, W. , Kuznetsov, Y. A. , and Dhooge, A. , 2005, “ Numerical Continuation of Bifurcations of Limit Cycles in MATLAB,” SIAM J. Sci. Comput., 27(1), pp. 231–252. [CrossRef]
Govaerts, W. , Kuznetsov, Y. A. , and Dhooge, A. , 2003, “ MATCONT: A MATLAB Package for Numerical Bifurcation Analysis of ODEs,” ACM Trans. Math. Software, 29(2), pp. 141–164. [CrossRef]
Sahin, O. , Magonov, S. , Su, C. , Quate, C. F. , and Solgaard, O. , 2007, “ An Atomic Force Microscope Tip Designed to Measure Time-Varying Nanomechanical Forces,” Nat. Nanotechnol., 2(8), pp. 507–514. [CrossRef] [PubMed]


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Fig. 1

Schematic of a microbeam with electrodes, where w̃(s̃,t) is the transverse beam response prior to scaling

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Fig. 2

One-mode frequency response curve with unstable (dashed) and stable (solid) branches of the two configurations for parameters in P1. Markers ×and+ show numerical validation of the asymptotic solution, which agrees well.

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Fig. 7

Period-doubled response in three-mode dynamical system with Ω=1 in phase space (w(1/2,τ),wτ(1/2,τ)) (gray) and its Poincaré map (black) for parameters in P1 and Γ=0.5

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Fig. 3

Three-mode frequency response curve for parameters in P1 and Γ=0.5 (case 1): (a) stable (solid) and unstable (dashed) equilibria: εa=εc=0 (very thick), εa (thick), εc (thin), and their fold points (circle) and (b) magnification of εc amplitude showing two maxima

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Fig. 4

Frequency response curve of the three-mode steady-state scenario with parameters in P2 and Γ=0.5 (case 2). Stable (solid line) and unstable (dashed line) branches undergo pitchfork (cross), fold (circle), and Hopf (pentagram) bifurcations. Region 4 comprising a supercritical Hopf bifurcation at Ω=0.958 and a subcritical one at Ω=0.960 is very narrow.

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Fig. 5

Brute force bifurcation (thick) and continuation (thin) of the single-mode ODE with Γ=0.73 and parameters in P2. Frequency response of stable (thick) and unstable branches (thin) undergoing pitchfork (square), fold (circle), and period-doubling bifurcations (triangle). The period-doubling bifurcation introduces additional orbits near Ω=1.04. Inset: Frequency response curve for Γ = 1.

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Fig. 6

Three coexisting stable orbits (gray) and their Poincaré map (cross) observed in Fig. 5 at Ω=1.039 for parameters in P2 and Γ=0.73

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Fig. 8

Quasi-periodic response in three-mode dynamical system with Ω=1.01 in phase space (w(1/2,τ),wτ(1/2,τ)) (gray) and its Poincaré map (black) for parameters in P1 and Γ=0.5

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Fig. 9

Chaoticlike response in three-mode dynamical system with Ω=1.01 in phase space (w(1/2,τ),wτ(1/2,τ)) (gray) and its Poincaré map (black) for parameters in P1, Γ=0.5, and the hardening stiffness relations β13=9β11, β33=81β11



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