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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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. 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. 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. 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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