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Research Papers

Excitation-Induced Stability in a Bistable Duffing Oscillator: Analysis and Experiments

[+] Author and Article Information
Z. Wu

Department of Mechanical Engineering,
University of Michigan,
Ann Arbor, MI 48109-2125
e-mail: wuzhen@umich.edu

R. L. Harne, K. W. Wang

Department of Mechanical Engineering,
University of Michigan,
Ann Arbor, MI 48109-2125

Manuscript received September 10, 2013; final manuscript received February 27, 2014; published online October 14, 2014. Assoc. Editor: D. Dane Quinn.

J. Comput. Nonlinear Dynam 10(1), 011016 (Oct 14, 2014) (7 pages) Paper No: CND-13-1217; doi: 10.1115/1.4026974 History: Received September 10, 2013; Revised February 27, 2014

The excitation-induced stability (EIS) phenomenon in a harmonically excited bistable Duffing oscillator is studied in this paper. Criteria to predict system and excitation conditions necessary to maintain EIS are derived through a combination of the method of harmonic balance, perturbation theory, and stability theory for Mathieu's equation. Accuracy of the criteria is verified by analytical and numerical studies. We demonstrate that damping primarily determines the likelihood of attaining EIS response when several dynamics coexist while excitation level governs both the existence and frequency range of the EIS region, providing comprehensive guidance for realizing or avoiding EIS dynamics. Experimental results are in good agreement regarding the comprehensive influence of excitation conditions on the inducement of EIS.

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Figures

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Fig. 1

Schematic of Mathieu's resonance tongue. Hatched areas correspond to stable domains of Mathieu's equation. B1, B2, and B3 are the first three transition curves on the δ-ɛ plane.

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Fig. 2

(a) Steady-state response amplitude. Red solid (dashed) lines are stable interwell (intrawell) responses whose stability are determined via Jacobian analysis; black dashed dotted lines are the first three approximated transition curves determined from Mathieu's resonance tongue. Blue circles (crosses) correspond to interwell (intrawell) responses computed via direct numerical integration. Gray lines are analytically predicted unstable responses. Criteria predict stable interwell response regions between S1 and S2 or above S3 (as indicated by arrows). Two interwell responses A and B coexist at ω = 2.3. Corresponding (b) phase portrait, (c) time series, and (d) frequency spectra are presented, with black and gray representing regular interwell and EIS, respectively. Blue dashed lines (dots) indicate the positions of the stable equilibria.

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Fig. 3

(a) EIS response amplitude as a function of damping and frequency with system parameter p = 5. Increasing lightness of the contour represents increasing response amplitude. Dashed lines correspond to estimated frequency boundaries using simplified form of Eqs. (10) and (12). (b) and (c) are two representative responses with damping γ = 0.05 and γ = 0.5. Solid black (gray) lines indicate interwell (intrawell) responses and dashed gray lines are analytically determined unstable responses.

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Fig. 4

(a) Basin of attraction map for system parameters p = 5, ω = 2.3 with white, gray, and black shading representing regular interwell response, EIS, and period-3 harmonic interwell responses, respectively. (b) Steady-state time domain responses (T = 2π/ω) and (c) corresponding phase portraits for different initial conditions with damping γ = 0.5. In (b) and (c) blue dashed lines and dots, respectively, represent positions of two stable equilibria.

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Fig. 5

(a) EIS response amplitude as a function of excitation level and frequency with system parameter γ = 0.1. Increasing lightness of the contour represents increasing in response amplitude. Dashed lines correspond to estimated frequency boundaries using simplified form of Eqs. (10) and (12). (b) and (c) are two representative responses with excitation level p = 0.2 and p = 2. Solid black (gray) lines indicate interwell (intrawell) responses and dashed gray lines are analytically determined unstable responses.

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Fig. 6

(a) Schematic of experimental test setup. (b) Photograph of test setup.

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Fig. 7

(a) Experimentally determined phase portraits with dashed black, solid gray, and black representing intrawell, EIS, and regular interwell responses, respectively. Two dots represent two stable equilibrium positions of the bistable oscillator. (b) Experimentally determined displacement frf of the bistable oscillator as a function of shaker input frequency and amplitude. Circles and triangles are experimentally determined frequency boundaries of the stable EIS region.

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