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Efficient potential well escape for bi-stable Duffing oscillators

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Abstract

The problem of escape from a potential well of bi-stable oscillators has attracted attention given the diversity of physical and engineering systems described by this mathematical model. Most previous studies have considered quasi-static dynamics leading to escape. In devising efficient escape strategies for structures, transient conditions have not yet received adequate consideration. In this study, the intra-well nonlinear resonant dynamics of bi-stable systems are studied and exploited, yielding a time-efficient strategy for triggering minimal amplitude escape by employing transient perturbations. The response characteristics of both, the symmetric and asymmetric double-well Duffing oscillators are explored analytically to identify the stable solution branches for any given forcing configuration. Based on the basins of attraction of the stable attractors, a novel actuation methodology employing controlled perturbations in the phase of the forcing for driving the system into a series of high-amplitude limit cycle oscillations and eventual escape to the desired stable solution is proposed. Additionally, accelerated settling to the desired configuration is achieved by implementing state feedback techniques. The proposed algorithm serves as a potential tool for implementing fast shape adaptation in bi-stable structural systems.

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Notes

  1. For the purposes of this study, the first instance of the system response crossing over the potential barrier is considered as escape from the initial potential well.

  2. All numerical simulations in this study are conducted using the ODE45 Solver in MATLAB based on the Runge–Kutta method of numerical integration.

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Acknowledgements

J. P. Udani and A. F. Arrieta gratefully acknowledge the support of the Purdue Research Foundation (PRF); this research was funded through start-up funds provided by PRF.

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  1. School of Mechanical Engineering, Purdue University, West Lafayette, IN, 47907, USA

    Janav P. Udani & Andres F. Arrieta

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  1. Janav P. Udani
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Correspondence to Andres F. Arrieta.

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Appendices

Comparison of forcing signals

See Fig. 14.

Fig. 14
figure 14

(Colour Online) Comparison of the forcing signals for a sample period of final 20 s before escape via quasi-static continuation dynamics (solid black line), transient dynamics (dashed red line) and excitation at linearized natural frequency of the potential well (dot blue line)

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Algorithm

See Fig. 15.

Fig. 15
figure 15

Detailed algorithm for efficient escape strategy from stable well of a Duffing system as described in Sect. 4.1. Here, \(\phi _{initial}\) is the initial phase of the forcing signal, \(t_{max}\) is the maximum time before which the system should jump between limit cycles, \(\phi _{rel}\) is the relative phase of the forcing signal and \(\phi _{ext}\) is the additional phase perturbation introduced

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Udani, J.P., Arrieta, A.F. Efficient potential well escape for bi-stable Duffing oscillators. Nonlinear Dyn 92, 1045–1059 (2018). https://doi.org/10.1007/s11071-018-4107-3

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