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Multiple equilibrium states in large arrays of globally coupled resonators

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Abstract

This work considers the response of an array of oscillators, each with cubic nonlinear stiffness, in the presence of global reactive and dissipative coupling. The interplay between the excitation, global coupling, and nonlinearity gives rise to steady-state solutions for the population in which the response of each individual element depends on that of every other component. The present analysis continues the work presented in Borra et al. (J Sound Vib 393:232–239, 2017. https://doi.org/10.1016/j.jsv.2016.12.021), in which a continuum formulation was introduced to study the steady-state response as the number of oscillators increases. However, that work considered only parameter values and excitation levels for which the equilibrium distribution was unique. In the present work, the individual resonators are excited to sufficient amplitude to allow for multiple coexisting equilibrium population distributions. The method of multiple scales is then applied to the system to describe evolution equations for the amplitude and phase of each resonator. Because of the global nature of the coupling, this leads to an integro-differential equation for the stationary populations. Moreover, the characteristic equation used to determine the stability of these states is also an integral equation and admits both a discrete and continuous spectrum for its eigenvalues. The equilibrium structure of the system is studied as the reactive and dissipative coupling parameters are varied. For specific families of the equilibrium distributions, two-parameter bifurcation sheets can be constructed. These sheets are connected as individual resonators transition between different branches for the corresponding individual resonators. The resulting one-parameter bifurcation curves are then understood in terms of the collections of these identified bifurcation sheets. The analysis is demonstrated for a system of \(N = 10\) coupled resonators with mass detuning and extended results with \(N = 100\) coupled resonators are illustrated.

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Data availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Notes

  1. With \(N = 10\) all eigenvalues arise from the discrete spectrum. However, it is convenient to associate them with either the continuous or discrete spectra that exist in the continuum limit as \(N \rightarrow \infty \).

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Funding

This material is based upon work supported by the National Science Foundation under Grant Nos. 1537701 and 1537988.

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Authors and Affiliations

  1. Department of Mechanical Engineering, The University of Akron, Akron, OH, 44325–3903, USA

    Chaitanya Borra & D. Dane Quinn

  2. Department of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh, PA, 15261, USA

    Nikhil Bajaj

  3. School of Mechanical Engineering, Birck Nanotechnology Center, West Lafayette, IN, 47907, USA

    Jeffrey F. Rhoads

  4. Ray W. Herrick Laboratories, Purdue University, West Lafayette, IN, 47907, USA

    Jeffrey F. Rhoads

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  1. Chaitanya Borra
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  4. D. Dane Quinn
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All authors contributed to the study conception and design. Formulation and analysis were performed by CB and DDQ, who also completed the initial draft. All authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to D. Dane Quinn.

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Borra, C., Bajaj, N., Rhoads, J.F. et al. Multiple equilibrium states in large arrays of globally coupled resonators. Nonlinear Dyn 111, 18585–18603 (2023). https://doi.org/10.1007/s11071-023-08870-4

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