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Intense cross-scale energy cascades resembling “mechanical turbulence” in harmonically driven strongly nonlinear hierarchical chains of oscillators

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Abstract

In the present study, the dynamics of one- and two-degree-of-freedom linear oscillators attached to hierarchical chains of oscillators coupled by strongly nonlinear cubic springs in parallel with viscous dampers are investigated. The mass, stiffness, and damping properties of the nonlinear oscillators are specified to follow a certain scaling rule, in such a way that their mass, stiffness, and damping properties decrease along the chain. The linear oscillators are driven harmonically near to their primary resonances. For sufficiently low excitation levels, linear behavior prevails leading to nearly harmonic responses, where the mechanical energy remains localized in the directly forced linear oscillator. For sufficiently high excitation level (that is, above a critical level of the forcing amplitude), however, the system responds in a chaotic way, where a substantial part of the energy is transferred across the hierarchical nonlinear chain in an energy cascade. Interestingly, there exist “chaotic bands” where all cubic oscillators are simultaneously activated or deactivated depending on the frequency; hence, a chaotic synchronization occurs in the system. When energy cascading is initiated the smaller-scale cubic oscillators become driven by the chaotic dynamics of the larger-scale cubic oscillators. Once the oscillator chain enters the chaotic response regime, the pattern of nonlinear energy cascading resembles (but is not identical to) the –5/3 Kolmogorov energy cascading power law observed in fully developed turbulence. Similarities and differences in the observed mechanical energy cascades and those realized in turbulent flows are discussed.

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Acknowledgements

This work was supported in part by Grant No. 201908120069 from China Scholarship Council which supported the sabbatical leave of Jian’en Chen to the University of Illinois. This support is greatly acknowledged.

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Authors and Affiliations

  1. Tianjin Key Laboratory for Advanced Mechatronic System Design and Intelligent Control, Tianjin University of Technology, Tianjin, People’s Republic of China

    Jian En Chen

  2. Institute for Aircraft Propulsion Systems, University of Stuttgart, Stuttgart, Germany

    Timo Theurich & Malte Krack

  3. Mechanical Engineering, Massachusetts Institute of Technology, Cambridge , USA

    Themistoklis Sapsis

  4. Aerospace Engineering, University of Illinois, Urbana, USA

    Lawrence A. Bergman

  5. Mechanical Science and Engineering, University of Illinois, Urbana, USA

    Alexander F. Vakakis

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  1. Jian En Chen
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  2. Timo Theurich
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  4. Themistoklis Sapsis
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  5. Lawrence A. Bergman
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  6. Alexander F. Vakakis
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Chen, J.E., Theurich, T., Krack, M. et al. Intense cross-scale energy cascades resembling “mechanical turbulence” in harmonically driven strongly nonlinear hierarchical chains of oscillators. Acta Mech 233, 1289–1305 (2022). https://doi.org/10.1007/s00707-022-03159-w

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  • DOI: https://doi.org/10.1007/s00707-022-03159-w

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