Global dynamics perspective on macro- to nano-mechanics

Abstract

In about the last two decades, global nonlinear dynamics has been evolving in a revolutionary way, with the development of sophisticated techniques employing concepts/tools of dynamical systems, bifurcation, and chaos theory, and applications to a wide variety of mechanical/structural systems. The relevant achievements entail a substantial change of perspective in dealing with vibration problems and are ready to meaningfully affect the analysis, control, and design of systems at different scales, in multiphysics contexts too. After properly framing the subject within some main stages of developments of nonlinear dynamics in solid/structural mechanics, as occurred over the last 40 years, the article focuses on highlighting the role played by global analysis in unveiling the nonlinear response and actual safety of engineering systems in different environments. Reduced order models of macro-/micro-structures are considered. Global dynamics of a laminated plate with von Kármán nonlinearities, shear deformability, and full thermomechanical coupling allows to highlight the meaningful effects entailed by the slow transient thermal dynamics on the fast steady mechanical responses. An atomic force microcantilever is referred to for highlighting the severe worsening of overall stability associated with the application of an external feedback control and the importance of global dynamics for conceiving and effectively implementing a control procedure aimed at enhancing engineering safety. The last part of the article dwells on the general role that a global dynamics perspective is expected to play in the safe design of real systems, in the near future, focusing on how properly exploiting concepts and tools of dynamical integrity to evaluate response robustness in the presence of unavoidable imperfections, and to improve the system’s actual load carrying capacity.

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Abbreviations

AFM:

Atomic force microscope

BC:

Boundary crisis

BS:

Basin stability

DI:

Dynamical integrity

GIM:

Global integrity measure

IF:

Integrity factor

LIM:

Local integrity measure

MEMS:

Micro-electro-mechanical system

nD:

n-dimensional

ODE:

Ordinary differential equation

OGY:

Ott–Grebogi–Yorke

PD:

Period doubling

Pm:

m-period solution

ROM:

Reduced order model

sdof:

single-degree-of-freedom

SN:

Saddle-node

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Acknowledgements

Contributions of Stefano Lenci and Paulo Gonçalves to the research background of this feature article are acknowledged.

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Correspondence to Giuseppe Rega.

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Rega, G., Settimi, V. Global dynamics perspective on macro- to nano-mechanics. Nonlinear Dyn 103, 1259–1303 (2021). https://doi.org/10.1007/s11071-020-06198-x

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Keywords

  • Multistability
  • Global bifurcation
  • Dynamical integrity
  • Control
  • Thermomechanics
  • Atomic force microscopy
  • Practical stability
  • Engineering design