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Chaotic vibrations of size-dependent flexible rectangular plates.

V A Krysko1, J Awrejcewicz1, I V Papkova2

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This study introduces a new mathematical model for nonlinear vibrations in size-dependent plates using Cosserat continuum theory. It analyzes plate dynamics and stability loss under harmonic loads, offering reliable results for complex behaviors.

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Area of Science:

  • Solid Mechanics
  • Continuum Mechanics
  • Vibrational Analysis

Background:

  • Size-dependent effects are crucial in micro/nanoscale plates, influencing their mechanical behavior.
  • Nonlinear vibrations and stability loss are complex phenomena requiring advanced modeling techniques.

Purpose of the Study:

  • To develop a comprehensive mathematical model for nonlinear vibrations of size-dependent rectangular plates.
  • To investigate the stability loss of these plates under harmonic loading.
  • To validate the accuracy and convergence of employed numerical methods.

Main Methods:

  • Cosserat continuum theory and von Kármán geometric nonlinearities.
  • Energetic Hamilton principle for deriving governing equations.
  • Galerkin-Krylov-Bogoliubov method (GKBM), Finite Difference Method (FDM), and Newmark method for solving PDEs.
  • Analysis of convergence for FDM and GKBM.

Main Results:

  • A robust model for nonlinear vibrations of size-dependent plates was established.
  • The study investigated the convergence properties of FDM and GKBM, confirming GKBM's reliability for complex dynamics.
  • Stability loss under harmonic load was analyzed, providing insights into plate behavior.

Conclusions:

  • The proposed mathematical model accurately captures the nonlinear dynamics of size-dependent plates.
  • The combination of GKBM with higher-order approximations provides reliable results for regular and chaotic plate vibrations.
  • The study offers a valuable framework for analyzing the stability and vibrational characteristics of advanced plate structures.