Advances in nano research

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Assessment of nonlocal nonlinear free vibration of bi-directional functionally-graded Timoshenko nanobeams

  • Elnaz Zare (Department of Civil Engineering, Yasouj University) ;
  • Daria K. Voronkova (Department of Mathematics and Natural Sciences, Gulf University for Science and Technology) ;
  • Omid Faraji (Department of Civil Engineering, Imam Hossein University) ;
  • Hamidreza Aghajanirefah (Department of Civil Engineering, Faculty of Engineering, Qazvin Branch Islamic Azad University) ;
  • Hamid Malek Nia (Department of of Civil Engineering, Arak Branch, Islamic Azad University) ;
  • Mohammad Gholami (Department of Civil Engineering, Yasouj University) ;
  • Mojtaba Gorji Azandariani (Department of Civil Engineering, Semnan University)
  • Received : 2023.02.24
  • Accepted : 2023.09.23
  • Published : 2024.05.25
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Abstract

The current study employs the nonlocal Timoshenko beam (NTB) theory and von-Kármán's geometric nonlinearity to develop a non-classic beam model for evaluating the nonlinear free vibration of bi-directional functionally-graded (BFG) nanobeams. In order to avoid the stretching-bending coupling in the equations of motion, the problem is formulated based on the physical middle surface. The governing equations of motion and the relevant boundary conditions have been determined using Hamilton's principle, followed by discretization using the differential quadrature method (DQM). To determine the frequencies of nonlinear vibrations in the BFG nanobeams, a direct iterative algorithm is used for solving the discretized underlying equations. The model verification is conducted by making a comparison between the obtained results and benchmark results reported in prior studies. In the present work, the effects of amplitude ratio, nanobeam length, material distribution, nonlocality, and boundary conditions are examined on the nonlinear frequency of BFG nanobeams through a parametric study. As a main result, it is observed that the nonlinear vibration frequencies are greater than the linear vibration frequencies for the same amplitude of the nonlinear oscillator. The study finds that the difference between the dimensionless linear frequency and the nonlinear frequency is smaller for CC nanobeams compared to SS nanobeams, particularly within the α range of 0 to 1.5, where the impact of geometric nonlinearity on CC nanobeams can be disregarded. Furthermore, the nonlinear frequency ratio exhibits an increasing trend as the parameter µ is incremented, with a diminishing dependency on nanobeam length (L). Additionally, it is established that as the nanobeam length increases, a critical point is reached at which a sharp rise in the nonlinear frequency ratio occurs, particularly within the nanobeam length range of 10 nm to 30 nm. These findings collectively contribute to a comprehensive understanding of the nonlinear vibration behavior of BFG nanobeams in relation to various parameters.

Keywords

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