Free in-plane vibration of heterogeneous nanoplates using Ritz method


1 Assistant Professor, Faculty of Mechanical Engineerng, Urmia University of Technology, Urmia, Iran

2 M.Sc. Student, Faculty of Mechanical Engineerng, Urmia University of Technology, Urmia, Iran


In this paper, the Ritz method has been employed to analyze the free in-plane vibration of heterogeneous (non-uniform) rectangular nanoplates corresponding to Eringen’s nonlocal elasticity theory. The non-uniformity is taken into account using combinations of linear and quadratic forms in the thickness, material density and Young’s modulus. Two-dimensional boundary characteristic orthogonal polynomials are applied in the Ritz method in order to examine the nonlocal effect, aspect ratio, length of nanoplate and non-uniformity parameters on the vibrational behaviors of the nanoplate. Results are verified with the available published data and good agreements are observed. The outcomes confirm apparent dependency of in-plane frequency of nanoplate on the small scale effect, non-uniformity, aspect ratio and boundary conditions. For instance, frequency parameter decreases by increasing the nonlocal parameter in all vibration modes; the frequency parameters increase with length and aspect ratio of nanoplates. Furthermore, the effect of nonlocal parameters on the frequency parameter is more prominent at the higher aspect ratios. Finally, the effect of nonlocal parameter on the in-plane modes is also presented in this analysis.


  • Free in-plane vibration of non-uniform nanoplates is studied using the Ritz method.
  • Effects of small-scale, non-uniformity and aspect ratio are investigated.
  • The in-plane frequencies are found significantly affected by nonlocal parameters.
  • Frequencies increase with Young’s modulus and decrease with density of nanoplates.
  • The evolution of mode shapes with increasing nonlocal parameters is visualized.


Main Subjects

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