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Natural Frequencies of an Inhomogeneous Square Thin Plate

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Abstract

The purpose of the study is to analyze the inhomogeneity effect on the natural transverse vibrations of square thin plates using asymptotic methods, when the thickness and/or stiffness of the plate can be assumed almost constant. The results of calculations by asymptotic formulas, which are determined by the perturbation method, are compared with the numerical results obtained by the finite element method using the COMSOL software package.

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REFERENCES

  1. A. W. Leissa, Vibration of Plates (U. S. Government Printing Office, Washington, DC, 1969).

  2. L. Roshan and K. Yajuvindra, “Transverse vibrations of nonhomogeneous rectangular plates with variable thickness,” Mech. Adv. Mater. Struct. 20, 264–275 (2013).

    Article  Google Scholar 

  3. A. Ya. Grigorenko and T. V. Tregubenko, “Numerical and experimental analysis of natural vibrations of rectangular plates with variable thickness,” Int. Appl. Mech. 36, 268–270 (2000).

    Article  Google Scholar 

  4. R. H. Guti‘errez, P. A. A. Laura, and R. O. Grossi, “Vibrations of rectangular plates of bilinearly varying thickness and with general boundary conditions,” J. Sound Vib. 75, 323–328 (1981).

    Article  Google Scholar 

  5. D. J. Dawe, “Vibration of rectangular plates of variable thickness,” J. Mech. Eng. Sci. 8, 42–51 (1966).

    Article  Google Scholar 

  6. R. B. Bhat, P. A. A. Laura, R. G. Gutierrez, V. H. Cortinez, and H. C. Sanzi, “Numerical experiments on the determination of natural frequencies of transverse vibrations of rectangular plates of non-uniform thickness,” J. Sound Vib. 138, 205–219 (1990).

    Article  Google Scholar 

  7. B. Singha and V. Saxena, “Transverse vibration of a rectangular plate with bidirectional thickness variation,” J. Sound Vib. 198, 51–65 (1996).

    Article  Google Scholar 

  8. L. Long-Yuan, “Vibration analysis of moderate-thick plates with slowly varying thickness,” Appl. Math. Mech. 7, 707–714 (1986).

    Article  Google Scholar 

  9. M. Huang, Y. Xu, and B. Cao, “Free vibration analysis of cantilever rectangular plates with variable thickness,” Appl. Mech. Mater. 130–134, 2774–2777 (2011). https://doi.org/10.4028/www.scientific.net/AMM.130-134.2774

    Article  Google Scholar 

  10. M. D. Olson and C. R. Hazell, “Vibrations of a square plate with parabolically varying thickness,” J. Sound Vib. 62, 399–410 (1979).

    Article  Google Scholar 

  11. S. M. Bauer, S. B. Filippov, A. L. Smirnov, P. E. Tovstik, and R. Vaillancourt, Asymptotic Methods in Mechanics of Solids (Birkhäuser, Basel, 2015).

    Book  Google Scholar 

  12. G. P. Vasiliev and A. L. Smirnov, “Free vibration frequencies of a circular thin plate with variable parameters,” Vestn. St. Petersburg Univ., Math. 53, 351–357 (2020).

    MATH  Google Scholar 

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Funding

The study was partially supported by the Russian Foundation for Basic Research, project nos. 18-01-00832-a and 19-01-00208-a.

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Authors and Affiliations

  1. St. Petersburg State University, 199034, St. Petersburg, Russia

    G. P. Vasiliev & A. L. Smirnov

Authors
  1. G. P. Vasiliev
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  2. A. L. Smirnov
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Correspondence to G. P. Vasiliev or A. L. Smirnov.

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Translated by N. Semenova

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Vasiliev, G.P., Smirnov, A.L. Natural Frequencies of an Inhomogeneous Square Thin Plate. Vestnik St.Petersb. Univ.Math. 54, 119–124 (2021). https://doi.org/10.1134/S1063454121020138

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  • DOI: https://doi.org/10.1134/S1063454121020138

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