Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)


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Smirnov A. L., Vasiliev G. P. Free vibration frequencies of a circular thin plate with nonlinearly perturbed parameters. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2021, vol. 21, iss. 2, pp. 227-237. DOI: 10.18500/1816-9791-2021-21-2-227-237, EDN: TSGRWC

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
31.05.2021
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Russian
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Article
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534.1:539.3
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TSGRWC

Free vibration frequencies of a circular thin plate with nonlinearly perturbed parameters

Autors: 
Smirnov Andrei L., St. Petersburg State University
Vasiliev Grigory P., St. Petersburg State University
Abstract: 

Transverse vibrations of an inhomogeneous circular thin plate are studied. The plates, which geometric and physical parameters slightly differ from constant ones and depend only on the radial coordinate, are analyzed. After separation of variables the obtained homogeneous ordinary differential equations together with homogeneous boundary conditions form a regularly perturbed boundary eigenvalue problem. For frequencies of free vibrations of a plate, which thickness and/or Young’s modulus nonlinearly depend on the radial coordinate asymptotic formulas are obtained by means of the perturbation method. As examples, free vibrations of a plate with parameters quadratically or exponentially depending on the radial coordinate, are examined. The effect of the small perturbation parameter on the behavior of frequencies is also analyzed under special conditions: i) for a plate, the mass of which is fixed, if the thickness is variable and ii) for a plate with the fixed average stiffness, if Young’s modulus is variable. Finally, effects of the boundary conditions and values of the wave numbers on the corrections to frequencies are studied. For a wide range of small parameter values, the asymptotic results for the lower vibration frequencies well agree with the results of finite element analysis with COMSOL Multiphysics 5.4 and the numerical results of other authors.

Acknowledgments: 
This work was supported by the Russian Foundation for Basic Research (projects No. 18-01-00832-a and No. 19-01-00208-a).
References: 
  1. Leissa A. W. Vibration of plates. Washington, US, Government Printing Office, 1969. 353 p.
  2. Bauer S. M., Filippov S. B., Smirnov A. L., Tovstik P. E., Vaillancourt R. Asymptotic methods in mechanics of solids. Basel, Birkhauser, 2015. 325 p. https://doi.org/10.1007/978-3-319-18311-4
  3. Vasiliev G. P., Smirnov A. L. Free vibration frequencies of a circular thin plate with variable parameters. Vestnik St. Petersburg University. Mathematics, 2020, vol. 53, no. 3, pp. 351–357. https://doi.org/10.1134/S1063454120030140
  4. Smirnov A. L. Free vibrations of annular circular and elliptic plates. COMPDYN Proceedings, 2019, vol. 2, pp. 3547–3555.
  5. Eisenberger M., Jabareen M. Axisymmetric vibrations of circular and annular plates with variable thickness. International Journal of Structural Stability and Dynamics, 2001, vol. 1, iss. 2, pp. 195–206. https://doi.org//10.1142/S0219455401000196
  6. Prasad C., Jain R. K., Soni S. R. Axisymmetric vibrations of circular plates of linearly varying thickness. Journal of Applied Mathematics and Physics (ZAMP), 1972, vol. 23, pp. 941–948. https://doi.org/10.1007/BF01596221
  7. Singh B., Saxena V. Axisymmetric vibration of a circular plate with exponential thickness variation. Journal of Sound and Vibration, 1996, vol. 192, iss. 1, pp. 35–42. https://doi.org/10.1006/jsvi.1996.0174
  8. Bauer S. M., Voronkova E. B. On natural frequencies of transversally isotropic circular plates. Vestnik St. Petersburg University. Mathematics, 2016, vol. 49, no. 1, pp. 77–80. https://doi.org/10.3103/S1063454116010027
  9. Anikina T. A., Vatulyan A. O., Uglich P. S. On the calculation of variable stiffness for a circular plate. Computational Technologies, 2012, vol. 17, no. 6, pp. 26–35 (in Russian).
  10. Bauer S. M., Voronkova E. B. On the unsymmetrical buckling of shallow spherical shells under internal pressure. Izvestiya of Saratov University. New Series. Series: Mathematics. Mechanics. Informatics, 2018, vol. 18, iss. 4, pp. 390–396 (in Russian). https://doi.org/10.18500/1816-9791-2018-18-4-390-396
Received: 
13.05.2020
Accepted: 
31.10.2020
Published: 
31.05.2021