Free vibrations of anisotropic rectangular plate laying on a heterogeneous viscouselastic basis

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The aim of the work. Free, transverse vibrations are considered heterogeneous along the three spatial coordinates of rectangular plates lying on an inhomogeneous viscoelastic base. It is assumed that the boundary conditions are homogeneous. A closed solution for the problem of free vibration of an inhomogeneous rectangular orthotropic plate based on an inhomogeneous viscoelastic foundation is developed in the article. Young's moduli and the density of the orthotropic plate continuously change with respect to three spatial coordinates, while the characteristics of a viscoelastic base change depending on the coordinates in the plane. Methods. The corresponding equation of motion is obtained using the classical theory of plates. The solution to the problem was constructed using the method of separation of variables and the Bubnov - Galerkin method. Results. Explicit formulas of the fundamental tone of the frequency of the transverse vibration of an anisotropic plate lying on an inhomogeneous viscoelastic base are determined. The influence of heterogeneity of orthotropic materials, viscosity inhomogeneities, inelastic and elastic substrates at dimensionless plate frequencies have been studied in detail.

About the authors

Vaqif C. Haciyev

National Academy of Sciences of Azerbaijan

Author for correspondence.

Doctor of Physical and Mathematical Sciences, Professor, Head of the Department of Theory of Elasticity and Plasticity, Institute of Mathematics and Mechanics

9 B. Wahabzadeh St., Baku, АZ1143, Republic of Azerbaijan

Gulnar R. Mirzoeva

National Academy of Sciences of Azerbaijan


Doctor of Philosophy in Mechanics, senior researcher of Department of Theory of Elasticity and Plasticity, Institute of Mathematics and Mechanics.

9 B. Wahabzadeh St., Baku, АZ1143, Republic of Azerbaijan

Matlab G. Agayarov

Sumgait State University


Doctor of Philosophy in Mathematics and Mechanics Sciences, Associate Professor, Head of Additional Education Center

43 quarter, Sumgait, AZ50008, Republic of Azerbaijan


  1. Lomakin V.A. (1976). Teoriya uprugosti neodnorodnyh tel [The theory of elasticity of inhomogeneous walked]. Moscow, Publishing House of Moscow State University. (In Russ.)
  2. Lehnitsky S.G. (1957). Anizotropnye plastinki [Anisotropic plates]. Gostekhizdat Publ. (In Russ.)
  3. Kravchuk A.S., Mayboroda V.V., Urzhumtsev Yu.S. (1985). Mekhanika polimernyh i kompozicionnyh materialov [Mechanics of polymer and composite materials]. Moscow, Nauka Publ. (In Russ.)
  4. Tornabene F. (2011). Free vibrations of anisotropic doubly-curved shells and plates of revolution with a free from meridian resting on Winkler – Pasternak elastic foundations. Compos. struct., (94), 186–206.
  5. Haciyev V.C., Sofiyev A.H., Kuruoglu N. (2018). On the free vibration of orthotropic and inhomogeneous with spatial coordinates plates resting on the inhomogeneous viscoelastic foundation. Mechanics of Advanced Materials and Structures, 26(10), 1–12. doi: 10.1080/15376494. 2018.1430271
  6. Sofiyev A.H., Schnack E., Haciyev V.C., Kuruoglu N. (2013). Effect of the two-parameter elastic, foundation on the critical parameters of non-homogeneous orthotropic shells. International Journal of Structural Stability and Dynamics, 12(05), 24. doi: 10.1142/S02194554125 00411
  7. Bajenov V.A. (1975). Izgib cilindricheskih obolochek v uprugoj srede [The Benching of the Cylindrical Shells in Elastic Medium]. Kiev, Vysshaya shkola Publ. (In Russ.)
  8. Sofiyev A.H., Hui D., Haciyev V.C., Erdem H., Yuan G.Q., Schnack E., Guldal V. (2017). The nonlinear vibration of orthotropic functionally graded cylindrical shells surrounded by an elastic foundation within first order shear deformation theory. Composites Part B: Engineering, 116, 170–185. 2017.02.006
  9. Haciyev V.C., Mirzeyeva G.R., Shiriyev A.I. (2018). Effect of Winkler foundation, inhomogenecity and orthotropic on the frequency of plates. Journal of Structural Engineering Applied Mechanics, I(1), 1–15.
  10. Haciyev V.C., Sofiyev A.H., Kuruoglu N. (2018). Free bending vibration analysis of thin bidirectional exponentially graded orthotropic rectangular plates resting on two-parameter elastic foundations. Composite Structures, 184, 372–277.
  11. Haciyev V.C., Sofiyev A.H., Mirzeyev R.D. (1996). Free vibration of non-homogeneous elastic rectangular plates. Proceedings of the Institute of Mathematics, (4), 103–108.
  12. Huang M., Sakiyama T., Matsuda H., Morita C. (2015). Free-vibration analysis of stepped rectangular plates resting on non-homogeneous elastic foundations. Engineering Analysis with Boundary Elements, 50, 180–187.
  13. Carnet H., Lielly A. (1969). Free vibrations of reinforced elastic shells. Journal of Applied Mechanics, 36(4), 835–844.

Copyright (c) 2019 Haciyev V.C., Mirzoeva G.R., Agayarov M.G.

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