DOI: https://doi.org/10.15587/2312-8372.2017.90520

### Investigation of natural oscillations of inhomogeneous orthotropic circular plate lying on an inhomogeneous viscoelastic foundation

Vaqif Haciyev, Aziz Shiriyev

#### Abstract

In the building of large engineering complexes, bridges and overpasses for various purposes and in many other areas the plates of widely different configurations are used. These plates are made of natural and artificial orthotropic materials. Among them, rectangular and circular plates are the most common. According to the above mentioned natural oscillations, engineer-designer and calculator need to properly assess real property of construction element and the influence of the environment, which is in contact during the operation. Therefore, the object of this study is inhomogeneous circular plate lying on inhomogeneous viscoelastic foundations.

It is assumed that the moduli of elasticity and the plate density are continuous functions of the current radius. In this case, unlike homogeneous plates, the motion equation is complex differential equation with variable coefficients. In this regard, there is need to build an approximate analytical solution method.

In the course of the study we used methods of separation of variables and Bubnov-Galerkin orthogonalization method, which gives effective results with homogeneous boundary conditions.

An axisymmetric form of natural oscillations of orthotropic circular plate with inhomogeneous radius lying on an inhomogeneous viscoelastic foundation is considered. The case, when the contour around the plate is rigidly clamped, is studied in detail. Numerical analysis for concrete values of the characteristic parameters is carried out.

The motion equation is obtained taking into account inhomogeneity of the plate and the foundation, as well as partial variable coefficients of the fourth order.

#### Keywords

plate; continuity; orthotropicity; density; foundation; frequency; elastic moduli; motion equation

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#### References

Lomakin, V. A. (1976). Teoriia uprugosti neodnorodnyh tel. Moscow: MSU, 376.

Hаciyev, V. G., Agamaliyrv, N. G., Mirzoeva, B. D. (2002). Stability of continuality non homogeneous, orthotropic rectangular plates render in plane compressions. In Sump. on Eng and Artutural sciences Balcan Caucassand Turkir. Perplic. Spart, Turkey, 74–78.

Sofiyev, A. H., Schnack, E., Haciyev, V. C., Kuruoglu, N. (2012). Effect of the two-parameter elastic foundation on the critical parameters of nonhomogeneous orthotropic shells. International Journal of Structural Stability and Dynamics, 12 (05), 1250041. doi:10.1142/s0219455412500411

Garnet, H., Levy, A. (1969). Free Vibrations of Reinforced Elastic Shells. Journal of Applied Mechanics, 36 (4), 835–844. doi:10.1115/1.3564779

Kovalenko, A. D. (1959). Kruglye plastiny peremennoi tolshchiny. Moscow, 294.

Zhemochkin, B. N., Sinitsyn, A. P. (1962). Prakticheskie metody rascheta fundamentnyh balok i plit na uprugom osnovanii. Ed. 2. Moscow: Gosstroiizdat, 239.

Pasternak, P. L. (1954). Osnovy novogo metoda rascheta fundamentov na uprugom osnovanii pri pomoshchi dvuh koeffitsientov posteli. Moscow: Gosstroiizdat, 89.

Klepnikov, S. N. (1967). Raschet konstruktsii na uprugogo osnovanii. Kyiv: Budivelnik, 184.

Lehnitskii, S. G. (1967). Anizotropnye plastinki. Moscow: Gosudarstvennoe izdatel'stvo tehniko-teoreticheskoi literatury, 463.

Timoshenko, S. P. (1967). Kolebaniia v inzhenernom dele. Moscow: Nauka, 444.

Rzhanitsyn, A. R. (1982). Stroitel'naia mehanika. Moscow: Vysshaia shkola, 400.

Haciyev, V., Jafarov, Kh. (2014). Lateral oscillations of a straight section of a two constant based heterogeneous pipeline. Eastern-European Journal Of Enterprise Technologies, 6(7(72)), 4–7. doi:10.15587/1729-4061.2014.31195

#### GOST Style Citations

Lomakin, V. A. Teoriia uprugosti neodnorodnyh tel [Text] / V. A. Lomakin. – Moscow: MSU, 1976. – 376 p.

Hаciyev, V. G. Stability of continuality non homogeneous, orthotropic rectangular plates render in plane compressions [Text] / V. G. Hаciyev, N. G. Agamaliyrv, B. D. Mirzoeva // In Sump. on Eng and Artutural sciences Balcan Caucassand Turkir. Perplic. – Spart, Turkey, 2002. – P. 74–78.

Sofiyev, A. H. Effect of the two-parameter elastic foundation on the critical parameters of nonhomogeneous orthotropic shells [Text] / A. H. Sofiyev, E. Schnack, V. C. Haciyev, N. Kuruoglu // International Journal of Structural Stability and Dynamics. – 2012. – Vol. 12, № (05). – P. 1250041. doi:10.1142/s0219455412500411

Garnet, H. Free Vibrations of Reinforced Elastic Shells. [Text] / H. Garnet, A. Levy // Journal of Applied Mechanics. – 1969. – Vol. 36, № 4. – P. 835–844. doi:10.1115/1.3564779

Kovalenko, A. D. Kruglye plastiny peremennoi tolshchiny [Text] / A. D. Kovalenko. – Moscow, 1959. – 294 p.

Zhemochkin, B. N. Prakticheskie metody rascheta fundamentnyh balok i plit na uprugom osnovanii [Text] / B. N. Zhemochkin, A. P. Sinitsyn. – Ed. 2. – Moscow: Gosstroiizdat, 1962. – 239 p.

Pasternak, P. L. Osnovy novogo metoda rascheta fundamentov na uprugom osnovanii pri pomoshchi dvuh koeffitsientov posteli [Text] / P. L. Pasternak. – Moscow: Gosstroiizdat, 1954. – 89 p.

Klepnikov, S. N. Raschet konstruktsii na uprugogo osnovanii [Text] / S. N. Klepnikov. – Kyiv: Budivelnik, 1967. – 184 p.

Lehnitskii, S. G. Anizotropnye plastinki [Text] / S. G. Lehnitskii. – Moscow: Gosudarstvennoe izdatel'stvo tehniko-teoreticheskoi literatury, 1967. – 463 p.

Timoshenko, S. P. Kolebaniia v inzhenernom dele [Text] / S. P. Timoshenko. – Moscow: Nauka, 1967. – 444 p.

Rzhanitsyn, A. R. Stroitel'naia mehanika [Text] / A. R. Rzhanitsyn. – Moscow: Vysshaia shkola, 1982. – 400 p.

Haciyev, V. Lateral oscillations of a straight section of a two constant based heterogeneous pipeline [Text] / V. Haciyev, Kh. Jafarov // Eastern-European Journal of Enterprise Technologies. – 2014. – № 6/7 (72). – P. 4–7. doi:10.15587/1729-4061.2014.31195

Copyright (c) 2017 Вагиф Джамал оглы Гаджиев, Азиз Интизар оглы Шириев