Structural Mechanics of Engineering Constructions and BuildingsStructural Mechanics of Engineering Constructions and Buildings1815-52352587-8700Peoples’ Friendship University of Russia (RUDN University)3156510.22363/1815-5235-2022-18-2-104-110Research ArticleForced oscillations of a multimodular beam on a viscous elastic baseRzayevNatig S.<p>Doctor of Philosophy in Mechanics (PhD), Associate Professor of the Department of Engineering Mechanics</p>nrzayev@beu.edu.azhttps://orcid.org/0000-0002-1159-9296Baku Engineering University2007202218210411020072022Copyright © 2022, Rzayev N.S.2022<p style="text-align: justify;">The aims of the research are to obtain and to solve equations of forced oscillations of beams made of different modular materials and located on a viscous elastic base. It is assumed that the beam, which has different resistance to expansion and compression and which is continuous and heterogeneous by thickness and length, performs forced oscillations under the action of a force that varies according to the cross-harmonic law. When solving the problem, the resistance of the environment is taken into account. Since the equation of motion is a complicated differential equation with partial derivatives with respect to bending, it is solved by approximate analytical methods. At the first stage, decomposition into variables is used, and at the second stage, the Bubnov - Galerkin orthogonalization method is used. Equations of dependence between the circular frequency and parameters characterizing the resistance of the external environment and heterogeneity are obtained. Calculations were carried out for the specific values of characteristic functions. Results are represented in the form of tables and curves of the corresponding dependencies. It is clear from the obtained equations that serious errors are made in solving problems of oscillating motion without taking into account the resistance of the environment and different modularity. In addition to this, as the values of parameters that determine the heterogeneity of the density increase, the value of the frequency difference changes significantly. The results can be used in reports on solidity, stability and gain-frequency characteristic of different modular beams, boards and cylindrical coatings, taking into account the resistance of the environment.</p>tensioncompressionbendingtorsionelasticfrequentlyvibrationрастяжениесжатиеизгибкручениеэластичностькруговая частотаколебание[Tolokonnikov L.A. On the relationship between tensions and deformations in different modular isotropic medium. Engineering Journal of Solid Mechanics. 1968;(6):108–110. (In Russ.)][Novatsky V. Dynamics of constructions. Moscow; 1963. (In Russ.)][GadjievV.D., Rzayev N.S. Lateral oscillations of a beam made of multi-modulus material lying on inhomogeneous visco-elastic foundation. Transaction of NAS of Azerbaijan. 2014;XXXIV(1):125–130. (In Russ.)][Gadjiev V.D., Rzayev N.S. Oscilllations of a nonhomogeneous different modulus beam with a load moving on it situated on nonhomogeneous viscoelastic foundation. Transaction of NAS of Azerbaijan. 2013;XXXIII(4):133–138. (In Russ.)][Rzaev N.S. A free oscillation of an heterogeneous different modular rod lying on a base of two constants. Building Mechanics of Engineering Structures and Constructions. 2016;(6):38–43. (In Russ.)][Rzaev N.S. On the stability of the flat shape of the bending of beams made of materials with different resistance to compression. Scientific Notes. 2016;1(3):172–176. (In Russ.)][Rzaev N.S. On the stability of an elastic-plastic rod lying on a heterogeneous elastic base. Journal of Theoretical and Applied Mechanics. 2014;(2):132–137. (In Russ.)][Pasternak P.L. Fundamentals of a new method for calculating the foundations on elastic base by means of two coefficients of poete. Moscow: Sroyizdat Publ.; 1954. (In Russ.)][Markin A.A., Sokolova M.Yu. Constitutive relations of nonlinear thermoelasticity of anisotropic bodies. Journal of Applied Mechanics and Technical Physics. 2003;44(1):141–145. https://doi.org/10.1023/A:1021702418574][Arbeloda-Monsalve L.G., Zapata-Medina D.G., Aristizabal-Ochoa J.D. Timoshenko beam-column with generalized end conditions on elastic foundation: dynamic-stiffness matrix and load vector. Journal of Sound and Vibration. 2008;310: 1057–1079.][Zhaohua F., Cook R.D. Beam elements on two-parameter elastic foundations. Journal of Engineering Mechanics. 1983;109:1390–1402.][Sofıyev A.H., Omurtag M.H., Schnack E. The vibration and stability of orthotropic conical shells with non-homogeneous material properties under a hydrostatic pressure. Journal of Sound and Vibration. 2009;319:963–983.][Gasymov G.M., Rzaev N.S. Transverse oscillation of the rod lying on a heterogeneous viscous-elastic base. Scientific Notes. 2013;1(3):41–45. (In Russ.)][Gadjiev V.D. A natural oscillation of the orthotropic circular plate lying on a heterogeneous viscous-elastic base. Bulletin of Modern Science. 2016;(5):20–24. (In Russ.)][Gasymov G.M. On a free oscillation of a continuous heterogeneous rectangular plate lying on structures and constructions with a heterogeneous viscous elastic bases. Building Mechanics of Engineering Structures and Constructions. 2017;(5):14–19. (In Russ.)]