Viscous friction modelling in material of a plate under its non-stationary loading with differential and integral operators


Voropay A. V.1*, Grishakin V. T.2**

1. National Technical University “Kharkiv Polytechnic Institute”, 2, Kirpicheva str., Kharkov, 61002, Ukraine
2. Moscow Automobile and Road Construction State Technical University (MADI), MADI, 64, Leningradsky Prospect, Moscow, 125319, Russia



The article presents results comparison of deflections computing in a certain point of a viscoelastic plate, obtained within the framework of classical Kirchhoff thin plates theory with the results of the similar computing based on the refined theory of Timoshenko type for medium thickness plates. Transversal pulse loading of a rectangular homogeneous isotropic viscoelastic plate, hinged along its contour, is considered while computations. Accounting for internal viscous friction based on Kelvin-Voigt model for Kirchhoff plates was realized by the well-known approach, employing differential operators.

The article proposes a new approach to transient analysis in a viscoelastic continuum caused by non-stationary force disturbances, based on integral transformations and the Efros theorem. This approach employs a smoothing linear integral operator and can be applied to any solutions, obtained in the framework of elasticity theory, which can be represented in the form of Duhamel’s integrals. The possibility of relatively fast recalculation of the plate deflections and deformations with account for the internal friction for various values dissipation and vibration coefficients is not excluded herewith.

The presented graphs allow making a conclusion that integral and differential operators application to account for energy dissipation while viscoelastic plates deforming (moreover, with various hypotheses of deformation by Kirchhoff and Timoshenko) gives close results. That is, the solutions, obtained by the two independent methods for different differential equations, practically coincide. This fact confirms the new developed approach propriety, correctness of smoothing integral operator application and veracity of performed calculations.

The indisputable advantage of the approach presented in the article, which employs smoothing linear integral operators for solving the deformed solid mechanics problems, as applied to viscoelastic continuum, consists in the fact that it does not utilize information on frequencies and shapes of free vibrations of the object under study. Thus, the suggested technique is not sensitive to the boundary conditions description errors and imperfection of the accepted hypotheses of deformation (Kirchhoff, Timoshenko, etc.). Due to the above-said property, the suggested approach can be effectively employed while data processing of full-scale experiments.


non-stationary loading, viscoelastic plate, Kirchhoff plate theory, refined Timoshenko theory, internal viscous friction, Kelvin-Voigt model, Efros theorem


  1. Mitin A.Yu., Tarlakovskii D.V., Fedotenkov G.V. Trudy MAI, 2019, no. 107, available at:

  2. Goldovskii A.A. Trudy MAI, 2019, 107, available at:

  3. Firsanov V.V., Fam V.T. Trudy MAI, 2019, no. 105, available at:

  4. Firsanov V.V., Zoan K.Kh. Trudy MAI, 2018, no. 103, available at:

  5. Firsanov V.V., Vo A.Kh., Chan N.D. Trudy MAI, 2019, no. 104, available at:

  6. Marco Amabili. Nonlinear vibrations of viscoelastic rectangular plates, Journal of Sound and Vibration, 2015, DOI: 10.1016/j.jsv.2015.09.035

  7. Litewka P., Lewandowski R. Nonlinear harmonically excited vibrations of plates with Zener material, Nonlinear Dynamics, 2017, vol. 89 (1). DOI: 10.1007/s11071-017-3480-7

  8. Ari M., Faal R.T., Zayernouri M. Vibrations Suppression of Fractionally Damped Plates using Multiple Optimal Dynamic Vibration Absorbers, International Journal of Computer Mathematics, 2019, DOI: 10.1080/00207160.2019.1594792

  9. Assaee H., Nazanin P. Development of a viscoelastic spline finite strip formulation for transient analysis of plates, Thin-Walled Structures, 2018, vol. 124, pp. 430 – 436. DOI: 10.1016/j.tws.2017.12.021

  10. Asnafi Alireza. Chaotic analysis of Kelvin–Voigt viscoelastic plates under combined transverse periodic and white noise excitation: an analytic approach, Acta Mechanica, 2019, pp. 1 – 16.

  11. Dastjerdi S., Abbasi M. A new approach for time-dependent response of viscoelastic graphene sheets embedded in visco-Pasternak foundation based on nonlocal FSDT and MHSDT theories, Mechanics of Time-Dependent Materials, 2019, pp. 1 – 33.

  12. Ivanov S.V., Mogilevich L.I., Popov V.S. Trudy MAI, 2019, no. 105, available at:

  13. Pisarenko G.S., Yakovlev A.P., Matveev V.V. Vibropogloshchayushchie svoistva konstruktsionnykh materialov (Vibration-absorbing properties of structural materials). Kiev, Naukova dumka, 1971, 375 p.

  14. Vasilenko N.V. Teoriya kolebanii (Theory of Vibration), Kiev, Vishcha shkola, 1992, 430 p.

  15. Panovko Ya.G. Vnutrennee trenie pri kolebaniyakh uprugikh system (Internal Friction at Elastic Systems Vibrations), Moscow, Nauka, 1960, 186 p.

  16. Yanyutin E.G., Bogdan D.I., Voropai N.I., Gnatenko G.A., Grishakin V.T. Identifikatsiya nagruzok pri impul’snom deformirovanii tel (Loads Identification at Pulse Deforming of Solids), Khar’kov, Izd-vo KhNADU, 2010, Ch. 1, 180 p.

  17. Yanyutin E.G., Voropai A.V., Povalyaev S.I., Yanchevskii I.V. Identifikatsiya nagruzok pri impul’snom deformirovanii tel (Loads Identification at Pulse Deforming of Solids), Khar’kov, Izd-vo KhNADU, 2010, Ch. II, 212 p.

  18. Voropai A.V., Grigor’ev A.L. Vestnik natsional’nogo tekhnicheskogo universiteta Khar’kovskii politekhnicheskii institut. Seriya: Matematicheskoe modelirovanie v tekhnike i tekhnologiyakh, 2017, no. 6 (1228), pp. 29 – 44.

  19. Voropai A.V., Grigor’ev A.L. Mekhanika i mashinostroenie, 2018, no. 1, pp. 3 – 22.

  20. Krasnov M.L., Kiselev A.I., Makarenko G.I. Funktsii kompleksnogo peremennogo. Operatsionnoe ischislenie. Teoriya ustoichivosti (Functions of a Complex Variable. Operational Calculus. Stability Theory), Moscow, Nauka, 2003, 208 p.

  21. Ditkin V.A., Prudnikov A.P. Integral’nye preobrazovaniya i operatsionnoe ischislenie (Integral Transformations and Operational Calculus), Moscow, Gosudarstvennoe izdatel’stvo fiziko-matematicheskoi literatury, 1961, 524 p.

  22. Korn G.A., Korn T.M. Mathematical Handbook for Scientists and Engineers, Dover, New York, 2000, 1152 p.

Download — informational site MAI

Copyright © 2000-2020 by MAI