Numerical analysis of nonlinear vibrations of a~plate on~a~viscoelastic foundation under the action of a moving oscillating load based on models with fractional derivatives

A. I. Krusser ; M. V. Shitikova

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 26 (2022) no. 4, pp. 694-714.

Résumé

Aim. In the present paper, nonlinear vibrations of an elastic simply supported plate on a viscoelastic foundation under the action of a moving oscillating load are studied in the case of the internal resonance $1{:}1$ accompanied by the external resonance. The properties of the viscoelastic foundation are given via the generalized Fuss–Winkler model with the damping term described by the standard linear solid model with the Riemann–Liouville fractional derivatives. The external load is presented by linear viscoelastic oscillator based on the Kelvin–Voigt model with a fractional derivative in the case when the viscosity of the oscillator is considered to be small value. The dynamic behavior of the plate is described by a set of nonlinear ordinary differential equations of the second order in time with respect to generalized displacements. Methods. To solve the resulting set of equations, the method of multiple time scales is used in combination with the method of expansion of the fractional derivative in a Taylor series. Results. Resolving equations for determining of the nonlinear amplitudes and phases of force driven vibrations of the plate are obtained. The governing set of equations allows one to control not only the damping properties of the environment and the foundation by changing the fractional parameters, but also to control the damping parameters of the external load. Conclusion. Numerical analysis has shown that in the system “a plate on a viscoelastic foundation + a moving oscillating load”, energy transfer between the interacting vibration modes is observed. A comparison of the results of numerical studies for various values of the external load is presented, and the dependence of the amplitudes of nonlinear vibrations on the values of the fractional parameters of the environment and the foundation is also shown.

Détail

Keywords: plate on a viscoelastic foundation, moving oscillator load, fractional derivative, method of multiple time scales, internal and external resonances.

Bibliographie

[1] Gerasimov S. I., Erofeev V. I., Kolesov D. A., Lissenkova E. E., “Dynamics of deformable systems carrying moving loads (review of publication and dissertation research)”, Vestnik nauchno-tekhnicheskogo razvitiya [Bulletin of Science and Technical Development], 2021, no. 160, 25–47 (In Russian) | DOI

[2] Frýba L., Vibration of Solids and Structures under Moving Loads, Mechanics of Structural Systems, 1, Dordrecht, Springer, 1973, 484+xxvii pp. | DOI

[3] Younesian D., Hosseinkhani A., Askari H., Esmailzadeh E., “Elastic and viscoelastic foundations: a review on linear and nonlinear vibration modeling and applications”, Nonlinear Dyn., 97:1 (2019), 853–895 | DOI

[4] Rajabi K., Kargarnovin M. H., Gharini M., “Dynamic analysis of a functionally graded simply supported Euler–Bernoulli beam subjected to a moving oscillator”, Acta Mech., 224:2 (2013), 425–446 | DOI

[5] Almbaidin A., Abu-Alshaikh I., “Vibration of functionally graded beam subjected to moving oscillator using Caputo–Fabrizio fractional derivative model”, Romanian Journal of Acoustics and Vibration, 16:2 (2019), 137–146

[6] Sawant V. A., Patil V. A., Deb K., “Effect of vehicle–pavement interaction on dynamic response of rigid pavements”, Geomech. Geoeng., 6:1 (2011), 31–39 | DOI

[7] Patil V. A., Sawant V. A., Deb K., “Finite element analysis of rigid pavement on a nonlinear two parameter foundation model”, Int. J. Geotech. Eng., 6:3 (2012), 275–286 | DOI

[8] Ding H., Yang Y., Chen L.-Q., Yang S.-P., “Vibration of vehicle–pavement coupled system based on a Timoshenko beam on a nonlinear foundation”, J. Sound Vib., 333:24 (2014), 6623–6636 | DOI

[9] Yang S., Chen L., Li S., “Modeling and dynamic analysis of vehicle-road coupled systems”, Dynamics of Vehicle-Road Coupled System, Springer, Berlin, 2015, 215–250 | DOI

[10] Meshkov S. I., Pachevskaya G. N., Postnikov V. S., Rossikhin Yu. A., “Integral representations of $\varepsilon_\gamma$-functions and their application to problems in linear viscoelasticity”, Int. J. Eng. Sci., 9:4 (1971), 387–398 | DOI

[11] Rossikhin Yu. A., Shitikova M. V., “Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids”, Appl. Mech. Rev., 57:1 (1997), 15–67 | DOI

[12] Ogorodnikov E. N., Yashagin N. S., “Forced oscillations of the fractional oscillator”, Proceedings of the Fifth All-Russian Scientific Conference with international participation (29–31 May 2008). Part 1, Matem. Mod. Kraev. Zadachi, Samara State Technical Univ., Samara, 2008, 215–221 (In Russian)

[13] Rossikhin Yu. A., Shitikova M. V., “New approach for the analysis of damped vibrations of fractional oscillators”, Shock and Vibration, 16:4 (2009), 387676 | DOI

[14] Rossikhin Yu. A., Shitikova M. V., “Application of fractional calculus for dynamic problems of solid mechanics: Novel trends and recent results”, Appl. Mech. Rev., 63:1 (2010), 010801 | DOI

[15] Ogorodnikov E. N., Radchenko V. P., Yashagin N. S., “Rheological model of viscoelastic body with memory and differential equations of fractional oscillator”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2011, no. 1(22), 255–268 (In Russian) | DOI

[16] Parovik R. I., Zunnunov R. T., “Analysis of forced vibrations of a fractional oscillator”, Probl. Mekhan., 2019, no. 4, 20–23 (In Russian)

[17] Kerr A. D., “Elastic and viscoelastic foundation models”, J. Appl. Mech., 31:3 (1964), 491–498 | DOI

[18] Winkler E., Die Lehre von der Elasticität und Festigkeit, Dominicius, Prague, 1867, 388 pp. (In German) | Zbl

[19] Zimmerman H., Die Berechnung des Eisenbahnoberbaues, Verlag von Ernst Korn, Berlin, 1888, 326 pp.

[20] Fuss N. I., “An experiment on the resistance caused to roads by all kinds of four-wheeled and two-wheeled carts, with the determination of the circumstances, in the presence of one of these carts is more useful than others”, Akademicheskiya sochineniya, 1801, Part. 1, 373–422, St. Petersburg (In Russian)

[21] Vlasov V. Z., Leontiev N. N., Balki, plity, obolochki na uprugom osnovanii [Beams, Plates and Shells on an Elastic Foundation], Fizmatlit, Moscow, 1960, 492 pp. (In Russian)

[22] Tsytovich N. A., Mekhanika gruntov [Soil Mechanics], Moscow, 1963, 637 pp. (In Russian)

[23] Lai J., Mao S., Qiu J., Fan H., Zhang Q., Hu Z., Chen J., “Investigation progresses and applications of fractional derivative model in geotechnical engineering”, Math. Probl. Eng., 2016:3 (2016), 9183296 | DOI

[24] Taheri M. R., Ting E. C., “Dynamic response of plate to moving loads: Structural impedance method”, Comput. Struct., 33:6 (1989), 1379–1393 | DOI

[25] Zaman M., Taheri M. R., Alvappillai A., “Dynamic response of a thick plate on viscoelastic foundation to moving loads”, Int. J. Numer. Analytical Methods Geomech., 15:9 (1991), 627–647 | DOI

[26] Yang S., Li S., Lu Y., “Investigation on dynamical interaction between a heavy vehicle and road pavement”, Int. J. Vehicle Mech. Mob., 48:8 (2010), 923–944 | DOI

[27] Li S., Yang S., Chen L., “A nonlinear vehicle-road coupled model for dynamics research”, J. Comput. Nonlinear Dynam., 8:2 (2013), 021001 | DOI

[28] Amabili M., “Nonlinear vibrations of viscoelastic rectangular plates validation”, J. Sound Vib., 362 (2016), 142–156 | DOI

[29] Amabili M., “Nonlinear damping in nonlinear vibrations of rectangular plates: Derivation from viscoelasticity and experimental validation”, J. Mech. Phys. Solids, 118 (2018), 275–292 | DOI

[30] Shitikova M. V., Kandu V. V., “Analysis of the nonlinear vibrations of an elastic plate on a viscoelastic foundation in the presence of the one-to-one internal resonance”, Izv. Vyzov. Stroitel'stvo, 2020, no. 3, 5–22 (In Russian)

[31] Shitikova M. V., Krusser A. I., “Nonlinear vibrations of an elastic plate on a viscoelastic foundation modeled by the fractional derivative standard linear solid model”, EURODYN 2020, Proc. of the XI International Conference on Structural Dynamics, National Techn. Univ. of Athens, Athens, 2020, 355–368 | DOI

[32] Shitikova M. V., Krusser A. I., “Force driven vibrations of nonlinear plates on a viscoelastic Winkler foundation under the harmonic moving load”, Int. J. Comput. Civil Struct. Eng., 17:4 (2021), 161–180 | DOI

[33] Volmir A. S., The Nonlinear Dynamics of Plates and Shells, Dept. of the Air Force, Dayton, 1974, 543 pp.

[34] Samko S. G. Kilbas A. A., Marichev O. I., Integraly i proizvodnye drobnogo poriadka i nekotorye ikh prilozheniia [Integrals and Derivatives of Fractional Order and Some of Their Applications], Nauka i tekhnika, Minsk, 1987, 688 pp. (In Russian) | Zbl

[35] Rossikhin Yu. A., Shitikova M. V., “Fractional operator models of viscoelasticity”, Encyclopedia of Continuum Mechanics, Springer, Berlin, 2020, 971–982 | DOI

[36] Rossikhin Yu. A., Shitikova M. V., “Centennial jubilee of Academician Rabotnov and contemporary handling of his fractional operator”, Fract. Calc. Appl. Anal., 17:3 (2014), 674–683 | DOI

[37] Shitikova M. V., “The fractional derivative expansion method in nonlinear dynamic analysis of structures”, Nonlinear Dyn., 99:1 (2020), 109–122 | DOI

[38] Nayfeh A. H., Perturbation Technique, Wiley, New York, 1973, 441 pp.

[39] Rossikhin Yu. A., Shitikova M. V., “Application of fractional calculus for analysis of nonlinear damped vibrations of suspension bridges”, J. Eng. Mech., 124:9 (1998), 1029–1036 | DOI

[40] Nayfeh A. H., Mook D. T., Nonlinear Oscillations, Wiley, 1995, 705 pp.

[41] Rossikhin Yu. A., Krusser A. I., Shitikova M. V., “Impact response of a nonlinear viscoelastic auxetic doubly curved shallow shell”, ICSV 2017, Proc. of the 24th International Congress on Sound and Vibration, Int. Inst. Acoust. Vibration, London, 2017

[42] Shitikova M. V., Kandu V. V., “Force driven vibrations of fractionally damped plates subjected to primary and internal resonances”, Eur. Phys. J. Plus, 134:9 (2019), 423 | DOI