Mathematical Simulation of Nonlinear Vibrations of a Channel Wall Interacting with a Vibrating die Via Viscous Liquid Layer

Abstract

The article considers the developed mathematical model and investigates the dynamics of the interaction of a channel wall supported by a nonlinear spring with a vibrating opposite wall through a viscous fluid layer filling the channel. A flat slotted channel formed by two absolutely rigid rectangular walls, parallel to each other was investigated. One of the channel dimensions in the plan was much larger than the other, which leads to the transition to a plane problem. The bottom channel wall rested on a spring with a cubic nonlinear characteristic, and the upper wall was a die oscillating according to a given law. The gap between the walls was assumed to be much smaller than the channel longitudinal dimension, and the amplitudes of wall vibrations were much less than the channel gap. The movement of the viscous fluid in the channel was considered to be creeping. The mathematical model of the channel under consideration consisted of an equation of the dynamics of a single-mass system with a spring having a cubic nonlinearity, as well as the Navier --- Stokes and continuity equations, supple-mented by the boundary conditions for fluid nonslip on the channel walls and its free outflow at the ends. The steady-state nonlinear vibrations of the bottom channel wall at the fundamental frequency were studied, and its hydroelastic response was determined. The proposed model can be used to study nonlinear vibrations of elastically fixed elements that are in contact with liquid and are parts of modern devices and assemblies

Please cite this article in English as:

Mogilevich L.I., Popov V.S., Popova A.A., et al. Mathematical simulation of nonlinear vibrations of a channel wall interacting with a vibrating die via viscous liquid layer. Herald of the Bauman Moscow State Technical University, Series Instrument Engineering, 2022, no. 2 (139), pp. 26--41 (in Russ.). DOI: https://doi.org/10.18698/0236-3933-2022-2-26-41

References

[1] Topilskaya S.V., Borodulin D.S., Kornyukhin A.V. Experimental estimation of allowable mechanical impact on dynamically adjusted gyroscope. Herald of the Bauman Moscow State Technical University, Series Instrument Engineering, 2018, no. 4 (121), pp. 69--79 (in Russ.). DOI: http://dx.doi.org/10.18698/0236-3933-2018-4-69-79

[2] Tinyakov Yu.N., Nikolaeva A.S. Computation of pressure sensor membrane. Herald of the Bauman Moscow State Technical University, Series Instrument Engineering, 2015, no. 6 (105), pp. 135--142 (in Russ.). DOI: http://dx.doi.org/10.18698/0236-3933-2015-6-135-142

[3] Abbakumov E.I., Gordeev B.A., Erofeev V.I., et al. Study on hydraulic vibrosupports with different working liquids. Problemy mashinostroeniya i nadezhnosti mashin, 2002, no. 2, pp. 33--36 (in Russ.).

[4] Andreichenko K.P., Smarun’ A.B. Simulation of the axial hydromechanical effect in gyroscopes with a spherical hydrodynamic suspension. J. Mach. Manuf. Reliab., 2011, vol. 40, no. 3, pp. 216--221. DOI: https://doi.org/10.3103/S1052618811020038

[5] Qin Z., Podchezertsev V.P. Influence of design features and gas filling parameters on dynamically tuned gyroscope characteristics. Herald of the Bauman Moscow State Technical University, Series Instrument Engineering, 2017, no. 2 (113), pp. 4--20 (in Russ.). DOI: http://dx.doi.org/10.18698/0236-3933-2017-2-4-20

[6] Dobryanskiy V.N., Rabinskiy L.N., Radchenko V.P., et al. Width evaluation of contact zone between flat-oval cooling channels and transmitting module case of active phased-array antenna. Trudy MAI, 2018, no. 101 (in Russ.). Available at: http://trudymai.ru/published.php?ID=98252

[7] Lamb H. On the vibrations of an elastic plate in contact with water. Proc. Roy. Soc. A, 1921, vol. 98, no. 690, pp. 205--216. DOI: https://doi.org/10.1098/rspa.1920.0064

[8] Amabili M., Kwak M.K. Free vibrations of circular plates coupled with liquids: revising the Lamb problem. J. Fluids Struct., 1996, vol. 10, no. 7, pp. 743--761. DOI: https://doi.org/10.1006/jfls.1996.0051

[9] Morozov D., Indeitsev D., Michailov A. Added mass study of plane structures at their various motions. Materials Physics and Mechanics, 2019, vol. 41, no. 1, pp. 116--124. DOI: https://doi.org/10.18720/MPM.4112019_19

[10] Amabili M. Vibrations of circular plates resting on a sloshing liquid: solution of the fully coupled problem. J. Sound Vib., 2001, vol. 245, no. 2, pp. 261--283. DOI: https://doi.org/10.1006/jsvi.2000.3560

[11] Askari E., Jeong K.-H., Amabili M. Hydroelastic vibration of circular plates immersed in a liquid-filled container with free surface. J. Sound Vib., 2013, vol. 332, no. 12, pp. 3064--3085. DOI: https://doi.org/10.1016/j.jsv.2013.01.007

[12] Vel’misov P.A., Pokladova Yu.V. Оn some mathematical models of mechanical system "pipeline--pressure" sensor. Vestnik SamGTU. Ser. Tekhnicheskie nauki [Vestnik of Samara State Technical University. Technical Sciences Series], 2011, no. 1, pp. 137--144 (in Russ.).

[13] Ankilov A.V., Vel’misov P.A., Tamarova Yu.A. Mathematical model of a vibration device. Avtomatizatsiya protsessov upravleniya [Automation of Control Processes], 2014, no. 3, pp. 58--67 (in Russ.).

[14] Kozlovsky Y. Vibration of plates in contact with viscous fluid: extension of Lamb’s model. J. Sound Vib., 2009, vol. 326, no. 1-2, pp. 332--339. DOI: https://doi.org/10.1016/j.jsv.2009.04.031

[15] Mogilevich L.I., Popov V.S., Popova A.A. Dynamics of interaction of elastic elements of a vibrating machine with the compressed liquid layer lying between them. J. Mach. Manuf. Reliab., 2010, vol. 39, no. 4, pp. 322--331. DOI: https://doi.org/10.3103/S1052618810040047

[16] Mogilevich L.I., Popov V.S. Investigation of the interaction between a viscous incompressible fluid layer and walls of a channel formed by coaxial vibrating discs. Fluid Dyn., 2011, vol. 46, no. 3, pp. 375--388. DOI: https://doi.org/10.1134/S0015462811030033

[17] Faria C.T., Inman D.J. Modeling energy transport in a cantilevered Euler --- Bernoulli beam actively vibrating in Newtonian fluid. Mech. Syst. Signal Process., 2014, vol. 45, no. 2, pp. 317--329. DOI: https://doi.org/10.1016/j.ymssp.2013.12.003

[18] Akcabay D.T., Young Y.L. Hydroelastic response and energy harvesting potential of flexible piezoelectric beams in viscous flow. Phys. Fluids, 2012, vol. 24, no. 5, art. 054106. DOI: https://doi.org/10.1063/1.4719704

[19] Mogilevich L.I., Popov V.S., Popova A.A. Longitudinal and transverse oscillations of an elastically fixed wall of a wedge-shaped channel installed on a vibrating foundation. J. Mach. Manuf. Reliab., 2018, vol. 47, no. 3, pp. 227--234. DOI: https://doi.org/10.3103/S1052618818030093

[20] Popov V.S., Popova A.A. Modeling of a channel wall interaction with an end seal flexibly restrained at the edge. Komp’yuternye issledovaniya i modelirovanie [Computer Research and Modeling], 2020, vol. 12, no. 2, pp. 387--400 (in Russ.). DOI: https://doi.org/10.20537/2076-7633-2020-12-2-387-400

[21] Popova A.A. Mathematical model of spring-disk system with hardening restoring force. Matematicheskoe modelirovanie, kom’yuternyy i naturnyy eksperiment v estestvennykh naukakh, 2018, no. 3, pp. 13--24 (in Russ.).

[22] Nayfeh A.H., Mook D.T. Nonlinear oscillations. John Wiley & Sons, 1979.

[23] Krack M., Gross J. Harmonic balance for nonlinear vibration problems. Nature Switzerland AG, Springer, 2020.

[24] Panovko Ya.G. Vvedenie v teoriyu mekhanicheskikh kolebaniy [Introduction to the theory of mechanical vibrations]. Moscow, Nauka Publ., 1991.

[25] Loytsyanskiy L.G. Mekhanika zhidkosti i gaza [Fluid mechanics]. Moscow, Drofa Publ., 2003.