Simulating Interaction Between a Plate on Elastic Foundation with the Softening Cubic Nonlinearity and a Vibrating Die Via the Viscous Fluid Layer

Abstract

The paper presents a mathematical model of interaction of a plate and a layer of viscous liquid located between the plate and the vibrating die taking into account nonlinearity in the plate elastic base properties. The plate forced nonlinear hydroelastic vibrations were studied satisfying the Kirchhoff’s hypotheses, it was placed on a base with the soft cubic nonlinearity. Related hydroelasticity problem was formulated for the vibration system under consideration. The mathematical model consisted of a system of equations for the viscous fluid dynamics and the Kirchhoff’s plate on the nonlinear elastic base. The system was supplemented with boundary conditions at the contact boundaries between liquid and the plate and die, as well as with conditions at the ends of the channel under consideration. Asymptotic analysis of the posed hydroelasticity problem was performed and, simplified equations of the viscous fluid motion were solved by the iteration method. Pressure distribution was determined, and nonlinear integral differential equation was obtained for the plate bending vibrations excited by the vibrating die. This equation was solved by the Bubnov --- Galerkin method. It was shown that the problem under consideration could be reduced to studying the generalized Duffing equation solved by the harmonic balance method. The main nonlinear hydroelastic response and the plate phase shift were determined. Those characteristics were numerically studied, which made it possible to make a conclusion on the importance of taking into account the viscous fluid motion inertia and the elastic properties of the plate base

This work was supported by the RSF project no. 23-29-00159

Please cite this article in English as:

Popov V.S., Popova A.A., Popova M.V., et al. Simulating interaction between a plate on elastic foundation with the softening cubic nonlinearity and a vibrating die via the viscous fluid layer. Herald of the Bauman Moscow State Technical University, Series Instrument Engineering, 2023, no. 4 (145), pp. 110--130 (in Russ.). DOI: https://doi.org/10.18698/0236-3933-2023-4-110-130

References

[1] Borisov R.A., Antonets I.V., Krotov A.V. Static and total pressure sensor development methodology based on elastic sensing elements and optical rules. Herald of the Bauman Moscow State Technical University, Series Instrument Engineering, 2021, no. 1 (134), pp. 33--50 (in Russ.). DOI: https://doi.org/10.18698/0236-3933-2021-1-33-50

[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: https://doi.org/10.18698/0236-3933-2015-6-135-142

[3] Antsiferov S.A., Kondratov D.V., Mogilevich L.I. Perturbing moments in a floating gyroscope with elastic device housing on a vibrating base in the case of a nonsymmetric end outflow. Mech. Solids, 2009, vol. 44, no 3, pp. 352--360. DOI: https://doi.org/10.3103/S0025654409030030

[4] Tsin T., 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: https://doi.org/10.18698/0236-3933-2017-2-4-20

[5] Shevtsova E.V. Gas damping in micromechanical instruments. Herald of the Bauman Moscow State Technical University, Series Instrument Engineering, 2006, no. 2 (63), pp. 100--111 (in Russ.).

[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] Liska J., Kunkel S. Localization of loose part impacts on the general 3D surface of the nuclear power plant coolant circuit components. Prog. Nucl. Energy, 2017, vol. 99, pp. 140--146. DOI: https://doi.org/10.1016/j.pnucene.2017.05.004

[8] Gorshkov A.G., Morozov V.I., Ponomarev A.T., et al. Aerogidrouprugost konstruktsiy [Aerohydroelasticity of structures]. Moscow, FIZMATLIT Publ., 2000.

[9] Amabili M. Nonlinear vibrations and stability of shells and plates. Cambridge, Cambridge University Press, 2008.

[10] Paidoussis M.P. Fluid-structure interactions: slender structures and axial flow. Vol. 1. New York, Academic Press, 2013.

[11] Velmisov P.A., Pokladova Y.V. Mathematical modelling of the "Pipeline--pressure sensor" system. J. Phys. Conf. Ser., 2019, vol. 1353, art. 01208. DOI: https://doi.org/10.1088/1742-6596/1353/1/012085

[12] Hasheminejad S.M., Mohammadi M.M. Hydroelastic response suppression of a flexural circular bottom plate resting on Pasternak foundation. Acta Mech., 2017, vol. 228, no. 12, pp. 4269--4292. DOI: https://doi.org/10.1007/s00707-017-1922-4

[13] Tulchinsky A., Gat A.D. Frequency response and resonance of a thin fluid film bounded by elastic sheets with application to mechanical filters. J. Sound Vib., 2019, vol. 438, pp. 83--98. DOI: https://doi.org/10.1016/j.jsv.2018.08.047

[14] Mogilevich L.I., Popov V.S., Popova A.A. Interaction dynamics of pulsating viscous liquid with the walls of the conduit on an elastic foundation. J. Mach. Manuf. Reliab., 2017, vol. 46, no. 1, pp. 12--19. DOI: https://doi.org/10.3103/S1052618817010113

[15] 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

[16] Barulina M., Santo L., Popov V., et al. Modeling nonlinear hydroelastic response for the endwall of the plane channel due to its upper-wall vibrations. Mathematics, 2022, vol. 10, no. 20, art. 3844. DOI: https://doi.org/10.3390/math10203844

[17] 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

[18] Reissner E. On postbuckling behavior and imperfection sensitivity of thin elastic plates on a nonlinear elastic foundation. Stud. Appl. Math., 1970, vol. 49, no. 1, pp. 45--57. DOI: https://doi.org/10.1002/sapm197049145

[19] Erofeev V.I., Kazhaev V.V., Lisenkova E.E., et al. Nonsinusoidal bending waves in Timoshenko beam lying on nonlinear elastic foundation. J. Mach. Manuf. Reliab., 2008, vol. 37, no. 3, pp. 230--235. DOI: https://doi.org/10.3103/S1052618808030059

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

[21] Howell P., Kozyreff G., Ockendon J. Applied solid mechanics. Cambridge, Cambridge University Press, 2008.

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

[23] Nayfeh A.H. Perturbation methods. Hoboken, John Wiley & Sons, 1973.

[24] Popov V.S., Popova A.A. Modeling of hydroelastic oscillations for a channel wall possessing a nonlinear elastic support. Kompyuternye issledovaniya i modelirovanie [Computer Research and Modeling], 2022, vol. 14, no. 1, pp. 79--92 (in Russ.). DOI: https://doi.org/10.20537/2076-7633-2022-14-1-79-92

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

[26] Krack M., Gross J. Harmonic balance for nonlinear vibration problems. Cham, Springer Nature, 2019.