Dynamics of a Plate with Elastically Attached Mass


Аuthors

Nigar E. S.

Lomonosov Moscow State University, 1, Leninskie Gory, Moscow, 119991, Russia

e-mail: nig-dig@unesp.co.uk

Abstract

The article regards the problem of a beam dynamic loading by an impact body in the presence of an intermediate damper, namelly, a spring of a given stiffness. The purpose of the study consisted in determining the joint motion of a mechanical system of a beam-spring-body type, neglecting herewith the spring mass. The beam movement is being modeled by the equations of the plate cylindrical vibrations. The obtained equations for the joint movement of the beam–spring–body system consist of equations for the beam deflection and the equation of the body motion, with account for the spring stiffness. The system of equations, modelling the motion, consists of a fourth-order partial differential equation in coordinate, and a second-order equation in time, one of the boundary conditions of which is an ordinary second-order differential equation in time. The problem is solved by the integral Laplace transform method in time. The Durbin numerical method is used for the obtained solution inversion. Graphs of solutions, allowing observing the body behavior and calculating the beam deflection at a time instant, were plotted using this method. The graphs of analytical and numerical solutions coincide for small initial times. The dependence of the sought functions on the main parameters of the problem, such as spring stiffness and the beam bending stiffness, is demonstrated as well. It can be seen from the illustrating graphs that the beam deflection and body motion functions are directly proportional to the spring stiffness, and inversely proportional to the bending stiffness of the beam.

Keywords:

beam deflection, beam vibration, spring, tension, deformation, system balance, Durbin’s method

References

  1. Barguev S.G. Kolebaniya neodnorodnoi balki s uprugo prisoedinennym telom s dvumya stepenyami svobody (Inhomogeneous beam vibrations with a rigidly attached body with two degrees of freedom), Moscow, Nauka, 2017, 358 p.

  2. Ryu J., Byeon H., Lee S.J., Sung H.J. Flapping dynamics of a flexible plate with Navier slip, Physics of Fluids, 2019, vol. 31, no. 9, article no. 091901.

  3. Prabavathi D., Selvaraj A., Jothi E., Shanmugan S. Rotating oscillations of solar cooker with a permeable bar plate in a couple stress of fluid dynamics, International Journal of Engineering and Advanced Technology, 2019, vol. 8, no. 5, pp. 876 - 879.

  4. Yakovleva T.V., Krysko V.A., Krysko V.A. Nonlinear dynamics of the contact interaction of a three-layer plate-beam nanostructure in a white noise field, Journal of Physics: Conference Series, 2019, vol. 1210, no. 1, article no. 012160.

  5. Tekin G., Kadloǧlu F. Viscoelastic behavior of shear-deformable plates, International Journal of Applied Mechanics, 2017, vol. 9, no. 6, article no. 1750085.

  6. Turkova V.A., Stepanova L.V. Vestnik Permskogo natsional'nogo issledovatel'skogo politekhnicheskogo universiteta. Mekhanika, 2016, № 3, pp. 207 - 221.

  7. Barguev S.G. Sbornik nauchnykh trudov po materialam VII Mezhdunarodnoi nauchno-prakticheskoi konferentsii, Ivanovo, Izd-vo IP Tsvetkov, 2016, pp. 18 - 21.

  8. Cha P.D. Free vibrations of a uniform beam with multiple elastically mounted two-degree-of-freedom systems, Journal of Sound and Vibration, 2007, vol. 307, no. 1 - 2, pp. 386 - 392.

  9. Wu J.-J., Whittaker A.R. The natural frequencies and mode shapes of a uniform cantilever beam with multiple two-DOF spring-mass systems, Journal of Sound and Vibration, 1999, vol. 227, no. 2, pp. 361 - 381.

  10. Midzhidon A.D., Barguev S.G. Sovremennye tekhnologii. Sistemnyi analiz. Modelirovanie, 2004, no. 1, pp. 32 - 34.

  11. Shurgal'skii E.F., Enikeev I.Kh., Danilenko N.V., Karepanov S.K., Bodzholyan V.A. Avtorskoe svidetel'stvo SU 1627219 A1, 15.02.1991.

  12. Dobryshkin A.Yu., Sysoev O.E., Sysoev E.O. Trudy MAI, 2019, no. 109, available at: http://trudymai.ru/eng/published.php?ID=111349. DOI: 10.34759/trd-2019-109-4

  13. Danilenko N.V., Kostin A.V., Shurgal'skii E.F., Enikeev I.Kh., Karepanov S.K. Avtorskoe svidetel'stvo SU 1526834 A1, 07.12.1989.

  14. Grushenkova E.D., Mogilevich L.I., Popov V.S., Popova A.A. Trudy MAI, 2019, no. 106, available at: http://trudymai.ru/eng/published.php?ID=105618

  15. Enikeev I.Kh., Kuznetsova O.F., Polyanskii V.A., Shurgal'skii E.F. Zhurnal vychislitel'noi matematiki i matematicheskoi fiziki, 1988, no. 28 (1), pp. 90.

  16. Sysoev O.E., Dobryshkin A.Yu., Nein S.N. Trudy MAI, 2018, no. 98, available at: http://trudymai.ru/eng/published.php?ID=90079

  17. Enikeev I.Kh. Chislennoe issledovanie obtekaniya zatuplennykh tel potokom gazovzvesi (Numerical study of blunt bodies flow-around by a gas suspension stream), Doctor's thesis, Moscow, 1984, 116 p.

  18. Enikeev I.Kh. Teoreticheskie osnovy khimicheskoi tekhnologii, 2006, no. 40 (1), pp. 85 - 94.

  19. Ruslantsev A.N., Dumanskii A.M. Trudy MAI, 2017, no. 97, available at: http://trudymai.ru/eng/published.php?ID=87163

  20. Shomakhov A.Yu. Vestnik Rossiiskogo universiteta druzhby narodov. Seriya: Matematika, informatika, fizika, 2013, no. 4, pp. 45 - 55.

  21. Shomakhov A.Yu. Vestnik Rossiiskogo universiteta druzhby narodov. Seriya: Matematika, informatika, fizika, 2012, no. 2, pp. 33 - 42.


Download

mai.ru — informational site MAI

Copyright © 2000-2021 by MAI

Вход