Mathematical modeling of three-layer beam hydroelastic oscillations
L. I. Mogilevich1 , V. S. Popov2 , A. A. Popova3 , A. V. Christoforova4 , E. V. Popova5
1, 2, 3Yuri Gagarin State Technical University of Saratov, Saratov, Russia
4, 5Saratov State University, Saratov, Russia
Vibroengineering PROCEDIA, Vol. 12, 2017, p. 12-18.
Received 11 April 2017; accepted 12 April 2017; published 30 June 2017
The problem of hydroelastic oscillations of three-layer beam interacting with viscous liquid layer is set up and analytically solved. The problem presents the equation system of a three-layer beam and Navier-Stokes equations. The following boundary conditions are chosen: the no-slip conditions, the conditions for pressure at the edges, the simply supported edges conditions. The problem is solved for the steady-state harmonic regime. The frequency dependent distribution functions of the beam deflection are constructed. The given function allows investigating the resonance hydroelastic oscillations of a three-layer beam, as well as its deflected mode.
Keywords: hydroelasticity, vibration, three-layer beam, viscous liquid, mathematical modeling.
The study was funded by Russian Foundation for Basic Research (RFBR) according to the Projects Nos. 15-01-01604-a, 16-01-00175-a and President of Russian Federation Grant No. MD-6012.2016.8.
- Indeitsev D. A., Polypanov I. S., Sokolov S. K. Calculation of cavitation life-time of ship engine liner. Problemy Mashinostraeniya i Nadezhnos’ti Mashin, Vol. 4, 1994, p. 59-64. [CrossRef]
- Avramov K. V., Strel’nikova E. A. Chaotic oscillations of plates interacting on both sides with a fluid flow. International Applied Mechanics, Vol. 50, Issue 3, 2014, p. 303-309. [CrossRef]
- Kerboua Y., Lakis, Thomas A. A. M., Marcouiller L. Vibration analysis of rectangular plates coupled with fluid. Applied Mathematical Modelling, Vol. 32, Issue 12, 2008, p. 2570-2586. [CrossRef]
- Bochkarev S. A., Lekomtsev S. V., Matveenko V. P. Hydroelastic stability of a rectangular plate interacting with a layer of ideal flowing fluid. Fluid Dynamics, Vol. 51, Issue 6, 2016, p. 821-833. [CrossRef]
- Önsay T. Effects of layer thickness on the vibration response of a plate-fluid layer system. Journal of Sound and Vibration, Vol. 163, 1993, p. 231-259. [CrossRef]
- Ageev R. V., Mogilevich L. I., Popov V. S., Popova A. A., Kondratov D. V. Mathematical model of pulsating viscous liquid layer movement in a flat channel with elastically fixed wall. Applied Mathematical Sciences, Vol. 8, Issue 159, 2014, p. 7899-7908. [CrossRef]
- Faria Cassio T., Inman Daniel J. Modeling energy transport in a cantilevered Euler-Bernoulli beam actively vibrating in Newtonian fluid. Mechanical Systems and Signal Processing, Vol. 45, 2014, p. 317-329. [CrossRef]
- 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 Dynamics, Vol. 46, Issue 3, 2011, p. 375-388. [CrossRef]
- 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. Journal of Machinery Manufacture and Reliability, Vol. 39, Issue 4, 2010, p. 322-331. [CrossRef]
- Akcabay D. T., Young Y. L. Hydroelastic response and energy harvesting potential of flexible piezoelectric beams in viscous flow. Physics of Fluids, Vol. 24, Issue 5, 2012. [CrossRef]
- Ageev R. V., Kuznetsova E. L., Kulikov N. I., Mogilevich L. I., Popov V. S. Mathematical model of movement of a pulsing layer of viscous liquid in the channel with an elastic wall. PNRPU Mechanics Bulletin, Vol. 3, 2014, p. 17-35. [CrossRef]
- Kuznetsova E. L., Mogilevich L. I., Popov V. S., Rabinsky L. N. Mathematical model of the plate on elastic foundation interacting with pulsating viscous liquid layer. Applied Mathematical Sciences, Vol. 10, Issue 23, 2016, p. 1101-1109. [CrossRef]
- Mogilevich L. I., Popov V. S., Popova A. A., Christoforova A. V. Mathematical modeling of hydroelastic walls oscillations of the channel on winkler foundation under vibrations. Vibroengineering Procedia, Vol. 8, 2016, p. 294-299. [CrossRef]
- 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. Journal of Machinery Manufacture and Reliability, Vol. 46, Issue 1, 2017, p. 12-19. [CrossRef]
- Gorshkov A. G., Starovoitov E. I., Yarovaya A. V. Mechanics of Layered Viscoelastoplastic Structural Elements. Moscow, Fizmatlit, 2005. (in Russian). [CrossRef]
- Kubenko V. D., Pleskachevskii Yu. M., Starovoitov E. I., Leonenko D. V. Natural vibration of a sandwich beam on an elastic foundation. International Applied Mechanics, Vol. 42, Issue 5, 2006, p. 541-547. [CrossRef]
- Starovoitov E. I., Leonenko D. V. Thermal impact on a circular sandwich plate on an elastic foundation. Mechanics of Solids, Vol. 47, Issue 1, 2012, p. 111-118. [CrossRef]
- Pradhan M., Dash P. R., Pradhan P. K. Static and dynamic stability analysis of an asymmetric sandwich beam resting on a variable Pasternak foundation subjected to thermal gradient. Meccanica, Vol. 51, Issue 3, 2016, p. 725-739. [CrossRef]
- Ageev R. V., Mogilevich L. I., Popov V. S. Vibrations of the walls of a slot channel with a viscous fluid formed by three-layer and solid disks. Journal of Machinery Manufacture and Reliability, Vol. 43, Issue 1, 2014, p. 1-8. [CrossRef]
- Panovko Y. G., Gubanova I. I. Stability and Oscillations of Elastic Systems. Consultants Bureau Enterprises, Inc., New York, 1965. [CrossRef]
- Vol’mir A. S. Shells in Fluid and Gas Flows: Aeroelasticity Problems. Nauka, Moscow, 1976, (in Russian). [CrossRef]
- Lamb H. Hydrodynamics, 6th Edition. Dover Publications Inc., New York, 1945. [CrossRef]