Analysis of spatial vibrations of coaxial cylindrical shells partially filled with a fluid
Keywords:
coaxial shells, viscous potential fluid, partial filling, natural vibrations, FEMAbstract
This work is devoted to a numerical study of natural vibrations of horizontally oriented elastic coaxial shells with the annular gap completely or partially filled with a compressible viscous fluid. The solution is searched for a three-dimensional formulation of the examined dynamic problem using the finite element method. The fluid motion is described in the acoustic approximation in terms of the velocity potential. The corresponding equations together with the boundary conditions, related to a full fluid-shell contact, are transformed using the Bubnov-Galerkin method. The hydrodynamic forces are determined from the viscous stress tensor. The mathematical formulation of the problem of thin-walled structure dynamics is based on the variational principle of virtual displacements, which includes the normal and tangential components of forces exerted by the fluid on the wetted parts of elastic bodies. Modeling of the shells is based on the assumption that their curvilinear surfaces are simulated quite accurately by a set of flat segments, the strains of which are determined by the relations of the classical thin plate theory. The reliability of the results obtained is substantiated by comparing them with the known data for the case when the annular gap is completely filled with an ideal fluid. The influence of the fluid level and the size of the annular gap on the eigenfrequencies and the corresponding vibration modes of coaxial shells under a variety of boundary conditions is analyzed. It has been demonstrated that partial filling leads to splitting of natural vibration frequencies, while a decrease in the filling level facilitates the growth of their minimum values. It is shown that the appearance of mixed vibrational modes, both in the meridional and circumferential directions depends on the size of the annular gap.
Downloads
References
Mnev E.N. Kolebaniya krugovoj cilindricheskoj obolochki pogruzhennoj v zamknutuyu polost’, zapolnennuyu szhimaemoj zhidkost’yu [Vibrations of a circular cylindrical shell submerged in a closed cavity filled with an ideal compressible liquid]. Teoriya plastin i obolochek: trudy VIII Vsesoyuznoy konferentsii po teorii obolochek i plastin – Theory of shells and plates: Proceedings of the II All-Union conference on the theory of shells and plates. Kiev, AN USSR, 1962, pp. 284-288.
Mnev E.N., Pertsev A.K. Gidrouprugost’ obolochek [Hydroelasticity of Shells]. Leningrad: Sudostroenie, 1970. 365 p.
Buyvol B.N., Guz’ A.N. O kolebaniyakh dvukh tsilindricheskikh ekstsentrichno raspolozhennykh obolochek v potoke nevyazkoy zhidkosti [Oscillations of two cylindrical eccentrically arranged shells in a stream of inviscid liquid]. Doklady AN USSR – Reports of the Academy of Sciences of the Ukrainian SSR, 1966, no. 11, pp. 1412-1415.
Balakirev Yu.G. K issledovaniyu osesimmetrichnykh kolebaniy soosnykh tsilindricheskikh sistem obolochek s zhidkim zapolnitelem [Axisymmetric vibrations of coaxial cylindrical shells filled with fluid]. Inzhenernyy zhurnal. Mekhanika tverdogo tela – Soviet Engineering Journal, 1968, no. 3, pp. 133-140.
Pshenichnov G.I. Sobstvennyye kolebaniya koaksial’nykh ortotropnykh tsilindricheskikh obolochek zapolnennykh zhidkost’yu [Free vibrations of coaxial orthotropic cylindrical shells filled with fluid]. Teoriya obolochek i plastin: trudy VIII Vsesoyuznoy konferentsii po teorii obolochek i plastin – Theory of shells and plates: Proceedings of the VIII All-Union conference on the theory of shells and plates.: Nauka, 1973. 798 p. Pp. 546-549.
Baghdasaryan G.Ye. Kolebaniya koaksial’nykh tsilindricheskikh obolochek s zazorom, chastichno zapolnennykh zhidkost’yu [The vibrations of coaxial cylindrical shells with a clearance partially filled with fluid]. Izvestiya Akademii nauk Armyanskoy SSR. Mekhanika – Proc. NAS RA. Mechanics, 1968, vol. 21, no. 4, pp. 40-47.
Baghdasaryan G.Ye., Marukhyan S.A. Dynamic behavior of a coaxial cylindrical shells, with a gap partially filled with fluid. Izvestiya Natsional’noy Akademii Nauk Armenii. Mekhanika – Proc. NAS RA. Mechanics, 2011, vol. 64, no. 3, pp. 10‑21.
Shvets R.N., Marchuk R.A. Sobstvennyye kolebaniya ortotropnoy tsilindricheskoy obolochki, soprikasayushcheysya s zhidkost’yu [Natural vibrations of an orthotropic cylindrical shell in contact with a fluid]. Matematicheskiye metody i fiziko-mekhanicheskiye polya – Mathematical Methods and Physicomechanical Fields, 1975, no. 2, pp. 63-67.
Shvets R.N., Marchuk R.A. Axisymmetric vibrations of coaxial orthotropic cylindrical shells filled with fluid. Soviet Appl. Mech., 1985, vol. 21, no. 8, pp. 763-767. DOI
Marchuk R.A. Kolebaniya ortotropnoy tsilindricheskoy obolochki, soprikasayushcheysya s zhidkost’yu [Oscillations of an orthotropic cylindrical shell in contact with a fluid]. Matematicheskiye metody i fiziko-mekhanicheskiye polya – Mathematical Methods and Physicomechanical Fields, 1979, no. 9, pp. 99-103.
Yudin A.S., Ambalova N.M. Forced vibrations of coaxial reinforced cylindrical shells during interaction with a fluid. Soviet Appl. Mech., 1989, vol. 25, no. 12, pp. 1222-1227. DOI
Brown S.J. A survey of studies into the hydrodynamic response of fluid-coupled circular cylinders. Pressure Vessel Technol., 1982, vol. 104, no. 1, pp. 2-19. DOI
Païdoussis M.P. Fluid-Structure interactions: Slender structures and axial flow. London: Academic Press, 2003. Vol. 2. 1040 p.
Krajcinovic D. Vibrations of two coaxial cylindrical shells containing fluid. Eng. Des., 1974, vol. 30, no. 2, pp. 242-248. DOI
Chen S.S., Rosenberg G.S. Dynamics of a coupled shell-fluid system. Eng. Des., 1975, vol. 32, no. 3, pp. 302-310. DOI
Au-Yang M.K. Free vibration of fluid-coupled coaxial cylindrical shells of different lengths. Appl. Mech., 1976, vol. 43, no. 3, pp. 480-484. DOI
Chung H., Turula P., Mulcahy T.M., Jendrzejczyk J.A. Analysis of a cylindrical shell vibrating in a cylindrical fluid region. Eng. Des., 1981, vol. 63, no. 1, pp. 109-120. DOI
Abramov V.V., Val’shonok L.S., Dodonov V.A., Dranchenko B.N., Sidorkin A.C., Sharyy N.B. Dinamicheskiye napryazheniya v elementakh konstruktsiy, rabotayushchikh v potokakh zhidkosti [Dynamic stresses in structural elements operating in fluid flows]. Eksperimental’nyye issledovaniya i raschët napryazheniy v konstruktsiyakh [Experimental studies and calculation of stresses in structures], ed. by N.I. Prigorovskiy. M.: Nauka, 1975. Pp. 149-160.
Tani J., Nozaki Y., Ohtomo K., Sugiyama H. of the Ninth World Conference on Earthquake Engineering. Tokyo-Kyoto, Japan, August 2-9, 1988. Vol. VI, pp. 619-624.
Guiggiani M. Dynamic instability in fluid-coupled coaxial cylindrical shells under harmonic excitation. Fluids Struct., 1989, vol. 3, no. 3, pp. 211-228. DOI
Chu M.L., Brown S. Experiments on the dynamic behavior of fluid-coupled concentric cylinders. Mech., 1981, vol. 21, no. 4, pp. 129-137. DOI
Yoshikawa S., Williams E.G., Washburn K.B. Vibration of two concentric submerged cylindrical shells coupled by the entrained fluid. Acoust. Soc. Am., 1994, vol. 95, no. 6, pp. 3273-3286. DOI
Chung H., Chen S.-S. Vibration of a group of circular cylinders in a confined fluid. Appl. Mech., 1977, vol. 44, no. 2, pp. 213-217. DOI
Wauer J. Finite oscillations of a cylinder in a coaxial duct subjected to annular compressible flow. Flow, Turbulence and Combustion, 1998, vol. 61, no. 1-4, pp. 161-177. DOI
Jeong K.-H. Natural frequencies and mode shapes of two coaxial cylindrical shells coupled with bounded fluid. Sound Vib., 1998, vol. 215, no. 1, pp. 105-124. DOI
Yeh T.T., Chen S.S. Dynamics of a cylindrical shell system coupled by viscous fluid. Acoust. Soc. Am., 1977, vol. 62, no. 2, pp. 262-270. DOI
Yeh T.T., Chen S.S. The effect of fluid viscosity on coupled tube/fluid vibrations. Sound Vib., 1978, vol. 59, no. 3, pp. 453-467. DOI
Vasin S.V., Mikolyuk V.V. Svobodnyye kolebaniya soosnykh tsilindricheskikh obolochek, razdelennykh vyazkoy zhidkost’yu [Free oscillations tolerable cylindrical shells separated by a viscous fluid]. Gidroaeromekhanika i teoriya uprugosti – Hydro aeromechanics and the theory of elasticity, 1983, no. 31, pp. 108-116.
Horáček J., Trnka J., Veselý J., Gorman D.G. Vibration analysis of cylindrical shells in contact with an annular fluid region. Struct., 1995, vol. 17, no. 10, pp. 714-724. DOI
Kondratov D.V., Mogilevich L.I. Matematicheskoye modelirovaniye protsessov vzaimodeystviya dvukh tsilindricheskikh obolochek so sloyem zhidkosti mezhdu nimi pri otsutstvii tortsevogo istecheniya v usloviya vibratsii [Mathematical modelling of processes of interaction of two cylindrical environments with the layer of the liquid between them under leakage absence in condition of vibration]. Vestnik SGTU – Bulletin of Saratov State Technical University, 2007, vol. 3, no. 2, pp. 15-23.
Bochkarev S.A., Lekomtsev S.V., Matveenko V.P. Numerical modeling of spatial vibrations of cylindrical shells, partially filled with fluid. Vychislitel’nyye tekhnologii – Computing technologies, 2013, vol. 18, no. 2, pp. 12-24.
Bochkarev S.A., Lekomtsev S.V., Matveenko V.P. Natural vibrations and stability of elliptical cylindrical shells containing fluid. J. Struct. Stabil. Dynam., 2016, vol. 16, no. 10, 1550076. DOI
Bochkarev S.A., Lekomtsev S.V., Matveenko V.P. Natural vibrations of loaded noncircular cylindrical shells containing a quiescent fluid. Thin Wall. Struct., 2015, vol. 90, pp. 12-22. DOI
Joseph D.D. Viscous potential flow. Fluid Mech., 2003, vol. 479, pp. 191-197. DOI
Amabili M. Free vibration of partially filled, horizontal cylindrical shells. Sound Vib., 1996, vol. 191, no. 5, pp. 757-780. DOI
Guz’ A.N. Problems of hydroelasticity for compressible viscous fluids. Soviet Appl. Mech., 1991, vol. 27, no. 1, pp. 1-12. DOI
Bochkarev S.A., Matveenko V.P. Numerical study of the influence of boundary conditions on the dynamic behavior of a cylindrical shell conveying a fluid. Solids, 2008, vol. 43, pp. 477-486. DOI
Zienkiewicz O.C. The finite element method in engineering science. London: McGraw-Hill, 1971. 535 p.
Reddy J.N. An introduction to nonlinear finite element analysis, 2nd edn. Oxford, Oxford University Press, 2015. 687 p. DOI
Lehoucq R.B., Sorensen D.C. Deflation techniques for an implicitly restarted Arnoldi iteration. SIAM J. Matrix Anal. Appl., 1996, vol. 17, no. 4, pp. 789-821. DOI