On the stability of solutions of certain classes of initial-boundary-value problems in aerohydroelasticity

P. A. Vel'misov ; A. V. Ankilov ; Yu. V. Pokladova

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the IV International Scientific Conference "Actual Problems of Applied Mathematics". Kabardino-Balkar Republic, Nalchik, Elbrus Region, May 22–26, 2018. Part I, Tome 165 (2019), pp. 34-46.

Résumé

We study the stability of solutions to initial–boundary-value problems for coupled systems of partial differential equations describing the dynamics of deformable structural elements interacting with a gas-liquid medium. The definitions of stability of deformable bodied adopted in this work correspond to the concept of the Lyapunov stability of dynamic systems. The stability of deformable elements of vibration devices interacting with subsonic and supersonic flows is examined. The effect of a gas or liquid (in the model of an ideal compressible medium) is determined from asymptotic equations of aerohydromechanics. For the description of the dynamics of elastic elements, we use nonlinear models of solid deformable bodies with transverse and longitudinal deformations. Models are described by coupled nonlinear systems of partial differential equations. The study of stability is based on the construction of positive-definite Lyapunov-type functionals corresponding to these systems; sufficient conditions for the stability of their solutions are obtained.

Détail

Keywords: aerohydroelasticity, mathematical modeling, dynamic stability, elastic plate, subsonic flow, supersonic flow, partial differential equation, functional.

Bibliographie

[1] Algazin S. D., Kiiko I. A., Flatter plastin i obolochek, Nauka, M., 2006

[2] Ankilov A. V., Velmisov P. A., Funktsionaly Lyapunova v nekotorykh zadachakh dinamicheskoi ustoichivosti aerouprugikh konstruktsii, UlGTU, Ulyanovsk, 2015

[3] Ankilov A. V., Velmisov P. A., Matematicheskoe modelirovanie v zadachakh dinamicheskoi ustoichivosti deformiruemykh elementov konstruktsii pri aerogidrodinamicheskom vozdeistvii, UlGTU, Ulyanovsk, 2013

[4] Ankilov A. V., Velmisov P. A., Dinamika i ustoichivost uprugikh plastin pri aerogidrodinamicheskom vozdeistvii, UlGTU, Ulyanovsk, 2009

[5] Ankilov A. V., Velmisov P. A., Gorbokonenko V. D., Pokladova Yu. V., Matematicheskoe modelirovanie mekhanicheskoi sistemy «truboprovod—datchik davleniya», UlGTU, Ulyanovsk, 2008

[6] Ankilov A. V., Velmisov P. A., “Issledovanie dinamiki i ustoichivosti uprugogo elementa konstruktsii pri sverkhzvukovom obtekanii”, Vestn. Saratov. gos. tekhn. un-ta., 3 (57):1 (2011), 59–67

[7] Ankilov A. V., Velmisov P. A., “Issledovanie ustoichivosti vyazkouprugogo elementa konstruktsii pri sverkhzvukovom obtekanii”, Zh. Srednevolzh. mat. ob-va., 18:3 (2016), 80–90

[8] Velmisov P. A., Sudakov V. A., Ankilov A. V., “Chislennyi eksperiment v zadache o dinamike zaschitnogo ekrana pri sverkhzvukovom obtekanii potokom gaza”, Vestn. Ulyanovsk. gos. tekhn. un-ta., 3 (2013), 38–44

[9] Guvernyuk S. V., Zubkov A. F., Simonenko M. M., “Eksperimentalnoe issledovanie sverkhzvukovogo obtekaniya osesimmetrichnoi koltsevoi kaverny”, Inzh.-fiz. zh., 89:3 (2016), 670–679

[10] Kiiko I. A., Pokazeev V. V., “K postanovke zadachi o kolebaniyakh i ustoichivosti polosy v sverkhzvukovom potoke gaza”, Mekh. zhidk. gaza., 1 (2009), 159–166

[11] Kollatts L., Zadachi na sobstvennye znacheniya, Nauka, M., 1968

[12] Mogilevich L. I., Popov V. S., Popova A. A., Khristoforova A. V., “Matematicheskoe modelirovanie dinamiki vzaimodeistviya silnovyazkoi zhidkosti so stenkami kanala, ustanovlennogo na uprugom osnovanii”, Dinam. sist. mekh. mashin., 3:1 (2016), 350–354

[13] Ryakhovskii A. I., Shmidt A. A., “Chislennoe modelirovanie MGD upravleniya sverkhzvukovym potokom v srede OpenFOAM”, Tr. ISP RAN., 28:1 (2016), 197–206

[14] Ankilov A. V., Vel'misov P. A., “Stability of solutions to an aerohydroelasticity problem”, J. Math. Sci., 219:1 (2016), 14–26 | MR | Zbl

[15] Askari E., Jeong K. H., Amabili M., “Hydroelastic vibration of circular plates immersed in a liquid-filled container with free surface”, J. Sound Vibration., 332:12 (2013), 3064–3085

[16] Aulisa E., Ibragimov A., Kaya-Cekin E. Y., “Fluid structure interaction problem with changing thickness beam and slightly compressible fluid”, Discr. Contin. Dyn. Syst. Ser. S., 7:6 (2014), 1133–1148 | DOI | MR | Zbl

[17] Baghdasaryan G. Y., Mikilyan M. A., Saghoyan R. O., “Influence of supersonic gas flow on the amplitude of non-linear oscillations of rectangular plates”, Mechanics. Proc. Natl. Acad. Sci. Armenia., 69:4 (2016), 20–40 | DOI | MR

[18] Brehm C., Housman J., Kiris C., “Noise generation mechanisms for a supersonic jet impinging on an inclined plate”, J. Fluid Mech., 797 (2016), 802–850 | DOI | MR | Zbl

[19] Faal R. T., Derakhshan D., “Flow-Induced Vibration of Pipeline on Elastic Support”, Proc. Eng., 14 (2011), 2986–2993

[20] Filippi A., Skews B., “Supersonic flow fields resulting from axisymmetric internal surface curvature”, J. Fluid Mech., 831 (2017), 271–288 | DOI | MR

[21] Gatica G. N., Heuer N., Meddahi S., “Coupling of mixed finite element and stabilized boundary element methods for a fluid-solid interaction problem in 3D”, Numer. Meth. Partial Differ. Equations., 30:4 (2014), 1211–1233 | MR | Zbl

[22] Gounko Y. P., “Patterns of steady axisymmetric supersonic compression flows with a Mach disk”, Shock Waves., 27:3 (2017), 495–-506 | DOI

[23] Kounadis A. N., “Flutter instability and other singularity phenomena in symmetric systems via combination of mass distribution and weak damping”, Int. J. Nonlin. Mech., 42:1 (2007), 24–35 | DOI | MR | Zbl

[24] Kontzialis K., Moditis K., Paidoussis M. P., “Transient simulations of the fluid-structure interaction response of a partially confined pipe under axial flows in opposite directions”, J. Pressure Vessel Tech., 139:3 (2017), 1–8

[25] Moditis K., Paidoussis M., Ratigan J., “Dynamics of a partially confined, discharging, cantilever pipe with reverse external flow”, J. Fluids Struct., 63 (2016), 120–139

[26] Vedeneev V. V., “Effect of damping on flutter of simply supported and clamped panels at low supersonic speeds”, J. Fluids Struct., 40 (2013), 366–372 | DOI

[27] Willems S., Gulhan A., Esser B., “Shock induced fluid-structure interaction on a flexible wall in supersonic turbulent flow”, Progr. Flight Phys., 5 (2013), 285–308 | DOI