List of issues > Series «Mathematics». 2018. Vol. 23
On Dynamic Stability of a Nonlinear Aeroelastic System
A nonlinear mathematical model of a device related to vibratory technology is considered. The device is intended for the intensification of technological processes, for example, the mixing process. The action of such devices is based on the vibrations of the elastic elements when they are flowed by the flow of the mixing medium. The dynamical stability of n elastic elements located inside the flow channel is studied. The subsonic flow of the gas-liquid medium (in the model of an ideal compressible medium) is considered. The definition of the stability of an elastic body corresponds to the concept of the stability of dynamical systems by Lyapunov. The model is described by a coupled nonlinear system of partial differential equations for unknown functions - the velocity potential of a gas-liquid medium and deformations of elastic elements. On the basis of the construction of the functional, the sufficient conditions for dynamic stability are obtained. The conditions impose restrictions on the flow velocity of the gas-liquid medium, flexural rigidity of elastic elements, and other parameters of the mechanical system.
About the Authors
Petr A. Velmisov, Dr. Sci. (Phys.–Math.), Prof., Ulyanovsk State Technical University, 32, Severny Venets st., Ulyanovsk, 432027, Russian Federation, e-mail: velmisov@ulstu.ru
Ankilov Andrey Vladimirovich, Cand. Sci. (Phys.–Math.), Assoc. Prof, Ulyanovsk State Technical University, 32, Severny Venets st., Ulyanovsk, 432027, Russian Federation, e-mail: ankil@ulstu.ru
For citation:
Velmisov P.A., Ankilov A.V. On Dynamic Stability of a Nonlinear Aeroelastic System. The Bulletin of Irkutsk State University. Series Mathematics, 2018, vol. 23, pp. 3-19. (In Russian). https://doi.org/10.26516/1997-7670.2018.23.3
1. Ageev R.V., Mogilevich L.I., Popov V.S., Popova A.A. Dvizhenie vyazkoy zhidkosti v ploskom kanale, obrazovannom vibriruyushchim shtampom i sharnirno opertoy plastinoy. Trudy MAI, 2014, no 78, pp. 1-13. (in Russian)
2. Ankilov A.V., Velmisov P.A. Funktsionaly Lyapunova v nekotorykh zadachakh dinamicheskoy ustoychivosti aerouprugikh konstruktsiy. Ul’yanovsk, UlGTU Publ., 2015. 146 p. (in Russian)
3. Ankilov A.V., Velmisov P.A., Tamarova Yu.A. Matematicheskaya model’ vibratsionnogo ustroystva [The dynamical stability of an elastic element of the flow channel] Avtomatizatsiya protsessov upravleniya, 2014, no 3(37), pp. 58–67. (in Russian)
4. Ankilov A.V., Velmisov P.A. Matematicheskoe modelirovanie v zadachakh dinamicheskoy ustoychivosti deformiruemykh elementov konstruktsiy pri aerogidrodinamicheskom vozdeystvii. Ul’yanovsk, UlGTU Publ., 2013. 322 p. (in Russian)
5. Ankilov A.V., Velmisov P.A., Semenova E.P. Issledovanie dinamicheskoy ustoychivosti uprugikh elementov stenok kanala. Vestnik Saratovskogo gosudarstvennogo tekhnicheskogo universiteta, 2009. no 2(38), iss. 1. pp. 7-17. (in Russian)
6. Ankilov A.V., Velmisov P.A. Dinamika i ustoychivost’ uprugikh plastin pri aerogidrodinamicheskom vozdeystvii. Ul’yanovsk : UlGTU Publ., 2009. 220 p. (in Russian)
7. Ankilov A.V., Velmisov P.A. Ustoychivost’ resheniy nekotorykh klassov integro-differentsial’nykh uravneniy v chastnykh proizvodnykh. Vestnik Samarskogo gosudarstvennogo universiteta, 2008, no 8/1(67), pp. 331-344. (in Russian)
8. Velmisov P.A., Gorshkov G.M., Ryabov G.K. Gidrodinamicheskiy izluchatel’. Patent 2062662 Rossiyskaya Federatsiya, MPK6 V 06 V 1/18, 1/20. zayavitel’ i patentoobladatel’ Ul’yanovskiy gos. tekhnich. un-t. № 5038746/28 zayavl.20.07.92 opubl. 27.06.96, Byul. №18. (in Russian)
9. Velmisov P.A., Loginov B.V. Metod gruppovykh preobrazovaniy v nekotorykh dvukhtochechnykh granichnykh zadachakh, opisyvayushchikh formy izgiba sterzhney. Matematicheskoe modelirovanie, 1995, vol. 7, no 5, pp. 37-38. (in Russian)
10. Kollatz L. Zadachi na sobstvennye znacheniya. M.: Nauka, 1968. 503 p. (in Russian)
11. Mogilevich L.I., Popov V.S., Popova A.A., Hristoforova A.V. Matematicheskoe modelirovanie dinamiki vzaimodeystviya sil’novyazkoy zhidkosti so stenkami kanala, ustanovlennogo na uprugom osnovanii. Dinamika sistem, mekhanizmov imashin, 2016, vol. 3, no 1, pp. 350-354. (in Russian)
12. Ankilov A.V., Vel’misov P.A. Stability of solutions to an aerohydroelasticity problem. Journal of Mathematical Sciences, 2016, vol. 219, no 1, pp. 14-26. https://doi.org/10.1007/s10958-016-3079-4
13. Ankilov A.V., Velmisov P.A. Mathematical Modelling of Dynamics and Stability of Elastic Elements of Vibration Devices. Proceeding of 1st IFAC Conference on Modelling, Identification and Control of Nonlinear Systems (MICNON 2015, Saint Petersburg, Russia, 24-26 June 2015). IFAC-PapersOnLine, 2015, vol. 48, is. 11, pp. 970-975. https://doi.org/10.1016/j.ifacol.2015.09.318
14. Askari E., Jeong K.-H., Amabili M. Hydroelastic vibration of circular plates immersed in a liquid-filled container with free surface. Journal of sound and vibration, 2013, vol. 332, no. 12. pp. 3064–3085. https://doi.org/10.1016/j.jsv.2013.01.007
15. Faal R.T., Derakhshan D. Flow-Induced Vibration of Pipeline on Elastic Support. Procedia Engineering, 2011, no 14, pp. 2986-2993. https://doi.org/10.1016/j.proeng.2011.07.376
16. 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. Methods Partial Differ. Equations, 2014, vol. 30, no 4, pp. 1211–1233. https://doi.org/10.1002/num.21866
17. 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. Journal of Pressure Vessel Technology, Transactions of the ASME, 2017, vol. 139, no 3, pp. 1-8. https://doi.org/10.1115/1.4034405
18. Loginov B.V., Trenogin V.A., Velmisov P.A. Bifurcation and stability in some problems of continua mechanics. Zeitschrift fur Angewandte Mathematik und Mechanik, 1996, vol. 76, supp. 2, pp. 241-244.
19. Moditis K., Paidoussis M., Ratigan J. Dynamics of a partially confined, discharging, cantilever pipe with reverse external flow. Journal of Fluids and Structures, 2016, no 63. pp. 120-139. https://doi.org/10.1016/j.jfluidstructs.2016.03.002
20. Mogilevich L.I., Popova A.A., Popov V.S. On the dynamic interaction of an elastic cylindrical shell with a fluid laminar stream inside in application to pipeline transportation. Science and technology in transport, 2007, no 2. pp. 69-72.
21. Paidoussis M. P. The Canonical problem of the fluid-conveying pipe and radiation of the knowledge gained to other dynamics problems across Applied Mechanics. J. Sound and Vibr, 2008, no 3, vol. 310, pp. 462-492. https://doi.org/10.1016/j.jsv.2007.03.065
22. Sidorov N., Loginov B., Sinitsyn A., Falaleev M. Lyapunov-Schmidt Methods in Nonlinear Analysis and Applications. Kluwer Academic Publischers, 2002. 547 p.