Stability of solutions of initial boundary value problems of aerohydroelasticity

P. A. Velmisov ; A. V. Ankilov

Contemporary Mathematics. Fundamental Directions, Proceedings of the Seventh International Conference on Differential and Functional-Differential Equations (Moscow, August 22–29, 2014). Part 2, Tome 59 (2016), pp. 35-52.

Résumé

At designing structures and devices interacting with the flow of gas or liquid, it is necessary to solve the problems associated with the investigation of the stability required for their functioning and operational reliability. The definition of stability of an elastic body, taken in the article, corresponds to the Lyapunov's concept of stability of dynamical system. On the base of a proposed nonlinear mathematical model the dynamic stability of the elastic aileron of the wing taking into account the incident subsonic flow of gas or liquid (in an ideal model of a incompressible environment) is investigated. Also a nonlinear mathematical model of the device relating to the vibration technique, which is intended for intensification of technological processes, for example, the process of mixing is considered. The action of these devices is based on the oscillations of elastic elements at the flowing around a of gas or liquid flow. The dynamic stability of the elastic element, located on one wall of the flow channel with the subsonic flow of gas or liquid (in an ideal model of a compressible environment) is investigated. The both models is described by coupled nonlinear system of differential equations for the unknown functions – the potential of the gas velocity and deformation of the elastic element. On the basis of the construction of functionals, the sufficient conditions of the stability, impose restrictions on the free-stream velocity of the gas, the flexural stiffness of the elastic element, and other parameters of the mechanical system is obtained. The examples of construction of the stability regions for particular parameters of the mechanical system are presented.

Bibliographie

[1] A. V. Ankilov, P. A. Velmisov, Stability of Viscoelastic Elements of Walls of Flowing Channels, UlGTU, Ul'yanovsk, 2000 (in Russian)

[2] A. V. Ankilov, P. A. Velmisov, V. D. Gorbokonenko, Yu. V. Pokladova, Mathematical Modeling of Mechanical System “Pipeline–Pressure Sensorm”, UlGTU, Ul'yanovsk, 2008 (in Russian)

[3] A. V. Ankilov, P. A. Velmisov, Dynamics and Stability of Elastic Plates at Aerohydrodynamic Influence, UlGTU, Ul'yanovsk, 2009 (in Russian)

[4] A. V. Ankilov, P. A. Velmisov, Mathematical Modelling in Problems of Dynamic Stability of Deformable Elements of Constructions at Aerohydrodynamic Influence, UlGTU, Ul'yanovsk, 2013 (in Russian)

[5] V. I. Van'ko, I. K. Marchevskii, G. A. Shcheglov, “Numerical and analytical method for studying the stability of airfoil equilibrium positions in flow”, Vestn. MGTU im. N. E. Baumana. Ser. Estestv. Nauki, 5 (2011), 3–10 (in Russian)

[6] P. A. Velmisov, G. M. Gorshkov, G. K. Ryabov, Hydrodynamic radiator, Patent 2062662, Russian Federation, MPK6 V 06V 1/18, 1/20, Applicant and patentee No. 5038746/28, claimed 20.07.92, published 27.06.96, Ulyanovsk State Tech. Univ., 1996 (in Russian)

[7] P. A. Velmisov, S. V. Kireev, Mathematical Modeling in Problems of Static Instability of Elastic Elements of Constructions at Aerohydrodynamic Influence, UlGTU, Ul'yanovsk, 2011 (in Russian)

[8] P. A. Velmisov, A. A. Molgachev, Mathematical Modeling in Problems of Dynamic Stability of Viscoelastic Elements of Flowing Channels, UlGTU, Ul'yanovsk, 2012 (in Russian)

[9] Yu. P. Barmetov, I. A. Dobrodeich, “To calculation of nonstationary currents of the compressed liquid in the pipeline”, Izv. vuzov. Aviatsionnaya tekhnika, 2006, no. 1, 18–21 (in Russian)

[10] A. V. Zvyagin, “The movement of viscous liquid in the channel with elastic boundaries”, Vestn. Mosk. un-ta. Ser. 1. Mat. Mekh., 2005, no. 1, 50–54 (in Russian) | Zbl

[11] B. A. Ershov, G. A. Kuteeva, “Fluctuations of ideal liquid in a rectangular vessel with an elastic insert on a wall. The accounting of internal friction in insert material”, Vestn. SPbU. Ser. 1, 2005, no. 2, 86–94 (in Russian)

[12] L. Kollatc, Problems on Eigenvalues, Nauka, Moscow, 1968 (in Russian)

[13] N. V. Naumova, V. A. Ershov, D. N. Ivanov, “Deformation of elastic spherical shell, fixed at the equator, in the flow of a viscous incompressible fluid”, Vestn. SPbU. Ser. 3, 2011, no. 3, 124–130 (in Russian)

[14] V. V. Ovchinnicov, V. M. Popov, S. V. Filimonov, “Application of the extended hypothesis of harmonicity for the calculation of flutter characteristics of the aircraft”, Nauchn. vestn. Mosk. gos. tekhn. un-ta grazhd. aviatsii, 2013, no. 195, 93–100 (in Russian)

[15] V. G. Sokolov, A. V. Bereznev, “The equations of the movement of a curvilinear site of the pipeline with a liquid stream”, Izv. vuzov. Neft' i gaz, 2004, no. 6, 76–80 (in Russian)

[16] V. G. Sokolov, I. O. Razov, “Parametrical vibrations and dynamic stability of long-distance gas pipelines at above-ground laying”, Vestn. grazhd. inzhenerov, 2014, no. 2, 65–68 (in Russian)

[17] Balakrishnan A. V., “Toward a mathematical theory of aeroelasticity”, System modeling and optimization, Proc. 21st IFIP TC7 Conf. (Sophia Antipolis, France, July 21–25, 2003), Kluwer Academic Publishers, Boston, 2005, 1–24 | DOI | MR | Zbl

[18] Bendiksen O. O., Seber G., “Fluid-structure interactions with both structural and fluid nonlinearities”, J. Sound Vibr., 315:3 (2008), 664–684 | DOI

[19] Dimitrienko Yu. I., Koryakov M. N., Zakharov A. A., Stroganov A. S., “Computational modeling of conjugated gasdynamic and thermomechanical processes in composite structures of high speed aircraft”, Math. Model. Num. Methods, 2:3-3(3) (2014), 3–24

[20] Florea R., Hall K. C., Dowell E. H., “Analysis of eigenvalues and reduced order model of nonstationary transonic potential flow of profiles to define the boundaries of flutter”, J. Aircraft, 37:3 (2000), 454–462 | DOI

[21] Masahide Ya., Koji I., Takefumi U., Itsuma Yu., “Shock-stall-flutter of a two-dimensional airfoil”, AIAA Journal, 42:2 (2004), 215–219 | DOI

[22] Mogilevich L. I., Popova A. A., Popov V. A., “On the dynamic interaction of an elastic cylindrical shell with a fluid laminar stream inside in application to pipeline transportation”, Sci. Tech. Transport, 2007, no. 2, 69–72

[23] Plyusnin A. V., “Boundary-element method modelling of inside and outside nonstationary interaction of aircraft body and liquid”, Math. Model. Num. Methods, 2:2-2(2) (2014), 77–100

[24] Soltani N., Esfahanian V., Haddadpour H., “Analytical prediction of panel flutter using unsteady potential flow”, J. Aircraft, 40:4 (2003), 805–807 | DOI

[25] 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 Vibr., 310:3 (2008), 462–492 | DOI

[26] Wu X.-Sh., Wu J.-Sh., “Numerical method for the calculation of the swept wing flutter”, Trans. Beijing Inst. Tech., 27:5 (2007), 385–389