Volume 8, no. 2Pages 24 - 35

Models of Multiparameter Bifurcations in Boundary Value Problems for ODEs of the Fourth Order on Divergence of Elongated Plate in Supersonic Gas Flow

T.E. Badokina, B.V. Loginov
At the application of bifurcation theory methods to nonlinear boundary value problems for ordinary differential equations of the fourth and higher order there usually arise technical difficulties, connected with determination of bifurcation manifolds, spectral investigation of the direct and conjugate linearized problems and the proof of their Fredholm property. For overcoming of this difficulty here the roots separation method is applied to the relevant characteristic equations with subsequent presentation of critical manifolds, that allows to investigate nonlinear problems in the precise statement. Such approach is applied here to two-point boundary value problem for the nonlinear ODE of the fourth order describing the buckling (divergence) of an elongated plate in a supersonic flow of gas, subjected to compressed or extended boundary stresses at the various boundary fastenings.
Full text
buckling of an elongated plate; bifurcation; Fredholm property.
1. Vol'mir A.S. Ustoychivost' deformiruemykh sistem [Stability of Deformated Systems]. Moscow, Nauka, 1967. 984 p.
2. Bolotin V.V. Nekonservativnye zadachi teorii uprugoy ustoychivosti [Nonconservative Problems of the Elastic Stability Theory]. Moscow, GIFML, 1961. 339 p.
3. Naimark M.A. Lineynye differentsial'nye operatory [Linear differential operators]. Moscow, Nauka, 1969. 528 p.
4. Loginov B., Badokina T., Makeeva O. Green Functions Construction for Divergence Problems in Aeroelasticity. ROMAI Journal, 2008, vol. 4, no. 2, pp. 33-44.
5. Mel'nikov Yu.A. Influence Functions and Matrices. Ser. Text and Reference Books. Mech. Engng. Vol. 119. N.Y., Basel, Marcel Dekker, 1999. 469 p.
6. Na T.Y. Computational Methods in Engineering Boundary Value Problems. London, Academic Press, 1979. 294 p.
7. Vel'misov P.A., Kireev S.V. Matematicheskoe modelirovanie v zadachakh staticheskoy neustoychivosti uprugikh elementov konstruktsiy pri aerogidrodinamicheskom vozdeystvii [Mathematical Modelling in Problems of Static Instability of Elastic Structural Elements under the Aerohydrodynamic Impact]. Ulyanovsk, UlGTU, 2011. 200 p.
8. Algazin S.D., Kiyko I.A. Flatter plastin i obolochek [Flutter of plates and shells]. Moscow, Nauka, 2006. 247 p.
9. Vainberg M.M., Trenogin V.A. Teoriya vetvleniya resheniy nelineynykh uravneniy [Branching Theory of Solutions to Nonlinear Equations]. Moscow, Nauka, 1969. 524 p.
10. Shafarevich I.R. O reshenii uravneniy vysshikh stepeney (metod Shturma) [On the Resolving of Higher Degrees Equations (Sturm Method)]. Moscow, GITTL, 1987. 24 p.