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Modeling aerodynamic discontinuities and onset of chaos in flight dynamical systems

Modélisation des discontinuitiés aérodynamiques et apparition du chaos dans les systèmes dynamiques de vol

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Abstract

Various representations of the aerodynamic contribution to the aircraft’s equations of motion are shown to be compatible within the common assumption of their Fréchet differentiability. Three forms of invalidating Fréchet differentiability are identified, and the mathematical model is amended to accommodate their occurrence. Some of the ways in which chaotic behavior may emerge are discussed, first at the level of the aerodynamic contribution to the equations of motion, and then at the level of the equations of motion themselves.

Analyse

Plusieurs représentations de la contribution aéro-dynamique aux équations du mouvement d’un avion sont compatibles entre elles si elles possèdent la propriété d’être différentiables selon Fréchet. Trois conditions de perte de cette propriété sont identifiées et le modèle mathématique est adapté à l’éventualité de leur occurrence. Certaines conditions d’apparition du comportement chaotique sont discutees, d’abord dans la représentation de la contribution aérodynamique aux équations du mouvement, puis dans les équations du mouvement elles-mêmes.

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References

  1. Bryan (G. H.). Stability in aviation.Mac Millan and Co., London (1911).

    MATH  Google Scholar 

  2. Jones (B.). Dynamics of the aeroplane.Sec. N. of Aerodynamic Theory,5, W. F. Durand, ed.,Dover Publications, New York (1963).

    Google Scholar 

  3. Tobak (M.). On the use of the indicial function concept in the analysis of unsteady motions of wings and wing- tail combinations.NACA Rep.1188 (1954).

  4. Tobak (M.), Pearson (W. E.). A study of nonlinear longitudinal dynamic stability.NASA TR R-209, USA (Sept. 1964).

    Google Scholar 

  5. Tobak (M.), Schiff (L. B.). On the formulation of the aerodynamic characteristics in aircraft dynamics.NASA TR R-456 (Jan. 1976).

  6. Tobak (M.), Schiff (L. B.). Aerodynamic mathematical modeling-basic concepts.AGARD Lecture Series No. 114, Paper 1 (March 1981).

  7. Chapman (G. T.), Tobak (M.). Nonlinear problems in flight dynamics. Proc. Berkeley-Ames Conference on nonlinear problems in control and fluid dynamics,Math. Sci. Press (1985). Also NASA TM-85940, USA (May 1984).

    Google Scholar 

  8. Tobak (M.), Chapman (G. T.), Schiff (L. B.). Mathematical modeling of the aerodynamic characteristics in flight dynamics. Proc. Berkeley-Ames Conference on nonlinear problems in control and fluid dynamics,Math. Sci. Press (1985). Also NASA TM-85880, USA (Jan. 1984).

    Google Scholar 

  9. Fliess (M.), Lamnahbi (M.), Lamnahbi-Lagarrigue (F.). An algebraic approach to nonlinear functional expansions.IEEE CAS, USA (Aug. 1983),30, No. 8, pp. 554–570.

    MATH  Google Scholar 

  10. Tobak (M.), Chapman (G. T.). Nonlinear problems in flight dynamics involving aerodynamic bifurcations. AGARD Symposium on unsteady aerodynamic- fundamentals and applications to aircraft dynamics,AGARD CP 386, USA (1985), pp. 25–1 to 25–15.

    Google Scholar 

  11. Tobak (M.), Ünal (A.). Bifurcations in unsteady aerodynamics.NASA TM-88316, USA (June 1986).

    Google Scholar 

  12. Volterra (V.). Theory of functionals and of integral and integro-differential equations.Dover Pub. Inc., New York (1959).

    MATH  Google Scholar 

  13. Coleman (B. D.), Noll (W.). An approximation theorem for functionals with applications in continuum mechanics.Arch. Rational. Mech. Anal., USA (1960),6, pp. 355–370.

    Article  MATH  MathSciNet  Google Scholar 

  14. Nowinski (J. L.). Note on the applications of the Fréchet derivative.Int. J. Non-Linear Mech., USA (1983),18, No. 4, pp. 297–306.

    Article  MATH  MathSciNet  Google Scholar 

  15. Brooke Benjamin (T.). Bifurcation phenomena in steady flows of a viscous fluid. I, Theory. II, Experiments.Proc. Roy. Soc. London, Ser. A, USA (1978),359, pp. 1–43.

    Article  Google Scholar 

  16. Joseph (D. D.). Stability of fluid motions I.Springer- Verlag, New York (1976).

    Google Scholar 

  17. Joseph (D. D.). Hydrodynamic stability and bifurcation. Hydrodynamic instabilities and the transition to turbulence: Topics inApplied Physics,45, Swinney (H. L.) and Gollub (J. P.), eds.,Springer-Verlag, New York, USA (1981), pp. 27–76.

    Google Scholar 

  18. Sattinger (D. H.). Bifurcation and symmetry breaking in applied mathematics.Bull. (New Series) Amer. Math. Soc,3, no 2, USA (1980), pp. 779–819.

    Article  MATH  MathSciNet  Google Scholar 

  19. Lanford (O. E.). Strange attractors and turbulence. Hydrodynamic instabilities and the transition of turbu- lence: Topics inApplied Physics,45, Swinney (H. L.) and Gollub (J. P.), eds.,Springer-Verlag, New York USA (1981), pp. 7–26.

    Google Scholar 

  20. Ladyzhenskaya (O. A.). The mathematical theory of viscous incompressible flow.Gordon and Breach, New- York USA (1969).

    MATH  Google Scholar 

  21. Iooss (G.). Bifurcation and transition to turbulence in hydrodynamics. Bifurcation theory and applications: Lecture Notes in Mathematics 1057, L. Salvadori, ed.,Springer-Verlag, New York USA (1984), pp. 152–201.

    Chapter  Google Scholar 

  22. Grosch (C. E.), Salwen (H.). The continuous spectrum of the Orr-Sommerfield equation, Part 1: The spectrum and the eigenfunctions.J. Fluid Mech., Part 1, USA (1978),87, pp. 33–54.

    Article  MATH  MathSciNet  Google Scholar 

  23. Herron (I. M.). A diffusion equation illustrating spectral theory for boundary layer stability.Stud. Appl. Math., USA (Dec. 1982),67, No. 3, pp. 231–241.

    MATH  MathSciNet  Google Scholar 

  24. Bouard (R.), Coutanceau (M.). The early stage of development of the wake behind an impulsively started cylinder for 40 < Re < 104,J. Fluid Mech., Part 3, USA (1980),101, pp. 583–607.

    Article  Google Scholar 

  25. Loc (T. P.), Bouard (R.). Numerical solution of the early stage of the development of the unsteady viscous flow around a circular cylinder: a comparison with experimental visualization and measurement.J. Fluid Mech., USA (1985),160, pp. 93–117.

    Article  Google Scholar 

  26. Nishioka (M.), Sato (H.). Mechanism of determination of the shedding frequency of vortices behind a cylinder at low Reynolds numbers.J. Fluid Mech., Part 1, USA (1978),89, pp. 49–60.

    Article  Google Scholar 

  27. Keener (E. R.). Flow separation patterns on symmetric forebodies.NASA TM-86016, USA (Jan. 1986).

    Google Scholar 

  28. Peake (D. J.), Tobak (M.). Three-dimensional interactions and vortical flows with emphasis on high speeds.AGARDograph No. 252 (1980). AlsoNASA TM-81169, USA (March 1980).

    Google Scholar 

  29. Tobak (M.), Peake (D. J.). Topology of three-dimensional separated flows.Ann. Rev. Fluid Mech., USA (1982),14, pp. 61–85.

    Article  MathSciNet  Google Scholar 

  30. Chapman (G. T.). Topological classification of flow separation on three-dimensional bodies.AIAA Paper 86-4085, USA (Jan. 1986).

    Google Scholar 

  31. Morkovin (M. V.). On the question of instabilities upstream of cylindrical bodies.NASA CR-3231, USA (1979).

    Google Scholar 

  32. Brooke Benjamin (T.). Theory of vortex breakdown phenomenon.J. Fluid Mech., Part 4, USA (1962),4, pp. 593–629.

    Article  Google Scholar 

  33. Hwang (C.), Pi (W. S.). Investigation of steady and fluctuating pressures associated with the transonic buffeting and wing rock of a one-seventh scale model of the F-5A aircraft.NASA CR-3061, USA (1978).

    Google Scholar 

  34. Sarpkaya (T.). Vortex-induced oscillations.J. Appl. Mech., (June 1979),46, No. 2, pp. 241–258.

    Google Scholar 

  35. Bearman (P. W.). Vortex shedding from oscillating bluff bodies.Ann. Rev. Fluid Mech., USA (1984),16, pp. 195–222.

    Article  Google Scholar 

  36. Farmer (J. D.), Ott (E.), Yorke (J. A.). The dimension of chaotic attractors.Physica, USA (1983),7D, pp. 153–180.

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Moffett Field, NASA Amer Research Center, 94035, California, USA

    M. Tobak & G. T. Chapman

  2. University of Santa Clara, 95053, Santa Clara, California, USA

    A. Ünal

Authors
  1. M. Tobak
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  2. G. T. Chapman
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  3. A. Ünal
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Tobak, M., Chapman, G.T. & Ünal, A. Modeling aerodynamic discontinuities and onset of chaos in flight dynamical systems. Ann. Telecommun. 42, 300–314 (1987). https://doi.org/10.1007/BF02995248

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  • DOI: https://doi.org/10.1007/BF02995248

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