Summary
A new method of calculating the flutter boundaries of undamped aeroelastic “typical section” models is presented. The method is an application of the weak transversality theorem used in catastrophe theory. In the first instance, the flutter problem is cast in matrix form using a frequency domain method, leading to an eigenvalue matrix. The characteristic polynomial resulting from this matrix usually has a smooth dependence on the system's parameters. As these parameters change with operating conditions, certain critical values are reached at which flutter sets in. Our approach is to use the transversality theorem in locating such flutter boundaries using this criterion:at a flutter boundary, the characteristic polynomial does not intersect the axis of the abscissa transversally. Formulas for computing the flutter boundaries of structures with two degrees of freedom are presented, and extension to multi degree of freedom systems is indicated. The formulas have obvious applications in, for instance, problems of panel flutter at supersonic Mach numbers. Substantial savings in computation resources are possible when this non-iterative method is used, compared to existing frequency domain methods which are essentially iterative.
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Bairstow, L.: Theory of wing flutter. British A. R. C., R & M 1041. London: HMSO 1925.
Frazer, R. A., Duncan, W. J.: The flutter of aeroplane wings. British A. R. C., R & M 1155, London: HMSO 1928.
Theodorsen, T.: General theory of aerodynamic instability and the mechanism of flutter. NACA Report496 (1935).
Johnson, W.: Helicopter theory. Princeton: University Press 1980.
Whitehead, D. S.: Force and moment coefficients for vibrating airfoils in cascades. BritishA. R. C., R & M 3254. London: HMSO 1960.
Whitehead, D. S.: Effect of mistuning on the vibration of turbomachine blades induced by wakes. J. Mech. Eng. Sci.8, 15–21 (1966).
Kaza, K. R. V., Kielb, R. E.: Flutter and response of a mistuned cascade in incompressible flow. AIAA J.20, 1120–1127 (1982).
Dugundji, J., Bundas, J.: Flutter and forced response of mistuned rotors using standing wave analysis. AIAA J.22, 1652–1661 (1984).
Bakhle, M. A., Reddy, T. S. R., Keith, T. G.: Time domain flutter analysis of cascades using a full-potential solver. AIAA J.30, 163–170 (1992).
Poincaré, H.: Thèse. Sur les propriétés des fonctions définies par les équations aux différences partielles, 1879, ∄urvres de Henri Poincaré, Tome I. Paris: Gauthier-Villars 1951.
Andronov, A. A., Pontryagin, L. S.: Systemes grossiers (Coarse systems). Dokl. Akad. Nauk SSSR14, 247–251 (1937).
Thom, R.: Structural stability and morphogenesis, reprint. Boston: Addison Wesley 1989.
Arnol'd, V. I.: Catastrophe theory. Berlin, Heidelberg, New York: Springer 1983.
Arnol'd, V. I.: Lectures on bifurcations in versal families. Russian Math. Surveys27, 54–123 (1972).
Arnol'd, V. I.: On matrices depending on parameters. Russian Math. Surveys26, 29–43 (1971).
Thom, R., Levin, H.: Singularities of differentiable mappings. Bonn Math. Schr.6 (1959). (reprinted in: Lecture Notes in Mathematics, vol.192. Berlin, Heidelberg, New York: Springer 1971).
Abraham, R., Robbin, J.: Transversal mappings and flows. New York: Benjamin 1967.
Brieskorn, E., Knörrer, H.: Plance algebraic curves. Boston: Birkhäuser 1986.
Zeeman, E. C.: The umbilic bracelet and the double cusp catastrophe. In: Structural stability, the theory of catastrophes, and its applications in the sciences. Lecture Notes in Mathematics, vol525, pp. 328–366. Berlin, Heidelberg, New York: Springer 1976.
Poston, T., Stewart, I.: Catastrophe theory and its applications. London: Pitman 1978.
Bisplinghoff, R. L., Ashley, H., Principles of aeroelasticity. New York: Dover 1962.
Dowell, E. H., Curtiss, H. C., Scanlan, R. H., Sisto, F.: A modern course in aeroelasticity. Rockville: Sijthoff & Noordhoff 1978.
Milnor, J.: Singular points of complex hypersurfaces. Ann. Math. Studies61. Princeton: University Press 1968.
Arnol'd, V. I., Gusein-Zade, S. M., Varchenko, A. N.: Singularities of differentiable mappings, vol II. Monodromy and asymptotics of integrals. Boston: Birkhäuser 1985.
Bolotin, V. V.: Nonconservative problems of elastic stability. Oxford: Pergamon 1963.
Duncan, W. J., Biot, M. A., Johnson, D. C., Bishop, R. E. D.: Receptances in mechanical systems. J. Roy. Aeronaut. Soc.58, 305 (1954).
Duncan, W. J.: Mechanical admittances and their applications to oscillation problems. British A. R. C., R & M 2000. London: HMSO 1946.
Bishop, R. E. D., Johnson, D. C.: The mechanics of vibration. Cambridge: University Press 1960.
Afolabi, D.: Sylvester's eliminant and stability criteria for gyroscopic systems. J. Sound Vib. (in press).
Turnbull, H. W.: Theory of equations. London: Oliver and Boyd 1939.
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Afolabi, D. Flutter analysis using transversality theory. Acta Mechanica 103, 1–15 (1994). https://doi.org/10.1007/BF01180214
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DOI: https://doi.org/10.1007/BF01180214