Asymptotics of aeroelastic modes and basis property of mode shapes for aircraft wing model. (English) Zbl 1064.74063

The author provides asymptotic and spectral analysis of two-dimensional strip model [A. V. Balakrishnan, Proc. SPIE 5th Annual Int. Symp. on Smart Structures, Intern. Soc. Opt. Eng. Vol. 3323, 44–54 (1998)] of an aircraft wing in subsonic air flow. This model is governed by a system of two coupled integro-differential equations and by a two-parameter family of boundary conditions. Spectral theory of non-selfadjoint polynomial operator pencils is used.


74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
74G25 Global existence of solutions for equilibrium problems in solid mechanics (MSC2010)
45M05 Asymptotics of solutions to integral equations
Full Text: DOI


[1] A.V. Balakrishnan, Aeroelastic control with self-straining actuators: continuum models in: V. Varadan (Ed.), Proceedings of the SPIE’s Fifth Annual International Symposium on Smart Structures, March’98, San Diego, CA, Intern. Soc. Optical Eng., Vol. 3323, pp. 44-54.
[2] R.L. Bisplinghoff, H. Ashley, R.L. Halfman, Aeroelasticity, Dover Publ. Inc., New York, 1996.
[3] A.V. Balakrishnan, Theoretical limits of damping attainable by smart beams with rate feedback, in: V. Varadan, J. Chandra (Eds.), Mathematical Control in Smart Structures, Proc. SPIE, Intern. Soc. Optical Eng. 3039 (1997) 204-215.
[4] Balakrishnan, A.V., Damping performance of strain actuated beams, Comput. appl. math., 18, 1, 31-86, (1999) · Zbl 0931.74030
[5] A.V. Balakrishnan, Vibrating systems with singular mass-inertia matrices, in: S. Sivasundaram (Ed.), First International Conference on Nonlinear Problems in Aviation and Aerospace, Daytona Beach, FL, Embry-Riddle University Press, 1996, pp. 23-32.
[6] Lee, C.K.; Chiang, W.W.; O’Sullivan, T.C., Piezoelectric modal sensor/actuator pairs for … vibration control, J. acoust. soc. amer., 90, 384-394, (1991)
[7] S.M. Yang, Y.J. Lee, Modal analysis of stepped beams with piezoelectric materials, J. Sound Vibration 176 (1994) 289-300. · Zbl 0945.74584
[8] Y.C. Fung, An Introduction to the Theory of Aeroelasticity, Dover Publ. Inc., New York, 1993. Prentice Hall, Englewood Cliff, NJ, 1962.
[9] M.A. Shubov, Mathematical analysis of problem arising in modelling of flutter phenomenon in aircraft wing in subsonic airflow, IMA J. Appl. Math., to appear. · Zbl 0990.76033
[10] M.A. Shubov, Riesz basis property of root vectors of nonself-adjoint operators generated by aircraft wing model in subsonic airflow, Math. Methods Appl. Sci. 23 (2000), to appear. · Zbl 0972.47039
[11] A.V. Balakrishnan, J.W. Edwards, Calculation of the transient motion of elastic airfoils forced by control surface motion and gusts, NASA TM 81351, 1980.
[12] Abramowitz, M.; Stegun (Eds.), I., Handbook of mathematical functions, (1972), Dover New York
[13] Magnus, W.; Oberhettinger, F.; Soni, R.P., Formulas and theorems for the special functions of mathematical physics, (1966), Springer New York · Zbl 0143.08502
[14] Adams, R.A., Sobolev spaces, (1975), Academic Press New York · Zbl 0186.19101
[15] M.A. Shubov, Asymptotic representations for root vectors of nonselfadjoint operators generated by aircraft wing model in subsonic air flow, J. Math. Anal. Appl., to appear. · Zbl 1068.76045
[16] Balakrishnan, A.V., Control of structures with self-straining actuators: coupled Euler/Timoshenko model, nonlinear problems in aviation and aerospace, (1998), Gordon and Breach Science Publishers Reading, United Kingdom
[17] V.I. Istratescu, Introduction to Linear Operator Theory, Pure Applied Mathematical Series of Monographs, Marcel Dekker Inc., New York, 1981.
[18] Weidmann, J., Spectral theory of ordinary differential operators, Lecture notes in math., Vol. 1258, (1987), Springer Berlin
[19] Marcus, A.S., Introduction to the spectral theory of polynomial pencils (translation of mathematical monographs), 71, (1988), AMS, Providence RI
[20] Shubov, M.A., Basis property of eigenfunctions of nonselfadjoint operator pencils generated by radial damped string, Integral equations oper. theory, 25, 289-328, (1996) · Zbl 0855.47010
[21] Shubov, M.A., Asymptotics of spectrum and eigenfunctions for nonselfadjoint operators generated by radial wave equations, Asymptotic anal., 16, 245-272, (1998) · Zbl 0938.35113
[22] Foias, C.; Sz-Nagy, B., Harmonic analysis of operators in Hilbert spaces, (1970), North Holland Amsterdam · Zbl 0201.45003
[23] Ivanov, S.A.; Pavlov, B.S., Carleson series of resonances and the Regge problem, Math. USSR izv., 12, 1, 21-51, (1978) · Zbl 0401.47020
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