Reduction of Boundary Value Problem to Possio Integral Equation in Theoretical Aeroelasticity
A. V. Balakrishnan and M. A. Shubov
Source: J. Appl. Math. Volume 2008
(2008), Article ID
846282, 27 pages.
Abstract
The present paper is the first in a series of works devoted to the solvability of the Possio singular integral equation. This equation relates the pressure distribution over a typical section of a slender wing in subsonic compressible air flow to the normal velocity of the points of a wing (downwash). In spite of the importance of the Possio equation, the question of the existence of its solution has not been settled yet. We provide a rigorous reduction of the initial boundary value problem involving a partial differential equation for the velocity potential and highly nonstandard boundary conditions to a singular integral equation, the Possio equation. The question of its solvability will be addressed in our forthcoming work.
First Page:
Show
Hide
Full-text: Access denied (no subscription
detected)
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.jam/1220969295
Digital Object Identifier: doi:10.1155/2008/846282
Mathematical Reviews number (MathSciNet): MR2415203
Zentralblatt MATH identifier: 1142.76028
References
C. Possio, ``Aerodynamic action on oscillating profile in compressible fluid at subsonic velocities,'' L'Aerotechnica, vol. 18, no. 4, pp. 441--458, 1938.
A. V. Balakrishnan and M. A. Shubov, ``Asymptotic behaviour of the aeroelastic modes for an aircraft wing model in a subsonic air flow,'' Proceedings of The Royal Society of London Series A, vol. 460, no. 2044, pp. 1057--1091, 2004.
Mathematical Reviews (MathSciNet): MR2133856
Zentralblatt MATH: 1109.76029
Digital Object Identifier: doi:10.1098/rspa.2003.1217
JSTOR: links.jstor.org
M. A. Shubov, ``Riesz basis property of mode shapes for aircraft wing model (subsonic case),'' Proceedings of The Royal Society of London Series A, vol. 462, no. 2066, pp. 607--646, 2006.
Mathematical Reviews (MathSciNet): MR2269680
Zentralblatt MATH: 1149.76636
Digital Object Identifier: doi:10.1098/rspa.2005.1579
M. A. Shubov, ``Flutter phenomenon in aeroelasticity and its mathematical analysis,'' Journal of Aerospace Engineering, vol. 19, no. 1, pp. 1--13, 2006.
E. Reissner, ``On the theory of oscillating airfoils of finite span in subsonic compressible flow,'' NACA Technical Note NACA-TN-1953, NACA-TR-1002, NASA Center, Cambridge, Mass, USA, 1953.
C. E. Watkins, H. L. Runyan, and D. S. Woolston, ``The kernel function of the integral equation relating the lift and downwash distributions of oscillating finite wings in subsonic flow,'' Tech. Rep. NACA-TN-3131, NASA Langley Research Center, Hampton, Va, USA, 1954.
Mathematical Reviews (MathSciNet): MR59081
A. V. Balakrishnan, ``The Possio integral equation of aeroelasticity theory,'' Journal of Aerospace Engineering, vol. 16, no. 4, pp. 139--154, 2003.
A. V. Balakrishnan, ``The Possio integral equation of aeroelasticity: a modern view,'' in System Modeling and Optimization, F. Ceragioli, A. Dontchev, H. Furuta, K. Marti, and L. Pandolfi, Eds., vol. 199 of IFIP International Federation for Information Processing, pp. 15--22, Springer, New York, NY, USA, 2006.
Mathematical Reviews (MathSciNet): MR2249318
Digital Object Identifier: doi:10.1007/0-387-33006-2_2
R. L. Bisplinghoff, H. Ashley, and R. L. Halfman, Aeroelasticity, Dover, New York, NY, USA, 1996.
M. Goland, ``The flutter of a uniform cantilever wing,'' Journal of Applied Mechanics, vol. 12, no. 4, pp. 197--208, 1945.
P. S. Beran, T. W. Strganac, K. Kim, and C. Nichkawde, ``Studies of store-induced limit cycle oscillations using a model with full system nonlinearities,'' Nonlinear Dynamics, vol. 37, no. 4, pp. 323--339, 2004.
Zentralblatt MATH: 1080.70012
D. Hodges and E. H. Dowell, ``Nonlinear equations of motion for the elastic bending and torsion of twisted nonuniform rotor blades,'' Tech. Rep. NASA-TN-D-7818, NASA Ames Research Center, Moffett Field, Calif, USA, 1974.
A. V. Balakrishnan, ``Subsonic flutter suppression using self-straining actuators,'' Journal of the Franklin Institute, vol. 338, no. 2-3, pp. 149--170, 2001.
Mathematical Reviews (MathSciNet): MR1828168
Zentralblatt MATH: 0981.74015
Digital Object Identifier: doi:10.1016/S0016-0032(00)00088-0
M. A. Shubov and C. A. Peterson, ``Asymptotic distribution of eigenfrequencies for a coupled Euler-Bernoulli and Timoshenko beam model,'' NASA Technical Publication NASA-CR-2003-212022, NASA Dryden Flight Research Center, Edwards, Calif, USA, 2003.
I. E. Garrick, H. Ashley, D. B. Hanson, A. E. Perry, and R. H. Scanlan, ``Critical essays,'' in A Modern View of Theodore Theodorsen, E. H. Dowell, Ed., AIAA, Washington, DC, USA, 1992.
R. Clark, D. Cox, H. C. Curtiss Jr., et al., Eds., A Modern Course in Aeroelasticity, vol. 116 of Solid Mechanics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 4th edition, 2004.
Mathematical Reviews (MathSciNet): MR2107428
M. A. Shubov, ``Mathematical modeling and analysis of flutter in long span suspension bridges and in blood vessel walls,'' Journal of Aerospace Engineering, vol. 17, no. 2, pp. 70--82, 2004.
M. A. Shubov, ``Mathematical modeling and analysis of flutter in bending-torsion coupled beams, rotating blades, and hard disk drives,'' Journal of Aerospace Engineering, vol. 17, no. 2, pp. 56--69, 2004.
M. J. Patil, D. H. Hodges, and C. E. S. Cesnik, ``Nonlinear aeroelastic analysis of complete aircraft in subsonic flow,'' Journal of Aircraft, vol. 37, no. 5, pp. 753--760, 2000.
M. J. Patil and D. H. Hodges, ``On the importance of aerodynamic and structural geometrical nonlinearities in aeroelastic behavior of high-aspect-ratio wings,'' Journal of Fluids and Structures, vol. 19, no. 7, pp. 905--915, 2004.
C. S. Ventres, ``Shear flow aerodynamics: lifting surface theory,'' AIAA Journal, vol. 13, no. 9, pp. 1183--1189, 1975.
Zentralblatt MATH: 0324.76009
E. H. Dowell and M. R. Chi, ``Variable thickness shear layer aerodynamics revisited,'' AIAA Journal, vol. 15, no. 5, pp. 745--747, 1977.
Zentralblatt MATH: 0361.76016
M. H. Williams, M. R. Chi, C. S. Ventres, and E. H. Dowell, ``Aerodynamic effects of inviscid parallel shear flows,'' AIAA Journal, vol. 15, no. 8, pp. 1159--1166, 1977.
Zentralblatt MATH: 0367.76002
S. G. Mikhlin, Multidimensional Singular Integrals and Integral Equations, Pergamon Press, New York, NY, USA, 1965.
Mathematical Reviews (MathSciNet): MR0185399
Zentralblatt MATH: 0129.07701
N. N. Lebedev, Special Functions and Their Applications, Dover, New York, NY, USA, 1972.
Mathematical Reviews (MathSciNet): MR0350075
Zentralblatt MATH: 0271.33001
D. Porter and D. S. G. Stirling, Integral Equations: A Practical Treatment, from Spectral Theory to Application, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, UK, 1990.
Mathematical Reviews (MathSciNet): MR1111247
Zentralblatt MATH: 0714.45001
F. G. Tricomi, Integral Equations, Dover, New York, NY, USA, 1985.
Mathematical Reviews (MathSciNet): MR809184
R. Estrada and R. P. Kanwal, Singular Integral Equations, Birkhäuser, Boston, Mass, USA, 2000.
Mathematical Reviews (MathSciNet): MR1728075
Zentralblatt MATH: 0945.45001
Journal of Applied Mathematics
