Abstract
The basics of static aeroelasticty, in contrast to dynamic aeroelasticity, are reviewed and some classic subjects such as divergence and control surface reversal are treated. The discussion starts with simple mathematical and physical models and progresses to more complex models and solution methods. Most of these models and methods prove to be useful in dynamic aeroelasticity as well.
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Notes
- 1.
See chapter ‘Aeroelasticity in Civil Engineering’, BA, especially pp. 189–200.
- 2.
For two dimensional, incompressible flow this is at the airfoil quarter-chord; for supersonic flow it moves back to the half-chord. See Ashley and Landahl [1]. References are given at the end of each chapter.
- 3.
Here in static aeroelasticity q plays the role of the eigenvalue; in dynamic aeroelasticity q will be a parameter and the (complex) frequency will be the eigenvalue. This is a source of confusion for some students when they first study the subject.
- 4.
For the reader with some knowledge of feedback theory as in, for example, Savant [2].
- 5.
Timoshenko and Gere [3].
- 6.
See, [3], pp. 197–200.
- 7.
Woodcock [4].
- 8.
See chapter ‘Aeroelastic Response of Rotorcraft’, BA, pp. 280–295, especially pp. 288–295.
- 9.
Housner, and Vreeland [5].
- 10.
Higher Order Terms.
- 11.
A more complete aerodynamic model would allow for the effect of an angle of attack at one spanwise location, say \(\eta \), on (nondimentional) lift at another, say y. This relation would then be replaced by \(C_{L}(y)=\int A(y-\eta )[\alpha _{0}(\eta )+\alpha _{e}(\eta )]d\eta \) where A is an aerodynamic influence function which must be measured or calculated from an appropriate theory. More will be said about this later.
- 12.
Note \(\lambda \equiv 0\) is not a divergence condition! Expanding (2.8) for \(\lambda \ll 1\), we obtain \(\alpha _{e}=\frac{K}{\lambda ^{2}}[1-\lambda ^{2}\tilde{y}-(1-\frac{\lambda ^{2}\tilde{y}^{2}}{2})+\cdots ] \rightarrow K[\frac{\tilde{y}^{2}}{2}-\tilde{y}]\) as \(\lambda \rightarrow 0.\)
- 13.
For a more detailed mathematical discussion of the above, see Hildebrand [6], pp. 224–234. This problem is one of a type known as ‘Sturm-Liouville Problems’.
- 14.
Duncan [7].
- 15.
For simplicity, \(\alpha _{0}\equiv 0\) in what follows.
- 16.
For additional discussion, see the following selected references: Hildebeand [6] pp. 388–394 and BAH, pp. 39–44.
- 17.
Bisplinghoff, Mar, and Pian [8], p. 247.
- 18.
This distinction between the two ways in which the aircraft may be restrained received renewed emphasis in the context of the oblique wing concept. Weisshaar and Ashley [9].
- 19.
Covert [10].
- 20.
For definiteness consider a rectangular wing divided up into small (rectangular) finite difference boxes. The weighting matrix[(W)] is for a given spanwise location and various chordwise boxes. The elements in the matrices, \(\{\partial w/\partial \xi \}\) and \(\{w\}\), are ordered according to fixed spanwise location and then over all chordwise locations. This numerical scheme is only illustrative and not necessarily that which one might choose to use in practice.
- 21.
Housner [11].
- 22.
See Sect. 4.
- 23.
Timoshenko and Gere [3].
- 24.
- 25.
Shute [12], p. 95.
- 26.
References
Ashley H, Landahl M (1965) Aerodynamics of wing and Bodies. Addison-Wesley, Boston
Savant CJ Jr (1958) Basic feedback control system design. McGraw-Hill, New York
Timoshenko SP, Gere J (1961) Theory of elastic stability. McGraw-Hill, New York
Woodcock DL (1959) Structural non-linearities, vol. I, chap 6, AGARD Manual on Aeroelasticity
Housner GW, Vreeland T Jr (1966) The analysis of stress and deformation. The MacMillan Co., New York
Hildebrand FB (1961) Advance calculus for engineers. Prentice-Hall, New Jersey
Duncan WJ (1937) Galerkin’s methods in mechanics and differential equations. Aeronaut Res Comm, Reports and memoranda, No 1798
Bisplinghoff RL, Mar JW, Pian THH (1965) Statics of deformable solids. Addison-Wesly, New York
Weisshaar TA, Ashley H (1974) Static aeroelasticity and the flying wing, revisited. J. Aircraft 11:718–720
Covert EE (1961) The aerodynamics of distorted surfaces. In: Proceedings of symposium on aerothermoelasticity. ASD TR 61-645, pp 369-398
Housner GW (1952) Bending vibrations of a pipe line containing flowing fluid. J Appl Mech 19:205
Shute N (1954) Slide rule. Wm. Morrow & CO., Inc., New York, p 10016
Dowell EH (1974) Aeroelasticity of plates and shells. Noordhoff International Publishing
Kornecki A (1974) Static and dynamic instability of panels and cylindrical shells in subsonic potential flow. J. Sound Vibr 32:251–263
Kornecki A, Dowell EH, O’Brien J (1976) On the aeroelastic instability of two-dimensional panels in uniform incompressible flow. J. Sound Vibr 47:163–178
Weisshaar TA (1978) Aeroelastic stability and performance characteristics of aircraft with advanced composite sweptforward wing structures, AFFDL TR-78-116
Weisshaar TA (1979) Forward swept wing static aeroelasticity, AFFDL TR-79-3087
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Dowell, E.H. (2022). Static Aeroelasticity. In: Dowell, E.H. (eds) A Modern Course in Aeroelasticity. Solid Mechanics and Its Applications, vol 264. Springer, Cham. https://doi.org/10.1007/978-3-030-74236-2_2
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