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'''Aeroelasticity''' is the branch of [[physics]] and [[engineering]] that studies the interactions between the [[inertial force|inertial]], [[elasticity (physics)|elastic]], and [[aerodynamic force|aerodynamic]] forces that occur when an elastic body is exposed to a [[fluid]] flow. The study of aeroelasticity may be broadly classified into two fields: ''static aeroelasticity'', which deals with the static or [[steady state]] response of an elastic body to a fluid flow; and ''dynamic aeroelasticity'', which deals with the body's [[Dynamics (mechanics)|dynamic]] (typically [[vibration]]al) response. 

Aircraft are prone to aeroelastic effects because of their requirement to be lightweight and produce large aerodynamic loads. Aeroelasticity problems are prevented by changing the mass or stiffness of an aircraft which can be determined and verified through the use of calculations, ''ground vibration tests'' and ''flight flutter trials''. ''Flutter'' can generally be eliminated by the careful placement of ''mass balances'' on [[Flight control surfaces |control surfaces]]. Aircraft are designed to avoid the key aeroelastic failures such as:
# '''divergence''' where the aerodynamic forces increase the angle of attack of a wing which further increases the force; 
# '''control reversal''' where control activation produces an opposite aerodynamic moment that reduces, or in extreme cases, reverses the control effectiveness; and 
# '''flutter''' which is the uncontained vibration that can lead to the destruction of an aircraft. 

The synthesis of aeroelasticity with [[thermodynamics]] is known as ''aerothermoelasticity'', and its synthesis with [[control theory]] is known as ''aeroservoelasticity''. 

== History ==
The second failure of [[Samuel Langley]]'s prototype plane on the Potomac has been attributed to aeroelastic effects (specifically, torsional divergence).<ref name="Bisplinghoff">Bisplinghoff, R. L., Ashley, H. and Halfman, H., ''Aeroelasticity''. Dover Science, 1996, {{ISBN|0-486-69189-6}}.</ref> An early scientific work on the subject was [[George H. Bryan|George Bryan]]'s ''Theory of the Stability of a Rigid Aeroplane'' published in 1906.<ref name="soundcloud.com">https://soundcloud.com/aerosociety-podcast</ref> Problems with torsional divergence plagued aircraft in the [[World War I|First World War]] and were solved largely by trial-and-error and ad-hoc stiffening of the wing. The first recorded and documented case of flutter in an aircraft was that which occurred to a [[Handley Page Type O|Handley Page O/400]] bomber during a flight in 1916, when it suffered a violent tail oscillation, which caused extreme distortion of the rear fuselage and the elevators to move asymmetrically. Although the aircraft landed safely, in the subsequent investigation [[Frederick W. Lanchester|F. W. Lanchester]] was consulted. One of his recommendations was that left and right elevators should be rigidly connected by a stiff shaft, which was to subsequently become a design requirement. In addition, the [[National Physical Laboratory (United Kingdom)|National Physical Laboratory]] (NPL) was asked to investigate the phenomenon theoretically, which was subsequently carried out by [[Leonard Bairstow]] and [[Arthur Fage]].<ref name="soundcloud.com"/>

In 1926, [[Hans Reissner]] published a theory of wing divergence, leading to much further theoretical research on the subject.<ref name="Bisplinghoff" /> The term ''aeroelasticity'' itself was coined by [[Roxbee Cox, Baron Kings Norton|Harold Roxbee Cox]] and [[Alfred Pugsley]] at the [[Royal Aircraft Establishment]] (RAE), [[Farnborough, Hampshire|Farnborough]] in the early 1930s.<ref name="soundcloud.com"/>

In the development of [[aeronautical engineering]] at [[Caltech]], [[Theodore von Kármán]] started a course "Elasticity applied to Aeronautics".<ref>[[Theodore von Kármán]] (1967) ''The Wind and Beyond'', page 155.</ref> After teaching the course for one term, Kármán passed it over to [[Ernest Edwin Sechler]], who developed aeroelasticity in that course and in publication of [[textbook]]s on the subject.<ref>[[Ernest Edwin Sechler]] and L. G. Dunn (1942) [https://archive.org/details/in.ernet.dli.2015.205355 Airplane Structural Analysis and Design] from [[Internet Archive]].</ref><ref>E. E. Sechler (1952) ''Elasticity in Engineering''.</ref>

In 1947, [[Arthur Roderick Collar]] defined aeroelasticity as ''"the study of the mutual interaction that takes place within the triangle of the inertial, elastic, and aerodynamic forces acting on structural members exposed to an airstream, and the influence of this study on design"''.<ref>{{cite journal |last=Collar |first=A. R. |title=The first fifty years of aeroelasticity |journal=Aerospace |year=1978 |volume=5 |series=2 |pages=12–20}}</ref>

== Static aeroelasticity ==
In an aeroplane, two significant static aeroelastic effects may occur. ''Divergence'' is a phenomenon in which the elastic twist of the wing suddenly becomes theoretically infinite, typically causing the wing to fail. ''Control reversal'' is a phenomenon occurring only in wings with [[aileron]]s or other control surfaces, in which these control surfaces reverse their usual functionality (e.g., the rolling direction associated with a given aileron moment is reversed).

=== Divergence ===
Divergence occurs when a lifting surface deflects under aerodynamic load so as to increase the applied load, or move the load so that the twisting effect on the structure is increased. The increased load deflects the structure further, which eventually brings the structure to the point of divergence. Divergence can be understood as a simple property of the [[differential equation]](s) governing the wing [[deflection (engineering)|deflection]]. For example, modelling the airplane wing as an [[isotropic]] [[Euler–Bernoulli beam theory|Euler–Bernoulli beam]], the uncoupled torsional [[equation of motion]] is

: <math>GJ \frac{d^2 \theta}{dy^2} = -M',</math>

where ''y'' is the spanwise dimension, ''θ'' is the elastic twist of the beam, ''GJ'' is the torsional stiffness of the beam, ''L'' is the beam length, and ''M''’ is the aerodynamic moment per unit length. Under a simple lift forcing theory the aerodynamic moment is of the form

: <math>M' = C U^2 (\theta + \alpha_0),</math>

where ''C'' is a coefficient, ''U'' is the free-stream fluid velocity, and α<sub>0</sub> is the initial angle of attack. This yields an [[ordinary differential equation]] of the form

: <math>\frac{d^2 \theta}{dy^2} + \lambda^2 \theta = -\lambda^2 \alpha_0,</math>

where

: <math>\lambda^2 = C \frac{U^2}{GJ}.</math>

The boundary conditions for a clamped-free beam (i.e., a cantilever wing) are

: <math>\theta|_{y=0} = \left.\frac{d\theta}{dy}\right|_{y=L} = 0,</math>

which yields the solution

: <math>\theta = \alpha_0 [\tan(\lambda L) \sin(\lambda y) + \cos(\lambda y) - 1].</math>

As can be seen, for ''λL'' = ''π''/2 + ''nπ'', with arbitrary integer number ''n'', tan(''λL'') is infinite. ''n'' = 0 corresponds to the point of torsional divergence. For given structural parameters, this will correspond to a single value of free-stream velocity ''U''. This is the torsional divergence speed. Note that for some special boundary conditions that may be implemented in a wind tunnel test of an airfoil (e.g., a torsional restraint positioned forward of the aerodynamic center) it is possible to eliminate the phenomenon of divergence altogether.<ref name="Hodges">Hodges, D. H. and Pierce, A., ''Introduction to Structural Dynamics and Aeroelasticity'', Cambridge, 2002, {{ISBN|978-0-521-80698-5}}.</ref>

=== Control reversal ===
{{main|Control reversal}}
Control surface reversal is the loss (or reversal) of the expected response of a control surface, due to deformation of the main lifting surface. For simple models (e.g. single aileron on an Euler-Bernoulli beam), control reversal speeds can be derived analytically as for torsional divergence. Control reversal can be used to aerodynamic advantage, and forms part of the [[Kaman servo-flap rotor]] design.<ref name="Hodges" />

== Dynamic aeroelasticity ==
'''Dynamic aeroelasticity''' studies the interactions among aerodynamic, elastic, and inertial forces. Examples of dynamic aeroelastic phenomena are:

=== Flutter ===<!-- This section is linked from [[Resonance]], and [[Aeroelastic flutter]] redirects here -->
'''Flutter''' is a dynamic instability of an elastic structure in a fluid flow, caused by [[positive feedback]] between the body's deflection and the force exerted by the fluid flow. In a [[linear system]], "flutter point" is the point at which the structure is undergoing [[simple harmonic motion]]—zero net [[damping]]—and so any further decrease in net damping will result in a [[self-oscillation]] and eventual failure. "Net damping" can be understood as the sum of the structure's natural positive damping and the negative damping of the aerodynamic force. Flutter can be classified into two types: ''hard flutter'', in which the net damping decreases very suddenly, very close to the flutter point; and ''soft flutter'', in which the net damping decreases gradually.<ref>G. Dimitriadis, University of Liège, [http://www.ltas-aea.ulg.ac.be/cms/uploads/Aeroelasticity06.pdf ''Aeroelasticity: Lectrue 6: Flight testing''].</ref> 

In water the mass ratio of the pitch inertia of the foil to that of the circumscribing cylinder of fluid is generally too low for binary flutter to occur, as shown by explicit solution of the simplest pitch and heave flutter stability determinant.<ref name="windeng372013">{{cite journal |url=http://multi-science.metapress.com/content/c553k773504m276x/?p=b99b20fe0d14408b9a01c901c4c31c05&pi=3 |archive-url=https://archive.is/20141029011643/http://multi-science.metapress.com/content/c553k773504m276x/?p=b99b20fe0d14408b9a01c901c4c31c05&pi=3 |url-status=dead |archive-date=2014-10-29 |title=Binary Flutter as an Oscillating Windmill – Scaling & Linear Analysis |journal=Wind Engineering |volume=37 |year=2013 }}</ref>

[[File:Tacoma Narrows Bridge destruction.ogv|thumb|Video of the Tacoma Narrows Bridge being destroyed through aeroelastic fluttering]]
Structures exposed to aerodynamic forces—including wings and aerofoils, but also chimneys and bridges—are designed carefully within known parameters to avoid flutter.  In complex structures where both the aerodynamics and the mechanical properties of the structure are not fully understood, flutter can be discounted only through detailed testing. Even changing the mass distribution of an aircraft or the [[stiffness]] of one component can induce flutter in an apparently unrelated aerodynamic component.  At its mildest, this can appear as a "buzz" in the aircraft structure, but at its most violent, it can develop uncontrollably with great speed and cause serious damage to or lead to the destruction of the aircraft,<ref name = "YouTube">{{youtube|nRit6tcNT4s|Visual demonstration of flutter which destroys an RC aircraft}}.</ref> as in [[Braniff Flight 542]], or the prototypes for the [[VL Myrsky]] fighter aircraft. Famously, the original [[Tacoma Narrows Bridge (1940)|Tacoma Narrows Bridge]] was destroyed as a result of aeroelastic fluttering.<ref name="billah91">The adequacy of comparison between flutter in aircraft aerodynamics and Tacoma Narrows Bridge case is discussed and disputed in Yusuf K. Billah, Robert H. Scanian, [http://www.ketchum.org/billah/Billah-Scanlan.pdf  "Resonance, Tacoma Bridge failure, and undergraduate physics textbooks"]; Am. J. Phys. 59(2), 118–124, February 1991.</ref>

==== Aeroservoelasticity ====
In some cases, automatic control systems have been demonstrated to help prevent or limit flutter-related structural vibration.<ref>{{cite web |url=https://history.nasa.gov/monograph39/mon39_b.pdf |title=Control of Aeroelastic Response: Taming the Threats |format=pdf}}</ref>

==== Propeller Whirl flutter ====
''Propeller whirl flutter'' is a special case of flutter involving the aerodynamic and inertial effects of a rotating propeller and the stiffness of the supporting nacelle structure. Dynamic instability can occur involving pitch and yaw degrees of freedom of the propeller and the engine supports leading to an unstable precession of the propeller.<ref> {{cite web |url=https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19670020798.pdf |first=Wilmer H. |last=Reed |title=Review of propeller-rotor whirl flutter |publisher=Nasa |access-date=2019-11-15}}</ref> Failure of the engine supports led to whirl flutter occurring on two Lockheed L-188 Electra in 1959 on [[Braniff Flight 542]] and again in 1960.<ref> {{cite web |url=https://www.intechopen.com/books/flight-physics-models-techniques-and-technologies/aeroelastic-stability-of-turboprop-aircraft-whirl-flutter |title=Aeroelastic Stability of Turboprop Aircraft: Whirl Flutter |first=Jiří |last=Čečrdle |access-date=2019-11-15}}</ref>

==== Transonic aeroelasticity ====
Flow is highly non-linear in the [[transonic]] regime, dominated by moving shock waves. It is mission-critical for aircraft that fly through transonic Mach numbers. The role of shock waves was first analyzed by [[Holt Ashley]].<ref>Holt Ashley. "Role of Shocks in the "Sub-Transonic" Flutter Phenomenon", Journal of Aircraft, Vol. 17, No. 3 (1980), pp. 187–197.</ref> A phenomenon that impacts stability of aircraft known as "transonic dip", in which the flutter speed can get close to flight speed, was reported in May 1976 by Farmer and Hanson<ref>Farmer, M. G. and Hanson, P. W., "Comparison of Super-critical and Conventional Wing Flutter Characteristics", NASA TM X-72837.</ref> of the [[Langley Research Center]].

=== Buffeting ===
'''[[Transonic buffet|Buffeting]]''' is a high-frequency instability, caused by airflow separation or shock wave oscillations from one object striking another. It is caused by a sudden impulse of load increasing. It is a random forced vibration. Generally it affects the tail unit of the aircraft structure due to air flow downstream of the wing.{{citation needed|date=June 2018}}

The methods for buffet detection are:
# Pressure coefficient diagram<ref>{{cite journal|last=Golestani |first=A.|title=An experimental study of buffet detection on supercritical airfoils in transonic regime|journal=Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering |year=2014}}</ref>
# Pressure divergence at trailing edge
# Computing separation from trailing edge based on [[Mach number]]
# Normal force fluctuating divergence

== Prediction and cure ==
Aeroelasticity involves not just the external aerodynamic loads and the way they change but also the structural, [[damping]] and mass characteristics of the aircraft. Prediction involves making a [[mathematical model]] of the aircraft as a series of masses connected by springs and dampers which are tuned to represent the [[structural dynamics|dynamic characteristics]] of the aircraft structure. The model also includes details of applied aerodynamic forces and how they vary.

The model can be used to predict the flutter margin and, if necessary, test fixes to potential problems. Small carefully chosen changes to mass distribution and local structural stiffness can be very effective in solving aeroelastic problems.

Methods of predicting flutter in linear structures include the ''p-method'', the ''k-method'' and the ''p-k method''.<ref name="Hodges" />

For [[nonlinear system]]s, flutter is usually interpreted as a [[limit cycle]] oscillation (LCO), and methods from the study of [[dynamical system]]s can be used to determine the speed at which flutter will occur.<ref>{{cite journal |last=Tang |first=D. M. |title=Effects of geometric structural nonlinearity on flutter and limit cycle oscillations of high-aspect-ratio wings |journal=Smart Materials and Structures |year=2004 |volume=19}}</ref>

== Media ==
These videos detail the [[X-53 Active Aeroelastic Wing|Active Aeroelastic Wing]] two-phase [[NASA]]-[[United States Air Force|Air Force]] flight research program to investigate the potential of aerodynamically twisting flexible wings to improve maneuverability of high-performance aircraft at transonic and [[supersonic]] speeds, with traditional control surfaces such as [[aileron]]s and leading-edge flaps used to induce the twist.

<gallery>
Image:Active Aeroelastic Wing time lapse.ogv|Time lapsed film of Active Aeroelastic Wing (AAW) Wing loads test, December, 2002
Image:F-18A Active Aeroelastic Wing flight test.ogg|F/A-18A (now X-53) Active Aeroelastic Wing (AAW) flight test, December, 2002
</gallery>

== Notable aeroelastic failures ==
*The original [[Tacoma Narrows Bridge (1940)|Tacoma Narrows Bridge]] was destroyed as a result of aeroelastic fluttering.<ref name="billah91"/>
*Propeller whirl flutter of the [[Lockheed L-188 Electra]] on [[Braniff Flight 542]].
*[[1931 Transcontinental & Western Air Fokker F-10 crash]].
* Body freedom flutter of the [[GAF Jindivik]] drone.<ref>{{cite report |publisher=[[Defence Science and Technology Organisation]] |title=Aircraft Accident Investigation at ARL-The first 50 years |year=1993 |first=J. L. |last=Kepert |url=https://apps.dtic.mil/dtic/tr/fulltext/u2/a267086.pdf}}</ref>

== See also ==
{{colbegin}}
*[[Adaptive compliant wing]]
*[[Aerospace engineering]]
*[[Kármán vortex street]]
*[[Mathematical modeling]]
*[[Oscillation]]
*[[Parker Variable Wing]]
*[[Vortex shedding]]
*[[Vortex-induced vibration]]
*[[X-53 Active Aeroelastic Wing]]
{{colend}}

== References ==
{{reflist}}

== Further reading ==
* Bisplinghoff, R. L., Ashley, H. and Halfman, H., ''Aeroelasticity''. Dover Science, 1996, {{ISBN|0-486-69189-6}}, 880 p.
* Dowell, E. H., ''A Modern Course on Aeroelasticity''. {{ISBN|90-286-0057-4}}.
* Fung, Y. C., ''An Introduction to the Theory of Aeroelasticity''. Dover, 1994, {{ISBN|978-0-486-67871-9}}.
* Hodges, D. H. and Pierce, A., ''Introduction to Structural Dynamics and Aeroelasticity'', Cambridge, 2002, {{ISBN|978-0-521-80698-5}}.
* Wright, J. R. and Cooper, J. E., ''Introduction to Aircraft Aeroelasticity and Loads'', Wiley 2007, {{ISBN|978-0-470-85840-0}}.
* Hoque, M. E., "Active Flutter Control", [[LAP Lambert Academic Publishing]], Germany, 2010, {{ISBN|978-3-8383-6851-1}}.
* Collar, A. R., "The first fifty years of aeroelasticity", Aerospace, vol. 5, no. 2, pp.&nbsp;12–20, 1978.
* Garrick, I. E. and Reed W. H., "Historical development of aircraft flutter", Journal of Aircraft, vol. 18, pp.&nbsp;897–912, Nov. 1981.
* {{cite magazine |url= http://aviationweek.com/business-aviation/low-speed-buffet-high-altitude-transonic-training-weakness-continues |title= Low-Speed Buffet: High-Altitude, Transonic Training Weakness Continues |date= Aug 23, 2018 |author= Patrick R. Veillette |magazine= Business & Commercial Aviation |publisher= Aviation Week Network}}

== External links ==
*[http://aeroelasticity.larc.nasa.gov/ Aeroelasticity Branch – NASA Langley Research Center]
*[http://www.dlr.de/ae/en DLR Institute of Aeroelasticity]
*[https://web.archive.org/web/20110811003313/http://www.nlr.nl/smartsite.dws?l=en&id=9041 National Aerospace Laboratory]
*[https://web.archive.org/web/20060210003204/http://aeweb.tamu.edu/aeroel/ The Aeroelasticity Group – Texas A&M University]
*[http://naca.larc.nasa.gov/ NACA Technical Reports – NASA Langley Research Center]

[[Category:Aerodynamics]]
[[Category:Aircraft wing design]]
[[Category:Aerospace engineering]]
[[Category:Solid mechanics]]
[[Category:Elasticity (physics)]]
[[Category:Articles containing video clips]]