12: Introduction to aeroelasticity
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Introduction to aeroelasticity
The Collar diagram of aeroelastic forces
The following paragraphs are excerpted from Aeroelasticity by R. L. Bisplinghoff, H. Ashley, and R. L. Halfman (1996).
Aeroelasticity is defined as a science which studies the mutual interaction between aerodynamic forces and elastic forces, and the influence of this interaction on airplane design. Aeroelastic problems would not exist if the airplane structure were perfectly rigid. Modern airplane structures are very flexible, and this flexibility is fundamentally responsible for the various types of aeroelastic phenomena. Structural flexibility itself may not be objectionable; however, aeroelastic phenomena arise when structural deformations induce additional aerodynamic forces. Such interactions may become smaller and smaller until a condition of stable equilibrium is reached, or they may tend to diverge and destroy the structure.
The term aeroelasticity, however, is not completely descriptive, since many important aeroelastic phenomena involve inertial forces as well as aerodynamic and elastic forces. We shall apply a definition in which the term aeroelasticity includes phenomena involving interactions among inertial, aerodynamic, and elastic forces, and other phenomena involving interactions between aerodynamic and elastic forces. The former will be referred to as dynamic and the latter as static aeroelastic phenomena.
Collar has ingeniously classified problems in aeroelasticity by means of a triangle of forces. Referring to Fig. 1-1 [figure [fig12.1] below], the three types of forces, aerodynamic elastic, and inertial are represented by the symbols A, E, and I, respectively, are placed at the vertices of a triangle. Each aeroelastic phenomenon can be located on the diagram according to its relation to the three vertices. For example, dynamic aeroelastic phenomena such as flutter F, lie within the triangle, since they involve all three types of forces and must be bonded to all three vertices. Static aeroelastic phenomena such as wing divergence, D, lie outside the triangle on the upper left side, since they involve only aerodynamic and elastic forces. Although it is difficult to define precise limits on the field of aeroelasticity, the classes of problems connected by solid lines to the vertices in Fig. 1-1 are usually accepted as the principal ones. Of course, other borderline fields of mechanical vibrations, V, and rigid-body aerodynamic stability, DS, are connected to the vertices by dotted lines. It is very likely that in certain cases the dynamic stability problem is influenced by airplane flexibility and it would therefore be moved within the triangle to correspond with DSA, where it would be regarded as a dynamic aeroelastic problem.
It would be convenient to state concise definitions of each aeroelastic phenomenon which appears on the diagram in Fig. 1-1.
Flutter, F. A dynamic instability occurring in an aircraft in flight at a speed called the flutter speed, where the elasticity of the structure plays an essential part in the instability.
Buffeting, B. Transient vibrations of aircraft structural components due to aerodynamic impulses produced by the wake behind wings, nacelles, fuselage pods, or other components of the airplane.
Dynamic response, Z. Transient response of aircraft structural components produced by rapidly applied loads due to gusts, landing, gun reactions, abrupt control motions, moving shock waves, or other dynamic loads.
Aeroelastic effects on stability, DSA & SSA. Influence of elastic deformations of the structure on dynamic and static airplane stability.
Load distribution, L. Influence of elastic deformations of the structure on the distribution of aerodynamic pressures over the structure.
Divergence, D. A static instability of a lifting surface of an aircraft in flight, at a speed called the divergence speed, where the elasticity of lifting surface plays an essential role in the instability.
Control effectiveness, C. Influence of elastic deformations of the structure on the controllability of an airplane.
Control system reversal, R. A condition occurring in flight, at a speed called the control reversal speed, at which the intended effects of displacing a given component of the control system are completely nullified by elastic deformations of the structure.
Mechanical vibrations, V. A related field.
Dynamic stability, DS. A related field.
Divergence analysis of a rigid wing segment
A model to illustrate the phenomenon of wing divergence consists of a uniform, rigid wing segment hinged to a fixed support in a wind tunnel as is shown in figure [fig12.2]. The hinge line is located at the elastic axis (E.A.) of the wing. The elastic axis coincides with the locus of shear centers of the wing sections.
Recall that the shear center of the cross section of a bar (wing) is a reference point in the cross section where the lateral deflections due to bending are de-coupled from the twist due to torsion (i.e., a shear force acting at the elastic axis results in bending deflections and no twist, and a torque acting at the elastic axis causes twist but no lateral deflection of the elastic axis due to bending).
The rigid wing segment is restrained against rotation, or twist, about the E.A. by a linear elastic rotational spring of stiffness
We assume two-dimensional, incompressible aerodynamics is applicable. Let
The angle of attack is written as
In the above equation
The pitching moment is given by
Moment equilibrium about the E.A. gives
Substituting for the elastic twist, lift, and pitching moment from eqs. ([eq12.1]) to ([eq12.4]), the moment equation becomes
From eq. ([eq12.10]) we see that
Responses of the rigid wing segment and the imperfect column
The response plots of the rigid wing segment model of article 1.2 and the geometrically imperfect column in article [sec11.4] on page are repeated in figure [fig12.4]. Comparing the two response plots reveals that these phenomena are the same. Both the column buckling and the wing divergence are static instabilities. The critical load
Divergence experiments
Experiments to measure the divergence dynamic pressure of an elastic wing confront the issue of damaging the wing and its supporting structure if the dynamic pressure is near or at its critical value. A nondestructive method to measure the critical dynamic pressure is accomplished by plotting the data on a Southwell plot, which was developed for elastic column buckling in article [sec11.4.1] on page . The Southwell plotting coordinates are determined from eq. ([eq12.10]) by formulating the change in the angle of attack
On the Southwell plot
Straight, uniform, unswept, high aspect ratio, cantilever wing in steady incompressible flow
Let
where
From eq. ([eq3.61]) on page the differential equation in torsion is
In reference to eq. ([eq3.121]) on page , St. Venant’s torsion theory relates the torque to the unit twist as
Aerodynamic strip theory
Strip theory assumes aerodynamic lift and moment at station
The differential lift and differential pitching moment acting at the A.C. on an typical element of the wing are shown in the figure [fig12.7], where
Differential equation of torsional divergence
Now substitute eq. ([eq12.25]) into ([eq12.20]) and rearrange the terms to get
The general solution of the ordinary differential eq. ([eq12.26]) is the sum of a particular solution and a homogenous solution.
The homogenous equation is
Hence, the general solution for the wing twist is
Hence from eqs. ([eq12.16]) and ([eq12.36]), the total wing incidence is
From eq. ([eq12.38]) we see that
The analogy between the divergence dynamic pressure for the rigid wing model and the elastic wing model is summarized in table [tab12.1].
Effect of wing sweep on divergence
Divergence of a slender straight wing that is approximately perpendicular to the airplane plane of symmetry is dependent on wing twist, referred to as torsional divergence, and bending is not a factor in the instability. For slender swept wings bending of the wing has an important and complicating affect on divergence and is referred to as bending-torsional divergence.
Let the angle
l180pt
Hence, streamwise segment
Consider a swept-forward wing with
From NASA Armstrong Fact Sheet: X-29 Advanced Technology Demonstrator Aircraft (Gibbs, 2015): The X-29’s thin supercritical wing was of composite construction. State-of-the-art composites permit aeroelastic tailoring, which allows the wing some bending but limits twisting and eliminates structural divergence within the flight envelope (deformation of the wing or breaking off in flight).
[sec12.5] Bisplinghoff, R.L., H. Ashley, and R.L. Halfman. Aeroelasticity. Mineola, NY: Dover Publications, Inc., 1996. pp. 1–3, 421–432, 474 & 475. (Originally published by Addison-Wesley, 1955.)
Gibbs, Y. “NASA Armstrong Fact Sheet: X-29 Advanced Technology Demonstrator Aircraft.” NASA.gov, November 5, 2015. https://www.nasa.gov/centers/armstrong/news/FactSheets/FS-008-DFRC.html.
Gordon, J. E. Structures, or Why Things Don’t Fall Down. Boston: Da Capo Press, 2003. (Originally published by Harmondsworth: Penguin Books. 1978.)
Practice exercises
An interesting historical account of wing torsional divergence is given by Gordon (2003); An excerpt follows. -5
During World War I Antony Fokker developed an advanced monoplane fighter—the Fokker D8—with performance better than available or in immediate prospect on the Allied side. As soon as the D8 was flown in combat conditions it was found out that, when the aircraft was pulled out of a dive in a dog fight, the wings came off. Since many lives were lost—including those of some of the best and most experienced German fighter pilots—this was a matter of very grave concern to the Germans at the time, and is still instructive to study the cause of the trouble.
Read pages 260–271 in the book by Gordon and answer the following questions.
For a given engine power, why is a monoplane generally faster than a biplane?
What was the material of wing skin on the D8? Is it effective in resisting shear?
What was the method of loading in the structural test of the wings of the D8?
What was the ultimate load factor from the structural test?
What was the first attempt to strengthen the rear wing spar?
What was the best method to strengthen the rear wing spar? and why did it work?
What is aileron reversal?
What common geometric feature do a tube and the torsion box of the old-fashioned biplane have that makes them so effective in resisting torsion?
The uniform wing sketched in figure [fig12.10] is fixed at both ends. Starting with the general solution eq. ([eq12.33]), derive the algebraic expression for
total incidence
, anddivergence dynamic pressure
.
Consider a rigid wing segment of weight W mounted on an elastic sting in a wind tunnel. The sting is modeled as a uniform, elastic, cantilever beam with bending stiffness
and length . Neglect the weight of the sting. The model is mounted in such a way to have the angle of attack when the beam is undeformed. Thus, the angle of attack , where is the nose-up rotation of the wing resulting from the bending of the sting. Denote the lift and the pitching moment acting at the aerodynamic center (A.C.) as L and , respectively.Assume
steady, two-dimensional incompressible flow at airspeed
and density ,the lift curve slope
is constant between stall points,and that the angle of attack is small.
Use the second theorem of Castigliano to determine the rotation
of the cantilever beam due to end force and moment as shown in the sketch above. Consider bending only.Determine the angle of attack
as a function of the dynamic pressure , , wing reference area S, flexural stiffness EI, chord length , , pitching moment coefficient , distance , and weight W.Determine the divergence dynamic pressure,
.
A uniform beam with a rectangular cross section rests on a knife edge at its left end, while the right end is clamped in rigid disk. This configuration is shown in figure [fig12.12]. The bending stiffness
, the distance between the knife edge and the beam’s connection to the disk is L, and the radius of the disk is . This disk rotates about a fixed smooth pin through its center under the action of applied moment as shown. Determine the relation between the applied moment and rotation angle of the disk under the assumption that the angle of rotation is small. In a wind tunnel test the disk is connected to a rigid airfoil, then this structural configuration is used to provide the rotational spring of stiffness depicted in figure [fig12.2].