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Mathematical Analysis of High Aspect-Ratio AircraftWing

in Subsonic Air Flow:Incompressible and Compressible Cases

Marianna Shubov

Department of Mathematics and Statistics,

University of New Hampshire,

Durham NH

Flutter Analysis

Flutter

Flutter Analysis

Definition

F-15B/FTF - II

Experiment

Facts

Models

Main Results

References

Aeroelastic Problem

IBVP

Operator Setting

SpectralAsymptotics

Analytic Mechanism

Possio Int. Eq.

2 / 45

• Flutter is an instability endemic to aircraft that occurs at high enoughairspeed in subsonic flight and sets a “flutter boundary” on attainableairspeed in the subsonic regime.The determination of aircraft flutter characteristics is one of the mostimportant safety aspects in the aircraft design and analysis process.Damage inflicted by flutter is extremely costly.

• Flutter analysis is ranked by NASA aeronautics as one of the topresearch projects

• “Some fear flutter because they do not understand it” said thefamous aerodynamist Theodore von Karman. “And some fear it,” headded “because they do.”

Flutter

Flutter Analysis

Definition

F-15B/FTF - II

Experiment

Facts

Models

Main Results

References

Aeroelastic Problem

IBVP

Operator Setting

SpectralAsymptotics

Analytic Mechanism

Possio Int. Eq.

3 / 45

“Flutter is the onset, beyond some speed -altitude combinations, of unstable

and destructive vibrations of a lifting surface in an airstream. Flutter most

commonly encountered on bodies subjected to large lateral aerodynamic loads

of lift type, such as wings, tails, and control surfaces”

Ψ = iLΨ+

t∫

0

F(t− σ)Ψ(σ)dσ

Evolution-convolution equation

Theodorsen - Garrick experiment

F-15B/FTF - II in Flight

Flutter

Flutter Analysis

Definition

F-15B/FTF - II

Experiment

Facts

Models

Main Results

References

Aeroelastic Problem

IBVP

Operator Setting

SpectralAsymptotics

Analytic Mechanism

Possio Int. Eq.

4 / 45

Flight Experiment Setup

Flutter

Flutter Analysis

Definition

F-15B/FTF - II

Experiment

Facts

Models

Main Results

References

Aeroelastic Problem

IBVP

Operator Setting

SpectralAsymptotics

Analytic Mechanism

Possio Int. Eq.

5 / 45

• Two Experiments conducted Simultaneously:

◦ Aeroelasticity Experiment with Flexible Wing

◦ Testing of New Gust Monitoring Device

Some Facts about the flights

Flutter

Flutter Analysis

Definition

F-15B/FTF - II

Experiment

Facts

Models

Main Results

References

Aeroelastic Problem

IBVP

Operator Setting

SpectralAsymptotics

Analytic Mechanism

Possio Int. Eq.

6 / 45

• Taking into account all four flights, the experiment flew almost 7hours.

• A total of 3.4 billion data points were sampled and recorded duringthe four flights by the high speed on-board recorder alone.

• Mach=0.8 @ 2000 feet were the maximum speed and altitudesachieved.

• Generated high angle of attack data & unique dynamic stall data ⇒sparked new research interest.

Flutter

Models

Main Results

References

Aeroelastic Problem

IBVP

Operator Setting

SpectralAsymptotics

Analytic Mechanism

Possio Int. Eq.

7 / 45

Models to be discussed

Flutter

Models

Main Results

References

Aeroelastic Problem

IBVP

Operator Setting

SpectralAsymptotics

Analytic Mechanism

Possio Int. Eq.

8 / 45

• Model of high aspect-ratio aircraft wing in subsonic, inviscid,incompressible air flow

• Same type wing in subsonic, inviscid, compressible air flow.Aerodynamic loads and Possio integral equation

Some of the main results to be presented

Flutter

Models

Main Results

References

Aeroelastic Problem

IBVP

Operator Setting

SpectralAsymptotics

Analytic Mechanism

Possio Int. Eq.

9 / 45

• Explicit asymptotic formulas for ground vibrations modes andcorresponding eigenfunctions

• Explicit asymptotic formulas for aeroelastic modes and correspondingmode shapes

• Analytic mechanism generating fluttering modes (“flutter matrix”)

• Practical efficiency of explicit asymptotic formulas for aeroelasticmodes; comparison with numerical results

References

Flutter

Models

Main Results

References

Aeroelastic Problem

IBVP

Operator Setting

SpectralAsymptotics

Analytic Mechanism

Possio Int. Eq.

10 / 45

1. Riesz basis property of mode shapes for aircraft wing model (subsonic case); Proceedings of the Royal Soc.,462, (2006), p.607-646.

2. Exact controllability of couple bending -torsion vibration model with two- end energy dissipation; IMA J.

Mathematical Control & Inform., 23, (2006), p.279-300.3. Flutter phenomenon in aeroelasticity and its mathematical analysis; Invited survey paper J. Aerospace

Engineering, 19, (1), (2006), p.1-13.4. Spectral operators generated by bending -torsion vibration model with two- end energy dissipation; Asymptotic

Analysis, 45, (1,2), (2005), p.133-169.5. Operator-valued analytic functions generated by aircraft wing model (subsonic case); In: Control of Partial

Differential Equations, Lecture Notes Pure & Applied Math., (2005); p.243 -259.6. Asymptotic distribution of eigenfrequencies for a coupled Euler –Bernoulli and Timosehnko beam model,

(jointly with C.A. Peterson); NASA Technical Publication NASA/CR–2003–212022;2003 (NASA DrydenCenter), p.1-78.

7. Asymptotic behavior of aeroelastic modes for aircraft wing model in subsonic air flow, (jointly with A.V.Balakrishnan), Proceedings of Royal Soc., 406, (2004), p.1057-1091.

8. Asymptotic and spectral properties of the operator–valued functions generated by aircraft wing model; (jointlywith A.V. Balakrishnan); Mathematical Methods for the Applied Sciences, 27 , (2004), p.329-362.

9. Spectral properties of nonselfadjoint operators generated by coupled Euler–Bernoulli and Timoshenko model,(jointly with A.V. Balakrishnan and C.A. Peterson); Zeitschrift fur Angewandte Mathematik und Mechanik,84, (2), (2004), p.291-313.

10. Asymptotic analysis of coupled Euler–Bernoulli and Timoshenko beam model; (jointly with C. Peterson);Mathematische Nachrischten, 267, (2004), p.88-109.

11. Asymptotic analysis of aircraft wing model in subsonic air flow, IMA Journal on Applied Mathematics, 66,(2001), p.319-356.

12. Asymptotic representation for root vectors of nonselfadjoint operators and pencils generated by aircraft wingmodel, J. Math. Anal. Appl., 206, (2001), p.341-366.

13. Riesz basis property of root vectors of nonselfadjoint operators generated by aircraft wing model in subsonicair flow, Math. Methods in Appl. Sci,, 23, (18), (2000), p.1585-1615.

14. Asymptotics of aeroelastic modes and the basis property of mode shapes for aircraft wing model; J. FranklinInst., 338, (2/3), (2001), p.171-185.

Aeroelastic Problem

Flutter

Models

Main Results

References

Aeroelastic Problem

Model

Recursion Relations

IBVP

Operator Setting

SpectralAsymptotics

Analytic Mechanism

Possio Int. Eq.

11 / 45

bending torsion angle

X(x, t) = (h(x, t), α(x, t))T , −L ≤ x ≤ 0, t ≥ 0,

Model: system of two coupled damped integro-differential equations

(Ms −Ma)X(x, t) + (Ds − uDa)X(x, t)+

(Ks − u2Ka)X = [f1(x, t), f2(x, t)]

T .

where u > 0 - stream velocity

Ms =

[m S

S I

], Ma = (−πρ)

[1 −a

−a (a2 + 1/8)

],

m - density of the structure, S - mass moment,I - moment of inertia; ρ≪ 1, a ∈ [−1, 1];

Flutter

Models

Main Results

References

Aeroelastic Problem

Model

Recursion Relations

IBVP

Operator Setting

SpectralAsymptotics

Analytic Mechanism

Possio Int. Eq.

12 / 45

Ds =

[0 0

0 0

], Da = (−πρ)

[0 1

−1 0

],

Ks =

[E ∂4

∂x4 0

0 −G ∂2

∂x2

], Ka = (−πρ)

[0 00 −1

],

E - bending stiffness; G - torsion stiffness

Flutter

Models

Main Results

References

Aeroelastic Problem

Model

Recursion Relations

IBVP

Operator Setting

SpectralAsymptotics

Analytic Mechanism

Possio Int. Eq.

13 / 45

f1(x, t) = −2πρ

t∫

0

[uC2(t− σ)− C3(t− σ)

]g(x, σ)dσ,

f2(x, t) = −2πρ

t∫

0

[1/2C1(t− σ)− auC2(t− σ) + aC3(t− σ)

+uC4(t− σ) + 1/2C5(t− σ)]g(x, σ)dσ

Aerodynamical functions: Ci, i = 1 . . . 5,

g(x, σ) = u α(x, σ) + h(x, σ) +

(1

2− a

)α(x, σ)

Recursion Relations

Flutter

Models

Main Results

References

Aeroelastic Problem

Model

Recursion Relations

IBVP

Operator Setting

SpectralAsymptotics

Analytic Mechanism

Possio Int. Eq.

14 / 45

C1(λ) =

∞∫

0

e−λt

C1(t)dt =u

λ

e−λ/u

K0(λ/u) +K1(λ/u), ℜλ > 0,

C2(t) =

t∫

0

C1(σ)dσ,

C3(t) =

t∫

0

C1(t− σ)(uσ −√

u2σ2 + 2uσ)dσ,

C4(t) = C2(t) +C3(t),

C5(t) =

t∫

0

C1(t− σ)((1+ uσ)√

u2σ2 + 2uσ − (1+ uσ)2)dσ,

K0(z) and K1(z) - the modified Bessel functions (McDonald functions)

Boundary Conditions

Flutter

Models

Main Results

References

Aeroelastic Problem

IBVP

Boundary Conditions

Initial Conditions

Energy of vibration

IBVP

Operator Setting

SpectralAsymptotics

Analytic Mechanism

Possio Int. Eq.

15 / 45

At the end x = −L:

h(−L, t) = h′(−L, t) = α(−L, t) = 0

At the end x = 0: self straining actuator action

Eh′′(0, t) + βh′(0, t) = 0, h

′′′(0, t) = 0,

Gα′(0, t) + δα(0, t) = 0, β, δ ∈ C+ ∪ {∞},

If β > 0, then β ≡ gh - bending control gain,If δ > 0, then δ ≡ gα - torsion control gain

Initial Conditions

Flutter

Models

Main Results

References

Aeroelastic Problem

IBVP

Boundary Conditions

Initial Conditions

Energy of vibration

IBVP

Operator Setting

SpectralAsymptotics

Analytic Mechanism

Possio Int. Eq.

16 / 45

h(x,0) = h0(x), h(x,0) = h1(x),

α(x,0) = α0(x), α(x,0) = α1(x)

Assumptions

(a)

det

[m S

S I

]> 0,

(b)

0 < u ≤√2G

L√πρ

(divergence speed)

Energy of vibration

Flutter

Models

Main Results

References

Aeroelastic Problem

IBVP

Boundary Conditions

Initial Conditions

Energy of vibration

IBVP

Operator Setting

SpectralAsymptotics

Analytic Mechanism

Possio Int. Eq.

17 / 45

E(t) = 1

2

0∫

−L

[E|h′′(x, t)|2 +G|α′(x, t)|2 + m|h(x, t)|2+

I|α(x, t)|2 + S(α ˙h+ αh)− πρu2|α(x, t)|2]dx

where I = I+ πρ(a2 + 1/8), m = m− πρ, S = S− πρa

Energy dissipation

E(t) = −E|h′(0, t)|2 ℜ β −G|α(0, t)|2ℜ δ

Initial - Boundary Value Problem

Flutter

Models

Main Results

References

Aeroelastic Problem

IBVP

Boundary Conditions

Initial Conditions

Energy of vibration

IBVP

Operator Setting

SpectralAsymptotics

Analytic Mechanism

Possio Int. Eq.

18 / 45

{mh+ Sα+Eh′′′′ + πρuα = f1(x, t), −L ≤ x ≤ 0,

Sh+ Iα−Gα′′ + πρu2α− πρuh = f2(x, t), t > 0

Boundary Conditions

at x = −L: h(−L, t) = h′(−L, t) = α(−L, t) = 0

at x = 0: h′′′(0) = 0;

{Eh′′(0, t) + βh′(0, t) = 0

Gα′(0, t) + δα(0, t) = 0

Initial Conditions

h(x,0) = h0, h(x,0) = h1, α(x,0) = α0, α(x,0) = α1

Operator Setting

Flutter

Models

Main Results

References

Aeroelastic Problem

IBVP

Operator Setting

SpectralAsymptotics

Analytic Mechanism

Possio Int. Eq.

19 / 45

State space ↔ Energy space

H - set of 4 component vector - valued functions

Ψ = (h, h, α, α)T ≡ (ψ0, ψ1, ψ2, ψ3)T

satisfyingψ0(−L) = ψ′

0(−L) = ψ2(−L) = 0

Norm of state space (Hilbert space):

||Ψ||2H = 1/2

0∫

−L

[E|ψ′′

0 (x)|2 +G|ψ′2(x)|2 + m|ψ1(x)|2 + I|ψ3(x)|2

+S(ψ3(x)ψ1(x) + ψ3(x)ψ1(x)) −πρu2|ψ2(x)|2]dx

Flutter

Models

Main Results

References

Aeroelastic Problem

IBVP

Operator Setting

SpectralAsymptotics

Analytic Mechanism

Possio Int. Eq.

20 / 45

Ψ = iLΨ+

t∫

0

F(t− σ)Ψ(σ)dσ

First order in time “evolution - convolution” equation in H

Ψ = iLβδΨ+ FΨ, Ψ = (ψ0, ψ1, ψ2, ψ3)T , Ψ|t=0 = Ψ0

Lβδ - matrix differential operator in HF - matrix integral operator in H

Flutter

Models

Main Results

References

Aeroelastic Problem

IBVP

Operator Setting

SpectralAsymptotics

Analytic Mechanism

Possio Int. Eq.

21 / 45

Lβδ - matrix differential operator in H

Lβδ = −i

0 1 0 0

−EI

∆

d4

dx4 −πρuS

∆− S

∆

(G d

2

dx2 + πρu2

)−πρuI

∆

0 0 0 1ES

∆

d4

dx4

πρum

∆

m

∆

(G d

2

dx2 + πρu2

)πρuS

∆

defined on the domain

D(Lβδ) ={Ψ ∈ H : ψ0 ∈ H4(−L,0), ψ1 ∈ H2(−L,0),

ψ2 ∈ H2(−L,0), ψ3 ∈ H1(−L,0);ψ1(−L) = ψ′

1(−L) = ψ3(−L) = 0; ψ′′′0 (0) = 0;

Eψ′′0 (0) + βψ′

1(0) = 0, Gψ′2(0) + δψ3(0) = 0}

∆ = mI− S2 > 0

Flutter

Models

Main Results

References

Aeroelastic Problem

IBVP

Operator Setting

SpectralAsymptotics

Analytic Mechanism

Possio Int. Eq.

22 / 45

F - matrix integral operator in H

F =

1 0 0 0

0 [I(C1∗)− S(C2∗)] 0 0

0 0 1 0

0 0 0 [−S(C1∗) + m(C2∗)]

×

0 0 0 0

0 1 u (1/2− a)0 0 0 0

0 1 u (1/2− a)

Spectral properties of both the differential operator Lβδ and the integral

operator F are of crucial importance for the representation of the solution

Flutter

Models

Main Results

References

Aeroelastic Problem

IBVP

Operator Setting

SpectralAsymptotics

Analytic Mechanism

Possio Int. Eq.

23 / 45

Ψ = iLβδΨ+ FΨ, Ψ|t=0 = Ψ0

Laplace transform representation for solution:

Ψ(λ) =(λI − iLβδ − λF(λ)

)−1

(I − F(λ))Ψ0

Goal: find the solution in space - time domain, i.e., “calculate” theinverse Laplace transform of Ψ

R(λ) =(λI − iLβδ − λF(λ)

)−1

Generalized resolvent operator ⇒analytic operator - valued function of λ on the complex plane having abranch - cut along the negative real semi - axis.

Poles of R(λ) - discrete spectrum ↔ aeroelastic modes,Branch cut - continuous spectrum

Spectral asymptotics of Lβδ

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Models

Main Results

References

Aeroelastic Problem

IBVP

Operator Setting

SpectralAsymptotics

Analytic Mechanism

Possio Int. Eq.

24 / 45

Two - branch spectrum: If |δ| 6=√

GI, then

β- branch

µβn = (sgnn)π2/L2

√EI/∆ (n− 1/4)2 + κn(ω),

ω = |δ|−1 + |β|−1, |n| → ∞,

where ∆ = mI− S2, and

supn∈Z

{|κn(ω)|} = C(ω), C(ω) −→ 0 as ω −→ 0

δ-branch

µδn =

πn

L

√I/G

+i

2L

√I/G

lnδ +

√GI

δ −√

GI+O(|n|−1/2)

Flutter

Models

Main Results

References

Aeroelastic Problem

IBVP

Operator Setting

SpectralAsymptotics

Analytic Mechanism

Possio Int. Eq.

25 / 45

Definition: ∃ a nontrivial Φn such that(λnI − iLβδ − λnF (λn)

)Φn = 0,

⇒ λn- aeroelastic mode, Φn- mode shape

Theorem 1.The set of aeroelastic modes {λn} is asymptotically close to the set {iµn}where {µn} are eigenvalues of Lβδ

Theorem 2.

a) Lβδ is a closed linear operator with compact resolvent;b) Lβδ is nonselfadjoint unless ℜβ = ℜδ = 0;c) If ℜβ > 0 and ℜδ > 0, then Lβδ - dissipative:

ℑ(LβδΨ,Ψ)H ≥ 0, Ψ ∈ D(Lβδ);

d) Adjoint operator L∗βδ, β, δ 7→ −β,−δ

Analytical mechanism generating fluttering modes

Flutter

Models

Main Results

References

Aeroelastic Problem

IBVP

Operator Setting

SpectralAsymptotics

Analytic Mechanism

Flutter Matrix

Numerical Results

Contour Integration

Circle of Instability

Possio Int. Eq.

26 / 45

Structure of matrix integral operator

S(λ) = λI− iLβδ − λF(λ)

F ⇒

1 0 0 0

0 [I(C1∗)− S(C2∗)] 0 0

0 0 1 0

0 0 0 [−S(C1∗) + m(C2∗)]

×

0 0 0 0

0 1 u (1/2− a)0 0 0 0

0 1 u (1/2− a)

, ∆ = mI− S2

Flutter

Models

Main Results

References

Aeroelastic Problem

IBVP

Operator Setting

SpectralAsymptotics

Analytic Mechanism

Flutter Matrix

Numerical Results

Contour Integration

Circle of Instability

Possio Int. Eq.

27 / 45

F(λ) is a Laplace transform of F(t) ⇒ (kernel of convolution operator)

F(λ) =

0 0 0 0

0 L(λ) uL(λ) (1/2− a)L(λ)0 0 0 0

0 N (λ) uN (λ) (1/2− a)N (λ)

L(λ) = −2πρu

λ∆

{−S/2+ [I+ (1/2+ a)S]T(λ/u)

},

N (λ) = −2πρu

λ∆

{−m/2+ [S+ (1/2+ a)m]T(λ/u)

}

Flutter

Models

Main Results

References

Aeroelastic Problem

IBVP

Operator Setting

SpectralAsymptotics

Analytic Mechanism

Flutter Matrix

Numerical Results

Contour Integration

Circle of Instability

Possio Int. Eq.

28 / 45

Theodorsen function:

T(z) =K1(z)

K0(z) +K1(z)

K0, K1 the modified Bessel functionsKn(z) = 1/2πieπni/2H

(1)n (iz)

Definitions:

K0(z) =

∞∑

m=0

z2m

22m(m!)2

(ψ(m+ 1)− 1

m+ 1ln(z/2)

),

K1(z) =1

z+ z

2

∞∑

m=0

z2m

22m(m!)2(m+ 1)

{ln(z/2)− 1

2[ψ(m+ 1)−

ψ(m+ 2)]} ,

where ψ(m) is the digamma function

Flutter

Models

Main Results

References

Aeroelastic Problem

IBVP

Operator Setting

SpectralAsymptotics

Analytic Mechanism

Flutter Matrix

Numerical Results

Contour Integration

Circle of Instability

Possio Int. Eq.

29 / 45

Asymptotics as |z| → ∞

T(z) = 1/2+ 1/16z+O(z−2)

V(z) = T(z)− 1/2 → 0 as |z| → ∞

S(λ) = λI− iLβδ − λF(λ)

“Flutter Matrix”

Flutter

Models

Main Results

References

Aeroelastic Problem

IBVP

Operator Setting

SpectralAsymptotics

Analytic Mechanism

Flutter Matrix

Numerical Results

Contour Integration

Circle of Instability

Possio Int. Eq.

30 / 45

λF(λ) = M+N(λ),

where

M =

0 0 0 0

0 A uA (1/2− a)A0 0 0 0

0 B uB (1/2− a)B

A = −πρu/∆[I+ (a− 1/2)S],

B = πρu/∆[S+ (a− 1/2)m]

Flutter

Models

Main Results

References

Aeroelastic Problem

IBVP

Operator Setting

SpectralAsymptotics

Analytic Mechanism

Flutter Matrix

Numerical Results

Contour Integration

Circle of Instability

Possio Int. Eq.

31 / 45

N(λ) =

0 0 0 0

0 A1(λ) uA1(λ) (1/2− a)A1(λ)0 0 0 0

0 B1(λ) uB1(λ) (1/2− a)B1(λ)

A1(λ) = −2πρu∆−1V(λ/u)[I+ (a+ 1/2)S],

B1(λ) = 2πρu∆−1V(λ/u)[S+ (a+ 1/2)m]

‖N(λ)‖ ≤ C|λ−1|, |λ| → ∞

Numerical results for the eigenvalues of the operatoriLβδ +M

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Models

Main Results

References

Aeroelastic Problem

IBVP

Operator Setting

SpectralAsymptotics

Analytic Mechanism

Flutter Matrix

Numerical Results

Contour Integration

Circle of Instability

Possio Int. Eq.

32 / 45

Recall: R(λ) = (λI − iLβδ −M−N(λ))−1

N(λ)-Asymptotically smallM-“Flutter matrix”

ℜλ

ℑλ

Flutter

Models

Main Results

References

Aeroelastic Problem

IBVP

Operator Setting

SpectralAsymptotics

Analytic Mechanism

Flutter Matrix

Numerical Results

Contour Integration

Circle of Instability

Possio Int. Eq.

33 / 45

S(λ) = λI− iLβδ −M−N(λ), R(λ) = S(λ)−1

Spectrum: {λβn}n∈Z ∪ {λδ

n}n∈Z (aeroelastic modes)

Mode shapes: {Φβn}n∈Z ∪ {Φδ

n}n∈Z

Properties of mode shapes

1. Minimality2. Completeness in H3. Riesz basis property

Contour Integration

Flutter

Models

Main Results

References

Aeroelastic Problem

IBVP

Operator Setting

SpectralAsymptotics

Analytic Mechanism

Flutter Matrix

Numerical Results

Contour Integration

Circle of Instability

Possio Int. Eq.

34 / 45

ℜλ

ℑλ

R

−R

Flutter

Models

Main Results

References

Aeroelastic Problem

IBVP

Operator Setting

SpectralAsymptotics

Analytic Mechanism

Flutter Matrix

Numerical Results

Contour Integration

Circle of Instability

Possio Int. Eq.

35 / 45

Ψ(x, t) =∑

n∈Z′

eλβnt

((I− F(λβ

n)(λβn))Ψ0,Φ

β∗n

)Φ

βn+

∑

n∈Z′

eλδnt

((I− F(λδ

n)(λδn))Ψ0,Φ

δ∗n

)Φ

δn+

1

π

∞∫

0

e−rt

(ℑR(−r)

(I− F(−r)

)Ψ0

)dr

“Circle of Instability”

Flutter

Models

Main Results

References

Aeroelastic Problem

IBVP

Operator Setting

SpectralAsymptotics

Analytic Mechanism

Flutter Matrix

Numerical Results

Contour Integration

Circle of Instability

Possio Int. Eq.

36 / 45

Main model equation: Ψ(t) = i (Lβδ − iM)Ψ(t) +

t∫

0

G(t− τ)Ψ(τ)dτ

Reduced model equation: Ψ(t) = iKβδΨ(t), Kβδ = Lβδ − iM.

mh+ Sα+Eh′′′′ − πρuh+ πρu(3

2− a

)α+ πρu2α = 0

Sh+ Iα−Gα′′ − πρu(1

2+ a

)h+ πρu

(1

2− a

)2α− πρu2

(1

2+ a

)α = 0

Heuristic derivation of the energy functional accounting air-structureinteraction;Non-local in time (or memory -type) functional

Flutter

Models

Main Results

References

Aeroelastic Problem

IBVP

Operator Setting

SpectralAsymptotics

Analytic Mechanism

Flutter Matrix

Numerical Results

Contour Integration

Circle of Instability

Possio Int. Eq.

37 / 45

E(t) =1

2

0∫

−L

{E|h′′|2 +G|α′|2 + m|h|2 + I|α|2 − πρu2

(1

2+ a

)|α|2+

S[α ˙h+ αh

]+ 2πρu

t∫

0

|h+

(1

2− a

)α+

u

2α|2dτdx

Here,

•1

2

0∫

−L

E|h′′(x, t)|

2dx - Potential energy due to bending displacement

•1

2

0∫

−L

m|h(x, t)|2dx - Kinetic energy due to bending displacement

•1

2

0∫

−L

G|α′(x, t)|

2dx - Potential energy due to torsional displacement

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Possio Int. Eq.

38 / 45

E(t) =1

2

0∫

−L

{E|h′′|2 +G|α′|2 + m|h|2 + I|α|2 − πρu2(1

2+ a)|α|2+

S[α ˙h+ αh] + 2πρu

t∫

0

|h+

(1

2− a

)α+

u

2α|2}dτ}dx

Here,

•1

2

0∫

−L

I|α(x, t)|2dx - Kinetic energy due to torsional displacement

•1

2

0∫

−L

S[α˙h + αh](x, t)dx - Kinetic energy due to bending-torsion coupling

• πρu

0∫

−L

dx

t∫

0

|

(h +

(1

2− a

)α +

u

2α

)(x, τ)|

2dτ - Energy of vibration due to air-structure

interaction

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Possio Int. Eq.

39 / 45

Theorem. For each value of the airspeed u, there exists R(u) > 0 suchthat the following statement holds. If an eigenvalue λn satisfies|λn| > R(u), then this eigenmode is stable, i. e., ℜλn < 0. Thefollowing estimate is valid for R(u) :

R(u) = C√

ρ

G ℜ δu3/2

with C being an absolute constant.

Corollary. For each u, all unstable modes are located inside the “circle ofinstability” |λ| = R(u). The number of these eigenmodes is finite.

Possio Integral Equation in Aeroelasticity

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Possio Int. Eq.

40 / 45

Physical assumptions on airflow

1. Flow ⇒ nonviscous fluid ⇒ Euler equation for stream velocity ~v

2. Compressible Flow: ρt +∇ • (ρ~v) = 0 (continuity equation)

3. Isentropic flow: P = kργ

4. Irrotational (or potential) flow: ~v = ∇Φ

5. Subsonic (0 < M < 1)

6. Coupling between structure and airflow

• Flow Tangency Condition(Flow is attached to wing surface)

• Kutta-Joukowsky Condition(Pressure drop off the wing and on trailing edge)

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Possio Int. Eq.

41 / 45

The Possio Integral Equation relates pressure distribution over a typicalsection of a slender wing to a normal velocity of the points on a wingsurface (“downwash”)

b−b~U

x

s

z Model: High aspect-ratio planar wing; all

cross-sections along wing-span are

identical; only one spatial variable along

cord

• Velocity field → potential U -free stream

velocity

Velocity potential:

U + φ(x, z, t),−∞ < x <∞, 0 < z <∞,

t > 0

Linearized version of Euler Equation for disturbancepotential

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Possio Int. Eq.

42 / 45

∂2φ

∂2t+ 2Ma∞

∂2φ

∂t∂x= a2∞(1−M2)

∂2φ

∂2x+ a2∞

∂2φ

∂2z

a∞-speed of sound in flight altitudeM = U/a∞-Mach number

Boundary conditions make the problem complicated

• Flow Tangency Condition (flow is attached)

• Kutta-Joukowsky Condition (zero pressure off the wing and at thetrailing edge)

• Far Field conditions

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Possio Int. Eq.

43 / 45

b−b~U

x

s

z

1. Flow Tangency Condition

∂

∂zφ(x, z, t)|z=0 = wa(x, t), |x| < b

wa-given normal velocity of wing

2. Kutta-Joukowsky ConditionAcceleration potential:

ψ(x, z, t) =∂φ

∂t+ U

∂φ

∂x

• ψ(x, 0, t) = 0, |x| > b• lim

x→b−0ψ(x, 0, t) = 0

3. Far Field conditionsDisturbance potential → 0 as |x| → ∞ or z → ∞

The Possio Integral Equation

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Possio Int. Eq.

44 / 45

Wa(·, λ) =

√1−M2

2

{HbA(·, λ)− 1√

1−M2×

[λHbL(λ)A(·, λ)− λg−(λ, x)e

−bλL(λ, A(·, λ)

)]−

α1∫

0

a(s)[λHbL(λs)A(·, λ)−

λg−(λs, x)e−λbsL

(λs, A(·, λ)

)]ds+

α2∫

0

a(−s)[λHbL∗(λs)A(·, λ) +

λg+(λs, x)e−λbsL

(−λs, A(·, λ)

)]ds

}

Asymptotical form of Possio Equation as λ → ∞

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Possio Int. Eq.

45 / 45

2TWa(·, λ) =[√

1−M2 − λL(λ)]F (·, λ) +

λe−bλh−(x, λ)L(λ, F (·, λ))−M {Hb[F (·, λ)]− T [F (·, λ)]}

Main difficulty:

L(λ) → Volterra integral operator

L(λ, ·) → Integral operator with degenerate kernel

Hb(·) → Finite Hilbert transform

T (·) → “Inverse” to Hb

Hb and T → Singular integral operators