Mathematical Analysis of High Aspect-Ratio Aircraft
Wing
in Subsonic Air Flow:
Incompressible and Compressible Cases
Marianna Shubov
Department of Mathematics and Statistics,
University of New Hampshire,
Durham NH
1 / 45
Flutter Analysis
Flutter
Flutter Analysis
Definition
F-15B/FTF - II
Experiment
Facts
Models
Main Results
• Flutter is an instability endemic to aircraft that occurs at high enough
airspeed in subsonic flight and sets a “flutter boundary” on attainable
airspeed in the subsonic regime.
The determination of aircraft flutter characteristics is one of the most
important safety aspects in the aircraft design and analysis process.
Damage inflicted by flutter is extremely costly.
References
Aeroelastic Problem
IBVP
• Flutter analysis is ranked by NASA aeronautics as one of the top
research projects
Operator Setting
Spectral
Asymptotics
Analytic Mechanism
Possio Int. Eq.
• “Some fear flutter because they do not understand it” said the
famous aerodynamist Theodore von Karman. “And some fear it,” he
added “because they do.”
2 / 45
Flutter
Flutter Analysis
Definition
F-15B/FTF - II
Experiment
Facts
“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”
Theodorsen - Garrick experiment
Models
Main Results
References
Ψ̇ = iLΨ +
Zt
F (t − σ)Ψ̇(σ)dσ
0
Aeroelastic Problem
IBVP
Evolution-convolution equation
Operator Setting
Spectral
Asymptotics
Analytic Mechanism
Possio Int. Eq.
3 / 45
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
Spectral
Asymptotics
Analytic Mechanism
Possio Int. Eq.
4 / 45
Flight Experiment Setup
Flutter
Flutter Analysis
Definition
F-15B/FTF - II
Experiment
Facts
• Two Experiments conducted Simultaneously:
◦ Aeroelasticity Experiment with Flexible Wing
◦ Testing of New Gust Monitoring Device
Models
Main Results
References
Aeroelastic Problem
IBVP
Operator Setting
Spectral
Asymptotics
Analytic Mechanism
Possio Int. Eq.
5 / 45
Some Facts about the flights
Flutter
Flutter Analysis
Definition
F-15B/FTF - II
Experiment
Facts
Models
Main Results
• Taking into account all four flights, the experiment flew almost 7
hours.
• A total of 3.4 billion data points were sampled and recorded during
the four flights by the high speed on-board recorder alone.
References
Aeroelastic Problem
IBVP
Operator Setting
Spectral
Asymptotics
Analytic Mechanism
• Mach=0.8 @ 2000 feet were the maximum speed and altitudes
achieved.
• Generated high angle of attack data & unique dynamic stall data ⇒
sparked new research interest.
Possio Int. Eq.
6 / 45
Flutter
Models
Main Results
References
Aeroelastic Problem
IBVP
Operator Setting
Spectral
Asymptotics
Analytic Mechanism
Possio Int. Eq.
7 / 45
Models to be discussed
Flutter
Models
Main Results
References
Aeroelastic Problem
IBVP
• Model of high aspect-ratio aircraft wing in subsonic, inviscid,
incompressible air flow
Operator Setting
Spectral
Asymptotics
Analytic Mechanism
Possio Int. Eq.
• Same type wing in subsonic, inviscid, compressible air flow.
Aerodynamic loads and Possio integral equation
8 / 45
Some of the main results to be presented
Flutter
Models
Main Results
References
• Explicit asymptotic formulas for ground vibrations modes and
corresponding eigenfunctions
Aeroelastic Problem
IBVP
Operator Setting
Spectral
Asymptotics
• Explicit asymptotic formulas for aeroelastic modes and corresponding
mode shapes
Analytic Mechanism
Possio Int. Eq.
• Analytic mechanism generating fluttering modes (“flutter matrix”)
• Practical efficiency of explicit asymptotic formulas for aeroelastic
modes; comparison with numerical results
9 / 45
References
Flutter
1.
Models
2.
Main Results
References
3.
4.
Aeroelastic Problem
IBVP
Operator Setting
5.
6.
Spectral
Asymptotics
7.
Analytic Mechanism
8.
Possio Int. Eq.
9.
10.
11.
12.
13.
14.
Riesz basis property of mode shapes for aircraft wing model (subsonic case); Proceedings of the Royal Soc.,
462, (2006), p.607-646.
Exact controllability of couple bending -torsion vibration model with two- end energy dissipation; IMA J.
Mathematical Control & Inform., 23, (2006), p.279-300.
Flutter phenomenon in aeroelasticity and its mathematical analysis; Invited survey paper J. Aerospace
Engineering, 19, (1), (2006), p.1-13.
Spectral operators generated by bending -torsion vibration model with two- end energy dissipation; Asymptotic
Analysis, 45, (1,2), (2005), p.133-169.
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.
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 Dryden
Center), p.1-78.
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.
Asymptotic and spectral properties of the operator–valued functions generated by aircraft wing model; (jointly
with A.V. Balakrishnan); Mathematical Methods for the Applied Sciences, 27 , (2004), p.329-362.
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.
Asymptotic analysis of coupled Euler–Bernoulli and Timoshenko beam model; (jointly with C. Peterson);
Mathematische Nachrischten, 267, (2004), p.88-109.
Asymptotic analysis of aircraft wing model in subsonic air flow, IMA Journal on Applied Mathematics, 66,
(2001), p.319-356.
Asymptotic representation for root vectors of nonselfadjoint operators and pencils generated by aircraft wing
model, J. Math. Anal. Appl., 206, (2001), p.341-366.
Riesz basis property of root vectors of nonselfadjoint operators generated by aircraft wing model in subsonic
air flow, Math. Methods in Appl. Sci,, 23, (18), (2000), p.1585-1615.
Asymptotics of aeroelastic modes and the basis property of mode shapes for aircraft wing model; J. Franklin
Inst., 338, (2/3), (2001), p.171-185.
10 / 45
Aeroelastic Problem
bending
Flutter
torsion angle
Models
Main Results
X(x, t) = (h(x, t), α(x, t))T ,
References
Aeroelastic Problem
Model
Recursion Relations
−L ≤ x ≤ 0, t ≥ 0,
Model: system of two coupled damped integro-differential equations
(Ms − Ma )Ẍ(x, t) + (Ds − uDa )Ẋ(x, t)+
IBVP
(Ks − u2 Ka )X = [f1 (x, t), f2 (x, t)]T .
Operator Setting
Spectral
Asymptotics
Analytic Mechanism
where u > 0 - stream velocity
Possio Int. Eq.
Ms =
m
S
S
I
,
Ma = (−πρ)
1
−a
−a
(a + 1/8)
2
,
m - density of the structure, S - mass moment,
I - moment of inertia; ρ ≪ 1, a ∈ [−1, 1];
11 / 45
Flutter
Models
Main Results
References
Aeroelastic Problem
Model
Recursion Relations
Ds =
0
0
Ks =
"
∂4
E ∂x
4
0
0
,
Da = (−πρ)
0
−1
1
0
,
IBVP
Operator Setting
Spectral
Asymptotics
Analytic Mechanism
0
0
∂2
−G ∂x
2
#
,
Ka = (−πρ)
0
0
0
−1
,
Possio Int. Eq.
E - bending stiffness; G - torsion stiffness
12 / 45
Flutter
Models
Main Results
References
Aeroelastic Problem
Model
Recursion Relations
f1 (x, t)
=
Zt h
i
−2πρ
uC2 (t − σ) − Ċ3 (t − σ) g(x, σ)dσ,
0
Operator Setting
Zt h
−2πρ
1/2C1 (t − σ) − auC2 (t − σ) + aĊ3 (t − σ)
Spectral
Asymptotics
+uC4 (t − σ) + 1/2Ċ5 (t − σ) g(x, σ)dσ
IBVP
f2 (x, t)
=
0
i
Analytic Mechanism
Possio Int. Eq.
Aerodynamical functions: Ci , i = 1 . . . 5,
g(x, σ) = u α̇(x, σ) + ḧ(x, σ) +
1
−a
2
α̈(x, σ)
13 / 45
Recursion Relations
Flutter
Models
Main Results
References
Aeroelastic Problem
Model
Recursion Relations
b 1 (λ)
C
=
C2 (t)
=
Zt
C1 (σ)dσ,
Zt
p
C1 (t − σ)(uσ − u2 σ 2 + 2uσ)dσ,
u
e−λ/u
C1 (t)dt =
, ℜλ > 0,
λ K0 (λ/u) + K1 (λ/u)
0
Operator Setting
C3 (t)
=
0
Analytic Mechanism
Possio Int. Eq.
e
−λt
0
IBVP
Spectral
Asymptotics
Z∞
C4 (t)
C5 (t)
=
C2 (t) + C3 (t),
=
Zt
0
C1 (t − σ)((1 + uσ)
p
u2 σ 2 + 2uσ − (1 + uσ)2 )dσ,
K0 (z) and K1 (z) - the modified Bessel functions (McDonald functions)
14 / 45
Boundary Conditions
Flutter
Models
Main Results
References
Aeroelastic Problem
IBVP
Boundary Conditions
Initial Conditions
Energy of vibration
IBVP
Operator Setting
Spectral
Asymptotics
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) + β ḣ′ (0, t) = 0,
Gα′ (0, t) + δ α̇(0, t) = 0,
h′′′ (0, t) = 0,
β, δ ∈ C+ ∪ {∞},
Analytic Mechanism
Possio Int. Eq.
If β > 0, then β ≡ gh - bending control gain,
If δ > 0, then δ ≡ gα - torsion control gain
15 / 45
Initial Conditions
Flutter
Models
Main Results
h(x, 0) = h0 (x),
ḣ(x, 0) = h1 (x),
References
α(x, 0) = α0 (x),
α̇(x, 0) = α1 (x)
Aeroelastic Problem
IBVP
Boundary Conditions
Initial Conditions
Energy of vibration
IBVP
Operator Setting
Assumptions
(a)
Spectral
Asymptotics
det
Analytic Mechanism
Possio Int. Eq.
(b)
0<u≤
√
2G
√
L πρ
m
S
S
I
> 0,
(divergence speed)
16 / 45
Energy of vibration
Flutter
Models
Main Results
References
E(t)
=
1
2
Z0
e ḣ(x, t)|2 +
[E|h′′ (x, t)|2 + G|α′ (x, t)|2 + m|
Aeroelastic Problem
−L
IBVP
Boundary Conditions
Initial Conditions
Energy of vibration
IBVP
e α̇h˙ + α̇ḣ) − πρu2 |α(x, t)|2 ]dx
Ĩ|α̇(x, t)|2 + S(
Operator Setting
Spectral
Asymptotics
e = m − πρ, S̃ = S − πρa
where Ĩ = I + πρ(a2 + 1/8), m
Analytic Mechanism
Possio Int. Eq.
Energy dissipation
Ė(t) = −E|ḣ′ (0, t)|2 ℜ β − G|α̇(0, t)|2 ℜ δ
17 / 45
Initial - Boundary Value Problem
Flutter
Models
Main Results
References
Aeroelastic Problem
IBVP
Boundary Conditions
Initial Conditions
Energy of vibration
IBVP
Operator Setting
Spectral
Asymptotics
e ḧ + S̃α̈ + Eh′′′′ + πρuα̇
m
S̃ḧ + Ĩα̈ − Gα′′ + πρu2 α − πρuh
=
=
f1 (x, t),
f2 (x, t),
−L ≤ x ≤ 0,
t>0
Boundary Conditions
at x = −L:
at x = 0:
h(−L, t) = h′ (−L, t) = α(−L, t) = 0
h′′′ (0) = 0;
′′
Eh (0, t) + β ḣ′ (0, t) = 0
Gα′ (0, t) + δ α̇(0, t) = 0
Analytic Mechanism
Initial Conditions
Possio Int. Eq.
h(x, 0) = h0 ,
ḣ(x, 0) = h1 ,
α(x, 0) = α0 ,
α̇(x, 0) = α1
18 / 45
Operator Setting
Flutter
State space ↔ Energy space
Models
Main Results
References
Aeroelastic Problem
H - set of 4 component vector - valued functions
Ψ = (h, ḣ, α, α̇)T ≡ (ψ0 , ψ1 , ψ2 , ψ3 )T
IBVP
Operator Setting
satisfying
ψ0 (−L) = ψ0′ (−L) = ψ2 (−L) = 0
Spectral
Asymptotics
Analytic Mechanism
Possio Int. Eq.
Norm of state space (Hilbert space):
||Ψ||2H
Z0 h
e 1 (x)|2 + Ĩ|ψ3 (x)|2
E|ψ0′′ (x)|2 + G|ψ2′ (x)|2 + m|ψ
= 1/2
−L
+S̃(ψ3 (x)ψ̄1 (x) + ψ̄3 (x)ψ1 (x)) −πρu2 |ψ2 (x)|2 dx
19 / 45
Flutter
Models
Main Results
References
Ψ̇ = iLΨ +
Aeroelastic Problem
Zt
F (t − σ)Ψ̇(σ)dσ
0
IBVP
Operator Setting
Spectral
Asymptotics
Analytic Mechanism
Possio Int. Eq.
First order in time “evolution - convolution” equation in H
Ψ̇ = iLβδ Ψ + F̃ Ψ̇,
Ψ = (ψ0 , ψ1 , ψ2 , ψ3 )T ,
Ψ|t=0 = Ψ0
Lβδ - matrix differential operator in H
Fe - matrix integral operator in H
20 / 45
Flutter
Models
Lβδ - matrix differential operator in H
Main Results
References
Aeroelastic Problem
IBVP
Operator Setting
Spectral
Asymptotics
Analytic Mechanism
Lβδ

0
1
 EĨ d
 − ∆ dx4
= −i

0

4
4
ES̃ d
∆ dx4
S̃
− πρu
∆
0
πρuf
m
∆
0
2
2
S̃
d
−∆
G dx
2 + πρu
f
m
∆
0
d2
G dx
2
2
+ πρu
0

Ĩ 
− πρu

∆


1

πρuS̃
∆
defined on the domain
Possio Int. Eq.
D(Lβδ )
=
Ψ ∈ H : ψ0 ∈ H4 (−L, 0), ψ1 ∈ H2 (−L, 0),
ψ2 ∈ H2 (−L, 0), ψ3 ∈ H1 (−L, 0);
ψ1 (−L) = ψ1′ (−L) = ψ3 (−L) = 0; ψ0′′′ (0) = 0;
Eψ0′′ (0) + βψ1′ (0) = 0, Gψ2′ (0) + δψ3 (0) = 0}
e Ĩ − S̃2 > 0
∆=m
21 / 45
Flutter
Models
Main Results
References
Fe - matrix integral operator in H
Aeroelastic Problem
IBVP
Operator Setting
Spectral
Asymptotics
Analytic Mechanism
Possio Int. Eq.
Fe =

1
 0

 0
0
0
[Ĩ(C̃1 ∗) − S̃(C̃2 ∗)]
0
0
0
 0

 0
0
0
1
0
1

0
u
0
u

0
(1/2 − a) 


0
(1/2 − a)
0
0
1
0

0

0
×

0
e C̃2 ∗)]
[−S̃(C̃1 ∗) + m(
Spectral properties of both the differential operator Lβδ and the integral
operator Fe are of crucial importance for the representation of the solution
22 / 45
Flutter
Models
Main Results
References
Aeroelastic Problem
IBVP
Operator Setting
Spectral
Asymptotics
Analytic Mechanism
Possio Int. Eq.
Ψ̇ = iLβδ Ψ + FeΨ̇,
Ψ|t=0 = Ψ0
Laplace transform representation for solution:
−1
b
b
Ψ(λ)
= λI − iLβδ − λF(λ)
(I − Fb(λ))Ψ0
Goal: find the solution in space - time domain, i.e., “calculate” the
b
inverse Laplace transform of Ψ
−1
R(λ) = λI − iLβδ − λFb(λ)
Generalized resolvent operator ⇒
analytic operator - valued function of λ on the complex plane having a
branch - cut along the negative real semi - axis.
Poles of R(λ) - discrete spectrum ↔ aeroelastic modes,
Branch cut - continuous spectrum
23 / 45
Spectral asymptotics of Lβδ
Flutter
Models
Two - branch spectrum: If |δ| =
6
Main Results
IBVP
Operator Setting
Spectral
Asymptotics
Analytic Mechanism
Possio Int. Eq.
GĨ, then
β- branch
References
Aeroelastic Problem
p
q
µβn = (sgn n)π 2 /L2 EĨ/∆ (n − 1/4)2 + κn (ω),
ω = |δ|−1 + |β|−1 , |n| → ∞,
e Ĩ − S̃2 , and
where ∆ = m
sup{|κn (ω)|} = C(ω), C(ω) −→ 0 as ω −→ 0
n∈Z
δ-branch
p
i
δ + GĨ
πn
p
q
+ O(|n|−1/2 )
+
ln
µδn = q
δ − GĨ
L Ĩ/G
2L Ĩ/G
24 / 45
Flutter
Models
Main Results
References
Aeroelastic Problem
IBVP
Operator Setting
Spectral
Asymptotics
Analytic Mechanism
Possio Int. Eq.
Definition: ∃ a nontrivial Φn such that
λn I − iLβδ − λn Fb(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)
b)
c)
Lβδ is a closed linear operator with compact resolvent;
Lβδ is nonselfadjoint unless ℜβ = ℜδ = 0;
If ℜβ > 0 and ℜδ > 0, then Lβδ - dissipative:
ℑ(Lβδ Ψ, Ψ)H ≥ 0,
d)
Adjoint operator L∗βδ ,
Ψ ∈ D(Lβδ );
β, δ 7→ −β̄, −δ̄
25 / 45
Analytical mechanism generating fluttering modes
Flutter
Structure of matrix integral operator
Models
Main Results
References
Aeroelastic Problem
b
S(λ) = λI − iLβδ − λF(λ)
IBVP
Operator Setting
Spectral
Asymptotics
Analytic Mechanism
Flutter Matrix
Numerical Results
Contour Integration
Circle of Instability
Possio Int. Eq.
Fb ⇒








1
0
0
0
0
0
0
0

0
0
0

[Ĩ(C̃1∗) − S̃(C̃2 ∗)] 0
0
×

0
1
0
0
 0 [−S̃(C̃1 ∗) + m̃(C̃2 ∗)]
0 0
0
1 u (1/2 − a) 
2
, ∆ = m
e
Ĩ
−
S̃

0 0
0
1 u (1/2 − a)
26 / 45
Flutter
Models
Main Results
References
Aeroelastic Problem
IBVP
Operator Setting
Spectral
Asymptotics
Analytic Mechanism
Flutter Matrix
Numerical Results
Contour Integration
Circle of Instability
Possio Int. Eq.
Fb(λ) is a Laplace transform of F (t) ⇒ (kernel of convolution operator)

0
0
b
F (λ) = 
0
0
0
L(λ)
0
N (λ)
0
uL(λ)
0
uN (λ)

0
(1/2 − a)L(λ) 


0
(1/2 − a)N (λ)
o
2πρu n
−S̃/2 + [Ĩ + (1/2 + a)S̃]T(λ/u) ,
L(λ) = −
λ∆
o
2πρu n
e
e
N (λ) = −
−m/2
+ [S̃ + (1/2 + a)m]T(λ/u)
λ∆
27 / 45
Flutter
Models
Theodorsen function:
Main Results
T(z) =
References
Aeroelastic Problem
IBVP
K1 (z)
K0 (z) + K1 (z)
K0 , K1 the modified Bessel functions
(1)
Kn (z) = 1/2πieπni/2 Hn (iz)
Operator Setting
Spectral
Asymptotics
Analytic Mechanism
Flutter Matrix
Numerical Results
Contour Integration
Circle of Instability
Possio Int. Eq.
Definitions:
K0 (z) =
∞
X
m=0
K1 (z) =
1
z
+
z
2
z2m
22m (m!)2
∞
X
m=0
1
ψ(m + 1) −
ln(z/2) ,
m+1
z2m
1
[ψ(m + 1)−
ln(z/2)
−
2
22m (m!)2 (m + 1)
ψ(m + 2)]} ,
where ψ(m) is the digamma function
28 / 45
Flutter
Models
Main Results
References
Asymptotics as |z| → ∞
Aeroelastic Problem
IBVP
Operator Setting
T(z) = 1/2 + 1/16z + O(z−2 )
Spectral
Asymptotics
Analytic Mechanism
Flutter Matrix
Numerical Results
Contour Integration
Circle of Instability
V(z) = T(z) − 1/2 → 0
as |z| → ∞
Possio Int. Eq.
b
S(λ) = λI − iLβδ − λF(λ)
29 / 45
“Flutter Matrix”
Flutter
Models
Main Results
References
Aeroelastic Problem
IBVP
Operator Setting
Spectral
Asymptotics
Analytic Mechanism
Flutter Matrix
Numerical Results
Contour Integration
Circle of Instability
Possio Int. Eq.
where
b
λF(λ)
= M + N(λ),

0
0
M =
0
0
0
A
0
B
0
uA
0
uB

0
(1/2 − a)A


0
(1/2 − a)B
A = −πρu/∆[Ĩ + (a − 1/2)S̃],
e
B = πρu/∆[S̃ + (a − 1/2)m]
30 / 45
Flutter
Models
Main Results
References
Aeroelastic Problem
IBVP
Operator Setting

0
0
N(λ) = 
0
0
0
A1 (λ)
0
B1 (λ)
0
uA1 (λ)
0
uB1 (λ)

0
(1/2 − a)A1 (λ)


0
(1/2 − a)B1 (λ)
Spectral
Asymptotics
Analytic Mechanism
Flutter Matrix
Numerical Results
Contour Integration
Circle of Instability
Possio Int. Eq.
A1 (λ) = −2πρu∆−1 V(λ/u)[Ĩ + (a + 1/2)S̃],
e
B1 (λ) = 2πρu∆−1 V(λ/u)[S̃ + (a + 1/2)m]
kN(λ)k ≤ C|λ−1 |,
|λ| → ∞
31 / 45
Numerical results for the eigenvalues of the operator
iLβδ + M
Flutter
Models
Main Results
References
Aeroelastic Problem
Recall: R(λ) = (λI − iLβδ − M − N(λ))−1
N(λ)-Asymptotically small
M-“Flutter matrix”
ℑλ
IBVP
Operator Setting
Spectral
Asymptotics
Analytic Mechanism
Flutter Matrix
Numerical Results
Contour Integration
Circle of Instability
ℜλ
Possio Int. Eq.
32 / 45
Flutter
Models
S(λ) = λI − iLβδ − M − N(λ),
Main Results
References
Aeroelastic Problem
IBVP
Spectrum:
Operator Setting
Mode shapes:
Spectral
Asymptotics
{λβn }n∈Z ∪ {λδn }n∈Z (aeroelastic modes)
{Φβn }n∈Z ∪ {Φδn }n∈Z
Analytic Mechanism
Flutter Matrix
Numerical Results
Contour Integration
Circle of Instability
Possio Int. Eq.
R(λ) = S(λ)−1
Properties of mode shapes
1.
2.
3.
Minimality
Completeness in H
Riesz basis property
33 / 45
Contour Integration
Flutter
Models
Main Results
References
ℑλ
R
Aeroelastic Problem
IBVP
Operator Setting
Spectral
Asymptotics
ℜλ
Analytic Mechanism
Flutter Matrix
Numerical Results
Contour Integration
Circle of Instability
Possio Int. Eq.
−R
34 / 45
Flutter
Models
Main Results
References
Aeroelastic Problem
IBVP
Operator Setting
Spectral
Asymptotics
Analytic Mechanism
Flutter Matrix
Numerical Results
Contour Integration
Circle of Instability
Possio Int. Eq.
Ψ(x, t) =
X
e
λβ
nt
e
λδ
nt
n∈Z′
X
n∈Z′
1
π
Z∞
0
e
β
β
β∗
I − F̂ (λn )(λn ) Ψ0 , Φn Φβn +
−rt
I−
F̂ (λδn )(λδn )
Ψ0 , Φδ∗
n
Φδn +
ℑR(−r) I − F̂(−r) Ψ0 dr
35 / 45
“Circle of Instability”
Flutter
Models
Main Results
References
Aeroelastic Problem
Main model equation: Ψ̇(t) = i (Lβδ − iM) Ψ(t) +
Zt
G(t − τ )Ψ(τ )dτ
0
IBVP
Operator Setting
Reduced model equation: Ψ̇(t) = iKβδ Ψ(t),
Spectral
Asymptotics
′′′′
Analytic Mechanism
Flutter Matrix
Numerical Results
Contour Integration
Circle of Instability
Possio Int. Eq.
Kβδ = Lβδ − iM.
e ḧ + S̃α̈ + Eh − πρuḣ + πρu
m
− a α̇ + πρu2 α = 0
2
′′
1
1
e
e
Sḧ + Iα̈ − Gα − πρu 2 + a ḣ + πρu 2 − a α̇ − πρu2
3
2
1
2
+a α=0
Heuristic derivation of the energy functional accounting air-structure
interaction;
Non-local in time (or memory -type) functional
36 / 45
Flutter
Models
Z0 1
1
e ḣ|2 + e
E|h′′ |2 + G|α′ |2 + m|
I|α̇|2 − πρu2
+ a |α|2 +
2
2
Main Results
References
E(t) =
−L
Aeroelastic Problem
Zt
i
u
1
e α̇h˙ + α̇ḣ + 2πρu |ḣ +
− a α̇ + α|2 dτ dx
S
2
2
h
IBVP
Operator Setting
Spectral
Asymptotics
Analytic Mechanism
Flutter Matrix
Numerical Results
Contour Integration
Circle of Instability
Here,
•
Z0
1
2
•
1
2
Possio Int. Eq.
•
−L
Z0
0
2
′′
E|h (x, t)| dx - Potential energy due to bending displacement
2
m|
e ḣ(x, t)| dx - Kinetic energy due to bending displacement
−L
Z0
1
2
′
2
G|α (x, t)| dx - Potential energy due to torsional displacement
−L
37 / 45
Flutter
Models
Main Results
1
2
E(t) =
References
−L
Aeroelastic Problem
Operator Setting
Analytic Mechanism
Flutter Matrix
Numerical Results
Contour Integration
Circle of Instability
Here,
•
1
2
•
1
2
Possio Int. Eq.
Z0
−L
Z0
−L
•
1
e ḣ|2 + e
{E|h′′ |2 + G|α′ |2 + m|
I|α̇|2 − πρu2 ( + a)|α|2 +
2
e α̇h˙ + α̇ḣ] + 2πρu
S[
IBVP
Spectral
Asymptotics
Z0
πρu
Zt
0
|ḣ +
1
u
− a α̇ + α|2 }dτ }dx
2
2
e α̇(x, t)|2 dx - Kinetic energy due to torsional displacement
I|
˙
e α̇h
+ α̇ḣ](x, t)dx - Kinetic energy due to bending-torsion coupling
S[
Z0
Zt 1
u
2
dx
| ḣ +
− a α̇ + α (x, τ )| dτ - Energy of vibration due to air-structure
2
2
−L
0
interaction
38 / 45
Flutter
Models
Main Results
References
Aeroelastic Problem
IBVP
Operator Setting
Spectral
Asymptotics
Analytic Mechanism
Flutter Matrix
Numerical Results
Contour Integration
Circle of Instability
Possio Int. Eq.
Theorem. For each value of the airspeed u, there exists R(u) > 0 such
that the following statement holds. If an eigenvalue λn satisfies
|λn | > R(u), then this eigenmode is stable, i. e., ℜλn < 0. The
following estimate is valid for R(u) :
R(u) = C
r
ρ
u3/2
G ℜ δ
with C being an absolute constant.
Corollary. For each u, all unstable modes are located inside the “circle of
instability” |λ| = R(u). The number of these eigenmodes is finite.
39 / 45
Possio Integral Equation in Aeroelasticity
Flutter
Models
Physical assumptions on airflow
Main Results
1.
Flow ⇒ nonviscous fluid ⇒ Euler equation for stream velocity ~v
2.
Compressible Flow: ρt + ∇ • (ρ~v) = 0 (continuity equation)
Operator Setting
3.
Isentropic flow: P = kργ
Spectral
Asymptotics
4.
Irrotational (or potential) flow: ~v = ∇Φ
Analytic Mechanism
5.
Subsonic (0 < M < 1)
6.
Coupling between structure and airflow
References
Aeroelastic Problem
IBVP
Possio Int. Eq.
• Flow Tangency Condition
•
(Flow is attached to wing surface)
Kutta-Joukowsky Condition
(Pressure drop off the wing and on trailing edge)
40 / 45
Flutter
Models
Main Results
References
The Possio Integral Equation relates pressure distribution over a typical
section of a slender wing to a normal velocity of the points on a wing
surface (“downwash”)
Aeroelastic Problem
Model: High aspect-ratio planar wing; all
cross-sections along wing-span are
identical; only one spatial variable along
cord
z
IBVP
Operator Setting
Spectral
Asymptotics
−b
Analytic Mechanism
Possio Int. Eq.
b
~
U
x
• Velocity field → potential U -free stream
velocity
Velocity potential:
s
U + φ(x, z, t),
−∞ < x < ∞, 0 < z < ∞,
t>0
41 / 45
Linearized version of Euler Equation for disturbance
potential
Flutter
Models
Main Results
References
2
2
∂2φ
∂2φ
2
2 ∂ φ
2 ∂ φ
+ 2M a∞
= a∞ (1 − M ) 2 + a∞ 2
∂2t
∂t∂x
∂ x
∂ z
Aeroelastic Problem
IBVP
Operator Setting
Spectral
Asymptotics
Analytic Mechanism
Possio Int. Eq.
a∞ -speed of sound in flight altitude
M = 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 the
trailing edge)
• Far Field conditions
42 / 45
Flutter
1.
Models
Main Results
Flow Tangency Condition
∂
φ(x, z, t)|z=0 = wa (x, t), |x| < b
∂z
References
Aeroelastic Problem
wa -given normal velocity of wing
IBVP
z
2.
Operator Setting
Spectral
Asymptotics
−b
Analytic Mechanism
b
~
U
Kutta-Joukowsky Condition
Acceleration potential:
x
ψ(x, z, t) =
Possio Int. Eq.
∂φ
∂φ
+U
∂t
∂x
• ψ(x, 0, t) = 0, |x| > b
•
lim ψ(x, 0, t) = 0
s
x→b−0
3.
Far Field conditions
Disturbance potential → 0 as |x| → ∞ or z → ∞
43 / 45
The Possio Integral Equation
Flutter
Models
Main Results
References
Aeroelastic Problem
IBVP
Operator Setting
Spectral
Asymptotics
Wa (·, λ)
=
√
1 − M2 n
1
Hb A(·, λ) − √
×
2
1 − M2
i
h
−bλ̃
λ̃Hb L(λ̃)A(·, λ) − λ̃g− (λ̃, x)e
L λ̃, A(·, λ) −
Zα1
0
Analytic Mechanism
h
a(s) λ̃Hb L(λ̃s)A(·, λ)−
−λ̃bs
λ̃g− (λ̃s, x)e
Possio Int. Eq.
Zα2
0
h
L λ̃s, A(·, λ)
i
ds +
a(−s) λ̃Hb L∗ (λ̃s)A(·, λ) +
−λ̃bs
λ̃g+ (λ̃s, x)e
L −λ̃s, A(·, λ)
i
ds
o
44 / 45
Asymptotical form of Possio Equation as λ → ∞
Flutter
Models
2T Wa (·, λ)
Main Results
=
References
hp
i
1 − M 2 − λ̃L(λ̃) F (·, λ) +
λ̃e−bλ̃ h− (x, λ̃)L(λ̃, F (·, λ)) −
Aeroelastic Problem
M {Hb [F (·, λ)] − T [F (·, λ)]}
IBVP
Operator Setting
Spectral
Asymptotics
Main difficulty:
Analytic Mechanism
L(λ)
Possio Int. Eq.
→
Volterra integral operator
L(λ̃, ·) →
Integral operator with degenerate kernel
Hb (·)
→
Finite Hilbert transform
T (·)
→
“Inverse” to Hb
Hb and T →
Singular integral operators
45 / 45