AAELecture 22 Typical dynamic instability problems and test review
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AAE 556
Aeroelasticity
Lecture 22
Typical dynamic instability problems and test
review
ARMS 3326
6:00-8:00 PM
Purdue Aeroelasticity
22-1
How to recognize a flutter problem in the making
Given: a 2 DOF system with a parameter Q that creates loads on the system that are linear
functions of the displacements
M1
0
0 &
x&
K1
1
+
M 2 &
x&
2
0
0 x1
0
=
Q
K 2 x2
p21
x x
1 = 1 ei ωt
x 2 x 2
p12 x1
0 x2
(
)
K
= 1
M1
ω 22 =
ω 4 − ω 2 ω 12 + ω 22 + ω12ω 22 = 0
Q is a real number
If p12 and p21 have the same
ω12
sign (both positive or both
negative) can flutter occur?
−ω 2 + ω 12
Q
− M p21
2
(
(
)
∆ = −ω +ω
2
2
1
)(
Q=0
K2
M2
Q
−
p12
M1
x 1 = 0
x 0
− ω 2 + ω 22 2
(
Q not zero
)
)
Q2
−ω +ω −
p12 p21 = 0
M 1M 2
2
2
2
23-2
Purdue Aeroelasticity
If flutter occurs two frequencies must merge
2
2
ω
+
ω
1
2
1
2
ωn =
±
2
2
(ω
2
1
−ω
)
2 2
2
Q2
+4
p12 p21
M 1M 2
For Flutter – Increasing Q must cause the term under the radical sign to become zero and then go negative. The zero
condition is:
(
K1
M1
K
ω 22 = 2
M2
ω12 =
ω 12
2
Q =−
− ω 22
)
2
Q2
= −4
p12 p 21
M1 M 2
(
M 1 M 2 ω 12 − ω 22
4 p12 p21
)
2
p12 p21 = −
(
M 1 M 2 ω 12
4Q 2
For frequency merging flutter to occur, p12 and p21 must have opposite signs.
23-3
Purdue Aeroelasticity
− ω 22
)
2
If one of the frequencies is driven to zero then we have divergence
M
−ω 2 1
0
0 x1 K1
+
M 2 x2 0
p12 x1
0 x2
( )( )
ωn = 0
∆ = 0 = ω 12 ω 22
( )( )
ω 12
0 x1
0
=
Q
K 2 x2
p21
ω 22
Q2
=
p12 p21
M 1M 2
KK
Q = 1 2
p12 p21
2
Q2
−
p12 p21
M 1M 2
M 1 M 2ω 12ω 22
Q =
p12 p21
2
p12 p21 =
M 1 M 2ω12ω 22
Q2
Divergence requires that the cross-coupling terms are of the same sign
23-4
Purdue Aeroelasticity
Aero/structural interaction model
TYPICAL SECTION
What did we learn?
L = qSCL α (α o + θ )
V
lift
torsion spring
KT
e
θ
GJ
KT ∝
span
qScCMAC
αo +
K
T
L = qSCLα
1 − qSeC Lα
K
T
23-5
Purdue Aeroelasticity
Divergence-examination vs. perturbation
L=
1−
qSCLα
qSeCLα
αo +
KT
Kh
0
1−
qSCLα
qSeC Lα
KT
0 h − L
=
KT θ MSC
∞
1
= 1 + q + q 2 + q 3 +...= 1 + ∑ q n
1− q
n=1
23-6
Purdue Aeroelasticity
qScC
MAC
K
T
Perturbations & Euler’s Test
V
KT ( ∆θ ) > ( ∆L )e
lift
torsion spring
KT
e
θ
...result - stable - returns -no static equilibrium in perturbed state
KT ( ∆θ ) < ( ∆L)e
...result - unstable -no static equilibrium - motion away from equilibrium state
KT ( ∆θ ) = ( ∆L)e
...result - neutrally stable - system stays - new static equilibrium point
23-7
Purdue Aeroelasticity
Stability equation is original equilibrium equation with R.H.S.=0.
∆θ ≠ 0
V
θ
lift
e
torsion spring
KT
(KT − qSeC Lα )= KT = 0
The stability equation is an equilibrium equation that represents an equilibrium state with no "external loads" –
Only loads that are deformation dependent are included
The neutrally stable state is called self-equilibrating
23-8
Purdue Aeroelasticity
Multi-degree of freedom systems
A
2KT
3KT
panel 2
panel 1
e
b/2
V
A
b/2
aero
centers
there is a solution to the
homogeneous equation only if the
determinant of the aeroelastic
stiffness matrix is zero
αο + θ2
αο + θ1
5
KT
−2
shear
centers
From linear algebra, we know that
view A-A
−2 θ1
−1 0 θ1
1
+ qSeC Lα
= qSeC Lα α o
2 θ2
0
−1
θ 2
1
23-9
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MDOF stability
Mode shapes? Eigenvectors and eigenvalues.
[KT ]{∆θ i }= {0}
KT = 0
Kij − qAij = 0
System is stable if the aeroelastic stiffness matrix determinant is positive. Then the system can absorb
energy in a static deformation mode. If the stability determinant is negative then the static system,
when perturbed, cannot absorb all of the energy due to work done by aeroelastic forces and must
become dynamic.
23-10
Purdue Aeroelasticity
Three different definitions of roll effectiveness
•
•
Generation of lift – unusual but the only game in town for the typical section
Generation of rolling moment –
•
•
•
contrived for the typical section – reduces to lift generation
Multi-dof systems – this is the way to do it
Generation of steady-state rolling rate or velocity-this is the information we really want for
airplane performance
•
Reversal speed is the same no materr which way you do it.
23-11
Purdue Aeroelasticity
Control effectiveness
q c CMδ
1+
qD e CLδ
L = qSCL δ δ o
=0
q
1−
qD
q c CM δ
1+
=0
qD e CLδ
KT CL δ
qR = −
ScCLα CMδ
Lift
α0+ θ
V
MAC t orsion spring KT
shear cent er
reversal is not an instability large input produces small
output
opposite to divergence
δ0
e
phenomenon
23-12
Purdue Aeroelasticity
Steady-state rolling motion
qScCMδ
v
L = 0 = qSCLα
δo −
+ qSC Lα δ o
KT
V
Lif t
α0+ θ
V
MAC t orsion spring KT
shear center
δ0
e
23-13
Purdue Aeroelasticity
Swept wings
α structural= θ − φ tan Λ
2
qn = qcos Λ
K1
f
K2
αo
V
C
V cosΛ
C
b
Kφ
0
−tb
0
− Q 2
Kθ
−te
b φ
b
Qαo
2 θ = cosΛ 2
e
e
23-14
Purdue Aeroelasticity
Divergence
bt
∆ = Kθ Kφ + Q Kθ − Kφe
2
Kθ
e c Kφ
tan Λ crit = 2
c b Kθ
2.0
nondimensional divergence
dynamic pressure
Seao
qD =
b K tan Λ
2
cos Λ 1− θ
e
K
2
φ
nondimensional divergence dynamic
pressure vs. wing sweep angle
1.5
sweep back
sweep f orward
1.0
5.72 degrees
0 .5
0 .0
-0 .5
-1.0
b/c=6
e/c=0.10
Kb/Kt=3
-1.5
-2.0
-90 -75 -60 -45 -30 -15
0
15
30
sweep angle
(degrees)
23-15
Purdue Aeroelasticity
45
60
75
90
Lift effectiveness
lif t ef f ect iveness
vs.
dynamic pressure
2.0
lif t ef f ect iveness
unswept
wing
1.5
unswept wing
divergence
1.0
15 degrees
sweep
0 .5
30 degrees
sweep
0 .0
0
50
10 0
150
20 0
250
30 0
350
dynamic pressure (psf)
23-16
Purdue Aeroelasticity
Flexural axis
Λ
θ E = θ − φ tanΛ
β
x
y
Flexural axis - locus of points where a concentrated force creates no stream-wise twist (or chordwise aeroelastic angle
of attack)
θE = 0
The closer we align the airloads with the
flexural axis, the smaller will be aeroelastic
effects.
23-17
Purdue Aeroelasticity
