L3_Static Aeroelasticity of Slender Wing
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Transcript of L3_Static Aeroelasticity of Slender Wing
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8/10/2019 L3_Static Aeroelasticity of Slender Wing
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Xie Changchuan2014 Autumn
3rd Static Aeroelasticityof Slender Wing
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Contents
1Model of slender wing
2Method of divergence
3Load redistribution
4Control efficiency and reversal
Main AimsUnderstand the basic mechanic principles and
analysis methods of static aeroelasticity fromthe simple slender wing model.
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Aiefoil: symmetry thin Spanl
Chord
C(y) Air force axis : ChordSlope of lifting curve Zero Moment coefficient:
Elastic axis: behind air force axis
Torsion stiffness: Twist angle: + nose up
Linear density of weight: Overload factor: NWeight center before elastic axis( )d y
( )y
( )e y
( )GJ y
( )LC y
0 ( )mC y
( )gm y
dy
TT dy
y
+
M
Nmg
e
w
y
y
l
d
C
V V
T mydy
Air force center line
Wei ht center
Elastic axis
L
Model of slender wing without swept
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Forces on slender wingAssumption of Small deformation
Wing bending would not produce air force
0ydT
T dy m dy T dy
+ + =
( )d
T GJ y
dy
=
Aerodynamics flat airfoil strip theory
Equilibrium equation
of small element
Elastic torque
( ) ( )L yC y C y
=
External Torque = Air force moment + weight moment
y
dTm
dy=
[ ] 20 0( ) ( ) ( ) ( ) ( ) ( ) ( )y L mm C y qC y e y C y qC y Ngm y d y = + +
dy
TT dy
y
+
V
T mydy
T Internal Torque
External Torqueym
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Differential equation of slender wing
20 0( ) ( )L L m
d dGJ C qCe C qCe C qC Ngmd
dy dy
+ = +
Boundarycondition
0, 0
, 0
y
dy l
dy
= =
= =
Twist divergence ---- Static stability problem
Static stability of equilibrium state:A system at it s staticequilibrium stategets an arbitrary external disturbance. When
the disturbance is eliminated, the system can go back to the
original static equilibrium state. Then, the static equilibriumstate of system is said to be static stable.
Unstable StableStable, but not
asymptotic
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Static stability of slender wing
= +
For general form of these functions,
the equation can not solved explicitly.
Obtain ahomogeneous
equation
L b u = =Note the equation as
operator form
0L 0 = =
( ) 0
L
d dGJ C qCe
dy dy
+ =
0, 0
, 0
y
dy l
dy
= =
= =
( ), ( ), ( ), ( )LGJ y C y C y e y
Attention
Let
L b u = = Equilibrium solution
Disturbed solution
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If there are any nonzero solutions
No! The disturbed solution is always zero. The system goes back
to its original state, then the system is stable.
Yes! The system can not go back to its original state, then the
system is stable.
Supposing the wing is uniform, the disturbed
equation is simplified as
2 2
2
2
0
L
ddy
qCe
CGJ
+ =
=
0, 0
, 0
y
dy l
dy
= =
= =
0 =Zero solution must exist
Static stability of slender wing
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22
2
0
d
dy
+ =
It means the equation has nonzero solutions
Equation have general solutions
sin cosA y B y = +
Substitute them into
boundary conditions s 0
0
co l
B
=
=
cos 0l = (2 1) ( 0,1,2, , )2
il i i = + =
2
L
qCeC
GJ
=2
24div L
GJ
q l CeC
=
Static stability of slender wing
When cos 0l =sin 0A y = Then for ally
From
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Eigenvalue problem in linear algebra
Discussion 1) Comparing with characteristic equation in linear algebra
22
2
d
dy
= Rewrite the
disturbed equation2 D u = =Abstract form
2
2D y
=
0, 0
, 0
y
d
y l dy
= =
= =
Ax px=,T T TA A y Ax x Ay= = 0Tx Ax >
Eigen vectors constitute a basement of linear space.
The homogeneous solution is combined linearly by the Eigen vectors.
Static stability of slender wing
is a linear partial difference operator
If A is symmetry and positive definite, then the
equation has nonnegative eigenvalues.
Symmetry: Positive definite
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Discussion 2) How to solve a non-uniform wing
( ) ( ) ( ) ( ) 0Ld dGJ y qC y C y e y
dy dy
+ =
[ ( ) ]d d
D GJ y
dy dy
=
0, 0
, 0
y
dy l
dy
= =
= =
0
( )( , ) ( ) ( )
l d dw yDw v GJ y v y dy
dy dy
=
Selected homework: Discuss the properties of the partial
differential operator in disturbed equation.
Inner production
2
20
( )( , ) ( )
ld w yDw v v y dy
dy
=
The eigenvalue theory of symmetric operator in Hilbert space
Static stability of slender wing
is a linear partial difference operator
Inner production
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Load redistribution of slender wing
The general
solutions
Considering an uniform wing, the un-homogeneous equation is
simplified as 22
2
2L
dK
dy
qCeC
GJ
+ =
=
0, 0
, 0
y
dy ldy
= = = =
2
24 Ldiv
Gq
Jq
l CeC
< = the equation has
unique stable solution
2
0 0
1( )L mK C qCe C qC Ngmd GJ
= +
2sin cos
KA y B y
= + +
2 unknows are determined by 2 boundary conditions
When
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Lift of rigid wing
2 2tan
K KA l B
= =
0( ) (cos tan sin 1) LL y y l y C qC = +
Then
2 (1 tan sin cos )
K
l y y =
2l
The condition of divergence:
At initial AOA , the lift distribution of uniform straight slender wing is0
[ ]0( ) ( ) ( )r
LL y C y qC L L y
= + = +
0r
LL C qC
=
cos tan sinr
r r
L L Ly l y
L L
+
= = +
Load redistribution of slender wing
Elastic increment
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Lift distribution of the wing with different stif fness
at given AOA and dynamic pressure
0 0.25 0.5 0.75 1.0
y
l
L/Lr
2.0
1.75
1.5
1.0
1.25
0l =8
l
=
4l
=
3l
=
Load redistribution of slender wing
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Model of swept slender wing
Swept angle:
Effective span:
Effective chord:
Aero axis1/4 chord line
Effective lifting slope:
Elastic axis: behind aero axis
Elastic twist: + nose upWeight axis: before elastic axisd
eL
C
cl
dw
dy
w
x
x
V
Vcos
A
B
l
l
y,
de
y,(elastic axis)
EffectiveRoot
Aero axis
Weight axis
Vsin
sin dw
Vdy
c
B
r +
Vcos sin dw
Vdy
tanr dw
dy +
tanidw
dy
=
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Divergence characters of swept slender wing
Lift of wing segment2( ) cosLL y qcC
=
Effective AOA
without bending r = +
r
tanr w
y
= + Bending induced AOA
2tan( ) ( ) cosrLL y qc w
Cy
= +
For sweep back wing, the elastic deformation decreases theeffectiveAOA and the lift coefficient. So the divergencedynamic pressure is increased.
For sweep forward wing, the elastic deformation increases the
effectiveAOA and the lift coefficient. So the divergencedynamic pressure is decreased.
Effective AOA
of rigid wing
Effective AOA
with bending
Nose down effect
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Divergence dynamics pressure changing
with sweep angle of slender wing
0-45
1.0
qD/qD|=0
Divergence characters of swept slender wing
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Factors to promote divergence speed of wings
Increase stiffness weight increasing
light structure with high stiffness
Increase sweep angle affects the aerodynamic
decrease the aerodynamic efficiency, especially in low air speed
Decrease the distance of elastic and aero axis
limitation on structure assignment
the aero axis is different in sub/supersonic Decrease aspect ratio, increase taper ratio
limitation on aerodynamic configuration and performance
decrease lift-drag ratio of the wing
Aeroelastic tailoring optimized design, utilize elastic
change the spanwise distribution of elastic and aero axis
change the stiffness coupling e.g. For sweep forward wing,
make it nose down when bending upward
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1Please give out divergence dynamic pressure ofuniform slender wing without sweep using energy
method.
2(selective) Considering the effective bending
stiffness EI of slender swept backward wing, please
solve the divergence dynamic pressure and load
distribution. Then discuss the possibility to eliminatethe divergence.
Load and divergence of slender wing
HOMEWORK
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py
V
B
d(y)e(y)
x
V
A
l1
l
y()
py
V
A -B
Rolling efficiency and reversal of
slender wing with aileron
Span: Chord:Aero axis:1/4 chord line Effective slope of lift:
Elastic axis behind aero axis elastic twist: + nose up
Weight axis before elastic axis
aileron position: from to wing tip
C
d
eL
Cl
1l
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0, 0
, 0
y
dy ldy
= =
= =
Boundary
conditoins
( )2 0 1mL L L
CC C Cd d pyGJ qCe qCe qC qCe mpyd y l
dy dy V
+ = +
( )1L LC Cd d pyGJ qCe qCe mpyd y l
dy dy V
+ = +
Simplify: Uniform rolling rate 0p =
1
1
1 ( )1 ( )
0 ( )
a
y ly
y l
>=
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22
)(12222
2
yKyV
p
dy
da
=+Amalgamate
equations 2
0
L
mL L
CqeC
GJ
CC CCK
e
=
= +
Solution: )()()( 12 yCyCV
ply +=
( ) 11 1sin ( )
( ) 1 ( ) 1 cos ( ) sincosa
l lC y K y y l yl
=
ll
y
l
yyC
cos
sin)(2 =
Rolling efficiency and reversal of
slender wing with aileron
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10
20
( ) 1 ( )
( )
lL
a
L
a l
CplC y y ydy
CV
yC y ydyl
+ = =
Rolling
efficiency
pl/V
l
Rl=1.44
1.61.20.80.40
pl
VSpiral angle
of wing tip
/pl
V
10
0.5,
0.6, 0.6
L L
m L
C C
lC C C
e l
=
= =
Rolling efficiency and reversal of
slender wing with aileron
Spiral angle by
unit deflection
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Reversal
Let Numerator be zero
10
20
( ) 1 ( )
( )
lL
a
L
a l
CplC y y ydy
CV
yC y ydy
l
+ = =
Rolling rate and acc. are both zerowhen the aileron deflects.
( ) 0cos2coscos0
21
222
1
0 =
+
+
l
C
e
Cllll
C
e
CC mmL
Transcendental equation Calculate the smallest satisfyingthe equation numerically. Note it as
rev
Reversal
dynamic pressure
2R rev
L
GJqC
eC
=
Rolling efficiency and reversal of
slender wing with aileron
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Thinking and trying
When there is connecting stiffness between
the wing and aileron, try to solve the control
efficiency and reversal dynamic pressure. Justconsider a wing segment with control surface.
k
Rolling efficiency and reversal of
slender wing with aileron