static aeroelasticity lift distribution 090120
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STATIC AEROELASTICITY
LIFT DISTRIBUTION
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Static Aeroelasticity Study of flexible aircraft structures under aerodynamic loads
Aerodynamic loads affect structural shape and vice versa
Forces and motions are considered to be independent of time
Only steady aerodynamics needs to be considered
Static aeroelastic deflections flight wing shape
Estimation of jig shape from desired flight shape
Loads in steady flight conditions Lift distribution
Drag forces (and hence range)
Effectiveness of the control surfaces
Aircraft trim behaviour
Static stability and control characteristics
Two critical phenomena
Divergence
Control reversal
Aerodynamic
loads
Wing bendi
and twist
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Static AeroelasticBehaviour of 2D Rigid
Aerofoil with Spring
Attachment
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Iterative Analysis (1)
Aerofoil initial incidence o and elastic twist Moment
Potential energy
Generalised moment
Lagrange
Assuming that pitching moment is not changed by twist
Twist causes new aerodynamic moment
Need to step between determining new load, new twist etc.
2 2 2 2
1 0 1 0 1 0
1 1M V c a ec V e c a qe c a
2 2
21U K2
2
1 0 2
1 0
qec aWQ qec a
22 1
1 0 0 0
q e c aK q ec a qR
K
2
1ec aRK
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Iterative Analysis (2) 1st iteration include initial incidence and elastic twist
potential energy term stays the same Moment
New Twist Angle
Further Iterations
Repeat above process
In the limit
Approach analogous to coupled CFD / FE models (time marching)
2
1 0 0M qe c a ( qR )
2
1 0 0
(1 qR)qec a qR(1 qR)
K
2 3 4
0qR 1 qR qR qR qR
0
qR
(1 qR)
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Single Step Analysis Same aerofoil as before let incidence include unknown twist
Moment
Potential energy
Generalised moment
Lagrange
Twist
Same result as iterative analysis will use direct approach
21U K2
2
1 0M qe c a ( )
2 1 0 2
1 0
qec a ( )WQ qec a ( )
2 2 2
1 0 1 1 0K q ec a ( ) K q ec a q ec a
2
10 02
1
q ec a qR K q ec a (1 qR)
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
2
4
6
8
10
ElasticTwist/InitialIncidence
q/qw
7
Divergence Consider elastic twist
Twist increases with q
As q 1/R twist goes to infinity
Physically, the wing twists off
Aerodynamic moment overcomes the restoring moment Divergence
Langleys Aerodrome failed due to divergence
How to increase the divergence speed?
0
qR
(1 qR)
div 21
K1q
R ec a
div0
div
qq
q1
q
2
1ec aRK
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Static Aeroelastic
Behaviour of Fixed
Root Flexible Wing
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Fixed Root Rectangular Wing (1) Rectangular wing
Flexural axis ec aft of aero centre
Assume linear twist
Lift of incremental strip
Total lift
Potential energy
Incremental WD
T
y
s
W 0 T
y
dL qca ( )dys
s
W 0 T W 0 T0
y sL qca ( )ds qca (s )
s 2
2 2s s
2TT
0 0
1 d 1 GJU GJ dy GJ dy
2 dy 2 s 2s
s s2 0 T
W 0 T W T0 0
s syW dL ec qca ( )dyec qec a
s 2 3
y
s
KE = 0
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Fixed Root Rectangular Wing (2) Total incremental WD
Lagranges equations
Elastic tip twist
Dynamic pressure at divergence
Observations
Reduce eccentricity or increase torsional rigidity to increase divergence speed If flexural axis = aerodynamic centre there is no twist and no divergence
If flexural axis forward of aero centre then tip twist downwards - divergence
Later two designs not usually possible for aircraft
s s2 2 0 T
W 0 T T W 0 0
s sy y
W dL ec qc a ( )dye qec as s 2 3
2 2 20 T TW W T W
s GJ s GJ sqec a qec a qec a
s 2 3 s 3
2 2
W
T 02 2
W
3qec s a
6GJ-2qec s a
W 2 2
W
3GJq
ec s a
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Lift Distribution Along Wing Combining and
Lift per unit strip becomes
In terms of the divergence speed
Total lift
As q L
2 2
WT 02 2
W
3qec s a
6GJ-2qec s a
W 0 T
ydL qca ( )dy
s
2 2
WW 0 T W 02 2
W
3qec s adL y yqca qca 1
dy s s6GJ-2qec s a
W
W 0
W
q3 qdL y
qca 1dy sq
2 1q
s
W
W 0
0
W
q3
qdLL dy qcsa 1
dy q4 1q
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
2
4
6
8
Liftperstrip/Liftper
stripatroot
Distance along semi-span
q/qw
= 0.2
q/qw
= 0.5
q/qw = 0.8
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Effect of Trim on Static
Aeroelastic Behaviour
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In practice, change of airspeed will require trim to be adjusted via
elevators to maintain height
Idealised rigid aircraft able to undergo heave and pitch motions
wings the same as considered previously, symmetric aerofoil
thrust and drag in-line
Generalised coordinates heavez, wing root incidence o, wing twist
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Effect of Trim (1)
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Effect of Trim (2) Potential energy
WD through incremental distances z, o and
Apply Lagrange to all 3 generalised coordinates
2 2s s
2T
T0 0
1 d GJU 2 GJ dy GJ dy
2 dy s s
KE = 0
s
T T 0 W 0 W 0
0
W L z l W z 2 qca ( )dy z l ec
Tz T W 0
( W)Q 0 L W 2qcsa
( z) 2
0T
T T W 0
0
( W)Q 0 L l 2qcsa
( ) 2
2 0 TT W
2GJQ 2qec sa
s 2 3
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Effect of Trim (3)
Eliminate LT
Tip twist
Divergence speed increases
Root incidence
Negative incidence beyond qw
Unlikely to get to divergence as aircraft will run out of trim first
TW W
0
W T2 2W W T
Wl2qcsa qcsa
l l2 GJqec sa qec sa 2
03 s
T T W TW
4GJ qWl /(l l ) 1
ecs 4q
T0 W
T W W W
Wl q q1 2qcsa 1
l l q 4q
A W2 2
W
12GJq q 4q
ec s a
0 0.5 1 1.5 2 2.5 3 3.5
0
10
20
30
TipTwist/TipTwist(q=0)
q/qw
0 0.5 1 1.5 2 2.5 3 3.5-20
-10
0
10
20
Theta0
/Theta0
(q=qw
/2)
q/qw
qW -ve incidence
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Combining expressions forand o
Lift per unit span
Linear variation along wing
Area under slope constant
Zero lift at root forq = qw Negative lift in-board forq > qw slope forq = 4qw = qA As lift moves outboard
root BM increases
For symmetric aerofoil, wing and tailplane lift constant with airspee
Tailplane lift
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9-5
0
5
10
LiftPerUnitSpan
Normalised Position on Chord
q/qw
= 0.5
q/qw
= 1.0
q/qw
= 2.0
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Effect of Trim on Lift Variation
T
T W W W
dL Wl q y q2 3 2 4s 1
l l q s 4qdy
T T T 0 EL qS a a
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Effect of Wing Sweep on
Static Aeroelastic
Behaviour
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Wing Sweep
Aircraft have swept-back wings
Increases speed at which shock waves are formed
Delays onset of associated drag
Reduces effective thickness to chord ratio
Swept forward wings
Similar drag reduction possible
Flow separation starts at wing root better than swept-back where flow
separation occurs near tip and diminishing aileron performance
Very few forward swept wing aircraft
Static aeroelastic problems
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Effect of Wing Sweep on Angle of Incidenc
Consider swept rectangular wing in uniform flow
Upwards bending of wing
Streamwise sections AC,AD,AB
No sweep (AC)
Bending doesnt effect incidence
Sweepback (AD)
Incidence reduces as bending moves D higher than A
Sweepforward (AB) Incidence increases as bending moves A higher than B
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Effective Streamwise Angle of Incidence
due to Flapping / Pitching
Consider rigid wing with two root springs Span and streamwise chord
constant with sweep angle
Consider flow over
streamwise strip
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Elemental Streamwise Strip Effective incidence depends upon
Difference in deflection of p and r Geometry of p,q,r
Pitch nose-up
Increase in incidence due to sweepback
Sweep in either direction decreases incidence
Flap downwards
Sweepback increases incidence / sweep forward decreases incidence
In practice flap is upwards (- ) so opposite effect occurs
Flap (bending) dominates changes in effective incidence
FLAP PITCH
pitch
c coscos
c
Flap
c sins in
c
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Effect of Sweep on Divergence Speed ( Consider rigid two spring wing
Lift on incremental strip
Incremental WD
Potential Energy
Lagrange
w 0dL qa cdy ( ) cos sin
s
w 0
0
s
w 0
0
y csinW qa cdy ( ) cos sin vertical movement of li
cos 4
ccosqa cdy ( ) cos sin moment ve nose up
4
2 21 1U K K2 2
2 2
w 0
2
w 0
cs c s sinK qa ( ) cos sin
2 cos 4
c scosK qa cos sin
4
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Effect of Sweep on Divergence Speed ( In matrix form
At divergence
Divergence speed
2 2 2 2
w w w
2 2 2 2 2w w w
s tan cs sin s cs sin cos s cs sin cosK qa c qa c qa c2 4 2 4 2 4
qa sc sin cos qa sc cos qa sc cosK
4 4 4
2 22 2 2
2 3ww w
qa sc coss tan cs sin sin cos s cs sin cosK qa c K qa sc
2 4 4 4 2 4
div 2 2 2 2 2
w
2K KVsc cos cs tan c sin
a K K4 2 4
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Divergence speed
Increases withsweepback
Decreases withsweepforward
Forward Swept Aircraft Divergence speed becomes limiting case
Very few aircraft with forward swept wings
Need to use aeroelastic tailoring to counteract effect
X29 / Sukhoi 47
-25 -20 -15 -10 -5 0 5 10 15 20 250.8
1
1.2
1.4
1.6
Sweep Angle (deg)
NormalisedDivergenceSpeed
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Effect of Sweep on Divergence Speed (
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Summary Lift Distribution
Lift distribution varies along wing due to flexibility
Divergence Speed Speed at which static instability occurs
Aerodynamic moment overcomes structural restoring force
Trim Consideration of trim angle increases divergence speed compared to single
wing case
Sweep Angle Wing bending and twist affect effective angle of attack
Sweepback increases divergence speed
Sweepforward reduces divergence speed Certification
Divergence and any undue loss of stability and control should be investigated