static aeroelasticity lift distribution 090120

7/27/2019 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

13

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

24

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

static aeroelasticity lift distribution 090120

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