L4_Matrix Method of Static Aeroelasticity

8/10/2019 L4_Matrix Method of Static Aeroelasticity

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4th Matrix Method ofStatic Aeroelasticity

Xie Changchuan2014 Autumn


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Content

1Finite Element Method(FEM)Brief introduction

2

Steady aeroedynamics and coefficient matrix3General equation of static aeroelasticity

4Divergence and load redistribution

5

Control efficiency and reversal

Main AimsUnderstand the engineering analysis method

and basic mathematic principles of elastic

aircraft from a simple slender wing model.


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Finite Element Method(FEM)

Considering bending problem of non-uniform Euler beam

w

y

x

P

2 2

2 2

( )( ) 0

w xEI x

x x

=

Equation

Boundarycondition

(0) (0) 0w w= =

[ ( ) ( )]x l

EI x w x Px =

=

Fixed end

Free end ( ) ( ) 0EI l w l = Bending moment

Shear force

Note as ( ) 0w =L

Note as ( ) 0w

=B

Definition domain : [0, ]x l

: 0,x x l = =


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Suppose the solution as1

( ) ( )n

i i

i

w x x q=

= And another trial functionsatisfying the boundary conditions as

1

( ) ( )n

i i

i

v x x p

=

=

Combine the different equation and boundary conditionsin a integral form (weak form)

( ) ( ) 0w vd w vd + = L B2 2

2 20

( )( ) ( ) ( )

l w xw vd EI x v x dx

x x

=

L

2 2 2 2

2 2 2 20

0 0

( ) ( ) ( )

l l

l w v w w vEI x dx EI x v EI x

x x x x x x = +

2 2

2 2

00

( ) ( ) ( ) 0

l l

w w vw vd EI x P v EI x

x x x x

= + =

B

integral by part


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2 2 2 2

2 2 2 20

0 0

( ) ( ) ( ) ( )

l l

l w v w w vw vd EI x dx EI x v EI x

x x x x x x

= +

L

Substitute the boundary conditions

2 2

2 20( ) ( )

l w vEI x dx Pv l

x x

=

Substitute the supposed solutions22

2 201 1 1

( )( )( ) ( )n n nl ji

i j j jli j j

xxEI x q p dx P x px x

= = =

=

22

2 20

1 1 1

( )( )( ) ( ) 0

n n nl

jii j j jl

i j j

xxEI x dxq p P x p

x x

= = =

= =

0 0jv p= Because

22

2 2

01 1 1

( )( )( ) ( ) 0

n n nlji

i j li j j

xxEI x dxq P x

x x

= = =

=


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Note

22

2 20

( )( )( )

lji

ij

xxK EI x dx

x x

=

1( )

n

i i li

F P x=

=

The general equation is written as

=Kq F

ijK = K The general stiffness matrix

[ ]iq=q The general coordination

[ ]iF=F The general force

( )i xLet be the FEM shape function ( ) 1, 2,3, 4iN x i =

2 3

1( ) 1 3 2N = +

2 3

3( ) 3 2N = ( )

2 3

2 ( ) 2 /N l = +

( )3 23( ) /N l =

/x l =

Then

2 2

3

2 2

12 6 12 6

6 4 6 2

12 6 12 6

6 2 6 4

e

l l

l l l lEI

l ll

l l l l

=

K

For uniform element

0

0

0

e

P

=

F


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Steady aerodynamics of slender wing

Strip theoryL

L qcC=

( )L

C Aspect ratio 5 >From aerodynamic theory

of infinite aspect ratio wing

Modification of

moderate aspect ratio ( )2

L LC C

=

+

3>

Aeroelastic

modification ( )

4L L

C C

=

+

Essentially, it is aerodynamic theory of 2D wing segment.

The interference between segments is ignored.

The engineering modification of strip theory.

From more precise load distribution results of wind tunnel pressure

test orcomputed fluid dynamics method at given states, replace theaerodynamic derivatives by local effective values.


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Lifting line theory Frederick W. Lanchester in 1907Ludwig Prandtl in 19181919

Steady, inviscid, irrotational,incompressible, potential flow

Unswept, undihedral, large aspect ratio single wing

Extension of l ifting line theory W.F. Phillips in 2000

Introducing compressible flow, swept and dihedral

angle, large aspect ratio multi wing

Mark Drela in 2007

Xie Changchuan in 2009

Introducing large aspect ratio multi wing

with large deformation



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Free vortexBound vortex

d

-d

x

z y

a. Horseshoe vortex b. S stem of horseshoe vortex

Using spanwise varying vortex(y) along 1/4 chordline

represents the complete wing, then after the trailing edge

there is a free vortex sheetdtending to infinite distance.



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= y F VKutta-Joukowski lawF yAero force vector acting on small segment

V Local inf low vector


In practice, the wing is

divided into several segments.On each segment a horseshoe

vortex is assigned.


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Prandtls Assumption: The aero forces acting on wing

section are equal to that on a section of infinite long

wing at same AOA.

2 21 1= ( )2 2

L a ind L eF V C S V C S + =

Geometric AOA of wing section, including the initial AOA,pre-twist of wing and torsion angle induced by elasticity.

a

ind

LC

e

Induced down wash angle at control point of wing section,Control point is at 1/4 chord point.

Lift line slope of airfoil, the value can be selected bypractice airfoil and itsAOA, which could introduce thenonlinearity of aerodynamics at some extent.

V SInflow speed Area of wing segment

Effective AOA

Lifting line theory


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21 [ (( ) ( )) ] ( )2

L a ii ind i iy yV y V C y yb y = +

( ) ( )ind i iy A y =

Circulation ofbound vortex Inducedangle Chord lengthof segment

AAerodynamic influence coefficient, calculated by

Biot-Savart law considering all the horseshoe vortex

[ ]21

2 LV V C

= ab A

aero forces on each segment in matrix form

11 1

2 2L L

VC VC

= +

abA b I

Lifting line theory


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121 1

2 2a L LV V C VC

= = + a

F b I b A Aeroforces

0( )a q= +F w wDD Aerodynamic coefficient matrix

Homework

v

Considering a un-swept wing

modeled by 3 horseshoes, the

control points are at 1/4 chord

points of middle line on eachsegment, please deduct the

aerodynamics influence matrix

A.

Lifting line theory


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Vortex circle method ---- a kind of panel method

z

O

y

x A

B

C

DA

B

C

D

There are many other panel methods,

including subsonic and supersonic panel

method, piston theory,


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Basic equation of static aeroleasticity

0( )q= + +w D w w f

q

K

w

Stifness matrix of wing

Displacement vector of wing

0w Initial displacement vector of wing

Aerodynamic coefficient matrix

Vector of other forces, like gravity, thrust,

Dynamics pressure of inflow


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Divergence and load redistribution

q=Kw Dw 0q+ +Dw f

Static aeroelasticdivergence

General Eigenvalue Problem ofhomogeneous equation

=Kw Dw mindiv iq =

Load redistribution Unique solution Problem ofun-homogeneous equation

1

0( ) ( )q q

= +w K D Dw f Aero forcedistribution 0

( )a q= +F D w w


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Control effectiveness and reversal

q q= +Kw Dw C

Basic equation of staticaeroelasticity with control surface

No relationship with the initialdisplacement and forces otherthan aerodynamics

1( )q q= w K D C

a

q q= +F Dw C

Specified sum loadof aerodynamics a

=F F

Control reversal / 0i =F

Control effectiveness/

( / )

i

i r

=

F

F

Aero load

distribution


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Homework

Please deduct the dynamic pressure

of control reversal and the control

effectiveness in matrix form

L4_Matrix Method of Static Aeroelasticity

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