L7_Engineering Method for Dynamic Aeroelasticity

8/10/2019 L7_Engineering Method for Dynamic Aeroelasticity

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7th Engineering Methodfor Dynamic Aeroelasticity

Xie Changchuan2014 Autumn


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Content1Elements of structural dynamics

2Solving method for flutter

3Dynamic aeroelastic responce4Aeroelastic design and specification

Main AimsBy a slender wing model, understanding the

methods of engineering analysis for flutteranddynamic response of elastic wing. Realize the

jobs in aeroelastic design and the requirements

in specifications.


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Elements of structural dynamics

h

x

Elasticline

yModeling the straight slender

wing as Euler beam along its

elastic line

Wing deformation = bending + torsion of elastic line

Free vibration equation of Euler beam bending

2 2 2

2 2 2

( , ) ( , )( ) [ ( ) ] 0

d h y t h y t m y EI ydt y y

+ =

Free vibration equation of column torsion2 2

2 2

( , ) ( , )( ) ( ) 0

d y t y t i y GJ y

dt y

+ =

mass of unit length

bending stiffness

of section

( )m y

( )I y

inertial moment

of unit length

torsion stiffness

of section

( )i y

( )GJ y

(0, ) (0, ) 0 ( , ) [ ( ) (0, )] 0h t h t h l t EI y h t y

= = = =

(0, ) (0, ) 0t t = =


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General equation and mathematical principle ofFEM

( , ) ( ) ( ) ( 1,2,3, , , , )i i

h y t f y q t i N = = Let

1All linear PDE 2Infinite DOF

3Separative space/time variables

4

Analytical solutions foruniform beam and column

Considering bending

Trial function (variation of solution, Virtual displacement)

( , ) ( ) ( ) ( 1,2,3, , , , )i ih y t f y q t i N = =

2

20

{ ( ) ( , ) [ ( ) ( , )]} ( , ) 0l

m y h y t EI y h y t h y t dyy

+ =

2 2 2

2 2 2( ) [ ( ) ]dm y EI y

dt y y = +

D

Operator

0[ ( , )] (, , 0)

lh y t h y t dyh h == D D

Then

2 2 2

2 2 2

( , ) ( , )( ) [ ( ) ] 0

d h y t h y t m y EI y

dt y y

+ =


Equation

characters


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Integrate by part and substitute the boundary conditions, noting the

trial function satisfies the boundary conditions too.

0 0, 1 , 1

( ) ( ) ( ) (( ) ( )) ( ) 0( )l l

i j i j

i j i j

m y f y f y dy EI y f y f yt q tdyq

= =

+ =

( ) ( ) 0q t q t + =M K

0 ( ) ( ) ( )

l

ij i jm y f y f y dy= 0

( ) ( ) ( )l

ij i jK EI y f y f y dy=

Matrix of general mass,real symmetry

Matrix of generalstiffness, real symmetry

2 ( ) ( ) 0q t q t + =M K

0( ) t

q t q e

=

0 0( , ) ( ) ( 1,2,3, , , , )it

i i

ih y t f y q e i N

= =



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Characters ofnormal modes

The homogeneous equation has general solution. The specificsolution can be expressed by linear superposition of eigenvectors.

Eigenvalues 1j ji i = =

Eigenvectors

jThejth natural

vibration frequency

Thejth natural mode shape

Orthogonality

of normal modes

0i j

ii

i j

i j

=

=M

2

0i j

ii ii i

i j

K M i j

=

= =K

For static problems

Solution of Initialvalue problem

Solution ofinhomogeneous problem

Modal

truncationThe equation is change to linear finite order

constant-coefficient ODE

According to the theory of linear system,

the general and specific solutions are given.

The PDE changes to linear finite order algebra equation



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FEM

0 0

, 1 , 1

( )( ) ( ) ( ) ( ) ) 0( ) (( )k kl l

i j i j

i j i

i

j

im y f y f y dy E q t I y f f qy y tdy

= =

+ =

For each beam element, still consider the general equation

( )if y be element shape function,Let ( )iq t be general displacement

0( ) ( ) ( )

kle

ij i jm y f y f y dy=

0( ) ( ) ( )

kle

ij i jK EI y f y f y dy=

Matrix ofelement mass,real symmetry

Matrix ofelement stiffness,real symmetry

For 2D

beam element

( )f y Usually selected as 3 order orthogonal polynomial,

in which there are 4 undetermined coefficients2 nodal displacements and 2 rotations linearly express 4coefficients. Then give out the element mass andstiffness matrix which is deducted by nodal freedoms.

The global stiffness and mass matrix are assembled by the elementmatrix according to the relationship of nodes.



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For a planar plate splineFunction values are 1D deflection along the normal of plate

Inter surface coupling of structure/aerodynamics

IPSInfinite Plate Spline

Spline function

[ ]1 2 1,2, ,T

i N i

i i

x x xi n

W

=

X

X

Given n grid coordinates in ND Eulerspace and function values at them.

3N =

2 2

1 1 1

1 1

( ) ln( )N n

p p N i i i

p i

W X c c x c r r + + += =

= + + + 2 2

1

( )N

i p pi

p

r x x=

= undetermined coefficients1 2 1, , , N nc c c + +

Precision parameter, set a small value based onthe smooth degree of surface

Introduce some supplement equations. 11

1

1

0

0 ( 1,2, , )

n

N i

i

n

N i pi

i

c

c x p N

+ +=

+ +=

=

= =

Then substitute the known grid coordinates

and function values to solve coefficients.

1

=C A W


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1st partial

differential

Used to calculate the normal vectors at given points

Give out the displacements at aero points (control points, )

from displacements at structural grids

22

1 1 2

1

2 [ln( ) ]( )n

ip N i i p pi

ip i

W rc c r x x

X r

+ + +

=

= + + +

+

( 1,2, , )p N=

a s =U P UIn matrix form

Give out the forces at structural grids from forces

at aero centers

Structural equivalence ---- Satisfy the equivalence of work, should not

use static equivalence method

T T

a a s s =U F U F T

s a=

F P F

Inter surface coupling of structure/aerodynamics

Given m grid coordinates to getfunction values at them.

1

m

= = =W BC BA W PW

Displacement

interpolation

Force

interpolation


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Solving method of flutter

Basic equation

of aeroelasticityFrom FEM, establish the equation in physical coordinates

( )q k+ =Mx Kx A x

Using fini te order modesas general coordinates =x qGeneral equation of aeroelasticity

q + =M q K q A q

( )q q q k q+ =M K A

M Matrix of general mass K Matrix of general sti ffness

Matrix of general aerodynanic coefficient

It is a function of Ma and reduced frequencyFor incompressible flow, it is not affected by Ma.

bk

V

=

But Mach number and frequency at flutter point are still unknown,the equation can not be solved directly.

Matrix ofmode shapes


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p-k

method

Assume the system

oscillated arbitrarily 0ptq q e=

p i = + frequency,

The aerodynamic coefficient matrix is still in harmonic form, whichis function of reduced frequency for incompressible flow.

1b

p p k ik ik iV

= = + = + = 2

22

[ ( )] 0V p q kb

+ + =M K A qThe equationwritten as

Solving process


decay ratio

END


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F

q F

Typicalq- andq-curves

From calculation

results, there are Ipairs of(Vi pi)


p i = +

( )i iV

( )i iV /( )i iq

/( )i iq


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Factors affecting flutter

Bending/torsion stiffnessincreasing proportionally, flutter speed increases;

increasing separately, can not determine the trends of flutter speed

Relative position ofweight, elastic and aero centerweight center moving forward, elastic centermoving backward,

flutter speed increases

Mass/inertial effects

wing store, fuel tank

Aero surface shapegiven wing span and area, increasing aspect ratio,

flutter speed increases

Flight altitude

air density decreasing, flutter speed/dynamic pressure increases


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Compressibility of air (Mach number)flutter concave in transonic

in supersonic, flutter speed increases with Ma increasing

Constant soundspeed line

0 Ma

VF/Ma

VF

Ma?1

Ma matching calculation

Factors affecting flutter


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Frequency domain method of gust responce

General aerodynamics by gust

Movement

equation

2 ( )g g

q i q q k q qq w + + = +A QM C K

g g

q wQ

1

g g

q q w= T Q

C General damping

2 ( )i q k = + + T M C K A

Modes superposition u q=1 ( )

g g gu q w G w = =T Q

Function of frequency response between gust and displacement( )G

Solving

equation


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Aeroelastic design of aircraft

Stage of concept designTheoretical analysis Experience of similar aircraft

Concept evaluation avoiding the failure in aeroelasticity

confirm the basic configuration, such as engine position

Stage of preliminary designRapid analysis approach Confirm basic stiffness distribution

make sure the extent of New Structure, MDO

Stage of detail designEngineering approach Components and complete aircraft analysis,

GVT and model update, detail structure by MDO,

such as aeroelastic tailoring, compensation control

high precision verification and nonlinearity estimate

Stage of finalization and airworthy approval

Verify the analysis method, new structures, new techniques, ground test,

wind tunnel test, flight test by government or airworthiness department

Stage of new status and redesign


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Jobs in aeroelastic design

Overall design

Researching test

Empirical formula

Simplified calculation

Parts & components

desi n

Flutter test model,Stiffness & GVT,

Flutter tesr

Normal modes

Flutter analysis

Verified by test

Prototype of

aircraft

Stiffness & GVT

Model test in doubt case

Update model

Flutter analysis to eliminateroblems in calculation

Limitations to flight

Flight vibration

& flutter test

Dynamic response calculation

verified by flight test

Solve the problems

in flight test

Prototype verified


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Requirements in specification

0

Damping

Equivalent air speed Vdl

Critical mode

Required dampingg= 0.03

Flutter speed

1.15VjxVjxNoncritical mode

The airplane strength and stiffness specification is a directive document

in airplane design, which should be obeyed.

civil aviation FAR25, utility airplane FAR23

strength and stiffness specification for military airplanestrength and stiffness specification for UAV


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Ma

H

Determined by flutter

Conservative

boundaryAeroelastic instability

Typical flight envelope of airplane

Requirements in specification

Documents

L7_Engineering Method for Dynamic Aeroelasticity