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
+ =
Elements of structural dynamics
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
= =
Elements of structural dynamics
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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
Elements of structural dynamics
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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.
Elements of structural dynamics
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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
Solving method of flutter
decay ratio
END
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F
q F
Typicalq- andq-curves
From calculation
results, there are Ipairs of(Vi pi)
Solving method of flutter
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
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