L5_Flutter and Responce of 2D Wing Segment

8/10/2019 L5_Flutter and Responce of 2D Wing Segment

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5th Flutter and Responseof 2D Wing Segment

Xie Changchuan2014 Autumn


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Content1Introduction of dynamics aeroelasticity

2Flutter model and mechanism of 2D wing segment

3

Solving method of flutter4Factors affect the flutter and wing/aileron system

5Dynamic aeroelastic response

Main AimsUnderstand the flutter----dynamic stability and

its mechanism, aeroelastic response of elasticaircraft from a 2D wing segment model.


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Dynamic aeroelasticity

Major problems

Inertial forceI

Aero forceL

Deformation

+I

Aerodynamics

Mass/

Inertia

Elastic

system

Block diagram ofdynamic aeroelastic feedback

Stability: Classical Flutter

Stall flutter

Buzz

Whirl flutter

Turbine flutter

Dynamic

Response:

Input fromoutside

Gust

Maneuver

Drop and ejection

Take off and landing


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+ + Mw Gw Kw =F

Linear constant-coefficient ODE for dynamics

Vibration: =F =G Free vibration without damping

=F G Damping free vibration

( )= tF F Forced vibrationAeroelasticity:

( , )= tF w

( )=F F w The equation dont include time explicitly,autonomous equation

No outside exciting(homogeneous).There could be dynamic solutions, called self-excited vibration

That is dynamic stability problem, FLUTTER

( , )= tF F wDynamic response problem

The unique solution of Eq.


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Dynamic stability

The relationship between static and dynamic stability

The dynamic stability covers the static stability. That meansthe system is static stable but is not always dynamic stable,otherwise the dynamic stability gives out static stability.

The static stability can be looked as the special case of dynamic

problem. The eigenvalues of static unstable system has non-negative real part and zero imagine part.

P

l

For a compress column with followerforce, the system will never losestability by static stability criterion, but

according to dynamic stability criterionthe system will flutter.

Stability theorem of linear constant-coefficient system

(Liapunovs first method)

[0, ) +tState equationThe necessary and sufficient condition of the zero solutions

exponential stabili ty is, all the eigenvalues of have negative

= Tx n

Treal part. The exponential stability orderis the smallest real part.


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Flutter model of 2D wing segment

E

C = 2b

V

Aero centere

h

G

Elastic

center

Weight

center

For a wing segment withpitch and plunge freedoms,analysis the system at itsstatic equilibrium state.

Airfoilsymmetry thin Spanunity(1)

Chordc=2b Air force center1/4 chord

Elastic centere behind air force centerWeight center: behind air force center

Coefficient of torsion spring

Slope of lift ing curve Speed of flowV

Elastic angle + nose up Plunging: h + downward

KyC


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Aero force is non-

conservative, the totalwork in a constant

amplitude vibrating

cycle should bediscussed carefully.

Mechanism of flutterPositive

work

(a)

2

Phase difference = 0

Total work is zero.Vibratingdirection

Aeroforce

0

V

Positivework

Negativework

Negativework

Energy balance

of aero force in a

vibrating cycle

Positive

Work

(b)

2

Phase difference = 90

Total work is positive.

V

0

Positive

Work

Elastic and inertial

forces are conservative,

the total work of them ina constant amplitude

vibrating cycle are aero.


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+ y

V

AOA changing of plunging vibrated airfoil.

y is plunging speed, is AOA

R e l a ti o n o f E x c i t i n g f o r c e L a n d d a m p i n g f o r c e L w i th

f li g h t s p e e d V

L

L

L

VFV

o

LSCVL 221=

Aero force by elastic pitching

Only consider the additionalaero force induced by elasticity

Positive work on plungingExciting effects

21 ( / )2

= h LV SC h V

Note: The phase difference is assumed to 90. Here it just to illustrate

the possibility of non-conservative aero force to do work. So it ispossible that the system could be excited continually.

Mechanism of flutter

Aero force by plunging speed

Negative work on plunging

Damping effects


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The forces and equations of wing segment

Using steady strip theory

Elastic restore forces = h hf K h = m K

Inertial forces = +

initf mh m

21

2 = LV SC

0 ( )= + +

initm I m mh

Steady---at every moving instant of wingsegment, take a photo to calculateaerodynamics ignoring speed(damping term).It can illustrate the mechanism of flutter.

Equations of plunging and pitch movements

0

0

+ + + =

+ + =

h L

L

mh S K h qSC

S h I K qSeC

=S m First moment of massabout elastic center2

0= +I I m Rotational inertial (second moment)about elastic center

Disturbance equations

of equilibrium state

21

2=q V

Dynamic pressureof inflow

Aero force by elastic pitching


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Method by equation analysis

Fundamental solutions of linearhomogeneous ordinary equations

0

0

= =

t

t

h h e

e

= + i Usually is complex number Circular frequency

Exponential decay order

0h 0 Determined by initial conditions

0 Divergence, static equilibrium solution is unstable.


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Substitute the fundamental solutions into equations

2 20

2

0

+ + =

+

h L

L

hm K S qSC

S I K qSeC

0

If the determinate of coefficient matrix equal to zero,the equation has arbitrary solutions.

0

24 =++ CBA

2= mI S

( )= + +h LB mK I K me S qSC

( )= h LC k k qSeC

ACBB

2

422 =Roots of one variable

quadratic equation



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are two negative roots

0B >

2 2

4 0B B AC> >

2 2

2 2 2 2 2 2

4 ( )

( ) ( ) ( ) ( )

=

= =

h

hh

B AC mK I K

K KmI mI

I m

2 2( ) ( )= + = + = +hh hK K

B mK I K mI mI

I m

2 1 = i 2 = hi

Equations degrade into free vibration problem.


0=qIf then


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0>q 0>B2 4 0B AC >

2 1 = qi 2 = hqi

There are 2 vibrations whose frequencyare different with free vibration.

2 4 0 =B AC determine the critical case

is one real root2

The smallest positive dynamicpressure is Fq

Discussing: If the negative flutter dynamics pressure has somephysic meaning? If it is possible equal to 0?


If and In case

are still two negative roots

2 4 0B AC

L5_Flutter and Responce of 2D Wing Segment

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