AAE556 v g flutter lecture 32 The Vg method

Airfoil dynamic motionθ t Khh KTθ xθ P=-L aero center e Ma V 2Purdue Aeroelasticity... This is what we’ll get when we use the V-g method to calculate frequency vs... When we do the V-g m

AAE 556 Aeroelasticity The V-g method

g

V/b ωθ

k decreasing

flutter point

mode 1

mode 2

1Purdue Aeroelasticity

Airfoil dynamic motion

θ (t)

Khh

KTθ

xθ P=-L

aero center

e

Ma

V

2Purdue Aeroelasticity

This is what we’ll get when we use the V-g method to calculate frequency vs airspeed and

include Theodorsen aero terms

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Velocity (V/ ωθ b)

/ ω θ

3Purdue Aeroelasticity

When we do the V-g method here is

damping vs airspeed

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -1

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Velocity (V/ ωθ b)

4Purdue Aeroelasticity

flutter

divergence

To create harmonic motion at all airspeeds we need an energy source or sink at all airspeeds

except at flutter

i Input energy when the aero damping takes energy out (pre-flutter)

i Take away energy when the aero forces put energy in (post-flutter)

5Purdue Aeroelasticity

2D airfoil free vibration with everything

but the kitchen sink

h g

ω

&

&& &&

Iθθ Mx h Kθ θ gθ g θ M M e ω

ω θ

&& &

&&

( − ω2M K + h   1 + i gh + g   ) h − ω2Mxθθ = P

( − ω2Iθ + Kθ   1 + i gθ + g   ) θ ω − 2Mx hθ = Ma

6Purdue Aeroelasticity

We will get matrix equations that

look like this

A(k, ω , g)E(k, ω , g) − B(k)D(k) = 0

A B

D E





   

h / b

θ











 =

0 0













…but have structural damping that

requires that …

7Purdue Aeroelasticity

2

m b

µ

πρ

=

The EOM’s are slightly different from those before (we also multiplied the previous

equations by µ)





   

h / b

θ











 =

0 0













/ ω2

)[1 + i(gh + g)]} + Lh

B = µ xθ + Lα = Lh (1 / 2 + a)

E = µ rθ2{1 − ( ωθ2

/ ω 2

)[1 + i(gθ + g)]}

−M h(1 / 2 + a) + Mα − Lα(1 / 2 + a)+ L h(1 / 2 + a)2

-8Purdue Aeroelasticity

Each term contains inertial, structural stiffness, structural damping and aero

information

1 2

h h

Look at the “A” coefficient and identify the eigenvalue – artificial damping is added to keep

the system oscillating harmonically

Ω2

= ( ωθ2

/ ω 2

)(1 + ig) = Ω2R

+ i Ω2I

9Purdue Aeroelasticity

We change the eigenvalue from a pure frequency term to a

frequency plus fake damping term So what?

2

1 h 1 h fa

ω

=  −    ÷ +  +

2

fakie

2

r

1

The three other terms are also

modified





   

h / b

θ











 =

0 0













B = µ xθ + Lα = Lh (1 / 2 + a)

D =+ µ xθ + Mh − Lh(1/ 2 + a)

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Purdue Aeroelasticity

Each term contains inertial, structural stiffness, structural damping and aero

information

( )

2

2

2

1

1

2

ig

θ

θ

ω ω

−  + ÷+ −  + ÷+  + ÷

  +

 ÷

 

 

To solve the problem we input k and

compute the two values of Ω2

Ω12 = ( ΩR2

)1 + i( Ω2I

)1

Ω22 = ( ΩR2

ω1 = ωθ / ( ΩR)1

ω2 = ωθ / ( ΩR)2

g1 = ( ΩI2

)1 / ( Ω2R

)1

g2 = (ΩI2

)2 / (ΩR2

)2

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Purdue Aeroelasticity

2

2

2

I

g

i θ

ω

ω ω

 ÷

 





=

The value of g represents the amount of damping that would be required to keep the system oscillating harmonically It should be negative for a stable system

Now compute airspeeds

using the definition of k

V 1 = b ω 1 / k

ω1 = ωθ / ( ΩR)1

ω2 = ωθ / ( ΩR)2

Remember that we always input k so the same value of k is used in both cases One k, two airspeeds and damping values

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Purdue Aeroelasticity

Typical V-g Flutter Stability Curve

g

V/bωθ

k decreasing

flutter point

mode 1

mode 2

gh ≈ gθ

Ω2 = ( ωθ2

/ ω 2

)(1 + i g ) ′

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Purdue Aeroelasticity

' h

g = g + = g gθ + g

Now compute the eigenvectors

(b θ / h)1 = − D / E( Ω1 ) ; h

b = 1 ( Ω2 = Ω12

)

(h / b θ )2 = − B / A( Ω2) ; θ = 1 ( Ω2 = Ω22

)

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Purdue Aeroelasticity

Example

Two-dimensional airfoil mass ratio, µ = 20

quasi-static flutter speed VF = 160 ft/sec

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Purdue Aeroelasticity

0.03

h

Example

1 / k = 3.1250

Mα = 0.37500 − i3.1250

Mh = 0.50000

ωh = 10 rad / sec

ωθ = 25 rad / sec.

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Purdue Aeroelasticity

The determinant

A = 19.8 96 − i4.0973 − 3.2 Ω2

B = − 11.3767 − i2.54 40

D = 2.5311 + i1.22919

E = 9.2380 − i2.3618 − 5.0 Ω2

A E− BD = 16(Ω)4 + (−129.043+ i28.044)Ω2 +199.794 − i64 418= 0

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Purdue Aeroelasticity

Final results for this k value – two

g’s and V’s

Ω2

= 4.0326 − i0.87638 ± 3.0067 − i3.0420

− 4.0326 − i0.87638 ± (1.9084 − i0.79702)

= 5.9410 − i1.67340

Ω22 = 2.1242 − i0.07936

ω2 = 17.153 rad / sec ( ωθ = 25 rad / sec)

V1 = 96.157 ft / sec

V2 = 160.810 ft / sec

b = 3.0 ft

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Purdue Aeroelasticity

g = + g gθ = −

g = + g gθ = −

Final results

g = 0.03 Flutter

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Purdue Aeroelasticity