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 Aeroelastic effects

Wind loading and structural response

Lecture 14 Dr. J.D. Holmes

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 Aeroelastic effects

• Very flexible dynamically wind-sensitive structures

• Motion of the structure generates aerodynamic forces

• Positive aerodynamic damping : reduces vibrations - steel lattice towers

• if forces act in direction to increase the motion : aerodynamic instability

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 Aeroelastic effects

• Example : Tacoma Narrows Bridge WA - 1940

• Example : ‘Galloping’ of iced-up transmission lines

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 Aeroelastic effects

• Aerodynamic damping (along wind) :

Relative velocity of air with respect to body = xU  

Consider a body moving with velocity in a flow of speed    Ux

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 Aeroelastic effects

• Aerodynamic damping (along wind) :

xU bρCU bρ2

1C

)U

x2(1U bρ

2

1C)xU b(ρ

2

1CD

aD

2

aD

2

aD

2

aD

Drag force (per unit length) =

U/xfor small

aerodynamic damping term

xU bρCxcaD   total damping term :

along-wind aerodynamic damping is  positive

transfer to left hand side of equation of motion : D(t)kxxcxm    

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 Aeroelastic effects

• Galloping :galloping is a form of aerodynamic instability caused by negative

aerodynamic damping in the cross wind direction

Motion of body in z direction will generate an apparent reduction in angle of attack,

From vector diagram : U/zΔα  

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 Aeroelastic effects

•Galloping :

Aerodynamic force per unit length in z direction (body axes) :

Fz = D sin + L cos = )cosCsinα b(CUρ2

1LD

2

a    

(Lecture 8)

)cosdα

dCsinαCsinα

dα

dCcosα b(CUρ

2

1

dα

dF LL

DD

2

az

 

For = 0 : )dα

dC b(CUρ21

dαdF L

D2

az

 )Δdα

dC b(CUρ

2

1ΔF LD

2

az  

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 Aeroelastic effects

•Galloping :

Substituting, U/zΔα  

)U

z)(

dα

dC b(CUρ

2

1F LD

2

az

z)dα

dC b(CUρ

2

1 LDa

 

For , Fz is positive - acts in same direction as0)dα

dC(C LD     z

negative aerodynamic damping when transposed to left-hand side

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 Aeroelastic effects

•Galloping :

den Hartog’s Criterion0)dα

dC(C LD  

critical wind speed for galloping,   Ucrit , occurs when total damping is zero

0z)dα

dC b(CU

2

1 zc LDcrita        

)dα

dC b(Cρ-

2c U

LDa

crit

)dα

dC b(Cρ-

mn8π U

LDa

1crit

   

Since c = 2(mk)=4mn1 (Figure 5.5 in book)

m = mass per unit length n1 = first mode natural frequency

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•Galloping :

Cross sections prone to galloping :

Square section (zero angle of attack)

D-shaped cross section

iced-up transmission line or guy cable

 Aeroelastic effects

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 Aeroelastic effects

•Flutter :

Consider a two dimensional body rotating with angular velocity   θ

Vertical velocity at leading edge : d/2θ

Apparent change in angle of attack : Ud/2θ

Can generate a cross-wind force and a moment

Aerodynamic instabilities involving rotation are called ‘flutter’

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 Aeroelastic effects

• Flutter :

General equations of motion for body free to rotate and translate :

 per unit massθHθHzHm

(t)Fzωzω2ηz 321

z2

zzz    

θAθAzAI

M(t)θωθω2ηθ 321

2

θθθ      per unit mass moment of inertia

θ

z

Flutter derivatives

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 Aeroelastic effects

• Flutter :

Types of instabilities :

Name Conditions Type of motion Type of section

Galloping H1>0 translational Square section‘Stall’ flutter   A2>0 rotational Rectangle, H-

section

‘Classical’ flutter   H2>0, A1>0 coupled Flat plate, airfoil

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• Flutter :

1

2

-0.1

-0.2

 A2*

0.1

0

1

2

6

4

2

0

8

1

2

H2*

-2 A

0.4

0.3

0.2

unstable

stable

stable

Flutter derivatives for two bridge deck sections :

 A1*3

2

1

00 2 4 6 8 10 12

1

2

-6

-4

-2

00

H1*

2 4 6 8 10 12

12

U/nd

U/nd

 Aeroelastic effects

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 Aeroelastic effects

• Flutter :

Determination of critical flutter speed for long-span bridges:

• Empirical formula (e.g. Selberg)

• Experimental determination (wind-tunnel model)

• Theoretical analysis using flutter derivatives obtained experimentally

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 Aeroelastic effects

• Lock - in :

Motion-induced forces during vibration caused by vortex shedding

Frequency ‘locks-in’ to frequency of vibration

Strength of forces and correlation length increased

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End of Lecture 14John Holmes

225-405-3789 JHolmes@lsu.edu