Aeroelastic Effects - Wind Engineering - Lecture Slides, Slides of Environmental Law and Policy

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• 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

• Aerodynamic damping (along wind) :

Relative velocity of air with respect to body = U^ x

Consider a body moving with velocity x in a flow of speed U

• Aerodynamic damping (along wind) :

ρ bU C ρ bUx 2

C

U

2 x ρ bU ( 2

ρ b(U x) C 2

D C

D a

2 D a

2 D a

2 D a

Drag force (per unit length) =

for small x/U

aerodynamic damping term

total damping term : cx^ ^ CD ρabUx

along-wind aerodynamic damping is positive

transfer to left hand side of equation of motion : mx  cxkxD(t)

• Galloping :

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

Fz = D sin  + L cos  = ρ U b(C sinα C cos )

D L

2

a ^ 

(Lecture 8)

cos ) dα

dC sinα C sinα dα

dC ρ U b(C cosα 2

dα

dF (^) L L

D D

2 a

z     

For  = 0 : )

dα

dC ρ U b(C 2

dα

dF (^) L D

2 a

z  

dα

dC ρ U b(C 2

ΔF

L D

2 z  a 

• Galloping :

Substituting, Δα^ z/U

U

z )( dα

dC ρ U b(C 2

F

L D

2 z a

)z dα

dC ρ Ub(C 2

1 L

 a D  

For ) 0 , Fz is positive - acts in same direction as

dα

dC (C

L D   z

negative aerodynamic damping when transposed to left-hand side

• Galloping :

Cross sections prone to galloping :

Square section (zero angle of attack)

D-shaped cross section

iced-up transmission line or guy cable

• Flutter :

Consider a two dimensional body rotating with angular velocity θ

Vertical velocity at leading edge : θd/ 

Apparent change in angle of attack : θd/2U

Can generate a cross-wind force and a moment

Aerodynamic instabilities involving rotation are called ‘flutter’

• Flutter :

Types of instabilities :

Name Conditions Type of motion Type of section

Galloping H 1 >0 translational Square section

‘Stall’ flutter A 2 >0 rotational Rectangle, H- section

‘Classical’ flutter H 2 >0, A 1 >0 coupled Flat plate, airfoil

• Flutter :

1

2

-0.

-0.

A 2 *

0

1

2

6

4

2

0

8

1

2

H 2 *

A

unstable

stable

stable

Flutter derivatives for two bridge deck sections :

3 A 1 *

2

1

0 (^0 2 4 6 8 10 )

1

2

0

0

H 1 *

(^2 4 6 8 10 )

1 2

U/nd

U/nd

• 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

End of Lecture 14

John Holmes

225-405-3789 [email protected]