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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

Aeroelastic effects

- Example : Tacoma Narrows Bridge WA - 1940

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

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

Relative velocity of air with respect to body =

Aeroelastic effects- Aerodynamic damping (along wind) :

transfer to left hand side of equation of motion : speed

for small

total damping term :

aerodynamic damping term

Aeroelastic effects- Aerodynamic damping (along wind) :

Drag force (per unit length) =

along-wind aerodynamic damping is positive

From vector diagram : speed

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,

F speed z = D sin + L cos =

Aeroelastic effects- Galloping :

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

(Lecture 8)

For = 0 :

Substituting, speed

For , Fz is positive - acts in same direction as

Aeroelastic effects- Galloping :

negative aerodynamic damping when transposed to left-hand side

Aeroelastic effects speed

- Galloping :

den Hartog’s Criterion

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

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

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

Aeroelastic effects speed

- Galloping :

Cross sections prone to galloping :

Square section (zero angle of attack)

D-shaped cross section

iced-up transmission line or guy cable

Consider a two dimensional body rotating with angular velocity

Apparent change in angle of attack :

Vertical velocity at leading edge :

Aeroelastic effects- Flutter :

Can generate a cross-wind force and a moment

Aerodynamic instabilities involving rotation are called ‘flutter’

Flutter derivatives velocity

Aeroelastic effects- Flutter :

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

per unit mass

per unit mass moment of inertia

U/nd velocity

U/nd

unstable

2

4

6

0

10

8

12

A1*

3

0

stable

stable

2

2

-2

1

1

-4

1

1

2

0

H1*

-6

0

10

4

6

2

8

12

H2*

0.4

A2*

8

2

0.3

6

1

0.2

4

2

2

2

0.1

0

0

A

-2

-0.1

-0.2

1

Aeroelastic effects- Flutter :

Flutter derivatives for two bridge deck sections :

Aeroelastic effects velocity

- 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

Aeroelastic effects velocity

- 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 velocityJohn Holmes225-405-3789 [email protected]

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