Wind loading and structural response

Lecture 14 Dr. J.D. Holmes

Aeroelastic effects
 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

 Aeroelastic effects

• Aerodynamic damping (along wind) :

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

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

 Aeroelastic effects

• Aerodynamic damping (along wind) :

Drag force (per unit length) =

1 1 2x
D  CD ρ a b(U  x ) 2  C D ρ a bU 2 (1  ) for small x /U
2 2 U
1
 CD ρ a bU 2  C Dρ a bUx
2

aerodynamic damping term

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

total damping term : cx  C Dρ a bUx

along-wind aerodynamic damping is positive

 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 : Δα  z /U

 Aeroelastic effects

• Galloping :

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

1
Fz = D sin  + L cos  = ρ a U 2 b(C Dsinα  C L cos )
2 (Lecture 8)

dFz 1 dC dC
 ρ a U 2 b(CD cosα  D sinα  C Lsinα  L cos )
dα 2 dα dα

dFz 1 dC
For  = 0 :  ρ a U 2 b(C D  L )
dα 2 dα

1 dC
ΔFz  ρ a U 2 b(C D  L )Δ
2 dα
 Aeroelastic effects

• Galloping :

Substituting, Δα   z /U
1 2 dC L z
Fz  ρ a U b(C D  )( )
2 dα U

1 dC
  ρ a Ub(C D  L )z
2 dα

dC L
For (C D  )  0 , Fz is positive - acts in same direction as z
dα

negative aerodynamic damping when transposed to left-hand side

 Aeroelastic effects

• Galloping :

dC L
(C D  )0 den Hartog’s Criterion
dα

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

1 dC
cz   a U crit b(C D  L )z  0
2 dα

2c 8πmn1
U crit  U crit 
dC L dC
- ρ a b(C D  ) - ρ a b(C D  L )
dα dα

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

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

 Aeroelastic effects

• 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

• Flutter :

Consider a two dimensional body rotating with angular velocity θ

Vertical velocity at leading edge : θ d/2

Apparent change in angle of attack :  θ d/2U

Can generate a cross-wind force and a moment

Aerodynamic instabilities involving rotation are called ‘flutter’

 Aeroelastic effects

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

z

θ

Fz (t)
z  2η z ω z z  ω 2z z   H1z  H 2θ  H 3θ per unit mass
m

θ  2η ω θ  ω 2θ  M(t)  A z  A θ  A θ per unit mass moment of inertia

θ θ θ 1 2 3
I

Flutter derivatives
 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
 Aeroelastic effects

• Flutter :
Flutter derivatives for two bridge deck sectionsU/nd
:
0 2 4 6 8 10 12 1
3 A1* 0
2
2 -2 stable
1
1 -4 1 2
2
0 -6 H1*
0 2 4 6 8 10 12

U/nd
0.4 A2* 8 H2*

0.3 6
unstable
0.2 4 1
2 2
0.1 2

0 0
A
-0.1 -2

-0.2
stable
1
 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

 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

 End of Lecture 14
John Holmes
225-405-3789 JHolmes@lsu.edu