aeroelastic galloping
TRANSCRIPT
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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 [email protected]