The Influence of Aerodynamic Damping in
the Seismic Response of HAWTs
Andrew T. Myers, PhD, PE, Assistant Professor
Vahid Valamanesh, Graduate Student
Department of Civil and Environmental Engineering
Northeastern University
Presentation Outline
• Motivation
• Dimensions of utility-scale HAWTs
• Vulnerability to earthquakes
• Derivation of aerodynamic damping
• Fore-aft direction
• Side-to-side direction
• Numerical example – 1.5 MW NREL baseline turbine
• Conclusions
Motivation: Exposure of HAWTs to Earthquakes
Installed wind capacity map as of Jan 2011
United States National Seismic Hazard Map
Dimensions and Period of HAWTs
Approximate dimensions of a utility-scale HAWT
First Period ~ 3 s
Vulnerability to Earthquakes
•
•
•
•
No redundancy in the support structure
Slender hollow sections (D/t as high as 280)
Farms consisting of many nearly identical structures
Large directional affect due to aerodynamic damping
Side-to-side
Fore-aft
Aerodynamic Damping of HAWTs in the Fore-Aft Direction
• Forces based on blade element
momentum theory (BEM)
• Flexibility of rotor is omitted
• Wind direction is along fore-aft
direction
• Steady wind
• First mode of vibration is considered
mx + cst x + kx = 𝑑Fx
1
𝐹𝑥 = ρ𝑁𝑏
2
2
[𝑉𝑟𝑒𝑙
𝐶𝐿 𝑐𝑜𝑠 ∅ + 𝐶𝐷 𝑠𝑖𝑛 ∅ 𝑐 𝑟 ]𝑑𝑟
𝑚𝑥 + [𝑐𝑆𝑇 + 𝑁𝑏 𝐴 + 𝐵 ]𝑥 + 𝑘𝑥 = 𝑁𝑏 (𝐴 + 𝐵)𝑉𝑤 (1 − 𝑎)
A=
r𝑡𝑖𝑝
rhub
ρ ∙ Vw ∙ (1 − a) CL cos ∅ + CD sin ∅ c r dr
𝑟𝑡𝑖𝑝
1
𝐵=
𝜌 ∙ 𝛺𝑟 ∙ (1 + 𝑎′) (𝐶𝐿𝛼 +𝐶𝐷 ) 𝑐𝑜𝑠 ∅ + (𝐶𝐷𝛼 −𝐶𝐿 ) 𝑠𝑖𝑛 ∅ 𝑐 𝑟 𝑑𝑟
𝑟ℎ𝑢𝑏 2
𝜉𝐴𝐷,𝑥 =
𝑁𝑏 (𝐴 + 𝐵)
2 𝑘𝑚
Aerodynamic Damping of HAWTs in the Side-to-Side Direction
1
Fy = ρ
2
Nb
rtip
i=1 rhub
2
Vrel
CL sin ϕ − CD cos ϕ c r ∙ cos(γi t )dr
B ′ − A′
my + cST + Nb
y + ky = 0
2
𝑟𝑡𝑖𝑝
1
𝐴 =
𝜌𝑉𝑤 1 − 𝑎
𝑟ℎ𝑢𝑏 2
′
𝐵′ =
𝑟𝑡𝑖𝑝
𝑟ℎ𝑢𝑏
𝐶𝐿𝛼 + 𝐶𝐷 sin ∅ + 𝐶𝐿 − 𝐶𝐷𝛼 cos ∅ 𝑐 𝑟 𝑑𝑟
𝜌Ω𝑟 1 + 𝑎′ 𝐶𝐿 sin ∅ − 𝐶𝐷 cos ∅ 𝑐 𝑟 𝑑𝑟
𝜉𝐴𝐷,𝑦 =
𝑁𝑏 (𝐵′ − 𝐴′ )
4 𝑘𝑚
Numerical Example – 1.5 MW Baseline Turbine by NREL
Nacelle
Power output
1.5 MW
Hub Height
84 m
Rotor Diameter
70 m
Number of Blades
3
Max Rotational Speed
20 rpm
Cut in wind speed
5 m/s
Cut out wind speed
25 m/s
Nacelle Mass
51 Ton
Hub Mass
15 Ton
Tower Mass
123 Ton
Rotor Mass
11 Ton
Active Pitch Control
Yes
Rotor
Tower
Foundation
[Base image from Nuta, 2010]
Numerical Example – 1.5 MW Baseline Turbine by NREL
Aerodynamic damping in the fore-aft direction with W=20 rpm and b=7.5ᵒ
77
66
55
44
33
22
11
00
-1
-1
-2
-2
7
6
AD,x(%)
(%)
AD,x
AD,x(%)
5
4
3
2
1
0
5
10
15
20
25
Vw(m/s)
𝜉𝐴𝐷,𝑥 =
𝑁𝑏 (𝐴 + 𝐵)
2 𝑘𝑚
A=
r𝑡𝑖𝑝
rhub
55
10
10
15
15
Vw(m/s)
Vw(m/s)
2020
2525
ρ ∙ Vw ∙ (1 − a) CL cos ∅ + CD sin ∅ c r dr
𝑟𝑡𝑖𝑝
1
𝐵=
𝜌 ∙ 𝛺𝑟 ∙ (1 + 𝑎′) (𝐶𝐿𝛼 +𝐶𝐷 ) 𝑐𝑜𝑠 ∅ + (𝐶𝐷𝛼 −𝐶𝐿 ) 𝑠𝑖𝑛 ∅ 𝑐 𝑟 𝑑𝑟
𝑟ℎ𝑢𝑏 2
Numerical Example – 1.5 MW Baseline Turbine by NREL
2
2
1.5
1.5
1
1
0.5
0.5
AD,y(%)
AD,y(%)
Aerodynamic damping in the side-to-side direction with W=20 rpm and b=7.5ᵒ
0
-0.5
0
-0.5
-1
-1
-1.5
-1.5
-2
-2
5
𝜉𝐴𝐷,𝑦 =
10
𝑁𝑏
′
(𝐵
15
Vw(m/s)
−
′
𝐴)
4 𝑘𝑚
20
25
𝑟𝑡𝑖𝑝
1
𝐴 =
𝜌𝑉𝑤 1 − 𝑎
𝑟ℎ𝑢𝑏 2
′
𝐵′ =
𝑟𝑡𝑖𝑝
𝑟ℎ𝑢𝑏
5
10
15
Vw(m/s)
20
25
𝐶𝐿𝛼 + 𝐶𝐷 sin ∅ + 𝐶𝐿 − 𝐶𝐷𝛼 cos ∅ 𝑐 𝑟 𝑑𝑟
𝜌Ω𝑟 1 + 𝑎′ 𝐶𝐿 sin ∅ − 𝐶𝐷 cos ∅ 𝑐 𝑟 𝑑𝑟
Numerical Example – 1.5 MW Baseline Turbine by NREL
7
7
6
6
5
5
4
4
AD,x(%)
AD,x(%)
Aerodynamic damping in the fore-aft direction with b=7.5ᵒ (left) and W=20 rpm (right)
3
2
W=20
W=15
W=10
1
0
3
5
7
9
b=0˚
b=5˚
b=7.5˚
b=10˚
b=15˚
3
2
1
11 13 15 17 19
0
3
5
7
9
11 13 15 17 19
Numerical Example – 1.5 MW Baseline Turbine by NREL
Aerodynamic damping in the side-to-side direction with b=7.5ᵒ (left) and W=20 rpm (right)
1
W=20
W=15
W=10
0.8
0.6
0.4
0.2
0
b=0˚
b=5˚
b=7.5˚
b=10˚
b=15˚
0.8
0.6
AD,y(%)
AD,y(%)
1
0.4
0.2
0
-0.2
-0.2
-0.4
-0.4
3
5
7
9
11 13 15 17 19
3
5
7
9
11 13 15 17 19
Numerical Example – 1.5 MW Baseline Turbine by NREL
Validation with FAST in the fore-aft direction with b=7.5ᵒ and W=20 rpm
8
FAST
FAST
7
Derivation
Equation #15.
AD,x(%)
6
5
4
3
2
1
0
10
15
Vw(m/s)
20
25
Numerical Example – 1.5 MW Baseline Turbine by NREL
Effect of aerodynamic damping on the seismic response with W=20 rpm
0.9
0.8
0.7
0.6
0.5
Side to Side
Fore-Aft
b = 0
b = 0
b = 5
b = 5
b = 7.5
b = 7.5
b = 10
b = 10
b = 15
b = 15
0.4
0.3
0.2
0.1
0
3
5
7
9
11
13
15
17
19
Conclusions
• Aerodynamic damping of operational wind turbines strongly depends on wind speed. For the
considered example (1.5 MW turbine, W = 20 rpm, b = 7.5˚, wind speed between cut-in and
cut-out):
• The fore-aft aerodynamic damping varies between 2.6% and 6.4%
• The side-to-side aerodynamic damping varies between -0.1% and 0.9%
• For this same operational case, the derivative of the lift coefficient with respect to the angle
of attack is the most influential parameter in aerodynamic damping in the fore-aft direction
• The blade pitch angle and rotational speed also influence the aerodynamic damping in both
the fore-aft and side-to-side directions
• The directional effect strongly influences the seismic response, with median spectral drift
predicted to be as much as 70% larger in the side-to-side direction than in the fore-aft
direction