1. 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

2. 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

3. Motivation: Exposure of HAWTs to Earthquakes
United States National Seismic Hazard Map
Installed wind capacity map as of Jan 2011

4. Dimensions and Period of HAWTs
Approximate dimensions of a utility-scale HAWT
First Period ~ 3 s

5. 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

6. 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
m x + c st x +kx= 𝑑F x
𝐹 𝑥 = 1 2 ρ𝑁 𝑏 [𝑉 𝑟𝑒𝑙 2 𝐶 𝐿 𝑐𝑜𝑠 ∅ + 𝐶 𝐷 𝑠𝑖𝑛 ∅ 𝑐 𝑟 ]𝑑𝑟
𝑚 𝑥 +[ 𝑐 𝑆𝑇 + 𝑁 𝑏 𝐴+𝐵 ] 𝑥 +𝑘𝑥= 𝑁 𝑏 (𝐴+𝐵) 𝑉 𝑤 (1−𝑎)
A= r hub r 𝑡𝑖𝑝 ρ∙ V w ∙(1−a) C L cos ∅ + C D sin ∅ c r dr
𝐵= 𝑟 ℎ𝑢𝑏 𝑟 𝑡𝑖𝑝 𝜌∙𝛺𝑟∙(1+𝑎′) (𝐶 𝐿𝛼 + 𝐶 𝐷 ) 𝑐𝑜𝑠 ∅ + (𝐶 𝐷𝛼 − 𝐶 𝐿 ) 𝑠𝑖𝑛 ∅ 𝑐 𝑟 𝑑𝑟
𝜉 𝐴𝐷,𝑥 = 𝑁 𝑏 (𝐴+𝐵) 2 𝑘𝑚

7. Aerodynamic Damping of HAWTs in the Side-to-Side Direction
F y = 1 2 ρ i=1 N b r hub r tip V rel 2 C L sin ϕ − C D cos ϕ c r ∙cos⁡( γ i t )dr
m y + c ST + N b B ′ − A ′ 2 y +ky=0
𝐴 ′ = 𝑟 ℎ𝑢𝑏 𝑟 𝑡𝑖𝑝 𝜌 𝑉 𝑤 1−𝑎 𝐶 𝐿𝛼 + 𝐶 𝐷 sin ∅ + 𝐶 𝐿 − 𝐶 𝐷𝛼 cos ∅ 𝑐 𝑟 𝑑𝑟
𝜉 𝐴𝐷,𝑦 = 𝑁 𝑏 ( 𝐵 ′ − 𝐴 ′ ) 4 𝑘𝑚
𝐵 ′ = 𝑟 ℎ𝑢𝑏 𝑟 𝑡𝑖𝑝 𝜌Ω𝑟 1+𝑎′ 𝐶 𝐿 sin ∅ − 𝐶 𝐷 cos ∅ 𝑐 𝑟 𝑑𝑟

8. Numerical Example – 1.5 MW Baseline Turbine by NREL
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
[Base image from Nuta, 2010]

9. Numerical Example – 1.5 MW Baseline Turbine by NREL
Aerodynamic damping in the fore-aft direction with W=20 rpm and b=7.5ᵒ
A= r hub r 𝑡𝑖𝑝 ρ∙ V w ∙(1−a) C L cos ∅ + C D sin ∅ c r dr
𝜉 𝐴𝐷,𝑥 = 𝑁 𝑏 (𝐴+𝐵) 2 𝑘𝑚
𝐵= 𝑟 ℎ𝑢𝑏 𝑟 𝑡𝑖𝑝 𝜌∙𝛺𝑟∙(1+𝑎′) (𝐶 𝐿𝛼 + 𝐶 𝐷 ) 𝑐𝑜𝑠 ∅ + (𝐶 𝐷𝛼 − 𝐶 𝐿 ) 𝑠𝑖𝑛 ∅ 𝑐 𝑟 𝑑𝑟

10. Numerical Example – 1.5 MW Baseline Turbine by NREL
Aerodynamic damping in the side-to-side direction with W=20 rpm and b=7.5ᵒ
𝐴 ′ = 𝑟 ℎ𝑢𝑏 𝑟 𝑡𝑖𝑝 𝜌 𝑉 𝑤 1−𝑎 𝐶 𝐿𝛼 + 𝐶 𝐷 sin ∅ + 𝐶 𝐿 − 𝐶 𝐷𝛼 cos ∅ 𝑐 𝑟 𝑑𝑟
𝜉 𝐴𝐷,𝑦 = 𝑁 𝑏 ( 𝐵 ′ − 𝐴 ′ ) 4 𝑘𝑚
𝐵 ′ = 𝑟 ℎ𝑢𝑏 𝑟 𝑡𝑖𝑝 𝜌Ω𝑟 1+𝑎′ 𝐶 𝐿 sin ∅ − 𝐶 𝐷 cos ∅ 𝑐 𝑟 𝑑𝑟

11. Numerical Example – 1.5 MW Baseline Turbine by NREL
Aerodynamic damping in the fore-aft direction with b=7.5ᵒ (left) and W=20 rpm (right)

12. 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)

13. 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
FAST
Derivation

14. Numerical Example – 1.5 MW Baseline Turbine by NREL
Effect of aerodynamic damping on the seismic response with W=20 rpm

15. 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