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**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

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**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

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**Motivation: Exposure of HAWTs to Earthquakes**

United States National Seismic Hazard Map Installed wind capacity map as of Jan 2011

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**Dimensions and Period of HAWTs**

Approximate dimensions of a utility-scale HAWT First Period ~ 3 s

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**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

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**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

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**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

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**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]

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**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

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**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

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**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)

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**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)

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**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

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**Numerical Example – 1.5 MW Baseline Turbine by NREL**

Effect of aerodynamic damping on the seismic response with W=20 rpm

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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

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