Kallesøe, Bjarne Skovmose

Larsen, Torben J.

Madsen Aagaard, Helge

Rasmussen, Flemming

Online Research Database In Technology

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Downloaded from orbit.dtu.dk on: Dec 17, 2017On the dynamics of a teeter hinge with delte3 angle on a two bladed multi MW windturbineKallesøe, Bjarne Skovmose; Aagaard Madsen , Helge; Larsen, Torben J.; Rasmussen, FlemmingPublished in:Conference proceedingsPublication date:2008Link back to DTU OrbitCitation (APA):Kallesøe, B. S., Madsen Aagaard, H., Larsen, T. J., & Rasmussen, F. (2008). On the dynamics of a teeter hingewith delte3 angle on a two bladed multi MW wind turbine. In Conference proceedings Reston, VA (US):American Institute of Aeronautics & Astronautics.On the Dynamics of a Teeter Hinge with δ3 Angle ona Two Bladed Multi MW Wind TurbineB. S. Kallesøe, H. Aa. Madsen, T. J. Larsen and F. RasmussenWind Energy Department, Risø National Laboratory, Technical University of Denmark, DK-4000Roskilde, Denmark.E-mail: bjarne.skovmose.kallesoee@risoe.dk1. IntroductionFor two bladed turbines the δ3 angle of the teeter hinge is said to stabilize the teeter angle de-flections. However the literature does not give a clear answer for what size and sign the δ3 angleshould have. Furthermore the upscaling of turbine blades has change the dynamics of the teeterhinge with δ3 and therefore the potentials for using δ3 angle on modern multi MW turbines. Thisstudy analysis the effect of δ3 angle on multi MW turbine using the analytical model for teetermotion suggested by [1] and aeroelastic simulations. The analysis shows that the model describesthe basic behavior of teeter motion well, but realistic loading from e.g. a yaw error is not described.The effect for yaw error and wind shear are analysis and incorporated in the analytical model. Thepotentials for using δ3 angle on modern multi MW turbines is discussed.2. Method and Preliminary Results2..1 Analytical ModelRef. [1] suggest that a teeter hinge with a δ3 angle can be described by a second order systemβ¨ + 2ξω0β˙ + ω20β = F (1)where the natural frequency is given byω0 = Ω√1 + λKp/8 (2)and the damping byξ =λ16√1 + λKp/8(3)where Kp = tan(δ3) and F is the forcing on the blade cause by e.g. changes in wind speed oroperating in yaw. The so called lock number λ is the ratio between aerodynamic forces and inertialforces and Ref [1] gives an analytical expression for a blade with a simple geometryλ =2ρR4CLαcI(4)where R is the radius of the blade, ρ is the air density, CLα is the slop of the lift curve, c is thechord and I is the polar moment of inertia of the blade around the teeter hinge. It is seen thatthis expression is dependent on the specific blade layout and linearly dependent on the air density.2..2 Determine Lock NumberFor a realistic blade layout the lock number can be determined by measuring the frequency anddamping of the teeter motion for the specific turbine setup at different air densities. Figure 1shows results from simulations with the aeroelastic code HAWC2 [2] of transient behavior of thegiven turbine setup at different air densities. The natural frequency is seen to be almost constantfor all air densities as predicted by the model (2) and the damping has a linearly dependency onthe air density as predicted by (3)10.970.980.991frequency [hz]0 0.05 0.1 0.15 0.200.050.10.150.2damping [%]air density ρFigure 1: Frequency and damping dependency of air density. Top graph: frequency, bottom graph:damping ratio. Blue line top graph: natural frequency, green line top graph: resonance frequency,blue line bottom graph: simulated damping, green line bottom graph: linear regression to fit locknumber to simulations.Since the damping is described byξ =λ(ρ)16√1 + λKp/8(5)and λ(ρ) = λ0ρ the lock number dependency of ρ can be found from the slope of the damping onFigure 1. For the particular turbine setup the lock number is found to be λ = 13.9ρThe lock number depends on the aerodynamic characteristics and the inertial of the blade, andthe up-scaling of turbine has changes the relations between these parameters of modern turbinesand therefore changes the effect and potentials for used of δ3 angle. This study will clarify the sizeeffect on the potentials for use of a δ3 angle.2..3 Effect of δ3 and Verification of Analytical ModelFigure 2 shows the frequency response curves for direct excitation of the teeter motion for twodifferent values of air density. The simulated frequency response is seen to agree very well withthe response predicted by the model, especially for the realistic air density (Figure 2(b)). The δ3angels effect on the natural frequency of teeter motion is clearly seen on the resonance peaks forthe simulations with low air density (Figure 2(a)). For the simulations with realistic air density(Figure 2(b)) the teeter motion is over damped and therefor it has no resonance peak. Figure 2(b)suggest a strong reduction of teeter motion for negative δ3 angels.2..4 Operation in YawTo analysis the effect of the δ3 angle under more realistic loading conditions the turbine is simulatedfor operation with a 30 deg yaw error with different δ3 angles. Figure 3 shows the amplitude forteeter angle oscillations at steady state conditions for two different air densities.Figure 3 shows only a small effect of a δ3 angle for alleviating teeter oscillations caused by yawerror for realistic operation conditions for a multi MW turbine. The model predict much lagereffect of the δ3 angle. The disagreement between the simulated response and the predicted teeterresponses is caused by the loading form yaw error is imposed through the aerodynamic and not20 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−10−5051015202530frequencyamplitude(a)0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−50−45−40−35−30−25−20−15−10−50frequencyamplitude(b)Figure 2: Frequency response functions for direct excitation of the teeter motion. Solid line is thesimulated response and the dotted line is the prediction of the model. Green line δ3 = −60 deg,red line δ3 = −30 deg, yellow line δ3 = −20 deg, black line δ3 = −10 deg, blue line δ3 = 0 deg,cyan line δ3 = 30 deg, magenta δ3 = 60 deg. a) ρ = 0.1kg/m3 b) ρ = 1.225kg/m3.directly at the teeter motion. The changes the yaw error impose on the aerodynamic also changesthe aerodynamic effect of the δ3 angle.The interaction between aerodynamic loading from yaw and wind shear with the δ3 angle willbe analyzed and described. The loading will be incorporated both in the analytical model andanalyzed through aeroelastic simulations.3. ConclusionThis work shows how the behavior of the teeter motion with δ3 angle can be described by themodel suggested by [1]. The results from the model are compare to aeroelastic simulations of amulti MW turbine showing good agreement. The prediction of teeter motion for operation in anyaw error is however not well described, since the loading from yaw interact with the effect of the δ3angle. This aerodynamic interaction is analyzed with aeroelastic simulations and by incorporatedthe oscillating aerodynamic loading into the analytical model. The aeroelastic simulations andanalysis of the analytical model clarify the potentials for using δ3 angles on two bladed MW wattturbines.References[1] M.B. Anderson, A.D. Garrad, and U. Hassan. Teeter excursions of a two-bladed horizontal-axiswind-turbine rotor in a turbulent velocity field. Journal of Wind Engineering and IndustrialAerodynamics, 17(1):71–88, 1984.[2] T.J. Larsen, H.Aa. Madsen, A.M. Hansen, and K. Thomsen. Investigations of stability effects ofan offshore wind turbine using the new aeroelastic code HAWC2. Proceedings of the conference”Copenhagen Offshore Wind 2005”, 2005.3−60 −40 −20 0 20 40 600.511.522.533.5teeter angle [deg]δ3 [deg]Figure 3: Amplitude on teeter angle oscillations for operation at 30 deg yaw at different δ3 angles.Blue line ρ = 1.225 kg/m3, glue line ρ = 0.1 kg/m3, red line predicted response for ρ = 1.225kg/m3, cyan line predicted response for ρ = 0.01 kg/m34

On the dynamics of a teeter hinge with delte3 angle on a two bladed multi MW wind turbine

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On the dynamics of a teeter hinge with delte3 angle on a two bladed multi MW wind turbine

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