Proceedings of 9th PhD Seminar on Wind Energy in Europe
September 18-20, 2013, Uppsala University Campus Gotland, Sweden
AERODYNAMIC DAMPING OF NONLINEARILY WIND-EXCITED WIND
TURBINE BLADES
P. van der Male1, K.N. van Dalen2, A.V. Metrikine2 1Offshore Engineering, Delft University of Technology 2Structural Mechanics, Delft University of Technology
Delft, the Netherlands e-mail: p.vandermale@tudelft.nl
ABSTRACT
This paper presents the first step of the derivation of an aerodynamic damping matrix that can be adopted for the foundation design of a wind turbine. A single turbine blade is modelled as a discrete mass-spring system, representing the flap and edge wise motions. Nonlinear wind forcing is applied, which couples the degrees of freedom. The structural response is determined by means of a Volterra series expansion. The contribution of the aerodynamic damping to the structural response is determined by comparing the response without structural feedback to the response that includes structural feedback.
The reduction of the structural response due to aerodynamic damping is significant. This also applies for the edge wise response and the cross response that results from the coupling. Due to the nonlinear forcing, higher order harmonics are excited. This study only presents the response to a single harmonic 1P forcing. To fully understand the response to the nonlinear forcing, a representative excitation spectrum needs to be adopted.
INTRODUCTION
In determining the wind forcing on a wind turbine, the interaction between the air flow and the structure cannot be neglected. The effective force due to a flow on a structure depends on the relative velocity of this flow with respect to the structure. If the structure responds dynamically, the relative flow velocity is affected by the structural vibration. This aspect is of particular importance for flexible structures like wind turbines, where the motion of the structure generally leads to a reduction of the effective wind force. Moreover, turbine blades are highly sensitive to perturbations in the angle of attack of the wind flow, reducing or increasing the forcing on the structure. The force reduction due to the structural feedback velocity is
commonly known as added damping, or – specifically for wind turbines – as aerodynamic damping.
Complex models, making use of computational fluid-dynamics techniques, can be adopted to estimate the effective forcing – and so the aerodynamic damping – of a wind flow on a wind turbine. For early design stages this approach is time-consuming, in both model construction and calculation processing. Besides, such complex models do not necessarily provide the physical insight that can be employed to improve the aero-elastic performance of the structure. On the other hand, when it comes to foundation design, the available techniques to determine the reduction in aerodynamic forcing due to structural response are rather simplified. Linearized expressions, neglecting the dependency of the force on the mean wind speed and the coupling between flap and edge wise blade motion – among other things, are adopted to estimate the effective wind forcing [1, 2, 3]. For offshore wind turbines, the aerodynamic damping of hydrodynamic forces is recognized [4], but the existing theories do not allow for a misalignment between the aerodynamic and hydrodynamic forcing.
This paper presents the first step in the derivation of an aerodynamic damping matrix that particularly serves the design of offshore wind support structures under combined wind and wave loading. A single blade is modelled as a mass-spring system representing the flap and edge wise motions. The blade is excited by a drag force that depends quadratically on the effective flow velocity. The forcing ensures coupling between the degrees of freedom. The modelling includes dependencies on the mean wind velocity and the pitch angle of the blade. In order to account for the nonlinear character of the loading, a Volterra series expansion is applied, a technique that enables the identification of higher-order transfer kernels in the frequency domain [5].
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he application assumed clam accounted fo ile uy express the pitch ang n be controlled ons describin an be written a 0 0 ; n x and y comp performed on tics [6]. First o nsfer function luate aerodyn nalyzed for a es of freedo analyze the w ass-spring sys g representatio mass-spring sys representing s kx and ky, x a ral damping cx alization is don edge wise f n of fixed inter mped at its root or. The flap
ses the edge gle with which d. ng the dyn as: (1) 0 (2) ponent: (3) n the order that namic a 1P om - wind-stem, on of stem, g the and y x and ne on fixed rface t and wise wise h the namic 0 ; a fre AE co ge , w – c fix dia co the int no Th of v, Li de Fi W ve ve tan co tur wi rep the ch co Th the Figure 1: ( part of the r eedom, repres ERODYNAM Since the oordinates ux eneralization to f where f repres consisting of t – and the xed interface agonal, wher omponents are e generalized tegrated over In order to onlinear flow-f f his relation de f the relative fl the flow fluct
w v
ike for the fo ependency. Al igure 2 depicts Within the fram ector can be elocity that ac ngential veloc omponent of t rbine, ̅ equa ind field. The presented by e generalized Other com hord width A, onsists of the li he wind flow w e rotor plane, a) the mass-sp rotor, (b) disc enting a single MIC FORCING blade is m and uy, the oo. This gener
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(5)
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ngle with res force f and blade as grees of eralized sents a as: ) g vector n and ge wise ts main o. Both o obtain to be llowing ) function an flow city u: ) spatial pendent. model. he flow n wind constant hat the y n idling s of the u can be x and e blade , which ) spect to f , see
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cos sin Figure 2: air drag forces. generalized fo blade length nition of the l he mean wind blade swept uations an us dependent f dius r to the b aero-elastic c e of attack , adius invariant endency is the NLINEAR BL The structura eover forces a ction, is estim ansion. With th be accounte eover, the tec ain. The theor d, and combin en limiting t uency-domain sform of u an be found from U U
ere the compo nd-order freq ws from the ut relation is d er to obtain sformation ma s sin n cos flow excited orcing vector h. Therefore, ength depend d velocity ̅ i t area. The nd . For the function is ado lade tip speed coefficients see Figure 2. t. The only re chord width A LADE ANALY al response to a coupling bet mated with t he help of this d for by hi chnique perm ry is already ap ned wind and
the expansion n response vec nd consisting m the following U onents of U quency respon well-known li defined by the force compon atrix is appl d blade model f follows from an importan ding componen s assumed to same applie e tangential ve opted, which r d. and are These functio maining comp A. YSIS the nonlinear tween the res the help of technique, th igher-order tr mits analyses pplied in analy wave excited n to the se ctor U, repres of the compo g relations: and U repre nses. The fir inear relation, transfer funct nents in x an ied: (8) l and resulting m integration nt aspect is nts of (1). Fir be constant w es for the w elocity ̅ a li relates the velo e functions of ons are assume ponent with le r excitation, w sponses in x a a Volterra s he nonlinear ef ransform ker in the frequ yzing respons d structures [7 econd order, enting the Fo onents and (9) esent the
first-rst-order resp , where the in tion matrix nd y g lift over the rst of within wind inear ocity f the ed to ength which and y series ffects rnels. uency es of 7, 8]. the ourier d , - and ponse nput-: Th of ma dir Vo fre se tra mu ex int Th co fu ap alt kn alg ou Vo rep sy un LI es lin ap wi the the an ex wh pit NR ex M wi to kx U he vector V , r f the frequenc atrix con rect or cross f olterra kernel equency , bu cond-order r ansforms, com ultiplied by t xcitation sign tegrated with he asterisks im onsistency with unctions of . Identificati pplying the h ternative exis nown partial gorithm has a utput systems olterra kernel i It can easi present the f ystem. Higher nique frequenc INEARIZED Currently, timated using nearized wind pproach, which ise motion of e rotating blad e flap wise fe ngle and there xpression for hich among ot tch angle and REL 5.0MW The struct xcitation is de MW reference ise structural p
the flap wise = ky = 1314 V representing th cy dependent ntains direct- frequency-resp ls are not ju ut of the frequ response U , mbined in the the excitation nal V∗. More respect to V∗V mply double h the matr , while the ion of the Vo harmonic pro sts in assem solutions [1 already been ex [11], this app identification. ily be shown t frequency-resp r-order Volter cy response fu BLADE RES aerodynamic g one-degree-o d flow-structu h is only vali the blade resu de only. Small edback veloci efore in the l the aerodyn ther things, ne the coupling o BLADE ANA tural respons etermined for turbine [6]. F properties hav e characteristic 41 N/m. It sh he Fourier tran excitation sig and cross-ke ponse function ust a functio uency too , the direct 2 4 matrix n signal V∗, eover, the re over an infinit V∗ d entries of rix. The entries
entries of V olterra kernels obing algorith mbling compli 10].Since the xtended for m proach is ado that the first o ponse function rra kernels ca unctions, but ar SPONSE damping due of-freedom sy ure interaction id for operatin ults from the t l fluctuations i ity, cause pert lift coefficient
amic dampin eglects the me of flap and edg ALYSIS se to the n a single blad For convenien ve been related cs, namely mx hould be noted (10 nsform of v, c gnals and ernel transfor ns. The secon n of the ex . In order to f - and cross need not o but as well elation needs te interval: (11 and , to m s of the vector V∗ are functi can be achie hm [9]. An icated kernel e harmonic p multi-input and
opted for the order Volterra n for the lin annot be defi re input-depen e to blade mo ystems as a re n. According ng turbines, t tangential velo in the wind fie turbations in th
t. Based on t ng can be ob ean wind veloc ge wise motio nonlinear win de of the NR nce’s sake, th d in a simple = my = 2242 d that an incr 0) consists , the rms, or nd-order citation find the s-kernel only be by the to be 1) maintain r V∗ are ions of eved by elegant s from probing d multi-current kernels nearized ined as ndent. otion is esult of to this the flap ocity of eld, and he flow this, an btained, city, the on. nd-flow REL 5.0 he edge manner kg and rease in
effective stiffness due to the rotation of the blade has not been accounted for. The natural frequencies corresponding to the motions are 2.42 rad/s and 3.42 rad/s. For structural damping a damping ratio of 0.01 is adopted.
The geometry of the blade is defined by its radius 63 m and the chord width, for which a radius independent average value of 3.0 m is adopted. The blade angle is set at 0o, the rated tip
speed is 80 m/s, and the mean wind velocity is 15 m/s. In order to define the generalized forcing, modal shapes proportional to a simplified expression for cantilever beams, 1
, are applied.
In order to derive first- and second-order Volterra kernels from the system described by equation (1-8), a number of assumptions have been adopted:
- The magnitude of the constant contribution to the relative flow w is larger than that of the fluctuating parts, i.e v v u . Moreover, the vector that follows from the multiplication w|w| is replaced by a vector with the two equal entries .
- The blade remains unstalled, implying that the contribution of the drag force f can be neglected and the relation between the lift coefficient and the angle of attack is linear.
- The local blade twist angle is constant in time. - Trigoniometric operations of the flow angle can be
approximated by the first term of the Taylor expansion around 0.
- The time-dependent contributions to the y component of the relative flow w - and – do not affect the fluctuation of the flow angle .
In comparison with the linearized approach, defined by the equation (13-15), the mean wind velocity, pitch angle and radius dependent tangential velocity have been taken into account explicitly. Moreover, coupling between flap and edge wise motion has been adopted and the quadratic flow-structure interaction has been accounted for by incorporating second-order Volterra kernels.
Figure 3 presents the linear direct and cross frequency- response functions of the nonlinear blade model, when neglecting the structural feedback u, see (6). Large peaks at the natural frequencies can be observed. The contribution of the cross-terms is significant; the response Uy is mostly affected by fluctuations of Vx. To verify the results, the transfer function of the linearized system is plotted too. Due to the fact that the pitch angle is set at 0o, this curve precisely follows the response
Ux to fluctuations of Vy.
By taking system feedback into account, the picture presented by Figure 3 gets disturbed, as can be seen in Figure 4. First of all, the height of the peaks has significantly decreased, which can be seen as the result of the added damping. Moreover, system response in ux direction affects the uy response, and vice versa.
Figure 3: First-order Volterra kernels, or frequency response functions, of the blade model without system feedback. Note the linearized system response, which equals the ux response to vy excitation.
Figure 4: First-order Volterra kernels, or frequency-response functions, of the blade model with system feedback.
In order to provide insight in the contribution of the aerodynamic damping, Table 1 presents the values of the transfer functions – with and without structural feedback – at the natural frequencies and the ratios of the magnitude of the transfer functions. The effect of the structural feedback is most significant for the flap wise response. The flap wise reduction ratios are in line with the ratio obtained with the linearized model. For the given configuration, the aerodynamic damping affects the structural response in y direction much less.
No feedback [s] Feedback [s] Ratio [-] Ux response to Vx excitation 35.8 0.651 55.0 Ux response to Vy excitation 20.4 0.365 55.9 Uy response to Vx excitation 21.7 0.977 22.2 Uy response to Vy excitation 7.22 0.476 15.2
Linearized system response 20.4 0.405 50.4 Table 1: Transfer function values at natural frequencies, with and without structural feedback.
0.01 0.1 1 10 100 0 1 2 3 4 5 6 7 8 9 10 H [s] [rad/s] First order Volterra kernels for a system without feedback H1xu H1xv H1yu H1yv H1 uxresponse to vxexcitation uxresponse to vyexcitation uyresponse to vxexcitation uyresponse to vyexcitation linearized system response 0.01 0.1 1 10 100 0 1 2 3 4 5 6 7 8 9 10 H [s] [rad/s] First order Volterra kernels for a system with feedback H1xu H1xv H1yu H1yv H1 uxresponse to vxexcitation uxresponse to vyexcitation uyresponse to vxexcitation uyresponse to vyexcitation linearized system response
Until now, only first-order responses have been considered. To analyze also the second-order response, which accounts for the nonlinearity of the system excitation, a specific forcing needs to be defined, since the second order Volterra kernels do not provide input independent transfer functions in the frequency domain, as was obtained for the first order Volterra kernels.
As system excitation, a harmonic fluctuation corresponding to a 1P frequency (0.84 rad/s) of the turbine blade is adopted. This excitation is thought to be active both in- and out-of-plane of the rotor, with equal amplitude in both directions. The structural response for the system with and without structural feedback is depicted in Figure 5. The first-order Volterra kernels provide system response at the excitation frequency. As expected, the second-order kernel gives a response that contains higher harmonics, the response frequency of which exactly doubles the excitation frequency. The effect of the added damping can be deduced from Figure 5 too. For each line, the higher data point indicates the structural response without the added damping. The lower data point shows the response including the beneficial effect of structural feedback.
Figure 5: First and second-order structural response due to harmonic in- and out-of-plane system excitation, corresponding to the rotation frequency of the blade.
To draw sound conclusions with respect to the aerodynamic damping, the frequency response to a representative wind power spectrum should be analyzed, since this provides real insight in the actual effect on the nonlinearity of the excitation to the response at different frequencies. The derivation of an excitation spectrum, containing additional peaks due to the harmonics of the rotating turbine will be the next step in this analysis.
CONCLUSIONS
The analysis performed on the basis of the discrete blade model show the significant impact of structural feedback on the response to fluctuations in the flow field. Moreover, the importance of coupling of the flap and edge wise motions has been observed, since the cross contributions to the structural
response cannot be neglected. Studying the effect of nonlinear excitation by means of second-order Volterra kernels has shown some higher-harmonic response. In order to get a clear picture of how the nonlinear forcing affects the structural motion, the response to a realistic excitation spectrum for a rotating turbine needs to be analyzed.
ACKNOWLEDGEMENTS
This work has been financially supported by the Far and Large Offshore Wind (FLOW) innovation program.
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201(1), p. 67-84. 0.001 0.01 0.1 1 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 U [m] [rad/s] Aerodynamic damping of first and second order structural response Series1 Series2 Series3 Series4 first order response Ux first order response Uy second order response Ux second order response Uy
