Applicability Limits of Operational Modal Analysis to Operational Wind Turbines

Transcription

1 Applicability Limits of Operational Modal Analysis to Operational Wind Trbines D. Tcherniak +, S. Chahan +, M.H. Hansen* + Brel & Kjaer Sond and Vibration Measrement A/S Skodsborgvej 37, DK-85, Naerm, Denmark * Wind Energy Division, Risø DTU National Laboratory for Sstainable Energy, Frederiksborgvej 399, DK-4, Roskilde, Denmark dtcherniak@bksv.com, schahan@bksv.com, mhha@risoe.dt.dk Nomenclatre f Freqency ω Circlar freqency t Time {F(ω)} Vector of excitation force spectra (inpt) {X(ω)} Vector of response spectra (otpt) [G FF (ω)], [G XX (ω)] Cross-spectra matrices of forces and responses respectively S (f) Ato- or cross-spectra of incident wind flctations U Mean wind speed σ Standard deviation of wind speed flctations r K Radis of point K on a blade κ(τ) Ato- or cross-correlation fnctions γ AB ( ω) Coherence of aerodynamic forces (or wind speed flctations) at points A and B Abstract Operational Modal Analysis (OMA) is one of the branches of experimental modal analysis which allows extracting modal parameters based on measring only the responses of a strctre nder ambient or operational excitation which is not needed to be measred. This makes OMA extremely attractive to modal analysis of big strctres sch as wind trbines where providing measred excitation force is an extremely difficlt task. One of the main OMA assmption concerning the excitation is that it is distribted randomly both temporally and spatially. Obviosly, closer the real excitation is to the assmed one, better modal parameter estimation one can expect. Traditionally, wind excitation is considered as a perfect excitation obeying the OMA assmptions. However, the present stdy shows that the aeroelastic phenomena de to rotor rotation dramatically changes the character of aerodynamic excitation and sets limitations on the applicability of OMA to operational wind trbines. The main prpose of the stdy is to warn the experimentalists abot these limitations and discss possible ways of dealing with them. Introdction The ability of Operational Modal Analysis to extract modal parameters of a strctre nder operation and withot measring excitation forces makes the techniqe extremely attractive to the application on wind trbines. First of all, a wind trbine is a hge strctre where providing measred excitation is a challenging task. Secondly, since

2 OMA can be applied to a strctre nder operation, it promises to provide experimental vales for operational natral freqencies and operational aerodynamic damping which differ from the ones at standstill and are otherwise difficlt to obtain. At the same time a wind trbine is natrally sbjected to aerodynamic forces de to wind trblence, which have a broadband spectrm and properly excite wind trbine modes. It s not incorrect to state that wind trbines have played a key role in developing OMA to a level where it crrently stands. Thogh tilization of otpt only measrements for system identification goes back to 97s, it wasn t till early 99s that the science of OMA developed to an extent where it started getting applied to hge civil strctres. This was reslt of research activities carried ot in early 99s at Wind Energy Research Organization, Sandia National Labs, USA; where by, while trying to nderstand the dynamics of wind trbines, James et al. proposed the NExT framework [], that laid down the fondations of Operational Modal Analysis as it is nderstood now. As a part of this research, NExT was applied to vertical and horizontal axis wind trbines [, ], with varying degree of sccess. It was following this research that immense potential of otpt-only system identification techniqes was realized, and sbseqently applied to variety of civil strctres inclding bildings, bridges, stadims etc. Having played sch an important role in shaping OMA techniqes, it s srprising to note that there wasn t mch follow p to this pioneering work in nderstanding wind trbine dynamics by tilizing OMA techniqes. It trns ot that application of OMA to wind trbines is not a straight forward task de to a nmber of reasons, inclding presence of considerable aeroelastic effects, presence of rotational components in the excitation forces and timevarying natre of the wind trbine strctre. These effects present considerable challenge for application of OMA techniqes to wind trbines as they psh the limits of these techniqes by stretching the very basic assmptions that form the core of OMA. A closer look onto OMA application to operational wind trbines reveals a nmber of inherent problems. First of all, the assmption of strctre time invariance is violated. This assmption is distinctive to any kind of experimental modal analysis methods, and states that the object nder test mst not change dring the test dration (or at least these changes shold not be significant). In case of operational wind trbine this is not tre: a wind trbine strctre consists of sbstrctres which move with respect to each other while the wind trbine operates: the nacelle rotates abot the tower (which is characterized by the yaw angle); the rotor rotates abot its axis; the blades pitch may change depending on wind trbine type and operating conditions. Different methods can be applied to deal with time varying natre of the strctre: For yaw and pitch, for example, periods of time when these angles do not change or change insignificantly can be selected for the analysis; a simple coordinate transformation can be performed in order to accont for yaw angle; averaged characteristic vale of pitch angle can be sed while accepting that the obtained modal parameters are smeared de to blades pitching dring the test dration. It is more difficlt to deal with rotor rotation. Inclding rotor rotation into the eqations of motion of entire wind trbine cases the mass, stiffness and gyroscopic matrices to be dependent on time. Formlating and solving the corresponding eigenvale problem yields to time-dependent eigenvales and eigenvectors, which do not have a meaning as modal freqencies, damping and mode shapes in traditional sense. Fortnately, by applying Coleman transformation [4-6], one can get rid of time dependencies in the eqation of motion and obtain meaningfl modal parameters. The combination of Coleman transformation and Operational Modal Analysis is demonstrated in [7]. Another inherent problem of application of OMA to operational wind trbines is the violation of OMA assmption concerning the operational excitation forces. OMA sets three qite specific reqirements to the excitation: the forces shold have broadband freqency spectra; they have to be distribted over entire strctre, and they have to be ncorrelated. Seemingly, the excitation de to wind trblence is ideal for OMA. Indeed, the forces de to wind trblence have almost niform broadband spectra; they are ncorrelated and obviosly excite entire strctre. However, the effect of rotor rotation dramatically changes the natre of aerodynamic forces. First of all, the shape of the spectra transforms from being flat to a crve with prominent peaks at rotation freqency and its harmonics; the peaks shape is not sharp bt characterized by thick tails [8, 9]. Secondly, the forces appear to be qite correlated arond the rotation freqency and its harmonics. The abovementioned reasons make the aerodynamic excitation mch less sitable for OMA. The presented paper addresses this phenomenon. The paper is bilt as follows: Section briefly reminds the reader the main ideas and assmptions OMA is based on. Section ses the analytical model of wind excitation based on von Karman spectrm, and derives the spectra of wind excitation forces and coherence between them; Section 3 demonstrates the same sing the reslts of nmerical simlation of wind/blades interaction of wind trbine nder operation. In Section 4, the reslts are discssed and some conclsions are drawn.

3 . Operational Modal Analysis: theory and main assmptions OMA is a system identification techniqe based only on measred otpt responses. It does not reqire inpt force information bt makes certain assmptions abot the natre of the inpt excitation forces. Like any other techniqe, adherence to basic assmptions is the key to sccessfl application of OMA techniqes. Importance of these assmptions can be gaged by the fact that modal parameters obtained sing OMA are obviosly affected depending on how closely the actal conditions resemble the one spported by these basic assmptions. These assmptions are listed below:. Power spectra of the inpt forces are assmed to be broadband and smooth;. The inpt forces are assmed to be ncorrelated; 3. The forces are distribted over entire strctre In other words, the excitation is assmed to be randomly distribted both temporally and spatially. Expression () gives the mathematical relationship between the vector of measred responses, {X(ω)} and vector of inpt forces, {F(ω)} in terms of the freqency response fnction (FRF) matrix [H(ω)] [3]: Now mltiplying () and () or with averaging, { ( ω) } [ H ( ω) ]{ F( ω) } X = ; () { X ( )} H { F( ω) } H [ H ( ω) ] H ω =. () { X ( )} { X ( ω) } H [ H ( ω) ]{ F( ω) }{ F( ω) } H [ H ( ω) ] H ω =, (3) [ G ( )] [ H ( ω) ][ G ( ω) ][ H ( ω) ] H XX ω =, (4) where [G XX (ω)] is the matrix of otpt power spectra and [G FF (ω)] is the inpt force power spectra matrix. From the first two assmptions, it follows that FF [ ( )] [ I ] G FF ω, (5) therefore the otpt power spectra [G XX (ω)] is proportional to the prodct [H(ω)][H(ω)] H and the order of otpt power spectrm is twice that of the freqency response fnctions. Ths [G XX (ω)] can be expressed in terms of freqency response fnctions as [ G ( ω )] [ H ( ω) ][ I ][ H ( ω) ] H Partial fraction form of G XX for particlar locations p and q is given as G pq ( ) XX. (6) N R pqk R pqk S pqk S pqk + + k= jω λk jω λk jω λk ω =, (7) + ( λk ) jω ( ) where R pqk and S pqk are k th mathematical reside terms and are not to be confsed with reside terms obtained sing FRF based model as these term do not contain modal scaling information (since inpt force is not measred). It is important to note that this expression shows that power spectra contains all information needed to define the modal model of the system (except for modal scaling factor), provided the loading assmptions are tre.. Freqency analysis of aerodynamic forces In wind trbine engineering [8], it is typical to separate the loading de to the steady wind and loading de to wind speed flctation (i.e. trblence). The first one is often called deterministic load component while the second is stochastic. The first component is characterized by the mean wind speed, which is considered to be time-constant

4 in the time scale over abot mintes. In this stdy, we consider it not possessing any broadband freqency content. The second component has broadband freqency content, and therefore has a key importance for Operational Modal Analysis de to its ability to provoke modal behavior of the wind trbine strctre. The presented paper concerns only the stochastic loading. The sorce of the stochastic loading is trblence, a flctation in wind speed on a relatively fast time-scale typically less than abot mintes [8]. In other words, we assme the flctations of wind speed have a zero mean when averaging over mintes. There are two main sorces of trblence: flow distrbance de to topographic featres and thermal effects casing air masses to move vertically. These are obviosly complex processes; therefore the description of trblence is typically developed sing its statistical properties. Trblence freqency spectrm is an important property of trblence. There are several trblence models sed in wind trbine engineering; in this stdy we will refer to two of them; the first one is described by Kaimal spectrm (sed in Section 3), the second one by von Karman spectrm. In this section we se the formlation sggested by von Karman []: f S ( f ) 4L = σ f L U + 7 U x x 5 / 6, (8) where f is freqency, S (f) is the atospectral density of the trblence (its longitdinal component which is denoted by symbol ), σ is the standard deviation of wind speed variations abot mean wind speed in longitdinal x direction U. Standard deviation σ and mean speed U are linked via trblence intensity I = σ / U. L is a trblence length scale. Different national standards provide empiric expressions defining the vales for the trblence intensity and length scale; these vales mainly depend on the elevation above the grond and the srface roghness. The ble line on Figre shows a typical spectrm calclated according to () with L = 64.4 m (corresponding to the elevation greater then 3 m above the grond) and U = 8 m/s. x As it can be seen from the plot, most of the energy is distribted on the very lower freqencies, mch below typical rotation freqencies (here set eqal to.hz and denoted by p on the freqency axis). On the higher freqencies the trblent energy tend to dissipate as a heat, and the spectral density asymptotically approaches the limit f -5/3 Hz. As it was mentioned in Section, Operational Modal Analysis sets a nmber of reqirements on strctre excitation. In the case of wind trbine, the main excitation is de to wind-strctre interaction; therefore it is important to nderstand the relation between the trblence and the reslting aerodynamic forces. As it is stated in [8], it is sal to assme a linear relation between the flctation of the wind speed incident on the aerofoil and the reslting loading flctations L. This assmption is correct for low wind speeds and becomes inaccrate for pitch reglated trbines as wind speed approaches ct-ot vale and breaks down for stallreglated machines when wind speed can case stall. In this section we will assme low wind speeds and therefore the linear relation between the wind speed flctation and the reslting loading, so the reslts will be given for wind speed flctation, implying the forces behave similarly. Spectral density S(f) /σ - von Karman Kaimal (p) Freqency, Hz Figre : Normalized spectra: according to von Karman (ble) and Kaimal (green) Following [8] and assming that the flid is incompressible, the trblence is homogeneos and isotropic and sing Taylor s hypothesis of frozen trblence, it is possible to obtain an analytical expressions for cross-correlation fnction of wind speed compted for points A and B on the blades (Figre ). In brief, the derivation is based on the se of Wiener-Khinchin theorem on power spectrm (8) which yields to analytical expression for the atocorrelation fnction at any fixed point in space; and then compting the cross-correlation fnction between points A t and B t+τ (Figre ). Using the hypothesis of frozen trblence, instantaneos

5 wind flctation at B t+τ is assmed to be eqal to the wind flctation at point B t. The derived expression for correlation along vector s at t = approximates the cross-correlation fnction between points A t and B t+τ : κ s r o,) κ ( r, r, ). ( = τ o κ ( r, r, τ ) = σ = Γ s / / 3 K s s / r + r r r cos( Ωτ + k π / 3) K ( ) x / 3 x x / 3 x 3.34L.34L.34L s.34l s (9) where interpretations for s, r, r can be fond on Figre a,b; K v is Bessel fnction of second kind; and k = if points A and B are on the same blade, k = - if blade with point A is moving before the blade with B and k = for the opposite sitation. As it can be seen from the triangle on Figre c, ( s = Uτ ) + r + r r r cos( Ωτ + k π / 3). () Figre 3a shows atocorrelation fnction compted for the fixed point on the blade (i.e. points A and B coincide, r = r and k = ) for different radii. The peaks at mltiples of the period of rotor revoltion T are de to high correlation of the incident wind speed with itself when a blade passes the same region of the rotor swept area. Spectra of aerodynamic forces Application of the Wiener-Khinchin theorem to expression (9) yields the power spectrm density (PSD) of the incident wind speed flctations; there is no implicit expression for the reslting integral, instead it can be approximated and estimated nmerically. The PSDs of the corresponding atocorrelation fnctions are shown on Figre 3b. a) b) c) Figre. Points A and B are located at radii r and r respectively. A t, B t are the positions of the corresponding points at time t. a) Geometry of the wind trbine; b) View along the wind speed; c) Position triangle sed for cross-spectrm calclation.

6 a) b) Norm. atocorrelation, κ o (r, r, τ )/σ Power density spectra, S o (r, r, f )/σ.6. T T 3T 4T 5T τ, Rotor revoltions - r = r = 6.4m r = 54.4m r = r = 6.4m r = 54.4m -... p p 3p 4p Freqency f, Hz Figre 3. a) Normalized atocorrelation fnctions compted for the fixed point on the blade (r = r and k = ), on the different radii (, 6.4 and 54.4 m); b) Normalized PSD calclated for the atocorrelation fnctions shown above. The following observations can be made:. The spectra have peaks at the fndamental freqency and its harmonics;. The peaks become more prononced with increasing of radis r (with no peaks at r = ); 3. The peaks are not sharp bt have rather thick tails. The physics of these phenomena can be explained by imagining a blade passing regions with higher or lower wind speed that generates periodicity of the incident wind speed and hence of the aerodynamic loading; increasing the ratio between the tangential speed of the point on a blade and the mean wind speed Ω r / U will generate higher peaks and deeper troghs between the peaks. The center of the rotor experiences only mean wind therefore the spectrm there coincides with the spectrm of the trblence (Figre, the ble crve). The third listed featre of the PSD is the peak s thick tails. It is difficlt to get a feeling abot the natre of this phenomenon. The following explanation can be sggested: one can roghly approximate the atocorrelation fnction at r (Figre 3a) by a prodct of a decaying fnction like the ble crve (the atocorrelation at r = ) and some periodic fnction. The latter consists of the infinite nmber of terms (de to the Bessel fnctions in (9)) which are periodic (de to periodicity of s, ()) and have different periods (de to the different powers of the cosine fnction). In freqency domain, a prodct of two atocorrelation fnctions becomes a convoltion of their spectra []. One has to imagine the spectrm of the first fnction (the ble crve on Figre 3b) convolved with the spectrm of the second one, which will have infinite nmber of sharp peaks at different freqencies. The reslt of the convoltion will be smearing of the peaks into the smooth spectra shaped as thick tails. As it was mentioned above, the flctations of the aerodynamic loading is linearly proportional to the trblence, meaning that the PSD of the aerodynamic forces will have the same shape as crves shown on Figre 3b. Ths, we can see that the excitation spectra are not flat in the freqency range of interest, and this therefore violates the first OMA assmption.

7 Correlation between aerodynamic forces The second important OMA assmption is that the forces acting at different points of a strctre mst be ncorrelated (see Section ). In this sbsection we will estimate the correlation of the excitation forces by calclating the coherence between the wind speed flctations at different points on the same and different blades. Coherence γ AB between incident wind speed flctation at points A and B on the rotating blades is AA BB S ( ) f γ AB ( f ) =, () AA BB S ( f ) S ( f ) where S and S are atospectra of wind trblence at points A and B respectively and S is cross-spectra between these points. The ato- and cross-spectra can be calclated from ato- and cross-correlation fnctions (9) applying Wiener-Khinchin theorem. Figre 4 shows the ato- and cross-correlation fnctions calclated at different points on different blades, corresponding ato- and cross-spectra, and the coherence. The correlation fnctions are presented on Figre 4a. Cross-correlation fnctions compted for the points on different blades pass ahead the atocorrelation fnction: indeed, if for example, blade # experiences a gst at some segment of the rotor swept area, it will take it a fll revoltion (and time T) to experience the same gst again. However, for blade # which follows blade #, it will take only T/3 to experience the same gst, and for blade #3 it will happen after T/3. This explains the corresponding time lags of the black and magenta crves on Figre 4a. The ato- and cross-spectra are obtained nmerically for the corresponding ato- and cross-correlation fnctions; their magnitdes are shown on Figre 4b. The cross-spectra for the cases when blade A follows blade B and B follows A are the same. Coherence fnctions between the wind trblence at points A and B if the points are located on different blades, are presented on Figre 4c. Similar to the spectra, the coherences have peaks at rotation freqency and its harmonics; the peaks are more prononced for low harmonics and gradally decay at higher freqencies. Frther, the peaks are not sharp bt have rather thick tails; this makes the troghs between them qite narrow. Ths, here one can claim the violation of the second OMA assmptions concerning the correlation between the excitation forces. As one can see, the assmption is strongly violated in the freqency range between.5p - 4.5p, leaving only narrow deeps between the peaks where the assmption is valid. AB AB 3. Analysis of the synthesized time data As it was mentioned in Section, the presented analytical approach is only sitable for weak wind where the linear relation between aerodynamic force flctations and wind trblence is valid. For higher wind speeds the analysis is sally performed in time domain by nmerical simlation of wind-blade interaction. Similar to [], the simlations were performed sing the nonlinear aeroelastic mlti-body code HAWC [3]. A trblence box with resoltion 3 x 3 x 6384 was generated based on Kaimal trblence spectrm with mean wind speed 8 m/s and trblence intensity 5%. Figre (green line) shows the corresponding spectrm calclated for this speed and L = 5 m. The trblence box corresponded to s of simlation. Aerodynamic forces at radii 6.4, and 54.4 m in X- and Y- directions were calclated as fnctions of time; the analysis was done on the generated time histories, for X (along the wind) direction only. Note that the nmerical approach sggested in [8] is sitable here since it assmes the symmetry of the correlation fnctions abot t =. This is correct for ato-correlation fnctions and cross-correlations between the points on the same blade. The cross-correlation fnctions between the points on different blades (i.e. for k ) are not symmetric, see Figre 4a.

8 a) b) c) Norm. correlations, κ o (ra, r B, τ )/σ Power density spectra, S o (ra, r B, f )/σ.6. Atocorrelation at r = 6.4m Atocorrelation at r = 54.4m Cross-correlation (r A = 6.4m and r B = 54.4m, same blade) Cross-correlation (r A = 6.4m and r B = 54.4m, blade A follows B) Cross-correlation (r A = 6.4m and r B = 54.4m, blade B follows A) T T 3T 4T 5T τ, Rotor revoltions p p 3p 4p Freqency f, Hz γ (r = 6.4m, r = 54.4m, same blade) AB A B Coherence γ AB.6 γ (r = 6.4m, r = 54.4m, diff. blades) AB A B γ (r = 6.4m, r = 6.4m, diff. blades) AB A B γ (r = 54.4m, r = 54.4m, diff. blades) AB A B. p p 3p 4p 6p 7p 8p 9p p.5 Freqency f, Hz Figre 4. a) Ato- and cross-correlation fnctions; b) ato- and cross-spectra; c) coherence. Figre 5 shows the atospectra of the aerodynamic forces acting in X-direction, and Figre 6 shows the coherence between aerodynamic forces simlated at different points on the blades. Obviosly, the reslts obtained analytically and nmerically are not directly comparable. First of all, for analytical case the spectra and coherence of wind flctations are shown where for the nmerical case the reslting aerodynamic forces are presented. Secondly, the wind speed is different (modest 8 m/s for analytical case vs. strong 8 m/s wind for the nmerical case). Thirdly, two different wind trblence spectra are sed: von Karman for the analytical case and Kaimal for the nmerical simlations. Despite this, qalitatively, the shape of the spectra and coherence fnctions look very similar. In both case they are characterized by peaks at rotation freqency and its harmonics, with their heights decreasing with freqency increase. At low freqencies (p-4p) the peaks clearly exhibit the thick tails. The shape of the coherence fnctions compted for the two cases is more different: in the nmerical case the sharp peaks present even on high harmonics. Otherwise, there are similarities, too. For example, the coherence between the points on the same radis on the different blades (Figre 6c,d) is generally higher compare to the one between the points at different radii (Figre 6a,b).

9 Aerodynamic force PSD, F x - -4 Atospectrm at r = 6.4 m Atospectrm at r = 54.4 m p p 3p 4p Freqency f, Hz Figre 5. Power spectral density of the simlated aerodynamic forces in X direction. 4. Discssion and Conclsion In the stdy it is demonstrated both analytically and nmerically, for weak and strong wind excitation that for operational wind trbine: ) The spectra of aerodynamic forces are not flat bt are characterized by peaks at rotational freqency and few lower harmonics. The peaks are not sharp bt have thick tails. ) The forces acting at different points on the blades are highly correlated on the rotational freqency and its lower harmonics. This means that two important OMA assmptions are not satisfied in a nmber of freqency regions arond the rotational freqency and its lower harmonics. Ths one mst not expect OMA to provide correct reslts in these freqency regions. Some important modes of modern wind trbines (e.g. drivetrain mode, first tower modes, first flapwise modes) are located in these regions for the nominal rotational speeds [4], so one has to be carefl trying to identify them sing OMA. a) b) c) p p 3p 4p 6p 7p 8p 9p p.5 d) p p 3p 4p 6p 7p 8p 9p p p p 3p 4p 6p 7p 8p 9p p.5 p p 3p 4p 6p 7p 8p 9p p.5 Figre 6. Coherence of the simlated aerodynamic forces. Same color scheme as on Figre 4c.

10 It is important to note that application of tone removing methods (e.g. based on synchronos averaging, [5]) cannot be considered as a proper soltion, as these methods work well only for sharp peaks bt will not help in this case de to the thick tails phenomenon. From a first glance, a se of rn-p and rn-down events looks attractive bt, first of all, these events are rather short compare to the acqisition time reqired for data collection for proper OMA (abot mintes if the lowest freqency of interest is.- Hz). Secondly, a wind trbine engineer is typically interested in the dependency of modal parameter on the rotor speed; in the case of rn-p/rn-down events, only averaged modal characteristics can be obtained. In order to solve the problem, one can consider a carefl planning of the experiment, constrcting the test matrix in a way to avoid the modal freqencies (which are typically known from FEA) to be in the vicinity of rotor speed and its lowest harmonics. This means that only few modes can be estimated with higher degree of confidentiality for a given rotor speed, while another rotor speed will be sitable for another set of modes. Amongst recently sggested methods, operational modal analysis based on transmissibility fnctions [6] appears very appealing. One of the main advantages of this method is its insensitivity to colored excitation spectra. However, so far this method is still nder development and is not ready for indstrial applications. Literatre [] G.H. James, T.G. Carne, J.P. Laffer (995), The Natral Excitation Techniqe (NExT) for Modal Parameter Extraction from Operating Strctres, Modal analysis: The International Jornal of Analytical and Experimental Modal Analysis, [] G.H. James (994), Extraction of Modal Parameters from an Operating HAWT sing the Natral Excitation Techniqe (NExT), Proceedings of the 3 th ASME Wind Energy Symposim, New Orleans, LA. [3] J. Bendat, A. Piersol (986), Random Data: Analysis and Measrement Procedres, nd edition, Wiley, New York. [4] M.H. Hansen (3), Improved Modal Dynamics of Wind Trbines to Avoid Stall-indced Vibrations, Wind Energ. 3; 6: [5] M.H. Hansen (6), Two Methods for Estimating Aeroelastic Damping of Operational Wind Trbine Modes from Experiment, Wind Energ. 6; 9: 79-9 [6] G. Bir (8), Mltiblade Coordinate Transformation and its Application to Wind Trbine Analysis, Proceedings of 8 ASME Wind Energy Symposim, Reno, Nevada, USA, Jan. 7-, 8 [7] D. Tcherniak, S. Chahan, M. Rosseti, I. Font, J. Basrko, O. Salgado (), Otpt-only Modal Analysis on Operating Wind Trbines: Application to Simlated Data, Sbmitted to Eropean Wind Energy Congress, Warsaw Apr. [8] T.Brton, D. Sharpe, N. Jenkins, E.Bossanyi (), Wind Energy Handbook, John Wiley & Sons, West Sssex, England [9] L. Kristensen, S. Frandsen (98), Model for Power Spectra of the Blade of a Wind Trbine Measred from the Moving Frame of Reference, Jornal of Wind Engineering and Indstrial Aerodynamics, (98) 49-6 [] T. von Karman (948), Progress in the statistical theory of trblence, Proc. Natl. Acad. Sci. (U.S.), 34 (948) [] R.B. Randall (987), Freqency Analysis, 3 rd edition, Brel and Kjaer [] S. Chahan, M. H. Hansen, D. Tcherniak (9), Application of Operational Modal Analysis and Blind Sorce Separation /Independent Component Analysis Techniqes to Wind Trbines, Proceedings of XXVII International Modal Analysis Conference, Orlando (FL), USA, Feb. 9 [3] T.J. Larsen, H.A. Madsen, A.M. Hansen, K. Thomsen (5), Investigations of stability effects of an offshore wind trbine sing the new aeroelastic code HAWC, Proceedings of Copenhagen Offshore Wind 5, Copenhagen, Denmark, p.p.5 8. [4] M.H. Hansen (7), Aeroelastic Instability Problems for Wind Trbines, Wind Energ. 7; :

11 [5] B. Peeters, B. Cornelis, K. Janssens, H. Van der Aweraer (7), Removing distrbing harmonics in operational modal analysis, Proceedings of International Operational Modal Analysis Conference, Copenhagen, Denmark, 7 [6] C. Devriendt, T. De Troyer, G. De Sitter, P. Gillame (8), Atomated operational modal analysis sing transmissibility fnctions, Proceedings of International Seminar on Modal Analysis, Leven, Belgim Sep. 8