Identification of structural non-linearities due to large deflections on a 5MW wind turbine blade

Identification of structural non-linearities due to large deflections on a 5MW wind turbine blade V. A. Riziotis and S. G. Voutsinas National Technical University of Athens 9 Heroon Polytechniou str., 578, Athens, Greece vasilis@fluid.mech.ntua.gr, spyros@fluid.mech.ntua.gr E. S. Politis and P. K. Chaviaropoulos Centre for Renewable Energy Sources 9 th km Marathonos Av., 99, Pikermi, Greece vpolitis@cres.gr, tchaviar@cres.gr A.M. Hansen, H.A. Madsen, F. Rasmussen Risø National Laboratory DTU DK-4, Roskilde, Denmark anders.melchior.hansen@risoe.dk helge.aagaard.madsen@risoe.dk flemming.rasmussen@risoe.dk Abstract The paper presents the main results from the work carried out within work package (Task.) of the European Project UPWIND, aiming at identifying important nonlinear structural effects that are relevant to large, flexible multi MW turbines. In the analyses presented focus is given to non-linear effects owed to large deflections. They are analysed on the basis of two different types of non-linear aeroelastic models. The first is a model (GAST) while the second is a multi-body model (HAWC) based on Timoshenko beam elements. Assessment of the important nonlinear couplings is performed by means of nonlinear aero-elastic simulations on the 5MW Reference wind turbine (a paper case machine defined within IEA Wind Annex XXIII activities) which are compared to those of a conventional model. Through these comparisons the effect of various nonlinear higher order structural couplings is quantified. It is also identified that certain nonlinearities - especially those related with the coupling of the blade torsion with the blade bending - are of great importance and should be taken into account in the modeling of large flexible blades. Keywords: non-linear aeroelasticity, large deflections, multi-body dynamics Overview The overall objective of the work carried out under Work Package of the UPWIND project is to develop a basis for aerodynamic and aeroelastic design of large flexible multi MW wind turbines. An important step towards this goal is to identify the most significant nonlinear structural effects that should be taken into account in new generation aero-elastic tools; then to estimate their relevance to fatigue loads and stability. To this end, first, existing state of the art aero-elastic tools must be upgraded in order to include nonlinear, higher order terms due to large deformations and rotations. At a second level, improvement of the structural modelling capabilities is sought in the use of advanced structural models that include the detailed inner structure (e.g. complex laminates or sandwich skins and webs) and account for the different structural couplings originating from the in-homogenous and anisotropic character of the real material. In analysing large deflection problems two modelling approaches are considered. The first consists in formulating the beam equations with reference to the deformed blade state. Then, through proper scaling, higher-order terms up to second order are retained. The formulation follows closely the development of the nd order theory derived by Hodges []. One advantage this

formulation has is that it allows identifying the different terms and understanding their origin. Otherwise the particular implementation used in the present work has been incorporated into GAST, a multi-body aero-elastic tool for the modelling of complete wind turbine configurations []. In the second approach the multi-body formulation is extended at the component level. The major components of the wind turbine are divided in a number of bodies each considered as an assembly of Timoshenko beam elements. Each body has its own coordinate system which follows the body in its motion. Large rotations and translations of the body are accounted for when the internal inertial loads are calculated. With respect to the body system, deflections and rotations are assumed small. This means that if for example the blade is modeled by a number of interconnected bodies, at their connections large deflections and rotations will be gradually built by imposing to each body as rigid motions the deflections and rotations of the preceding body. The above implementation has been introduced in HAWC [3] which is also a full aero-elastic code. Both models use the blade element momentum theory for calculating the aerodynamic loads while they are also equipped with dynamic inflow and dynamic stall sub models. Although there are differences in this respect, previous cross comparisons have indicated that they will not affect the final predictions at an appreciable way [4], [5]. In the present work, assessment of important nonlinear effects is carried out by means of nonlinear aero-elastic simulations. Results from higher order analysis are cross-compared to first order results (similar to those currently used by the wind industry) for an isolated rotor in uniform inflow as well as for the full wind turbine in turbulent inflow over the whole range of the wind turbine normal operation. Identification of non-linearities In the present section certain aspects of nd order beam theory are discussed mainly in support of the subsequent analysis. Reference to the deformed state of the beam, O ξηζ instead of the undeformed Ο xyz, involves a non-linear transformation matrix Eu ( ): ˆ u cos( ) sin( ˆ u θτ uw u θτ uw ) cos ˆ sin ˆ u w Ε u θ w θ ( usin ˆ θ wcos ˆ Τ+ Τ Τ+ θτ) sin ˆ w cos ˆ w θτ w θτ which depends on the elastic displacements and rotations. Primes denote differentiation with respect to y, θ is the local twist angle and t ˆ θ = ˆ θ + Τ θ with ˆ t θ being the local torsion angle about η axis: y ˆ θ = θ + u w dy () θˆ + θ t Figure : Beam co-ordinate system definition. Eu ( ) appears in the definition of the position vector, here defined without including warping: u u ξ v r = y+ v + E( u ), u = () w w ζ θ which is used in the derivation of the dynamic equations. The derivation is too comprehensive to include here (see [],[6] for details), so only the most important terms will be discussed.

Large bending deflection will modify the tensile (radial) displacement: y ( ) v = v u w dy e + (3) where v is the local radial displacement along e the deformed beam axis. This leads to tensionbending couplings. In the two bending directions the corresponding terms become important if there is an offset of the tension axis with respect to the elastic axis. Tension is also coupled with torsion giving a tensile stiffness term of the form: θ ( θ θ + ) (4) EI t t and a torsional stiffness term of the form: u w EIt ( θt + θ) v + + (5) where EI t is the summation of the two bending stiffnesses EI ξξ, EI ζζ. It originates from the fact that torsion generates axial stress proportional to θ t θ + θ / which contributes to the pitching moment. Torsion is also coupled with bending leading to torsional stiffness terms of the form: ( EI ) cos( ( ˆ ξξ EIζζ θt + θ)) u w + ˆ ( EIξξ EIζζ ) sin( ( θt θ)) ( u w ) (6) As the bending displacements of the blade increase the local bending moments M ξ and M ζ generate torsion moment about the undeformed blade axis y (see [6]). The magnitude of this pitching moment primarily depends on the bending curvatures u, w. Extra terms will appear if there is offset of the tension axis. 3 The Reference Wind Turbine (RWT) The importance of the nonlinear structural effects is identified by means of aeroelastic simulations based on the Reference Wind Turbine (RWT) defined in the UPWIND project. As RWT a 5 MW paper case machine developed by NREL that resembles the REpower 5M machine (a 5 MW prototype) is used. In building the complete aeroelastic model of this 5 MW wind turbine, NREL used existing data from the REpower 5M machine, sharing also information from the DOWEC [7], RECOFF [8] and WindPact [9] projects to fill in missing data. As the size of wind turbines increases, the requirement of reduced weights calls for more slender, lighter and therefore more flexible blades. On the other hand, structural nonlinearities are expected to become significant in situations where large deflections are encountered. The question that naturally arises is how flexible this RWT is, as compared to smaller scale existing commercial wind turbines and whether this wind turbine can form the basis for analyzing nonlinearities and structural coupling effects triggered by large deflections. In Table the natural frequencies of the RWT at standstill are presented. The first tower bending frequencies (fore-aft and lateral) are found in the vicinity of.5 P as in most commercial wind turbines. Also, the first rotor flapwise and edgewise natural frequencies are within the frequency range according to usual design practice (3-4 P for the asymmetric rotor flap frequencies, 3.5 P for the symmetric rotor flap frequency and near 5.5 P for the asymmetric rotor edgewise frequencies). So, as concerns natural frequencies the wind turbine considered is a rather typical one. However, since the weight of the blades is relatively low (7.74 kg), they are also expected to be very flexible. In Figure the mean blade flap and lag deflections, as functions of the wind speed are shown. The maximum mean flapwise deflection (at rated conditions) is found to be almost % of the rotor radius. Also in full load conditions, torsion angles of about.5 are encountered (see Figure 3). These are way too high compared to those usually encountered by commercial smaller scale blades. Mode Description Natural frequency [Hz] [P] st tower lateral.76.37 st tower longitudinal.8.4 st shaft.598.97 st rotor asymmetric flap (yaw).64 3.5 st rotor asymmetric flap (tilt).654 3.4 st rotor collective flap.699 3.46 st rotor asymmetric edge.89 5.4 st rotor asymmetric edge.5 5.48 nd rotor asymmetric flap (yaw).63 8.8 nd rotor asymmetric flap (tilt).784 8.85 nd rotor collective flap.974 9.79 Table : Natural frequencies at standstill. So, it is clearly demonstrated that the RWT can establish a basis for analyzing nonlinear effects

due to large deflections. Structural nonlinearities are expected to be predominant near rated conditions where maximum flapwise deflections are obtained or at high wind speeds where blade torsion angles become large. displacement at blade tip [m] 6 5 4 3-6 8 4 6 8 4 wind speed (m/s) Figure : Mean blade tip displacements. torsion angle at blade tip [deg].5 -.5 - -.5 - -.5 lead-lag flap 6 8 4 6 8 4 wind speed (m/s) Figure 3: Mean blade tip torsion angle. 4 Identification of nonlinear effects through aeroelastic simulations The results provided by GAST and HAWC are compared to those obtained using a conventional first order (linear) beam model. The analysis of the effect of the nonlinear terms triggered by the large deflections of the blade is performed in the context of the isolated rotor under uniform inflow conditions as well as for the full wind turbine under turbulent inflow conditions. In the latter case, a preliminary assessment of the effect on fatigue loads of the blade is also performed. 4. Isolated rotor analysis under uniform inflow conditions As starting point for the analysis the isolated rotor case is considered. In order to eliminate the uncertainties related to the spatially or time varying inflow a simple uniform inflow with no wind shear or yaw misalignment, no rotor tilt, no turbulence and no tower shadow effects is assumed. In this way the response of the blade to the rotational frequency (driven only by the periodic gravitation loading) becomes smaller and therefore the differences associated with the higher order, nonlinear terms are clarified. The rotor is operated at fixed rotational speed and pitch angle (open loop, i.e. the controller is not active) corresponding to the average operating conditions for a certain wind speed. Computations are presented for two wind speeds and 8 m/s, the first near to rated operation and the second well in the full load conditions. As indicated in section 3 the first test point is characterised by the large flapwise deflections due to high thrust while the second is selected because of the large torsion angles encountered at high wind speeds. Comparisons at the level of the blade deflections and loads are provided among: the baseline first order beam simulations, referred to as st order beam in the plots, and the two non-linear models referred to as and multi-body. In the multi-body model, each blade is divided into bodies. Results are presented, in the form of time-series (after periodicity has been reached) over four revolutions and for the following quantities: Deflections: Flapwise displacement at the blade tip (in the direction perpendicular to the blade chord at the tip). Lead-lag displacement at the blade tip (in the direction parallel to the blade chord at the tip). Angle of attack (including torsional deformation) at 75% radius. Radial displacement at the tip (with respect to the un-deformed blade axis). Loads: Flapwise bending moment (parallel to the chord at the tip) at the blade root (at r =.5 m from rotor centre). Torsion moment at the blade root (at r =.5 m from rotor centre).

5.4 multi-body ( bodies) multi-body ( bodies) flapwise displacement at blade tip [m] 5.3 5. 5. 5 4.9 lead-lag displacement at blade tip [m] -. -.4 -.6 -.8 - -. 4.8 -.4 Figure 4: Blade tip flapwise displ. at m/s. Figure 6: Blade tip lead-lag displ. at m/s. 5.5 multi-body ( bodies). angle of attack at r/r=.75 [deg] 5 4.5 4 extension at blade tip [m] -. -. -.3 multi-body ( bodies) -.4 3.5 -.5 Figure 5: Local incidence at 75% radius at m/s. Figure 7: Blade tip radial displ. at m/s. At the wind speed of m/s (see Figure 5) both non linear models predict higher amplitudes of the angle of attack variation (and therefore of the torsion angle variation). Although the increase in amplitude predicted by the multi-body model is lower than that of the model the phase characteristics of the two signals perfectly match. The increase in the amplitude of the torsion angle also affects the flapwise response of the blade (Figure 4). The amplitude of the variation of the flapwise deflection increases and the phase of signal is shifted in such a way that it almost coincides with the phase of signal of the angle of attack. Both non-linear models provide similar results which differ only in the mean level. The change in mean value originates from the reduction of the effective rotor radius resulting from high bending (see equation (3) ). There is a reduction of the effective rotor radius of.35 m (see Figure 7). This reduction gives rise to lower thrust loads on the rotor disk and consequently leads to lower flapwise deflections. The above effect is only accounted for in the multi-body model which explains the different levels in the flapwise displacements. In the lead-lag direction (see Figure 6), where the stiffness is high and therefore deflections are small, the difference between the first order and the higher order approximations is almost negligible. The lower level predicted by the multi-body model is again associated to the reduction of the effective rotor radius.

flapwise displacement at blade tip [m]..9.8.7.6 multi-body ( bodies) lead-lag displacement at blade tip [m].6.4. -. -.4 multi-body ( bodies).5 -.6.4 -.8 Figure 8: Blade tip flapwise displ. at 8 m/s. Figure : Blade tip lead-lag displ. At 8 m/s. - multi-body ( bodies).3. multi-body ( bodies) angle of attack at r/r=.75 [deg] -.5 - -.5 extension at blade tip [m]. -. -. -3 -.3 Figure 9: Local incidence at 75% radus at 8 m/s. Figure : Blade tip radial displ. At 8 m/s. At the wind speed of 8 m/s the differences between the first order and the higher order models are much smaller (see Figure 8 - Figure ). At this wind speed the flapwise deflections are considerably lower. On the contrary torsion angles are expected to be high as shown in Figure 3. The similarity of the predictions provided by the low and higher order models indicates that non linearities are mainly triggered by the large bending deflections rather than high torsion deformations. In Figure -Figure 5 the blade root flapwise bending and torsion moments are presented for the same two wind speeds. At the wind speed of m/s the amplitude of the torsion moment considerably increases due to non-linear effects (see Figure ). The increase in amplitude is consistently predicted by both non-linear models. As in the case of the angle of attack variation, the multibody model provides slightly lower amplitudes than the second order beam model. The increase in the range of variation of the effective incidence gives rise to slightly higher amplitudes of the flapwise bending moment also predicted by both non-linear models. The effect on the flapwise bending moment is indirectly produced as a result of the change of the blade local incidence and therefore of the localised blade loads. It then follows that depending on the phase difference between the blade flapwise deflection and torsional deformation, a reduction of the ranges could also be obtained. At the wind speed of 8 m/s the effect of nonlinearities is marginal. Only a slight increase in the amplitude of the

flapwise bending moment at blade root [KNm] 95 9 85 8 multi-body ( bodies) flapwise bending moment at blade root [KNm] 35 3 5 multi-body ( bodies) Figure : Blade root flapwise bending mom. at m/s. Figure 4: Blade root flapwise bending mom. at 8 m/s. multi-body ( bodies) 8 6 multi-body ( bodies) torsion moment at blade root [KNm] 8 6 4 - -4 torsion moment at blade root [KNm] 4 - -4-6 -8-6 - Figure 3: Blade root torsion mom. at m/s. torsion moment is noted while the flapwise moment remains almost unchanged. As regards the lead-lag moment (not shown in the figures) all models have provided identical predictions. In the sequel an attempt is made to provide an interpretation of the differences already noted between the st order linear model and the higher order non-linear models. This is done on the basis of the nd order beam model which as already discussed offers a ground for identifying non-linear effects by gradually switching on various non-linear terms. It is noted that the nd order beam model reduces to the standard st order beam model when higher order non-linear terms are neglected. Figure 5: Blade root torsion mom. at 8 m/s. Four variants of the beam model are examined ranging from the baseline st order beam model to the complete nd order beam model (referred to as - ). The first intermediate variant ( - ), is formulated with respect to the deformed state; but only keeps the structural couplings appearing in the baseline first order model. The second variant ( nd order beam- ) includes the tension-torsion coupling terms already discussed in section but not the bending-torsion coupling terms which are switched on in -. (see equation (6)). In Figure 6-Figure 9 the blade tip flapwise displacements and torsion angles are shown for the two wind speeds of and 8 m/s. The tip deflections predicted by the st order beam model as compared to those predicted by the - model at the wind speed

flapwise displacement at blade tip [m] 5.6 5.5 5.4 5.3 5. 5. 5 - - - sections leads to higher effective angles of attack and consequently higher loads and deflections in the flapwise direction. On the contrary, at 8 m/s wind speed (see Figure 8 and Figure 9) the differences between and - models are almost negligible. This is because at high wind speeds (well above the rated speed) the blade flapwise deflections are relatively small (about % of the blade length) and therefore second order effects almost vanish. 4.9 4.8 Figure 6: Blade tip flapwise displ. at m/s. torsion angle at blade tip [deg].5 -.5 - - - flapwise displacement at blade tip [m]..9.8.7.6.5.4 - - - Figure 8: Blade tip flapwise displ. at 8 m/s. - -. -.4 - - - Figure 7: Blade tip torsion angle at m/s. of m/s (see Figure 6 and Figure 7), are found to be similar in shape but different in level. The mean level of the torsion angle is.5 higher in the - model. As explained in Figure, when the mass centre of the section is offset with respect to the elastic axis, the increase of bending deflections will increase the centrifugal force component perpendicular to the deformed beam axis which produces extra twisting moment. If the mass centre is located aft (towards the trailing edge) of the elastic axis then this twisting moment will generate a pitch up (corresponding to higher incidence) rotation of the section. The opposite occurs when the mass centre is fore (towards the leading edge) of the beam axis. An increase in the torsion angle of the torsion angle at blade tip [deg] -.6 -.8 - -. -.4 -.6 -.8-3 Figure 9: Blade tip torsion angle at 8 m/s.

Figure : Centrifugal force effect on blade pitch due to large bending Figure : Bending-torsion coupling effect due to high bending wind speed [m/s] 3 9 8 7 6 5 6 8 3 3 34 36 38 4 torsion angle at the tip [deg].5 -.5-6 8 3 3 34 36 38 4 Figure : Time-series of wind speed and torsion angle at blade tip (wind speed 8 m/s) The introduction of the tension torsion coupling terms (model variant - ) produces an effect on the torsional deformations that is counter to the effect of the centrifugal force for the RWT blade. By introducing this coupling effect, the tip torsion angles are reduced and shifted (and also the flapwise deflections because of the lower effective incidence) towards the predictions of the model. It is also found that the effect of the tension torsion coupling is nearly independent of the wind speed. Hence, both at and 8 m/s a reduction of appr.. is obtained. The nonlinear effect with the highest significance is that of the bending torsion coupling. As a result of the introduction of the relevant terms ( - ), the amplitudes of the torsional deformations at m/s become significantly higher (see Figure 7) while the mean value of the torsion angles slightly drops as compared to - model. As mentioned in section the torsion moment produced by the internal bending loads, acting on the deformed blade section, depends on the blade bending curvatures (second spatial derivative with respect to the y axis). The first

term in equation (6) is responsible for the increase in the amplitude of the torsion angles variation while the second dominates the mean value of the torsion angle. Between the two, the first one is more significant when the local twist and torsion angles are relatively small. Since the variation of the torsion angles is originally not in phase with the variation of the flapwise displacements (compare Figure 6 and Figure 7) the increase in the amplitude of the first, affects the phase characteristics of the latter. Besides the change in the phase characteristics of the flap response, also the amplitudes increase. At 8 m/s, where the bending deflections are small and therefore the bending curvatures are expected to be also small, the effect of the bending-torsion coupling is almost negligible. Both the torsion angles and the flapwise deflections are almost identical as obtained with - and - (see Figure 8 and Figure 9). 4. Full Wind Turbine Analysis with Stochastic Inflow The full wind turbine configuration, under turbulent inflow conditions, is analysed in this section. The aim of the analysis is to preliminary assess the effect nonlinearities have on fatigue loads. The modelling of the stochastic wind inflow is based on the specifications of the IEC standard [] for normal turbulent conditions (NTM) and for wind turbine class A (highest wind speeds and turbulence levels). As opposed to the simulations presented in section 4. the effects of wind shear, tower shadow and rotor tilt are now taken into account. Also, the wind turbine is operated in closed loop. Ten-minute simulations are performed for wind speeds of 8, and 8 m/s. Predictions of the blade deflections and loads, obtained with the baseline model and the model are compared, in view to highlight the effect of the various nonlinearities addressed in the previous section. In Figure the time-series of the blade tip torsion deflection is shown for the simulation at 8 m/s mean wind speed. The comparison of the two models indicates that for low and moderate wind speeds (below and near rated) the variation of the torsion angles considerably increase due to non-linear effects. The increase becomes higher as we get closer to the rated wind speed. This is clearly seen in Figure by comparing the torsion angles in the time interval 8-3 s (wind speed varying between and m/s) with those in the time interval 36-38 s (wind speed varying between 7 and 9 m/s). flapwise bending moment at blade root (knm) 9 8 7 6 5 4 3 4 6 8 4 6 8 4 wind speed (m/s) Figure 3: Equivalent fatigue loads of flapwise bending moment (Hz, m=). torsion moment at blade root (knm) 5 5 4 6 8 4 6 8 4 wind speed (m/s) Figure 4: : Equivalent fatigue loads of torsion moment (Hz, m=). As regards loads, in Figure 3 and Figure 4 the Hz equivalent fatigue loads of the flapwise bending moment and the torsion moment at the blade root are shown for the three wind speeds. Slightly lower bending loads are predicted by the second order model at low and moderate wind speeds. The reduction of the flapwise bending moment is almost constant up to the rated speed. As indicated in section 4., the increase in the variation of the torsion angles (resulting from the bending-torsion coupling effect) indirectly affects the flapwise deflections and loads by changing the effective angles of attack seen by the blade. Depending on the phase difference between the torsion angle and the flapwise deflections, the increase in amplitude of the torsion angle can

either lead to higher or lower amplitudes of the flapwise bending moments. The equivalent torsion moment substantially increases. The increase is higher at the lowest wind speed (almost % increase). This is because at a mean wind speed of 8 m/s the rotor operates at all times below rated conditions. At m/s wind speed the rotor frequently enters variable pitch operation in which case the flapwise loads decrease and the non linear effects become less pronounced. As expected the increase in the equivalent torsion moment dramatically drops at 8 m/s wind speed. 5 Conclusions The structural nonlinearities due to large deflections are analysed in the present paper. In identifying their effect on the response of multi- MW wind turbines two different modeling approaches are followed; a model formulated in the deformed state and a multi-body model based on linear Timoshenko beam elements. From the analyses presented, it is concluded that both types of models, although formulated in a completely different way, produce similar results as regards the aeroelastic response of a highly deflected large scale blade. Comparison with the results of a lower order model (similar to those typically employed by the wind industry) indicates that suppression of certain higher order nonlinear couplings can lead to a noticeable underprediction of the blade loads. The bending-torsion coupling is identified as the main non-linear effect that should be taken into account in state-of-the-art aeroelastic tools when dealing with large flexible blades. Inclusion of the above effect results in predictions of the blade torsion loads that are even three times higher than those provided by a standard linear beam model. Acknowledgements This work was supported by the European Commission under contract SES6, 9945. prediction tool for wind turbines, EWEC 97 Dublin Ireland, (997). [3] Larsen, T. J., Hansen, A. M. and Buhl, T., Aeroelastic Effects of Large Blade Deflections for Wind Turbines, Proceedings of The Science of making Torque from Wind Conference, DELFT University of Technology, The Netherlands, 9- April, 4, Edited by G. A. M. van Kuik, Published by DUWIND Delft University of Technology. [4] Schepers, J.G., (), VEWTDC: Verification of European Wind Turbine Design Codes, Final Report for JOR3-CT98-67 Joule III project, ECN. [5] Politis, E. S. and Chaviaropoulos, P. K. (Eds.), Benchmark Calculations on the NM8 Wind Turbine, Technical report of STABCON project (NNK5-CT -67 contract), 5. [6] Politis, E. S. and Riziotis V.A., The Importance of Nonlinear Effects Identified by Aerodynamic and Aero-elastic Simulations on the 5MW Reference Wind Turbine, Technical report of UPWIND project, ( SES6, 9945 contract number), 7 [7] Kooijman, H. J. T., Lindenburg, C., Winkelaar, D., and van der Hooft, E. L., DOWEC 6 MW Pre-Design: Aero-elastic modeling of the DOWEC 6 MW pre-design in PHATAS, ECN-CX---35, DOWEC 46_9, Energy Research Center of the Netherlands, September 3. [8] Tarp-Johansen, N. J., RECOFF Home Page, http://www.risoe.dk/vea/recoff/, Roskilde, Denmark: Risø National Laboratory, July 4. [9] Malcolm, D. J. and Hansen, A. C., WindPACT Turbine Rotor Design Study, NREL/SR-5-3495, National Renewable Energy Laboratory, August. [] IEC 64-, IEC 3, 8884CDV, edited by TC88-MT, 5-6May 4, pp. 6 9, 3rd edition. References [] Hodges, D. H. and Dowell, E. H., Nonlinear Equations of Motion for the Elastic Bending and Torsion of Twisted Non-uniform Rotor Blades, NASA TN D-788, 974. [] Riziotis V.A. and Voutsinas S.G., GAST: A general aerodynamic and structural