Dissertation Larwood

Dynamic Analysis Tool Development for Advanced Geometry Wind Turbine BladesBySCOTT MICHAEL LARWOODB.S. (California Polytechnic State University, San Luis Obispo) 1988M.S. (Stanford University) 1993DISSERTATIONSubmitted in partial satisfaction of the requirements for the degree ofDOCTOR OF PHILOSOPHYinMechanical and Aeronautical Engineeringin theOFFICE OF GRADUATE STUDIESof theUNIVERSITY OF CALIFORNIADAVISApproved:Committee in charge2009iDonald MargolisFidelis EkeCase van DamDynamic Analysis Tool Development for Advanced Geometry Wind Turbine BladesCopyright 2009byScott Michael LarwoodAcknowledgmentsThe California Energy Commission partially funded this work through the PIEREnergyInnovations Small Grant Program, grant number 54905A/06-16.I would like to thank the men and women of the wind energy industry, especially myformer colleagues at Kenetech Windpower, who believed that wind could become a com-petitivepowersource. IwouldalsoliketothankCasevanDam, myPhDadviser, foraccepting me into his research group (friends Eddie Mayda, Jon Baker, Henry Shiu, andRay Chow) at UC Davis. His wise and gentle guidance made my transition into academia(from industry) very smooth. Kevin Jackson at Dynamic Design planted the seeds for thisdissertation, and encouraged me to apply for external funding. Mike Zuteck, one of themost talented engineers I know, pushed the concept of the swept blade into reality, alongwith leadership from Gary Kanaby at Knight and Carver and Tom Ashwill at Sandia. Thewind turbine code developers at NREL, Marshall Buhl and Jason Jonkman, laid excellentgroundwork for my code development, and tolerated my pesky questions.Finally, I would like to acknowledge my family. My parents whole-heartily supportedmy shocking decision to leave a good job in my middle age to go back for my doctorate.It was the best decision of my life. At Davis I met my wife Brenna, and we now have awonderful son Paul. Thank you Brenna and Paul, for giving me the time to nish this work.Our journey together has just begun.iiContentsList of Figures viList of Tables viiiNomenclature ixAbstract 11 Introduction 21.1 Current Status of Wind Energy . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Advanced Wind Turbine Rotors . . . . . . . . . . . . . . . . . . . . . . . 31.3 Wind Turbine Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Research Objectives and Motivation . . . . . . . . . . . . . . . . . . . . . 82 Blade Finite Element Modeling 102.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.1 Coordinate Systems and Transformations . . . . . . . . . . . . . . 112.2.2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2.3 Lagranges Equation . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.4 Steady State Equations and Axial Force . . . . . . . . . . . . . . . 252.2.5 Oscillation about Steady State . . . . . . . . . . . . . . . . . . . . 282.2.6 Beam Element Matrices . . . . . . . . . . . . . . . . . . . . . . . 292.2.7 Solution for Blade Frequencies and Mode Shapes. . . . . . . . . . 302.3 Finite Element Method Verication Results . . . . . . . . . . . . . . . . . 322.3.1 Curved Beam Deection Verication . . . . . . . . . . . . . . . . 322.3.2 Non-Rotating Curved Beam Modes Verication . . . . . . . . . . . 352.3.3 Rotating Tapered Beam Verication . . . . . . . . . . . . . . . . . 352.3.4 Rotating Beams from Leung and Fung 1988 Verication . . . . . . 382.3.5 Rotating Curved Beam Verication . . . . . . . . . . . . . . . . . 452.4 Conclusions and Recommendations . . . . . . . . . . . . . . . . . . . . . 47iii3 Dynamic Analysis of Curved Wind Turbine Blades 483.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.2 Coordinate Systems and Transformations . . . . . . . . . . . . . . . . . . 493.2.1 Local Undeected Axis . . . . . . . . . . . . . . . . . . . . . . . . 493.2.2 Local Deected Coordinate System . . . . . . . . . . . . . . . . . 523.2.3 Local Aerodynamics Coordinate System . . . . . . . . . . . . . . . 543.3 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.3.1 Positions and Displacements . . . . . . . . . . . . . . . . . . . . . 573.3.2 Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.3.3 Accelerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.4 Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.4.1 Generalized Inertia Forces . . . . . . . . . . . . . . . . . . . . . . 693.4.2 Generalized Active Forces . . . . . . . . . . . . . . . . . . . . . . 743.5 Blade Loads. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.5.1 Blade Root Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.5.2 Blade Gage Moment Outputs . . . . . . . . . . . . . . . . . . . . . 953.6 Verication with FAST and Adams . . . . . . . . . . . . . . . . . . . . . . 953.7 Validation with Field Test Data. . . . . . . . . . . . . . . . . . . . . . . .1063.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1103.8.1 Verication with Adams . . . . . . . . . . . . . . . . . . . . . . .1103.8.2 Validation with Field Test Data . . . . . . . . . . . . . . . . . . . .1113.9 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1124 Design Studies 1134.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1134.2 Baseline Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1144.2.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1144.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1184.2.3 Conclusions and Discussion . . . . . . . . . . . . . . . . . . . . .1214.3 Scaling to Larger rotors. . . . . . . . . . . . . . . . . . . . . . . . . . . .1214.3.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1214.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1264.3.3 Conclusions and Recommendations . . . . . . . . . . . . . . . . .128Bibliography 129A Blade Finite Element Matrices 134A.1 Mass Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .134A.2 Elastic Stiffness Matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . .136A.3 Gyroscopic Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .137A.4 Spin Stiffness Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . .139A.5 Axial Force Stiffness Matrix . . . . . . . . . . . . . . . . . . . . . . . . .142A.6 Axial Reduction Matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . .143B CurveFEM Program Structure 145ivC Summary of FAST Modications 147vList of Figures1.1 Swept blade concept. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Analysis ow diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1 First transformation of element . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Second transformation of element with built-in structural twist . . . . . . . 142.3 Beam element coordinate system. . . . . . . . . . . . . . . . . . . . . . . 182.4 Beam element axial force (after Leung [1]) . . . . . . . . . . . . . . . . . 272.5 Nodal degrees of freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.6 Curved beam for deection verication (after Kosmatka [2]) . . . . . . . . 322.7 Curve beam deection and slope for Fx= 100 lb . . . . . . . . . . . . . . 332.8 Curve beam deection and slope for Fy= 100 lb . . . . . . . . . . . . . . 342.9 Curved beam for natural response (after Kosmatka [2]) . . . . . . . . . . . 352.10 Rotating tapered beam (all dimensions in inches) . . . . . . . . . . . . . . 362.11 Tapered beam verication results. . . . . . . . . . . . . . . . . . . . . . . 372.12 Horizontal Cantilever (after Leung [1]) . . . . . . . . . . . . . . . . . . . . 382.13 Horizontal Cantilever Verication Results . . . . . . . . . . . . . . . . . . 392.14 Inclined Cantilever (after Leung [1]) . . . . . . . . . . . . . . . . . . . . . 402.15 Inclined Cantilever Verication Results . . . . . . . . . . . . . . . . . . . 412.16 L-beam (after Leung [1]) . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.17 L-beam with = 90 Verication Results . . . . . . . . . . . . . . . . . . 432.18 L-beam with = 30 Verication Results . . . . . . . . . . . . . . . . . . 442.19 Rotating curved beam verication results . . . . . . . . . . . . . . . . . . 463.1 First transformation of element . . . . . . . . . . . . . . . . . . . . . . . . 503.2 Second transformation of element with built-in structural twist . . . . . . . 513.3 Undeected blade element offsets . . . . . . . . . . . . . . . . . . . . . . 573.4 Sweep angle for swept blade . . . . . . . . . . . . . . . . . . . . . . . . . 783.5 Blade tip twist verication . . . . . . . . . . . . . . . . . . . . . . . . . . 983.6 Blade tip deection verication . . . . . . . . . . . . . . . . . . . . . . . . 993.7 Generator power verication . . . . . . . . . . . . . . . . . . . . . . . . .1003.8 Flap bending moment verication . . . . . . . . . . . . . . . . . . . . . .1013.9 Edge bending moment verication . . . . . . . . . . . . . . . . . . . . . .1023.10 Out-of-plane deection for STAR7d rst ap bending mode . . . . . . . .1043.11 Torsional deection for STAR7d rst ap bending mode . . . . . . . . . .105vi3.12 Generator power validation. . . . . . . . . . . . . . . . . . . . . . . . . .1073.13 Blade pitch validation. . . . . . . . . . . . . . . . . . . . . . . . . . . . .1083.14 Edge bending validation . . . . . . . . . . . . . . . . . . . . . . . . . . .1093.15 Flap bending validation. . . . . . . . . . . . . . . . . . . . . . . . . . . .1104.1 Parametric study of damage equivalent loads . . . . . . . . . . . . . . . . .1184.2 Parametric study of annual energy production . . . . . . . . . . . . . . . .1194.3 Parametric study of blade tip deection . . . . . . . . . . . . . . . . . . .1204.4 Flap bending comparison for scaled design . . . . . . . . . . . . . . . . . .127viiList of Tables2.1 XYZ Component Values . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Curved beam modes verication . . . . . . . . . . . . . . . . . . . . . . . 353.1 CurveFAST Blade 1 Degrees of Freedom . . . . . . . . . . . . . . . . . . 533.2 CurveFAST Degrees of Freedom. . . . . . . . . . . . . . . . . . . . . . . 563.3 Turbulent wind le parameters . . . . . . . . . . . . . . . . . . . . . . . . 963.4 Maximum percentage differences for the Adams/CurveFAST verication . 973.5 Extreme Load Verication, Normalized Maximum Values . . . . . . . . . .1034.1 Parametric Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1154.2 WindPACT model wind turbine parameters . . . . . . . . . . . . . . . . .1224.3 WP1500 blade properties . . . . . . . . . . . . . . . . . . . . . . . . . . .1234.4 WP3000 blade properties . . . . . . . . . . . . . . . . . . . . . . . . . . .1244.5 Scaled model results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .126viiiNomenclatureRomana acceleration vectorA sectional areaC matrix for Kanes methodCDdrag coefcientCDdrag coefcient modied for sweepCLlift coefcientCLlift coefcient modied for sweepCMmoment coefcientD damping matrixEA extensional stiffnessEI bending stiffnessf vector in equation of motion for Kanes methodF element axial forceF force vectorixFrgeneralized active forcesFrgeneralized inertia forcesg gravitational accelerationg

hub primed system unit vectorsG gyroscopic matrixGJ torsional stiffnessh undeected blade element offsetH angular momentum vectori coned system unit vectorsI identity matrixI inertia dyadicj pitched system unit vectorsJ

polar moment of inertia about centroidal axisK combined stiffness matrixKmodied stiffness matrixKeelastic stiffness matrixKgaxial force stiffness matrixKspin stiffness matrixl direction cosine, element lengthL beam lengthxLj local undeected element system unit vectorsm inverse of the W ohler (or S-N) curve, direction cosinem element aerodynamic system unit vectorsM mass matrixM momentM moment vectorMmodied mass matrixn number of degrees of freedom, number of cycles for a particular stress level, direc-tion cosinen local deected element system unit vectorsN allowable number of cycles for a particular stress levelN shape function matrixq CurveFAST degree of freedomq nodal displacement vectorQ apex of coned systemr radial positionr position vectors distance along beam elementS coefcient in axial reduction equationte element trailing edge system unit vectorsxiT kinetic energyTCH hub (primed) to coned system transformationTDEundeected to deected element system transformationTEPpitched to undeected element system transformationTPCconed to pitched system transformationu deection in x-directionu displacement vectorU strain energyv deection in y-directionv velocity vectorV wind velocityw deection in z-directionxreigenvectoryrreal part of eigenvector xrzrimaginary part of eigenvector xrz tower system unit vectorsGreek direction cosine angular velocity pitch anglexii modal damping ratio angle, twistreigenvalue transformation matrix angle between relative velocity and chord line lineal density non-dimensional position along element mass density interpolation function eigenvector element matrix of eigenvector elements angular velocity vector rotational speed spinning matrixSubscripts0 value at inboard element nodea applied, aerodynamicAero aerodynamicave averageAxRedaxial reductionxiiiB1 blade 1Dampdampinge elementElastic elasticeq EquivalentE1 rst edge modeF1 rst ap modeF2 second ap modeGrav gravitationall value at outboard element noder partial velocity derivatives structural, steady stateT1 rst torsion modeTeet teeter degree of freedomu ultimatev oscillatory componentSuperscriptsB1Tipblade 1 tip nodeE inertial frameH hub bodyxivM1 blade 1 element bodyS1 blade 1 element center of massOther Notation( )

differentiation with respect to s() time derivative( )TtransposeAcronymsAEP annual energy productionBEMblade element momentum methodCFD computational uid dynamicsCOE cost of energyDEL damage equivalent loadDOE United States Department of EnergyDOF degree of freedomEISGEnergy Innovations Small GrantIEC International Electrotechnical CommissionLWSTLow Wind Speed Technology programNRELNational Renewable Energy LaboratoryNTMnormal turbulence modelRPS renewable portfolio standardxvSTARswept-tip adaptive rotorWindPACTwind partnerships for advanced component technologyxvi1AbstractThis dissertation describes work to develop a dynamic analysis code for swept wind turbineblades. Because of their aeroelastic behavior, swept blades offer the potential to increaseenergy capture and lower fatigue loads. This work was an outgrowth of United States De-partment of Energy contract on swept blades, where the author used the AdamsTMdynamicsoftware. The author based the new code on the National Renewable Energy LaboratorysFAST code. The new code would allow for lower cost analysis and faster computationtimes for swept blades compared to Adams. The FAST revisions included the geometryand mode shapes required for the bending and twisting motion of the swept blade. The au-thor also developed a nite-element program to determine mode shapes for the swept blade.The author veried the new code with Adams. The comparisons were favorable; however,the Adams model exhibited more twist. The differences may be attributed to differences inmodeling approach. The author attempted to validate the code with eld test data; however,uncertainties in the test wind speed and the turbine controller made comparison difcult.The author used the new code to perform preliminary designs of swept rotors for 1.5 MWand 3.0 MW wind turbines. The designs showed a 5% increase in annual energy productionand a decrease in ap-bending fatigue over the baseline straight-blade designs.2Chapter 1Introduction1.1 Current Status of Wind EnergyAt the beginning of the 21st century, wind energy has become the fastest-growing newenergy source. Worldwide installation continues at an exponential rate [3]. Large corpora-tions have recently entered the business, such as BP, General Electric, Shell, and Siemens.Wind energy in California was 1.5% of the gross system energy production in 2005 [4].Under the Renewables Portfolio Standard (RPS), California has mandated that renewableswill account for 20% of the energy production by 2010. The majority of this production isexpected to come from wind energy.The growth in the wind energy industry can be attributed to competitive cost of energy(COE) in comparison to traditional energy supplies. Wind energy is consistently cited asbeing on par with the cost of generation by natural gas. Tangible benets of wind energyinclude zero fuel costs in addition to zero greenhouse gas and airborne pollutant emissions.The dramatic decrease in COE since the 1980s has coincided with the increase in sizeof wind turbines. In the Solano area, for instance, ninety 1980s vintage Kenetech 100 kWmachines have each been replaced by six GE 1.5 MW units. The diameter of the old 100kW turbine is 17 m, while the new 1.5 MW turbine diameter is 77 m. The 1.5 MW turbine3benets from an increase in available wind power with the square of the rotor diameter, andalso from increased wind speeds at higher hub heights due to wind shear. Larger diameterrotors are also necessary for increasing the wind energy potential at low wind speed sitesthat are closer to major metropolitan load centers.However, these added benets of larger diameter rotors potentially come at increasingCOE. With an increase in rotor diameter, if all dimensions remain proportional the masswill increase with the cube of the diameter due to the volume increase. Mass is directlyrelated to cost. This relationship is commonly referred to as the square cube law. Inaddition to the mass increasing by the cube of the diameter, the stress on the blades due togravity increases in proportion to the rotor diameter increase [5]. These two effects, massand stress increase, require that blades must be proportionally lighter and produce lowerstress in order for the COE to not increase with increasing rotor diameter.With this engineering challenge to continue the reduction of COE, the U.S. Departmentof Energy (DOE) initiated the Low Wind Speed Technology (LWST) program to moveresearch in the direction of capturing more wind energy in lowwind speed areas that requirelarger rotor diameters, thus positively affecting the economics for these regions.1.2 Advanced Wind Turbine RotorsOne thrust of the LWST program is in advanced rotor control concepts. The idea isto reduce loads for larger rotor diameters so that they can be made proportionally lighter.Because the rotor typically accounts for 20%of the capital costs of a turbine [6], an increasein the cost of the rotor for these advancements would be more than offset by increasedenergy capture.One concept [7] is to incorporate ap/twist coupling into the rotor with off-axis berorientationintothebladeconstruction. Asthebladedeectsintheapwisedirection,the tip twists toward feather and reduces the aerodynamic loading. A different physical4mechanism, leading to similar aerodynamic load reduction, is analyzed by Zuteck [8] andproposed earlier by Liebst [9]. This concept is to sweep the outboard-region rotor planformin the plane of rotation aft of the pitch axis. The loads generated at the tip then introducea moment about the pitch axis. With sufcient blade torsional exibility, the tip twiststoward feather, thus reducing the loads.This concept is illustrated below in Figure 1.1.Awire-frame model of a turbine with swept blades is shown in the left hand corner of Figure1.2 Undeflected Blade Pitch Axis RotationWind UnloadedSection Tip View Wind PitchAxisTwists tofeather Load Control through passive meansutilizing blade geometry Aero LoadsFigure 1.1: Swept blade concept.In 2004 the Blade Division of Knight & Carver won an LWST contract administeredthrough Sandia National Labs. Knight & Carver assembled a team to design, manufacture,and test a rotor based on the swept-blade concept. The rotor would be designed for atmo-spheric testing on a Zond Z-50 wind turbine with 750 kW rating. The project team calledthe concept the STAR (Sweep-Twist Adaptive Rotor). The author was responsible for thedynamic analysis of the prototype rotor. Preliminary results show the load reduction effects[10].The design goal of the project was to increase annual energy capture of the baselineturbine by 5%-10%. The rotor swept area was increased by 25% to increase the below-rated energy capture by 25% for a straight-bladed rotor. For the Z-50 turbine, the rotor5radius was increased from 25 m to 28 m. With sweep and twist, it was expected that theincrease in below-rated energy capture would be 15%-20% and that therefore the overallannual energy capture would increase by 5%-10%. The turbine power rating would not beincreased; therefore, there would be no increase in above-rated energy capture.1.3 Wind Turbine AnalysisThe design and analysis of wind turbines is a well-established eld with several textson the subject, such as Burton, Sharpe, Jenkins, and Bossanyi [6], Gasch and Twele [5],and Manwell, McGowan, and Rogers [11]. Molenaar [12] extensively surveys the specictopic of dynamic analysis for wind turbines. The wind energy industry has matured toa level such that full-aeroelastic dynamic modeling is used to design and analyze windturbines. Wind turbine dynamic modeling is a complicated interaction of turbine structuraldynamics, rotor/wake aerodynamics, and atmospheric boundary-layer uid dynamics. Theprimary use of dynamic modeling is design and loads evaluation to ensure proper sizing ofcomponents. Engineers perform the evaluation according to industry accepted standards,such as the International Electrotechnical Commission (IEC) [13] or Germanischer Lloyd[14].Figure 1.2 shows a typical process to be followed for wind turbine dynamic analysis.The National Renewable Energy Laboratory (NREL) developed and maintains the tools[15] in this diagram for wind turbine analysis. In the process, the analyst develops inputles of the turbine properties. The FAST [16] code uses for input the machine proper-ties, such as dimensions, masses, and inertias. FAST is a wind turbine specic dynamicanalysis code. Aerodynamic and geometric properties of the blade are needed as input forthe AeroDyn [17] aerodynamics subroutines. Simulated wind environments are built usingTurbSim [18] and IECWind [19] codes. IECWind produces gust-type simulations, whereasTurbSim produces full-eld three-dimensional turbulence. FAST is then executed to build6 FAST AeroDyn ADAMS System Properties Wind Field FAST Input FilesAeroDyn Input FilesADAMS Datasets Time Series Data Baseline Turbine Model TurbSim/ IECWind Preprocessing Figure 1.2: Analysis ow diagram.input les for MSC/AdamsTM. Currently, FAST does not have the capability of analyz-ing rotors with blade sweep. Adams is a commercial code used for multi-body dynamicsimulation in many industries, including the aerospace and automotive sectors. NREL hasworked on interfacing to Adams for wind turbine analysis [20]. The Adams models includelumped-parameter representation of the rotor blades.During execution, Adams solves theequations of motion at each time step, and the AeroDyn subroutines are called to computethe aerodynamic forces. The output is time series data of requested parameters that can beprocessed as desired.Previous work has primarily focused on straight wind turbine blades. Recent interestinbladeswithsweephavepushedthelimitsofwindturbineanalysistools. Currentlythe tool with the most modeling delity is the commercial code Adams. The author usedAdams for modeling the Knight and Carver swept blade for the LWST program[10]. Liebst[9] performed an early analysis of a swept-blade wind turbine rotor;however, the blade7model had constant cross-section, an assumption that would not work with modern bladeconstruction. Neither analysis is validated with test data.While Adams is a sophisticated modeling tool, it has several drawbacks that will bediscussed below. Several wind-turbine specic modeling tools have been developed forthe industry, including NRELs FAST code, Garrad Hassans Bladed code from the UnitedKingdom, and Stig yes FLEX5 code from Denmark. These codes are recognized byGermanischer Lloyd for loads evaluation of wind turbines. The primary modeling differ-ence between Adams and these other codes is that Adams models the blade with lumpedparameters, whereas the other codes use prescribed mode shapes to model the blade.For the LWST project, the Knight and Carver team obtained great condence in theswept blade design concept through the use of Adams. However, during the analysis theauthor determined that continuing to use Adams would be a hindrance to the further de-velopment of the concept. At the time Adams was being used under an academic license,and it was expected that a commercial license would have to be obtained for further work.A recent quote for Adams was $32,000 with a 20% annual fee [21].The other two codes,Bladed and FLEX5, had capabilities of modeling swept blades. Garrad Hassan quotedBladed at 25,000 British pounds. The project team was unable to obtain a quote for FLEX5.Another major problem with Adams was slow run times, with a typical 10-minute tur-bulence simulation taking 3.5 hours (using Dell Latitude D600 with Pentium M 1400 MHzprocessor and 512 MB RAM). NREL has mentioned the run-time problem in the literature[20]. The run times were exacerbated by rotor-torque simulation problems, which requiredshorter time steps and longer run-times for convergence. The Adams models also showedsensitivity to certain turbulent wind le random-seeds. These slow run times make fordifcult rapid design evaluation and automated loads analysis.The author decided that modifying an existing wind turbine code with public-domainsource code would be the best option for further work on the swept blade. The only code inthe public domain with Germanischer Lloyd certication is NRELs FAST [22]; therefore,8the author chose this code for new development. Wilson, Freeman, Walker, and Harman[23] at Oregon State University originally developed FAST, which has recently been up-dated by NREL [24]. The code uses Kanes Method [25] for derivation of the equations ofmotion. Both the blade and tower are modeled as exible elements with prescribed modeshapes [26]. Bladed and FLEX5 use similar methods. Currently, FAST includes three blademode shapes: rst ap-bending, second ap-bending, and rst edge-bending. Use of modeshapes reduces the modeling complexity compared to the lumped-parameter representationin Adams. In addition, developing the program specically for wind turbine analysis addsto the computational efciency. The author has found that a model with straight blades inFAST can take seven times longer with Adams.1.4 Research Objectives and MotivationThe objectives for this doctoral thesis work were to increase the modeling capabilityof the FAST code for the analysis of swept rotor blades, and to study the feasibility of theswept-blade concept for turbines of greater size than the STAR rotor. As mentioned above,these advanced-geometry blades offer the possibility of extending the rotor diameter ofan existing straight-blade design, allowing increased energy capture without exceeding thedesign load envelope.The motivation behind increasing the modeling capability of FAST, besides being ofeducational interest to the author, was to lower the cost and to increase the computationspeed of analyzing the swept rotor. Along with high cost, Adams simulation times forconventional wind turbine designs were seven times the run time of FAST. Similar concernsare raised by NREL [20]. An example design case is to run all of the operating conditionsfor a particular design with 10-minute turbulent wind simulations according to the IECStandard [13]. There would be 11 different wind speeds if 2 m/s steps were taken between4m/s and24m/s. For eachwindspeed, sixturbulent simulationseachwould haveto9run according to the standard. The total for this design case is 66 simulations. With ananticipatedFASTruntimeof30minutes, thisdesigncasewouldtake33hours. WithAdams at approximately seven times the run time of FAST, the design case would takeover nine days. Increasing the capability of FAST, without increasing the simulation timesubstantially, shows great promise as a design optimization tool.The modication of FAST for swept blade required solution of the blade mode shapes.The author developed a nite element code for this purpose, called CurveFEM that is de-scribed in Chapter 2. Chapter 3 describes the equation of motion for the modications toFAST, called CurveFAST. The author veried the changes for the swept blade with Adams.The author also compared CurveFAST model runs with eld test data from the STAR rotor.A prototype of the STAR rotor ran on a turbine in Tehachapi, California, in the winter of2008.The author also studied the feasibility of scaling the swept blade concept to larger windturbine rotors. Current production wind turbines vary from 1.5 MW to 3 MW, with offshoreprototypes up to 5 MW. Key geometric parameters of the blade were studied for optimumdesign, such as the shape of the blade sweep curve, the torsional exibility of the blade, andthe maximum amount of sweep at the blade. There are important design limitations dueto transportation of the blades from the manufacturer to the wind plant. It may be that theconcept would not be feasible for larger blades due to these constraints. With a modiedFAST, these design studies could be automated.Withanincreasingrotordiameter, andlighterbladesasrequiredbytheeconomicsdiscussed above, comes the potential for utter stability problems as discussed by Lobitz[27]. The author studied the utter boundary were studied for the STAR rotor.Chapter 4 describes these design and utter studies.10Chapter 2Blade Finite Element Modeling2.1 IntroductionNRELs FAST program [26] models wind turbine blades and towers as exible bodies.The blade response is a linear summation of the lower bending modes. Authors [26][28]commonly call this technique the normal mode summation (or superposition) method. Cur-rently, FAST uses three bending modes:rst ap, rst edge, and second ap. The modescouple through built-in twist. NREL provides the Modes program [29] to generate thesemode shapes for straight, rotating, pitched, and tapered blades. Modes uses a Rayleigh-Ritzmethod as described in Jonkman [26] to determine mode shapes.For blade sweep to be incorporated into FAST, the program requires new mode shapesthat allow for twist deections to occur during transverse bending motion.The author de-cided to retain the mode summation method in FAST, and therefore investigated methodsfor determining mode shapes of rotating, tapered, twisted, and curved beams. The propellerand rotorcraft (helicopter) research community has generated work on curved blades. Theprimary goal of the research is to use sweep to delay the onset of drag divergence. Sec-ondary goals of the rotorcraft community are to potentially reduce hub loads. Bielawa [30]and Kosmatka [2][31] have works related to high-speed propellers. Bauchau and Hong11[32], Bir [33], Celi and Friedmann [34], Hodges [35][36], Rosen and Rand [37][38][39],and Tarzanin and Vlaminick [40] have works related to rotorcraft. Many of these worksmodel non-isotropic properties that are important for very exible composite rotor blades.Most of the researchers also include non-linear behavior, which is important for rotorcraftstability analysis. Although Adams allows non-linear motion, most of the capabilities de-scribed by these researchers is beyond that currently modeled in Adams for wind turbines.The author found that Leung and Fung [1] produced the most applicable and best doc-umented work for the current project, titled Spinning Finite Elements.Leung and Fungproduced this work for general aerospace applications. Their model uses standard linearnite elements of constant properties with six degrees of freedom (DOF) at each node. Theauthor adapted their work for tapered properties into the new program CurveFEM, as ex-plained in Section 2.2 below. This section describes the coordinate systems and kinematicsof the nite elements, and the use of Lagranges equation to build the equations of motion(EOM). The nite element matrices derive from this development. The EOMs then becomean eigenvalue problem. The solution to the eigenvalue problem are the natural frequenciesand mode shapes of the blade. Section 2.3 describes the verication of CurveFEM withseveral published examples.2.2 Method2.2.1 Coordinate Systems and TransformationsThis section documents the coordinate systems used by CurveFEM, and are commonto FAST. Some of the coordinate systems have the superscript B1 for blade one; there aresimilar coordinate systems for the remaining blades. The rst coordinate system is the hub(primed) coordinate system (gB1). This system rotates with the hub, with thegB11unitvector parallel to the hub rotation axis. The hub rotates at constant speed , which is thesame assumption in Modes. The gB13unit vector would align to the blade pitch axis given12zero coning. The next system is the coned iB1system related to the gB1by:___iB11iB12iB13___= [TCH ]___gB11gB12gB13___(2.1)with the transformation matrix:[TCH ] =__cos[PreCone(1)] 0 sin[PreCone(1)]0 1 0sin[PreCone(1)] 0 cos[PreCone(1)]__where PreCone(1) is the hub pre-coning angle for blade 1, which is positive downwind.Currently there are no provisions for coning in CurveFEM; therefore, the TCH matrix isequal to the identity matrix [I]. Coning could be added later to CurveFEM. The transfor-mation to the blade pitched coordinate system (jB1) is:___jB11jB12jB13___= [TPC]___iB11iB12iB13___(2.2)with the transformation matrix:[TPC] =__cos[BlPitch(1)] sin[BlPitch(1)] 0sin[BlPitch(1)] cos[BlPitch(1)] 00 0 1__where BlPitch(1) is the blade 1 pitch. The pitch angle is relative to the chord line at zeroaerodynamic twist and is positive toward feather (leading edge upwind). Currently thereis no provision for pitch (can be added later) in CurveFEM; therefore the transformation13matrix TPC is equal to the identity matrix [I].The next coordinate systemis the blade element systemaligned with the local undeected-axis. It has the unit vectors LjB1i(i=1, 2, 3). The transformation from the pitch systemis:___LjB11(r)LjB12(r)LjB13(r)___= [TEP(r)]___jB11jB12jB13___(2.3)The transformation development follows a similar method to Rao [28] p. 328, for niteelements, butwithacoordinatesystemchangetomatchtheFASTelements. Therststage involves a transformation matrix [1] between the pitched coordinates XY Z and thecoordinates x y z by assuming the y axis to be parallel to the Y Z plane (Fig. 2.1): j3 j1 j2 X Z Y 3j2j1jxzyl i j Figure 2.1: First transformation of element___ x y z___= [1]___XYZ___The next step is between the local coordinates xyz (principal axes) and the coordinates x y z14as (Fig. 2.2):___xyz___= [2]___ x y z___ Lj2Lj1x y 2j1jxys

Figure 2.2: Second transformation of element with built-in structural twistThe desired transformation between the xyz and the XY Z system is therefore:[TEP(r)] = [2][1]where:___xyz___= [TEP(r)]___XYZ___15From Fig. 2.1, the direction cosines of the longitudinal axis ( z or z) are:lo z= loz=Xj Xilmo z= moz=Yj Yilno z= noz=Zj Zilwhere the length of the element is:l = {(Xj Xi)2+ (Yj Yi)2+ (Zj Zi)2}1/2As in the FAST2ADAMS.f90 preprocessor, the interior element nodes coincide withthe analysis nodes, which are interpolated from the properties in the blade input le. Thelast node uses the values at the tip. Table 2.1 shows values of the XY Z components andtheir FAST variable names.Table 2.1: XYZ Component ValuesComponent Inner Elements Tip ValueX PrecrvRef(i) PrecrvRef(TipNode)Y PreswpRef(i) PreswpRef(TipNode)Z BlFract(i) BldFlexL +BldFlexL + HubRadHubRadBecause the unit vector j2 (parallel to the y axis) is normal to both the unit vectors j1(parallel to the X axis) andj3 (parallel to the z axis) the following vector analysis relationholds:j2=j3 j1__j3 j1__=1dj1j2j3lozmoznoz1 0 0=1d(j2 noz j3 moz)where:d = (m2oz + n2oz)1/216Therefore, the direction cosines of the y axis with respect to the XY Z system are:lo y= 0, mo y=nozd, no y= mozdBecause the x axis (unit vectorj1) is orthonormal to the y axis (j2) and the z axis (j3),j1 is:j1= j2 j3=j1j2j3lo ymo yno ylozmoznoz=j1j2j30nozdmozdlozmoznoz=1d_j1(m2oz + n2oz) +j2(lozmoz) +j3(loznoz)The direction cosines for the x axis are therefore:lo x=m2oz + n2ozd, mo x= lozmozd, no x= loznozdTherefore, the [1] matrix is:[1] =__lo xmo xno xlo ymo yno ylo zmo zno z__=__(m2oz + n2oz)/d (lozmoz)/d (loznoz)/d0 noz/d moz/dlozmoznoz__The principal axes of the element (xyz axes) makes an angle s (built-in structural twist)17about the negative z axis. This direction is such that a positive s results in a lower angleof attack. The tranformation between the systems is:___xyz___=__cos ssin s0sin scos s00 0 1_____ x y z___= [2]___ x y z___so that:[2] =__cos ssin s0sin scos s00 0 1__Note that this procedure breaks down when the z (or z) axis is aligned to the Xaxis. Inthis case,mox=nox=0 and therefored=0. However, this situation is not realizablefor a horizontal-axis wind turbine because it implies that two stations are at the same radialposition.2.2.2 KinematicsThis section draws on the work of Leung and Fung [1],mentioned by the author inthe introduction. They develop the element matrices for rotating, linear, constant property,space-frame elements with six DOF at each node. The author changed their coordinatesystems to match the alignment in FAST, in addition to adding tapered element properties.The beam element is located in space with three sets of orthogonal axes:1. xyz along the local principal axes of the beam (with unit vectors Lji)2. XY Z that is the pitched blade system for no pre-cone or pitch (with unit vectors ji)with X parallel to the spinning axis3.X YZ, an inertial frame withX parallel to X18 x Y y z {rh} {r0} {rg} {s} X Z Figure 2.3: Beam element coordinate systemFigure 2.3 shows the rst two coordinate systems.The undeformed element is at rest relative to frame XY Z and the position vector forany point on the element is:{r0} = {rg} + ({rh} {rg}), 0 1or:{r0} = {rg} + s{s}, 0 s l (2.4)where {r0}, {rg}, and {rh} are respectively the position vectors of the point and the endsof the beam with respect to frame XY Z, {s} is the unit vector along the beam axis with{s} = {rgh}/ |rgh| and {rgh} = {rh} {rg}.A spinning matrix [] used in the ensuing analysis for the element is:[] = __0 0 00 0 10 1 0__(2.5)19The absolute position vector for any point on the beam is:{r} = [X, Y, Z]TThe absolute velocity vector with respect to the inertial frame is:{v} = { r} + []{r} (2.6)The absolute acceleration vector with respect to the inertial frame is:{a} = {r} + 2[]{ r} + [][]{r}The displacement vector {u} in the principal beam axis with reference to the local coordi-nates xyz is:{u} = [u, v, w]Twhere u, v, w are displacements along the xyz axes respectively. The displacement vector{ u} in the principal beam axis with reference to the moving coordinates XY Z is:{ u} = [R]T{u}where [R] is the transformation between the global coordinates XY Z and the local coordi-nates xyz. The elements in [R] are:[R] =__111213212223313233__(2.7)For the case of zero coning and zero pitch, [R] is equal to [TEP], as in Eq. 2.3. Also, the20unit vector {s} is:{s} = [31, 32, 33]T(2.8)The position vector of a point on the deformed beam with reference to frame XY Z is:{r} = {r0} + { u} = {r0} + [R]T{u} (2.9)and:{ r} = [R]T{ u}, because { r0} = 0 (2.10)2.2.3 Lagranges EquationLagranges equation in vector form is:ddt_T{ u}_T{u}+U{u}= {F} (2.11)where Tand U are the kinetic and strain energies, respectively, and {F} is the generalizedforce vector. For the free vibration problem, {F} = 0. The kinetic energy of the beam is:T=12_A{v}T{v}ds (2.12)where is the mass density and A is the sectional area. The strain energy of the beam is:U =12_EA_ws_2ds +12_EIx_2vs2_2ds +12_EIy_2us2_2ds +12_F(s)_us_2ds + (2.13)12_F(s)_vs_2dsAt this point the analysis does not include torsion of the element, which is assumed in Le-ung and Fung [1] to be uncoupled from rotation. The rst three terms in the strain energy21equation 2.14 are the standard beam extension and bending terms, which are found in niteelement texts such as Raos [28]. The last two terms account for the work done by theelement axial force F(s), which arises from rotation. They account for the beams resis-tance to bending due to centrifugal forces, and are sometimes called axial reduction factors.Their derivation, described in Section 3.3.1, are from the theory of vibrating strings, whichMeirovitch [41] describes in the context of rotating beams.Substituting Eqs. 2.9 and 2.10 into Eq. 2.6 the absolute velocity vector is:{v} = [R]T{ u} + []({r0} + [R]T{u}) (2.14)The term in the kinetic energy equation 2.12 is therefore:{v}T{v} = { u}T[R][R]T{ u} + {u}T[R][]T[][R]T{u} +{r0}T[]T[]{r0} + 2{ u}T[R][]{r0} + (2.15)2{ u}T[R][][R]T{u} + 2{r0}T[]T[]{u}From Eqs. 2.7 and 2.5:[R][R]T= [I] and []T[] = [2] = 2__0 0 00 1 00 0 1__(2.16)For the nite element method, the element displacement vector {ue} is interpolatedfrom the nodal coordinate vector {qe} as in:{ue} = [N]{qe}and { ue} = [N]{ qe} (2.17)22where N is the shape function matrix:[N] =__N1N2N3__(2.18)Section 2.2.6 describes the development of theseN terms. The nodal coordinate vector{qe} is:{qe} = [{q1}T, {q2}T, {q3}T]T(2.19)where:{q1}T= [u1, y1, u2, y2] (2.20)are the bending nodal displacements in the xz plane, and:{q2}T= [v1, x1, v2, x2] (2.21)are the bending nodal displacements in the yz plane, and:{q3}T= [w1, w2] (2.22)are the axial nodal displacements.From Eqs. 2.12 and 2.16 the kinetic energy for an element is:Te=12{ qe}T[Me]{ qe} +12{qe}T[Ke]{qe} + T0e +{ qe}T{fe} + { qe}T[Ge]{qe} + {Fe}T{qe} (2.23)23where (following Leung and Fung [1]):[Me] =_l0A[m]ds, [Ke] =_l0A[k]ds,[Ge] =_l0A[g]ds, {fe} =_l0A[N]T[R][]{r0}ds, (2.24){Fe} =_l0A[N]T[fx, fy, fz]Tds,T0e=12_l0A{r0}T[2]{r0}dswhere Me is the element mass matrix, Ke is the element spin-stiffness matrix, and Ge isthe element gyroscopic matrix. The vectors {fe} and {Fe} and the scalar T0e drop out ofthe ensuing analysis. The terms in the element matrices are:[m] =__NT1N1NT2N2NT3N3__[g] = __0 b1NT1N2b2NT1N30 b3NT2N3skew symmetric 0__(2.25)[k] = 2__a11NT1N1a12NT1N2a13NT1N3a22NT2N2a23NT2N3symmetric a33NT3N3__where:a11= 212 + 213, a12= 1222 + 1323,a13= 1232 + 1333, a22= 222 + 223, (2.26)a23= 2232 + 2333, a22= 222 + 22324and:b1= 1322 1223,b2= 1332 1233, (2.27)b3= 2332 2233Thetotalkineticenergycomesfromassemblyoftheniteelementmatricesintypicalfashion, as described in Rao [28], with the matrices transformed to theXY Zcoordinatesystemsandthecommonnodesaddedforthecompatibilityrelation. Thetotalkineticexpression is:T=

eTe=12{ q}T[M]{ q} +12{q}T[K]{q} + T0 +{ q}T{f } + { q}T[G]{q} + {F}T{q}, (2.28)where {q} is the global nodal displacement vector in the rotating reference frame XY Z.Similar to the kinetic energy, the total strain energy for an element is:Ue=12{qe}T([Kee] + [Kge]){qe}, (2.29)where:[Kee] =__EIx[Ka]EIy[Kb]EA[Kc]__[Kge] =__[Kga][Kgb]0__, (2.30)25where Kee is the element elastic stiffness matrix and Kge is the element axial force stiffnessmatrix. The terms in the element matrices are:[Ka] =_l0[N

1]T[N

1]ds, [Kb] =_l0[N

2]T[N

2]ds,[Kc] =_l0[N

3]T[N

3]ds, (2.31)[Kga] =_l0F(s)[N

1]T[N

1]ds, [Kgb] =_l0F(s)[N

2]T[N

2]ds,where the prime denotes differentiation with respect to s.The total strain energy is:U=

eUe=12{q}T([Ke] + [Kg]){q}. (2.32)2.2.4 Steady State Equations and Axial ForceThe nodal displacement vector {q} is divided into the steady state displacement {qs}and the oscillation about steady state, {qv}, so that:{q} = {qs} + {qv} (2.33)For steady state the oscillatory component and the derivatives are zero, as in:{qv}, { qv}, { qs}, and{ q} = 0 (2.34)Inserting Eqs. 2.33 and 2.34 into Eqs. 2.28 and 2.32 the steady state kinetic energy be-comes:T=12{qs}T[K]{qs} + T0 + {F}T{qs}, (2.35)26and the steady state strain energy becomes:U=12{qs}T([Ke] + [Kg]){qs}. (2.36)The Lagrangian equation 2.11 then reduces to:T{qs}+U{qs}= 0, (2.37)and with Eqs. 2.35 and 2.36:([Ke] + [Kg] [K]){qs} = {F}. (2.38)Thebladeisassumedtobeacantileverbeamandthereforetheinternalforcesarepredetermined. The axial force stiffness matrix [Kg] is therefore independent of the steadystate deformation. The derivation of the axial force F(s) assumes that the centrifugal forcedoes not change with deection. From Leung and Fung [1] the centrifugal force per unitlength along the beam is:= A{s}T[][]{r0}. (2.39)From Eqs. 2.4, 2.5, and 2.8, the above equation becomes:= A(a + bs) (2.40)where:a = 2(32Y0 + 33Z0) (2.41)and:b = 2(232 + 233) (2.42)and Y0 and Z0 are the coordinates of the element inboard endpoint {rg}. The axial force at27a point along the element, F(s) (Figure 2.4) is:F(s) = F0 _s0(s)ds (2.43)where F0 is the axial force at the inboard node of the element. F0F(s) s (s) Figure 2.4: Beam element axial force (after Leung [1])Substituting Eq. 2.40 into Eq. 2.43:F(s) = F0 _s0A(a + bs)ds= F0 _s0(s)(a + bs)ds (2.44)where(s) is the lineal density (kg/m) along the element. The model assumes linearlytapered properties, so the lineal density is:(s) =_0 +l 0ls_(2.45)where the zero subscript represents the property at the inboard node, and thel subscriptrepresents the property at the outboard node. Substituting Eq. 2.45 into Eq. 2.44 andintegrating:F(s) = F0 cs ds22es33(2.46)28where:c = 0ad =(l 0)a + 0bll(2.47)e =(l 0)bl2.2.5 Oscillation about Steady StateFor a small oscillation about steady state, the nodal velocity is:{ q} = { qv}, (2.48)and the kinetic and strain energies are:T =12{ qv}T[M]{ qv} +12{qs +qv}T[K]{qs +qv} + T0 +{ qv}T{f } + { qv}T[G]{qs +qv}{F}T{qs +qv}, (2.49)and:U=12{qs +qv}T([Ke] + [Kg]){qs +qv}. (2.50)Substituting these into Eq. 2.11, the Lagrangian becomes:[M]{ qv} + 2[G]{ qv} + ([Ke] + [Kg] [K]){qv} = {0}. (2.51)292.2.6 Beam Element MatricesThe beam element matrices are assembled using Eqs. 2.24 and 2.30. The inertia andstiffness properties vary linearly along the element as in Eq. 2.45 for the mass and:EA(s) =_EA0 +EAl EA0ls_, (2.52)for the extensional stiffness, and:EI(s) =_EI0 +EIl EI0ls_, (2.53)for the bending stiffnesses, and:GJ(s) =_GJ0 +GJl GJ0ls_, (2.54)for the torsional stiffness, and:J

(s) =_J

0 +J

l J

0ls_, (2.55)for the polar moment of inertia about the centroidal axis.The shape function matrices are:[N1] =_(2s3 3ls2+ l3)/l3, (s3 2ls2+ l2s)/l2, (2.56)(2s3 3ls2)/l3, (s3 ls2)/l2and:[N2] =_(2s3 3ls2+ l3)/l3, (s3 2ls2+ l2s)/l2, (2.57)(2s3 3ls2)/l3, (s3 ls2)/l230and:[N3] = [(1 s/l), s/l] (2.58)where l as the length of the element. These are standard shape (or interpolation) functionsfor a space frame element, which can be found in Raos nite element text [28]. The nodaldegrees of freedom are ordered as in Figure 2.5 Lj3q1Lj1 q3 q2 Lj2 q4q5 q6q7q10q12 q9q11q8Figure 2.5: Nodal degrees of freedomAppendix A list the entries for the beam element matrices.2.2.7 Solution for Blade Frequencies and Mode ShapesThemodel frequenciesandmodeshapesarefromaneigenvaluesolutionofEqua-tion 2.51. This equation cannot be solved with standard techniques because of the skew-symmetricgyroscopicmatrix. Meirovitch[42]outlinesasolutionthatavoidscomplexarithmetic, which combines the matrices into symmetric ones. Given the stiffness matrix:K = [Ke] + [Kg] [K], (2.59)31Meirovitch makes a modied stiffness matrix:K=__KM1K KM1KGTM1K K+GTM1G__, (2.60)and a modied mass matrix:M=__K 00 M__, (2.61)which assembles into a standard eigenvalue problem with 2n entries:K{xr} = rM{xr}, (2.62)where n is the number of degrees of freedom, {xr} is an eigenvector (r=1, 2, . . . , 2n),andris an eigenvalue. The solutions consist ofn pairs of repeated eigenvalues andnpairs of associated eigenvectors yr and zr, where yr is the real part and zr is the imaginarypart of eigenvector xr.The author used Meirovitchs method for verication with Leungs results,which isdescribed in the Section 2.3.4. However, to use the modes from Meirovitchs method forsolving the response, a dynamics code would require twice the number of degrees of free-dom; one mode for the real part and one mode for the complex part. Therefore the nalmethod in CurveFEM assumed that the gyroscopic terms for the wind turbine blade areconsidered small. This is standard practice in the industry. The blade motions perpendicu-lar to the axis of rotation are small in comparison to motions parallel to the axis, as in bladeap motion. Therefore, the analysis neglects the gyroscopic matrix and the eigensolutionfor Equation 2.51 is:K{xr} = rM{xr}, (2.63)with K from Eq. 2.59. The solutions consist of n eigenvalues and n real eigenvectors. Theauthor veried this method for a rotating, tapered beam with published results from Baner-32jee, Su, and Jackson [43], which Section 2.3.3 describes. Other verications describedbelow are the deection and vibration of a non-rotating curved beam, and the vibration ofa rotating curved beam. Appendix B describes the nite element program structure.2.3 Finite Element Method Verication Results2.3.1 Curved Beam Deection VericationIn following Kosmatkas work [2], the author veried the nite element stiffness matri-ces (Ke) with an example curved beam under deection. Figure 2.6 shows the cantileverbeam. The beam had six elements with applied forces at the end in the x and y directions. 1 in. 2 in. y z Material: Aluminum 6061-T6 20 in. 0 90 y x Finite element representation (6 elements) Figure 2.6: Curved beam for deection verication (after Kosmatka [2])The nite element results compare well with analytical results from Roarks [44] for beams33of varying curvature, shown in Figure 2.7 for thex-force direction and Fig. 2.8 for they-force direction.0.00E+002.00E-034.00E-036.00E-038.00E-031.00E-021.20E-021.40E-021.60E-021.80E-020 10 20 30 40 50 60 70 80 90Beam Curvature, degreesDisplacement, in.0.00E+002.00E-044.00E-046.00E-048.00E-041.00E-031.20E-031.40E-031.60E-03Slope (note negative)Finite ElementRoark'sy-displacmentx-displacmentslopeFx = 100 lbsFigure 2.7: Curve beam deection and slope for Fx= 100 lb340.00E+005.00E-031.00E-021.50E-022.00E-022.50E-023.00E-023.50E-024.00E-024.50E-020 10 20 30 40 50 60 70 80 90Beam Curvature, degreesDisplacement, in.0.00E+005.00E-041.00E-031.50E-032.00E-032.50E-033.00E-033.50E-03Slope (note negative)Finite ElementRoark'sy-displacmentx-displacmentslopeFy = 100 lbsFigure 2.8: Curve beam deection and slope for Fy= 100 lb352.3.2 Non-Rotating Curved Beam Modes VericationAgain following Kosmatkas work [2], the author veried the nite element stiffnessand mass matrices (KeandM) with an example of curved beam free-response. Figure2.9showsthecurvedbeamwithxed-xedboundaryconditions. Table2.2showsthe 1 in. 2 in. y z Material: aluminum 6061-T6 Six finite elements 20 in. 90 x y Figure 2.9: Curved beam for natural response (after Kosmatka [2])nite element results compared to Kosmatka [2] and Blevins [45]. The results are in goodagreement with previous results for curved beams.Table 2.2: Curved beam modes vericationDescription Frequency from Frequency from FiniteDescription Blevins [45] (Hz) Kosmatka [2] elementFirst out-of-plane 473.2 472.2 475Second out-of-plane - 1343.8 1355First in-plane 2277.5 2237.3 22312.3.3 Rotating Tapered Beam VericationTheauthorthenconductedsimulationswithataperedrotatingbeamtocomparetowork by Banerjee, Su, and Jackson [43]. Banerjee et al uses the dynamic stiffness method36for solution, neglects the Coriolis terms and only considers the apping (or out-of-plane)motion. TheauthorsvericationalsoneglectedtheCoriolismatrix, anddidnotusedMeirovitchs method for solution. Instead, the model used the standard eigenvalue solutionas in Eq. 2.63.The verication used the beam with dimensions and properties shown in Figure 2.10. yxz E = 1.00 107 psi = 0.00305 slugs/in3 20 2 4 2 1 Figure 2.10: Rotating tapered beam (all dimensions in inches)The author also compared with results for a non-spinning tapered beam by Mabie andRogers [46] and with an equivalent model with the NREL MODES program [29], describedin Section 2.1. The verication was conducted at rotational speeds representing the rangeof Banerjees et al analysis. Figure 2.11 shows the verication results for the rst threemodes, which show good agreement amongst the methods.370100020003000400050000 100 200 300 400 500 600 700 800 900Rotational RPMNatural frequency, rad/sBanerjee (2006)CurveFEMModes ProgramMabie (1972)Mode 1Mode 2Mode 3Figure 2.11: Tapered beam verication results382.3.4 Rotating Beams from Leung and Fung 1988 VericationThe author compared the nite element program against results from Leung [1] forseveral examples. The rst example was for a rotating horizontal cantilever (Figure 2.12)of four elements. Y Z X Figure 2.12: Horizontal Cantilever (after Leung [1])Thevericationrunswereatzerorotationalspeedandanon-dimensionalspinningrate of 20 shown in Figure 2.13, with the lines digitized from Leung [1] and the trianglesrepresenting the authors results.The results show good agreement at zero rotational speed, which is comparable to re-sults available in Roarks [44]. For the maximum rotational speed, the rst two bendingmodes match well, but for CurveFEM the third and fourth modes have slightly lower natu-ral frequency.3901020304050600 2 4 6 8 10 12 14 16 18 20Spinning rate (Al4/EI )1/2 Natural frequency (Al4/EI)1/2Lines from Leung and FungSymbols are CurveFEMFigure 2.13: Horizontal Cantilever Verication Results40The next example is a rotating beam inclined45from the rotational axis, shown inFigure 2.14. 45 Y Z X Figure 2.14: Inclined Cantilever (after Leung [1])Again the results show good agreement at zero rotational speed. For the maximum ro-tational speed, for CurveFEM the second and fourth modes show lower frequencies (Figure2.15).4104812162024280 1 2 3 4 5Spinning rate (Al4/EI )1/2Natural frequency (Al4/EI)1/2Lines from Leung and FungSymbols are CurveFEMFigure 2.15: Inclined Cantilever Verication Results42The next two examples are for an L- beam shown in Figure 2.16. A B C L1 L2 Figure 2.16: L-beam (after Leung [1])The results show good agreement at zero rotational speed. For the90 beam (Figure2.17), the second and third modes at the highest rotational speed show lower natural fre-quency for CurveFEM.For the 30 beam (Figure 2.18), the results showgood agreement for the rst and secondmodes at the highest rotational speed, and slightly lower frequency for the third mode forCurveFEM.430204060801000 4 8 12 16 20 24 28Spinning rate (rad/s)Natural frequency (rad/s)Lines from Leung and FungSymbols are CurveFEMFigure 2.17: L-beam with = 90 Verication Results440204060801000 4 8 12 16 20 24 28Spinning rate (rad/s)Natural frequency (rad/s)Lines from Leung and FungSymbols are CurveFEMFigure 2.18: L-beam with = 30 Verication Results452.3.5 Rotating Curved Beam VericationThe author veried CurveFEMwith rotating curved beamresults fromWang and Mahren-holtz [47]. This is the only reference the author found for modes of rotating curved beams.WangandMahrenholtzdevelopedamodelforbendingoftheminoraxisofarotatingcurved beam with the Bernoulli-Euler approximation and with Coriolis terms neglected.The authors used a Galerkin method for solution. The author compared results for straightbeams and beams with inplane curvature of 0.6 (arclength/radius) and zero hub radius. Thebeam properties were according to the following relation:EIL4= 1,whereL is the total beam length. The author used 20 elements of equal length for theCurveFEM model. The major axis bending, torsional, and extensional stiffnesses were1000 times the value of the inplane bending stiffness. Figure 2.19 shows the comparison.AsinWangandMahrenholtz[47], theCurveFEMresultsshowlittledifferenceinfre-quency between straight and curved beams. The results match well for the rst mode. Forthe second mode, both analysis predict lower frequencies for the curved blade; however,CurveFEM predicts lower frequencies overall.46051015202530350 1 2 3 4 5 6 7 8 9 10Rotational rate, rad/sNatural frequency, rad/sCurveFEM straightCurveFEM curvedWang straightWang curvedSecond ModeFirst ModeFigure 2.19: Rotating curved beam verication results472.4 Conclusions and RecommendationsThe verication results show excellent agreement between the authors nite elementanalysis and previously published results for curved beam deection and natural frequen-cies. The results also show excellent agreement with previous work on rotating taperedbeams. Forrotatingbeamsthat arenot straight, theresultsshowgoodagreement forthe fundamental mode; however, CurveFEM underpredicts the second bending mode fre-quency. This may be due to differences in the solution method. Wang and Mahrenholtz[47] do not mention their numerical procedure, and Leung and Fung [1] use a Newtonianprocedure. Without experimental results, it is not clear which method is most correct.For future work, the author recommends the following:Add pitch angle to the analysisAdd coning angle to the analysisAdd center of mass and elastic axis offsets to the analysisVerify natural frequencies and mode shapes with experimental data on rotating curvedbeamsThefollowingchapterdescribeshowtheauthorusedtheCurveFEM-derivedmodeshapes and frequencies in a modied wind-turbine dynamics code.48Chapter 3Dynamic Analysis of Curved WindTurbine Blades3.1 IntroductionThis chapter presents the development of the equations of motion for the curved bladeto be used in the wind turbine analysis code FAST. The new program is called CurveFASTto distinguish it from the existing version of FAST. This development is separate from thenite element modeling of the blade that is used to solve for the blade mode shapes, asexplained in Chapter 2. These mode shapes enter in the equations of motion, shown below.This development follows the theoretical development of FAST. Currently there is noFAST theory manual, however, the author obtained from the developer, Jason Jonkmanof NREL, several documents outlining the equations of motion for FAST. Much of thebackground is also documented in Jonkmans masters thesis [26], which worked from theoriginal code developed at Oregon State University [23]. The basis of the method is knownas Kanes Method, which can be found in Kane and Levinsons text [25].FAST uses the aerodynamic subroutines of AeroDyn [17] for calculation of the aero-dynamic forces.Moriarity and Hansen [48] cover the theory for AeroDyn.AeroDyn uses49blade element momentum (BEM) theory but has several improvements to account for un-steadiness and asymmetric loading in wake. For CurveFAST, the author modied version6.10a of FAST that used version 12.58 of AeroDyn.The following development begins with dening coordinate systems, then moves sys-tematically to kinematics and then kinetics.3.2 Coordinate Systems and TransformationsThe section describes the coordinate systems unique to CurveFAST. They are similar tothose from Section 2.2.1; however, the transformations are different in order to match thoseused by the FAST2ADAMS.f90 pre-processor. The author retained the FAST2ADAMS.f90transformations because CurveFAST would be veried with Adams.3.2.1 Local Undeected AxisThe rst coordinate system unique to CurveFAST is the blade element system alignedwith the local undeected-axis with unit vectorsLjB1i(i =1, 2, 3). The superscript B1refers to blade number one, and there are similar coordinate systems for the other blades.This coordinate systemis transformed fromthe pitched coordinate system, with unit vectorsjB11, jB12, and jB13. The transformation is:___LjB11(r)LjB12(r)LjB13(r)___= [TEP(r)]___jB11jB12jB13___(3.1)The rst stage involves a transformation matrix[1] between the pitched coordinatesXY Z and the coordinates x y z (Fig. 3.1):50 j3 j1 j2 X Z Y 3j2j1jxzyl i j Figure 3.1: First transformation of element___ x y z___= [1]___XYZ___The next step is between the local coordinates xyz (principal axes) and the coordinates x y zas (Fig. 3.2):___xyz___= [2]___ x y z___The desired transformation between the xyz and the XY Z system is therefore:[TEP(r)] = [2][1]51 Lj2Lj1x y 2j1jxys

Figure 3.2: Second transformation of element with built-in structural twistwhere:___xyz___= [TEP(r)]___XYZ___The rst transformation, 1 assumes small rotations about the X and Yaxis, with:1=_TransMat(1= B1X (r), 2= B1Y(r), 0)where TransMatis the orthonormal transformation matrix subroutine used in FAST forsmall rotations about the three axes. The rotations at a particular analysis node i are:B1Xi=Xi+1 Xi1Zi+1 Zi1B1Yi=Yi+1 Yi1Zi+1 Zi1where X is the blade precurve (RefAxisxb), Yis the blade presweep (RefAxisyb) andZ is the length along the j3 axis RNodes.52After the rst transformation, the principal axes of the element (xyzaxes) make anangles (built-in structural twist) about the negative z axis. This direction is such that apositive s results in a lower angle of attack. The transformation between the systems is:___xyz___=__cos ssin s0sin scos s00 0 1_____ x y z___= [2]___ x y z___so that:[2] =__cos ssin s0sin scos s00 0 1__3.2.2 Local Deected Coordinate SystemThe next transformation is from the undeected element to the deected coordinatesystem, which is:___nB11(r)nB12(r)nB13(r)___= [TDE(r)]___LjB11(r)LjB12(r)LjB13(r)___with[TDE(r)] =_TransMat(1= B1Lx(r), 2= B1Ly(r) 3= B1Lz(r)The rotations, which are angular deections, are:___B1Lx(r)B1Ly(r)B1Lz(r)___= [TEP(r)]___B1jx(r)B1jy(r)B1jz(r)___(3.2)53The rotations in the j system (jx, jy, jz) are:B1jx(r) = B141(r)4(r) qB1F1 + B142(r)4(r) qB1F2+B143(r)4(r) qB1E1 + B144(r)4(r) qB1T1B1jy(r) = B151(r)5(r) qB1F1 + B152(r)5(r) qB1F2(3.3)+B153(r)5(r) qB1E1 + B154(r)5(r) qB1T1B1jz(r) = B161(r)6(r) qB1F1 + B162(r)6(r) qB1F2+B163(r)6(r) qB1E1 + B164(r)6(r) qB1T1In these rotation equations, as an example, the symbol B141(r) is the eigenvector com-ponent (subscript-prex 4) in the pitched (j) system for the x-component of slope (sweepdeection) for the rst blade ap mode (subscript-sufx 1) at a particular radial station forblade 1. The symbol4(r) represents the interpolation function for the x-component ofslope. The CurveFAST analysis nodes coincide with the nite element nodes and theinterpolation functions are equal to 1. In the current version of the code the variable wasnot be implemented. The symbol qB1E1 is the rst ap mode degree of freedom for blade 1.For reference, Fig. 2.5 shows the numbering of the element degrees of freedom. Table 3.1shows CurveFASTs degrees of freedom for blade 1.Table 3.1: CurveFAST Blade 1 Degrees of FreedomName DescriptionqB1F1Blade 1, First Flap ModeqB1F2Blade 1, Second Flap ModeqB1E1Blade 1, First Edge ModeqB1T1Blade 1, First Torsion ModeTo reduce the number of multiplications during the simulation, the program uses Eq.543.2 in the form:___B1Lx(r)B1Ly(r)B1Lz(r)___= [TEP(r)] []___qB1F1qB1F2qB1E1qB1T1___with the matrix [] as:[] =__B141(r) B142(r) B143(r) B144(r)B151(r) B152(r) B153(r) B154(r)B161(r) B162(r) B163(r) B164(r)__ThesubroutineCalcTEPintheleFASTIOSML.f90pre-multipliesthematrices[TEP(r)] and [] at initialization.3.2.3 Local Aerodynamics Coordinate SystemThe coordinate system for calculating and returning the aerodynamic loads is:___mB11(r)mB12(r)mB13(r)___=__cos_B1p+ B1s(r)sin_B1p+ B1s(r)0sin_B1p+ B1s(r)cos_B1p+ B1s(r)00 0 1_____nB11(r)nB12(r)nB13(r)___55This is the same transformation as in FAST, with p representing the blade pitch and s rep-resenting the built-in structural twist (both positive to lower angle of attack). The transfor-mation does not include the Lz (elastic twist) term. The transformation to the trailing-edgecoordinate system, used in aeroacoustic calculations, is also the same as FAST, with:___teB11(r)teB12(r)teB13(r)___=__cos_B1p+ B1a(r)sin_B1p+ B1a(r)0sin_B1p+ B1a(r)cos_B1p+ B1a(r)00 0 1_____mB11(r)mB12(r)mB13(r)___where a is the local aerodynamic twist, a user input, which is usually equal to the structuraltwist.3.3 KinematicsThere are several references to the model degrees of freedom in CurveFAST in thissection; for example, the indices in the partial velocity derivatives refer to the degrees offreedom. For reference, Table 3.2 shows these degrees of freedom for 2- and 3-bladed windturbines.56Table 3.2: CurveFAST Degrees of FreedomNo. 2 Bladed 3 Bladed Description1 Sg Sg Platform Surge2 Sw Sw Platform Sway3 Hv Hv Platform Heave4 R R Platform Roll5 P P Platform Pitch6 Y Y Platform Yaw7 TFA1 TFA1 First Tower Fore-Aft8 TSS1 TSS1 First Tower Fore-Aft9 TFA2 TFA2 Second Tower Fore-Aft10 TSS2 TSS2 Second Tower Fore-Aft11 Yaw Yaw Yaw12 RFrl RFrl Rotor Furl13 GeAz GeAz Generator Azimuth14 DrTr DrTr Drivetrain Flexibility15 TFrl TFrl Tail Furl16 B1F1 B1F1 Blade 1 First Flap Mode17 B1E1 B1E1 Blade 1 First Edge Mode18 B1F2 B1F2 Blade 1 Second Flap Mode19 B1T1 B1T1 Blade 1 First Torsion Mode20 B2F1 B2F1 Blade 2 First Flap Mode21 B2E1 B2E1 Blade 2 First Edge Mode22 B2F2 B2F2 Blade 2 Second Flap Mode23 B2T1 B2T1 Blade 2 First Torsion Mode24 Teet B3F1 Teeter or Blade 3 First Flap Mode25 B3E1 Blade 3 First Edge Mode26 B3F2 Blade 3 Second Flap Mode27 B3T1 Blade 3 First Torsion Mode573.3.1 Positions and DisplacementsThe position vector from the apex of the coned system (point Q) to the blade 1 node(S1) is:rQS1(r) =_hB11(r), hB12(r), hub radius + hB13(r)____jB11jB12jB13___+_uB1(r), vB1(r), wB1(r) + wB1AxRed(r)____jB11(r)jB12(r)jB13(r)___(3.4)where h1,h2, and h3 are the undeected blade element offsets shown in Fig. 3.3. In thecode, the offsets are given by RefAxisxb(i), RefAxisyb(i), and RNodesNorm(i) BldFlexL respectively. Also in the equation,u,v, and w are the deections in the jsystem. The additional axial term wAxRed is the axial reduction term from blade bending. j3 h1 j1 h3 h2 j2 Figure 3.3: Undeected blade element offsetsSimilar to the angular deections (Eq.3.3 and Fig.2.5), the above deections in the j58system are:uB1(r) = B111(r)1(r) qB1F1 + B112(r)1(r) qB1F2+B113(r)1(r) qB1E1 + B114(r)1(r) qB1T1vB1(r) = B121(r)2(r) qB1F1 + B122(r)2(r) qB1F2(3.5)+B123(r)2(r) qB1E1 + B124(r)2(r) qB1T1wB1(r) = B131(r)3(r) qB1F1 + B132(r)3(r) qB1F2+B133(r)3(r) qB1E1 + B134(r)3(r) qB1T1with the values of the interpolation functions () again equal to one.The axial reduction term is more complex. The innitesimal change in element length(see Fig. 2.3) is:dwAxRed= ds dlAxRed, (3.6)where dlAxRed is the displaced length due to axial reduction, which is:dlAxRed=_(ds)2+_usds_2+_vsds_2_1/2(3.7)The binomial expansion of Eq. 3.7 with the rst two terms is:dlAxRed ds_1 +12_us_2+12_vs_2_(3.8)From Eq. 3.6, the innitesimal change in element length is therefore:dwAxRed ds_12_us_2+12_vs_2_(3.9)59The total change in element length is:wAxRed=_l0dwAxRed= _l0_12_us_2+12_vs_2_ds (3.10)This equations resembles the terms in the strain energy relationship (Eq. 2.14), without theaxial force term. In the same manner as Eqs. 2.30 and 2.31:[KAxRede] =__[KAxReda][KAxRedb]0__, (3.11)and:[KAxReda] =_l0[N

1]T[N

1]ds, [KAxRedb] =_l0[N

2]T[N

2]ds. (3.12)Appendix A lists the components of KAxRede. With the eigenvector matrix:[] = [{1}, {2}, {3}, {4}]and the vector of degree of freedom displacements:{q} =___qB1F1qB1F2qB1E1qB1T1___the deection due to axial reduction from Eq. 3.10 is:wAxRed= 12[[]{q}]T[KAxRed][[]{q}]60where [KAxRed] is the global axial reduction matrix. For a particular nodes axial reduction,the eigenvectors and the axial reduction matrix include the element degrees of freedom atthe node in addition the degrees of freedom of all the inboard nodes. Using the transposeproperty:[[A][B]]T= [B]T[A]Tthe above relation becomeswAxRed= 12{q}T[]T[KAxRed][[]{q}]Expanding:wAxRed=12{{1}qB1F1 + {2}qB1F2 + {3}qB1E1 + {4}qB1T1}T[KAxRed]{{1}qB1F1 + {2}qB1F2 + {3}qB1E1 + {4}qB1T1}Multiplying the terms through:wAxRed=12_{1}T[KAxRed]{1}q2B1F1 + {2}T[KAxRed]{2}q2B1F2+{3}T[KAxRed]{3}q2B1E1 + {4}T[KAxRed]{4}q2B1T1 +2{1}T[KAxRed]{2}qB1F1qB1F2 + 2{1}T[KAxRed]{3}qB1F1qB1E1 +2{1}T[KAxRed]{4}qB1F1qB1T1 + 2{2}T[KAxRed]{3}qB1F2qB1E1 +2{2}T[KAxRed]{4}qB1F2qB1T1 + 2{3}T[KAxRed]{4}qB1E1qB1T1_61or:wAxRed=12_S11q2B1F1 + S22q2B1F2 + S33q2B1E1 + S44q2B1T1+2S12qB1F1qB1F2 + 2S13qB1F1qB1E1 + 2S14qB1F1qB1T1 + (3.13)2S23qB1F2qB1E1 + 2S24qB1F2qB1T1 + 2S34qB1E1qB1T1_where:Sij= {i}T[KAxRed]{j}; i, j= 1, 2, 3, 43.3.2 VelocitiesThe angular velocity of the blade 1 element body (M1) in the inertial frame is:EM1(r) =EH+HM1(r) (3.14)The analysis uses this angular velocity with the aerodynamic moments in the developmentof the generalized active forces and for the blade twist in the generalized inertia force. Thesimulation does not calculate the ap and edge rotation generalized inertia forces, which isconsistent with the Bernoulli hypothesis.The angular velocityEHis the angular velocity of the hub in the inertial frame andthe angular velocity of the element body in the hub frame is:HM1(r) =_B1jx(r),B1jy(r),B1jz(r)____jB11jB12jB13___(3.15)The time derivatives of the element angular deections come from taking the time deriva-62tive of Eq. 3.3:B1jx(r) = B141(r)4(r) qB1F1 + B142(r)4(r) qB1F2+B143(r)4(r) qB1E1 + B144(r)4(r) qB1T1B1jy(r) = B151(r)5(r) qB1F1 + B152(r)5(r) qB1F2+B153(r)5(r) qB1E1 + B154(r)5(r) qB1T1B1jz(r) = B161(r)6(r) qB1F1 + B162(r)6(r) qB1F2+B163(r)6(r) qB1E1 + B164(r)6(r) qB1T1The velocity of a blade element node in the inertial frame is:EvS1(r) =EvQ+HvS1(r) +EHrQS1(r) (3.16)The velocity in the hub frame comes from the time derivative of Eq. 3.4:HvS1(r) =HdrQS1(r)dt=_ uB1(r), vB1(r), wB1(r) + wB1AxRed(r)____jB11jB12jB13___(3.17)Pitch dynamics are not included in FAST, therefore the derivative of the j unit vectors arezero. The time derivatives of the element deections come from the time derivatives of Eq.633.5: uB1(r) = B111(r)1(r) qB1F1 + B112(r)1(r) qB1F2+B113(r)1(r) qB1E1 + B114(r)1(r) qB1T1 vB1(r) = B121(r)2(r) qB1F1 + B122(r)2(r) qB1F2+B123(r)2(r) qB1E1 + B124(r)2(r) qB1T1 wB1(r) = B131(r)3(r) qB1F1 + B132(r)3(r) qB1F2+B133(r)3(r) qB1E1 + B134(r)3(r) qB1T1and the time derivative of Eq. 3.13: wAxRed=_S11qB1F1 qB1F1 + S22qB1F2 qB1F2 + S33qB1E1 qB1E1 + S44qB1T1 qB1T1+S12( qB1F1qB1F2 + qB1F1 qB1F2) + S13( qB1F1qB1E1 + qB1F1 qB1E1) +S14( qB1F1qB1T1 + qB1F1 qB1T1) + S23( qB1F2qB1E1 + qB1F2 qB1E1) +S24( qB1F2qB1T1 + qB1F2 qB1T1) + S34( qB1E1qB1T1 + qB1E1 qB1T1)_The partial angular velocity, used for Kanes method [25], comes from the partial ve-locity derivative of Eq. 3.14:EM1r(r) =EHr+HM1r(r) (3.18)with:HM1r(r) =_B1jx(r),B1jy(r),B1jz(r)_r___jB11jB12jB13___64where the r subscript represents the partial velocity derivative. Expanding Eq. 3.18:EM1r(r) =EHr+___HM1B1F1(r) for r = B1F1HM1B1F2(r) for r = B1F2HM1B1E1(r) for r = B1E1HM1B1T1(r) for r = B1T10 otherwise(3.19)with the additional partial velocites coming from the partial velocity derivative of Eq. 3.15:HM1B1F1(r) =_B141(r)4(r), B151(r)5(r), B161(r)6(r)____jB11jB12jB13___HM1B1F2(r) =_B142(r)4(r), B152(r)5(r), B162(r)6(r)____jB11jB12jB13___HM1B1E1(r) =_B143(r)4(r), B153(r)5(r), B163(r)6(r)____jB11jB12jB13___HM1B1T1(r) =_B144(r)4(r), B154(r)5(r), B164(r)6(r)____jB11jB12jB13___65The partial linear velocity comes from the partial velocity derivative of Eq. 3.16:EvS1r(r) =EvQr+HvS1r(r) +EHrrQS1(r) (3.20)with:HvS1r(r) =_ uB1(r), vB1(r), wB1(r)_r___jB11jB12jB13___Expanding Eq. 3.20:EvS1r(r) =EvQr+___EHrrQS1(r) for r = 4,5, . . . ,14HvS1B1F1(r) for r = B1F1HvS1B1F2(r) for r = B1F2HvS1B1E1(r) for r = B1E1HvS1B1T1(r) for r = B1T1EHTeet rQS1(r) for r = Teet0 otherwise(3.21)66with the additional partial velocites coming from the partial velocity derivative of Eq. 3.17:HvS1B1F1(r) = B111(r)1(r)jB11+ B121(r)2(r)jB12+(B131(r)3(r) {S11qB1F1 + S12qB1F2 +S13qB1E1 + S14qB1T1})jB13HvS1B1F2(r) = B112(r)1(r)jB11+ B122(r)2(r)jB12+(B132(r)3(r) {S22qB1F2 + S12qB1F1 +S23qB1E1 + S24qB1T1})jB13HvS1B1E1(r) = B113(r)1(r)jB11+ B123(r)2(r)jB12+(B133(r)3(r) {S33qB1E1 + S13qB1F1 +S23qB1F2 + S34qB1T1})jB13HvS1B1T1(r) = B114(r)1(r)jB11+ B124(r)2(r)jB12+(B134(r)3(r) {S44qB1T1 + S14qB1F1 +S24qB1F2 + S34qB1E1})jB133.3.3 AccelerationsIn Kanes method [25], angular accelerations are:ENi( q, q, q, t) =NDOF

r=1ENir(q, t) qr +NDOF

r=1ddt_ENir(q, t)_ qr +ddt_ENit(q, t)_for each body Ni in the system.The term required for the angular acceleration comes from the time derivative of thepartial angular velocity (Eq. 3.19):67ddt_EM1r(r) =ddt_EHr_+___EHEM1B1F1(r) for r = B1F1EHEM1B1F2(r) for r = B1F2EHEM1B1E1(r) for r = B1E1EHEM1B1T1(r) for r = B1T10 otherwiseLinear accelerations are:EaXi( q, q, q, t) =NDOF

r=1EvXir(q, t) qr +NDOF

r=1ddt_EvXir(q, t)_ qr +ddt_EvXit(q, t)_(3.22)for each point Xi in the system.The term required for the linear acceleration is given by the time derivative of the partial68velocity (Eq. 3.21):ddt_EvS1r(r) =ddt_EvQr_+___EHr_HvS1(r) +EHrQS1(r)for r = 4,5,6ddt_EHr_rQS1(r)+EHr_HvS1(r) +EHrQS1(r)for r = 7,8, . . . ,14[S11 qB1F1 + S12 qB1F2 + S13 qB1E1S14 qB1T1]jB13+EHHvS1B1F1(r) for r = B1F1[S22 qB1F2 + S12 qB1F1 + S23 qB1E1S24 qB1T1]jB13+EHHvS1B1F2(r) for r = B1F2[S33 qB1E1 + S13 qB1F1 + S23 qB1F2S34 qB1T1]jB13+EHHvS1B1E1(r) for r = B1E1[S44 qB1T1 + S14 qB1F1 + S24 qB1F2S34 qB1E1]jB13+EHHvS1B1T1(r) for r = B1T1ddt_EHTeet_rQS1(r)+EHTeet _HvS1(r) +EHrQS1(r)for r = Teet0 otherwise3.4 KineticsKanes equations of motion [25] for a simple holonomic system are:Fr + Fr= 0 (r = 1, 2, . . . , NDOF)69where the generalized active forces are:Fr=w

i=1EvXir FXi+ENir MNiand the generalized inertia forces are:Fr=w

i=1EvXir (mNiEaXi) +ENir (EHNi)where the forces (FXi) are applied at the center of mass point (Xi) for each rigid body(Ni). The time derivative of the angular momentum of rigid bodyNi about its center ofmass Xi in the inertial reference frame (E) is:EHNi=INiENi+ENi INiENi(3.23)where INiis the inertia dyadic of the body.FAST uses Kanes equation of motion in matrix form, as in:[C(q, t)]{ q} + {f( q, q, t)} = {0}or:[C(q, t)]{ q} = {f( q, q, t)}3.4.1 Generalized Inertia ForcesFor the generalized inertia forces, the program determines the contribution from eachblade element and the tip-node body. In the ensuing analysis, the author has only developedthe equations for blade 1; there are similar equations for the other blades.70The generalized inertia force for a blade 1 element that includes torsion is:Fr |M1(r) = NodeMass(r)EvS1r(r) EaS1(r)dr (3.24)+EM1r(r) _EHM1(r)_where NodeMass is:NodeMass(r) = AdjBlMsB1 BMassDenB1(r) DRNodesB1(r)where AdjBlMs is the blade mass adjustment factor in the blade le, BMassDen is theblade mass density per unit length for the analysis node, andDRNodes is the elementlength. Note that, as in FAST2ADAMS.f90 pre-processor, the node mass was not adjustedfor additional length from blade sweep and curvature. Therefore, there is an error thatgrows with sweep angle. This can be countered by multiplying the blade mass densitywith:BMassDennew= BMassDenold Actual element lengthDRNodesWith the addition of twist, the model includes the generalized inertia force for the twistmotion. For the twist motion the blade element dyadic is:IM1(r) = ((InerBFLpB1(r) +InerBEdgB1(r)) DRNodes(r) +SmllNmbr)nB13(r)nB13(r)The variable InerBFlp is the ap inertia per unit length. Similarly, InerBEdg is theedge inertia per unit length. DRNodes is the element length, and SmllNmbr is a smallnumber (9.999E-4) so that an innite acceleration is not computed. This inertia is similarto the one computed in FAST2ADAMS.f90Using the Eq. 3.24 and previously derived terms, the total generalized inertia force for71blade 1 is:Fr |B1=_BldNodes

j=1NodeMass(j)EvS1r(j) __14

i=1EvS1i(j) qi_+_19

i=16EvS1i(j) qi_+EvS1Teet(j) qTeet +_14

i=4ddt_EvS1i(j)_ qi_+_19

i=16ddt_EvS1i(j)_ qi_+ddt_EvS1Teet(j)_ qTeet_+EM1r(j) _IM1(j) __14

i=4EM1i(j) qi_+_19

i=16EM1i(j) qi_+EM1Teet(j) qTeet +_14

i=7ddt_EM1i(j)_ qi_+_19

i=16ddt_EM1i(j)_ qi_+ddt_EM1Teet(j)_ qTeet_+EM1(j) IM1(j) EM1(j)__+ . . .Continuing with the blade 1 generalized inertia force:72Fr |B1= . . . +mB1TipEvS1r(BldFlexL) __14

i=1EvS1i(BldFlexL) qi_+_19

i=16EvS1i(BldFlexL) qi_+EvS1Teet(BldFlexL) qTeet +_14

i=4ddt_EvS1i(BldFlexL)_ qi_+_19

i=16ddt_EvS1i(BldFlexL)_ qi_+ddt_EvS1Teet(BldFlexL)_ qTeet_EM1r(BldFlexL) _IB1Tip__14

i=4EM1i(BldFlexL) qi_+_19

i=16EM1i(BldFlexL) qi_+EM1Teet(BldFlexL) qTeet +_14

i=7ddt_EM1i(BldFlexL)_ qi_+_19

i=16ddt_EM1i(BldFlexL)_ qi_+ddt_EM1Teet(BldFlexL)_ qTeet_+EM1(BldFlexL) IB1TipEM1(BldFlexL)_Rearranging the blade 1 generalized inertia forces (for a two-bladed turbine) into matrix73form:[C(q, t)]|B1(Row, Col) =_BldNodes

j=1NodeMass(j)EvS1Row(j) EvS1Col(j)+EM1Row(j) IM1(j) EM1Col(j)_+mB1TipEvS1Row(BldFlexL) EvS1Col(BldFlexL) +EM1Row(BldFlexL) IB1TipEM1Col(BldFlexL)(Row, Col = 1, 2, . . . , 14; 16, 17, 18, 19; 24)74and:{f( q, q, t)}|B1(Row) =_BldNodes

j=1NodeMass(j)EvS1Row(j) __14

i=4ddt_EvS1i(j)_ qi_+_19

i=16ddt_EvS1i(j)_ qi_+ddt_EvS1Teet(j)_ qTeet_+EM1Row(j) _IM1(j) __14

i=7ddt_EM1i(j)_ qi_+_19

i=16ddt_EM1i(j)_ qi_+ ddt_EM1Teet(j)_ qTeet_+EM1(j) IM1(j) EM1(j)__mB1TipEvS1Row(BldFlexL) __14

i=4ddt_EvS1i(BldFlexL)_ qi_+_19

i=16ddt_EvS1i(BldFlexL)_ qi_+ddt_EvS1Teet(BldFlexL)_ qTeet_EM1Row(BldFlexL) _IB1Tip__14

i=7ddt_EM1i(BldFlexL)_ qi_+_19

i=16ddt_EM1i(BldFlexL)_ qi_+ddt_EM1Teet(BldFlexL)_ qTeet_+EM1(BldFlexL) IB1TipEM1(BldFlexL)_(Row = 1, 2, . . . , 14; 16, 17, 18, 19; 24)3.4.2 Generalized Active ForcesThe blade 1 generalized active forces include the gravitational forces, the aerodynamicforces, the elastic forces, and the damping forces, as in:Fr= Fr|GravB1 + Fr|AeroB1 + Fr|ElasticB1 + Fr|DampB175Blade Gravity ForcesThe generalized forces due to gravity are in the same form as the current version ofFAST, and is:Fr|GravB1= _BldFlexL0B1(r)gEvB1r(r) z2dr mB1TipgEvB1r(BldFlexL) z2(subscript r = 3, 4, . . . , 14; 16, 17, 18, 19; 24)wheregis the gravitational acceleration andz2is the tower base unit vector that pointsvertical. In matrix form:[C(q, t)]GravB1= 0and:{f( q, q.t)}|GravB1(Row) = _BldFlexL0B1(r)gEvB1Row(r) z2dr mB1TipgEvB1Row(BldFlexL) z2(Row = 3, 4, . . . , 14; 16, 17, 18, 19; 24)Blade Aerodynamic ForcesThe application of the aerodynamic forces is the same as in FAST, with:Fr|AeroB1=_BldFlexL0_EvS1r(r) FS1AeroB1(r)+EM1r(r) MM1AeroB1(r)dr +EvS1r(BldFlexL) FS1TipDragB1(BldFlexL)(subscript r = 1, 2, . . . , 14; 16, 17, 18, 19; 24)76where FS1AeroB1(r) and MM1AeroB1(r) are aerodynamic forces and pitching moments applied topoint S1on blade 1 expressed per unit span. The pitching moment can include momentsdue to airfoil moment coefcients (CM) and moments due to offsets of the aerodynamiccenter to the element center of mass.In matrix form the generalized active aerodynamic forces are:[C(q, t)]AeroB1= 0and:{f( q, q.t)}|AeroB1(Row) =_BldFlexL0_EvS1Row(r) FS1AeroB1(r)+EM1Row(r) MM1AeroB1(r)dr +EvS1Row(BldFlexL) FS1TipDragB1(BldFlexL)(Row = 1, 2, . . . , 14; 16, 17, 18, 19; 24)As stated in Chapter 1, FAST uses the AeroDyn subroutines to compute the aerody-namic forces. However, there is no accounting for blade sweep in AeroDyn. The authortherefore made corrections to the two-dimensional airfoil tables that are used to computethe forces.The aerodynamic correction for sweep takes into account that the pressure forces onthe blade are dependent on the relative velocity normal to the rotor leading edge. This isthe same analogy used for xed wing aircraft (see Hoerner [49]). This assumption is notused by researchers in the rotorcraft community; however, researchers with the STAR pro-gram studied the use of this assumption. They used a lifting-surface wake code originallydeveloped for helicopter rotors and modied for wind turbines [50] to study the effect ofsweep on a model wind turbine. It was found that the spanwise loading closely matchedthe typical modication to the lift coefcient for sweep:77CL= CLcos2 (3.25)where is the angle between the relative velocity and the chord line of the blade section.The drag coefcient was not altered for sweep. The results showed the assumption wasacceptable to about 20 of sweep. For the STARanalysis, however, Adams provides relativevelocities already normal to the leading edge to AeroDyn. The author did not modify thelift coefcient; however, the drag coefcient was:CD=CDcos2(3.26)Figure 3.4 shows the angle for the swept blade, assuming no out-of-plane curvature(RefAxisxb). The sweep angle is the angle between the r velocity vector and theperpendicular to the reference axis. The formula for is: = where is the angle between the pitch axis and radial position vector, and is:= where is the pre-sweep which is machined at the root face the balance the root pitchingmoment and is the blade sweep angle. The formula is the arctangent of the deriva-tiveoftheformulafortheSTAR-bladelocalsweepdisplacementofthereferenceaxis(RefAxisyb):RefAxisyb = Tip Sweep (distance)_Blade Station Start of sweepBlade Length Start of sweep_Sweep Exponent(3.27)for blade stations outboard of the sweep onset station.78 RNodes Radial position vector (no curvature, RefAxisxb = 0) ) Swept reference axis RefAxisyb Local Tangent Pitch or j3 axis Figure 3.4: Sweep angle for swept blade79In addition to the lifting surface studies, van Dam and Saephan built a computationuid dynamics (CFD) model of the STAR [51] rotor, which showed good agreement to theBEM corrected for sweep as above. The researchers therefore had further condence inusing this assumption for sweep.The CurveFAST modication duplicates the current method used in the Adams model,by supplying the relative velocities to AeroDyn in the plane perpendicular to the leadingedge, and the drag coefcients in the two-dimensional airfoil tables are adjusted accordingto Eq. 3.26.Blade Elastic ForcesFor the blade elastic forces, the model uses the results from the blade nite elementanalysis. From the FAST kinetics development [23] the Blade 1 generalized elastic forceis:Fr|ElasticB1= VB1qr(3.28)The model approximates the blade elastic potential energy by using four mode shapes, asin:V =12q2B1F1 {1}T{Ke} {1} +12q2B1F2 {2}T{Ke} {2} + (3.29)12q2B1E1 {3}T{Ke} {3} +12q2B1T1 {4}T{Ke} {4}whereforexample {1}istherstmodeeigenvectorand {Ke}istheelasticstiffnessmatrix for the non-rotating blade. The eigenvectors and stiffness matrix are referenced tothe pitched j system. CurveFAST internally calculates the elastic stiffness matrix to reducethe truncation error in the matrix multiplications. The derivation of this matrix is in Chapter802. The eigenvectors are included at the end of blade data input le.Substituting Eq. 3.29 into Eq. 3.28:Fr|ElasticB1=___qB1F1 {1}T{Ke} {1} for r = B1F1qB1F2 {2}T{Ke} {2} for r = B1F2qB1E1 {3}T{Ke} {3} for r = B1E1qB1T1 {4}T{Ke} {4} for r = B1T10 otherwiseThe matrix multiplications ofandKeare computed at initialization in the subroutinecoeff in the le FASTSML.f90.In matrix form, the elastic forces for blade 1 are:[[C(q, t)]|ElasticB1= 0{f( q, q, t)}|ElasticB1(16) = qB1F1 {1}T{Ke} {1}{f( q, q, t)}|ElasticB1(17) = qB1E1 {3}T{Ke} {3}{f( q, q, t)}|ElasticB1(18) = qB1F2 {2}T{Ke} {2}{f( q, q, t)}|ElasticB1(19) = qB1T1 {4}T{Ke} {4}{f( q, q, t)}|ElasticB1(other rows) = 0As in FAST, blade damping is assumed to be proportional to the stiffness, which isa typical assumption in modal analysis. The damping, or dissipative function in the La-grangiandevelopment fromniteelement analysisasinRao [28], Eq. 12.20, isgivenas:R =12{ q}T[D]{ q} (3.30)where Q is the vector of nodal displacments and {C} is the damping matrix. For a partic-81ular mode (e.g. rst blade ap):{q}B1F1= qB1F1 {1} (3.31)Damping proportional to mass and stiffness is (Rao[28], Eq. 12.81):[D] = a [M] + b [Ke] (3.32)where a and b are constants and [M] is the mass matrix. For a particular mode, the modaldamping constant is (Rao[28], Eq. 12.85):i=a + b2i2iwhere is the square root of the mode eigenvalue in rad/s. Setting a to zero and solvingfor b:b =2ii(3.33)Substituting Eq. 3.33 into Eq. 3.32 for a particular mode:[D]i=2ii[Ke] (3.34)Substituting Eqs. 3.31 and 3.34 into Eq. 3.30 the dissipation becomes:R =12 q2B1F1211{1}T{Ke} {1} +12 q2B1F2222{2}T{Ke} {2} + (3.35)12 q2B1E1233{3}T{Ke} {3} +12 q2B1T1244{4}T{Ke} {4}82Similar to the elastic force the generalized damping force takes the following form:Fr|DampB1= RB1 qr(3.36)Substituting Eq. 3.36 into Eq. 3.35:Fr|DampB1=___211 qB1F1 {1}T{Ke} {1} for r = B1F1222 qB1F2 {2}T{Ke} {2} for r = B1F2233 qB1E1 {3}T{Ke} {3} for r = B1E1244 qB1T1 {4}T{Ke} {4} for r = B1T10 otherwiseIn matrix form, the damping forces for blade 1 are:[[C(q, t)]|DampB1= 0{f( q, q, t)}|DampB1(16) = 211 qB1F1 {1}T{Ke} {1}{f( q, q, t)}|DampB1(17) = 233 qB1E1 {3}T{Ke} {3}{f( q, q, t)}|DampB1(18) = 222 qB1F2 {2}T{Ke} {2}{f( q, q, t)}|DampB1(19) = 244 qB1T1 {4}T{Ke} {4}{f( q, q, t)}|DampB1(other rows) = 0833.5 Blade Loads3.5.1 Blade Root LoadsFAST constructs the equations of motion with loads that are typical quantities of inter-est. When the equations of motion are solved, these loads are readily available for outputand do not require further computations. For the blades, these loads are the blade rootforces and moments. Kanes method breaks the forces into the following (for a 2-bladedturbine):FXiSource( q, q, q, t) =_24

r=1FXiSourcer(q, t) qr_+FXiSourcet( q, q, t) (3.37)whereFXiSourcerare the partial forces andFXiSourcetis all the components ofFXiSource not ofthis form. For the moments:MNi@XiSource( q, q, q, t) =_24

r=1MNi@XiSourcer(q, t) qr_+MNi@XiSourcet( q, q, t) (3.38)where MNi@XiSourcer are the partial moments and MNi@XiSourcet are all the components of MNi@XiSourcenot of this form.Blade 1s generalized active force in terms of the loads acting on the hub center of mass(point C) is:Fr|B1=EvCr FCB1 +EHr MHB1(r = 1, 2, . . . , 24)Because the hub is rigid, the forces on the hubs center of mass are related to the bladeroot loads (FS1B1(0) and MHB1(0)) by:FS1B1(0) = FCB1and:MHB1= MHB1(0) +rCS1(0) FS1B1(0)84or:MHB1= MHB1(0) +_rQS1(0) rQCFS1B1(0)Because of the two-point velocity in Kane and Levinson [25]:EvCr=EvQr+EHrrQCthe generalized active force expands to:Fr|B1=_EvQr+EHrrQC_ FS1B1(0) +EHr_MHB1(0) +_rQS1(0) rQCFS1B1(0)_(r = 1, 2, . . . , 24)Applying the cyclic permutation law of the scalar triple product:a (b c) = (a b) cthe generalized active force becomes:Fr|B1=EvQr FS1B1(0) +EHr {rQCFS1B1(0)} +EHr {MHB1(0) +_rQS1(0) rQCFS1B1(0)}(r = 1, 2, . . . , 24)85which simplies to:Fr|B1=EvQr FS1B1(0) +EHr_MHB1(0) +rQS1(0) FS1B1(0)(3.39)(r = 1, 2, . . . , 24)This generalized active force must produce the same effects as the generalized active andinertia forces from blade 1. Therefore:Fr|B1= Fr |B1 + Fr|AeroB1 + Fr|GravB1 + Fr|ElasticB1 + Fr|DampB1(r = 1, 2, . . . , 24)BecauseEvQrandEHrare equal to zero unlessr=1, 2, . . . , 14; Teet, the blade elasticand damping forces do not contribute to the root loads. Therefore:Fr|B1= Fr |B1 + Fr|AeroB1 + Fr|GravB1(r = 1, 2, . . . , 14; Teet)Expanding:Fr|B1=_BldFlexL0EvS1r(r) _FS1AeroB1(r) B1(r)gz2 B1(r)EaS1(r)dr +_BldFlexL0EM1r(r) _MM1AeroB1(r) IM1(r) EM1(r)EM1(r) IM1(r) EM1(r)_dr +EvS1r(BldFlexL) _FS1TipDragB1mB1Tip_gz2 +EaS1(BldFlexL)_+EM1r(BladeFlexL) _IB1TipEM1(BldFlexL) EM1(BldFlexL) IB1TipEM1(BldFlexL)(r = 1, 2, . . . , 14; Teet)86Because:EvS1r(r) =EvQr+HvS1r(r) +EHrrQS1(r)the equation above becomes:Fr|B1=_BldFlexL0_EvQr+HvS1r(r)_FS1AeroB1(r) B1(r)gz2B1(r)EaS1(r)dr +_EvQr+HvS1r(BldFlexL)_FS1TipDragB1 mB1Tip_gz2 +EaS1(BldFlexL)_+_BldFlexL0_EHrrQS1(r)_FS1AeroB1(r) B1(r)gz2B1(r)EaS1(r)dr +_EHrrQS1(r)_FS1TipDragB1 mB1Tip_gz2 +EaS1(BldFlexL)_+_BldFlexL0EM1r(r) _MM1AeroB1(r) IM1(r) EM1(r)EM1(r) IM1(r) EM1(r)_dr +EM1r(BladeFlexL) _IB1TipEM1(BldFlexL) EM1(BldFlexL) IB1TipEM1(BldFlexL)(r = 1, 2, . . . , 14; Teet)BecauseHvS1r(r) is equal to zero (rigid hub) andEM1r(r) is equal toEHrwith the con-87straint (r = 1, 2, . . . , 14; Teet), the preceeding equation simplies to:Fr|B1=_BldFlexL0EvQr(r) _FS1AeroB1(r) B1(r)gz2 B1(r)EaS1(r)dr +EvQr _FS1TipDragB1 mB1Tip_gz2 +EaS1(BldFlexL)_+_BldFlexL0_EHrrQS1(r)_FS1AeroB1(r) B1(r)gz2B1(r)EaS1(r)dr +_EHrrQS1(r)_FS1TipDragB1 mB1Tip_gz2 +EaS1(BldFlexL)_+_BldFlexL0EHr_MM1AeroB1(r) IM1(r) EM1(r)EM1(r) IM1(r) EM1(r)_dr +EHr_IB1TipEM1(BldFlexL) EM1(BldFlexL) IB1TipEM1(BldFlexL)(r = 1, 2, . . . , 14; Teet)With the cyclic permutation law of the scalar triple product, the preceeding equation be-88comes:Fr|B1=_BldFlexL0EvQr(r) _FS1AeroB1(r) B1(r)gz2 B1(r)EaS1(r)dr +EvQr _FS1TipDragB1 mB1Tip_gz2 +EaS1(BldFlexL)_+_BldFlexL0EHr_rQS1(r) _FS1AeroB1(r) B1(r)gz2B1(r)EaS1(r)_dr +EHr_rQS1(BldFlexL)_FS1TipDragB1 mB1Tip_gz2 +EaS1(BldFlexL)__+_BldFlexL0EHr_MM1AeroB1(r) IM1(r) EM1(r)EM1(r) IM1(r) EM1(r)_dr +EHr_IB1TipEM1(BldFlexL) EM1(BldFlexL) IB1TipEM1(BldFlexL)(r = 1, 2, . . . , 14; Teet)The root force and moment come from the comparison of the preceeding equation with Eq.(3.39):FS1B1(0) =_BldFlexL0_FS1AeroB1(r) B1(r)gz2 B1(r)EaS1(r)dr +FS1TipDragB1 mB1Tip_gz2 +EaS1(BldFlexL)89and:MHB1(0) +rQS1(0) FS1B1(0) =_BldFlexL0_MM1AeroB1(r) IM1(r) EM1(r) EM1(r) IM1(r)EM1(r)dr +_BldFlexL0rQS1(r) _FS1AeroB1(r) B1(r)gz2B1(r)EaS1(r)dr +rQS1(BldFlexL) _FS1TipDragB1mB1Tip_gz2 +EaS1(BldFlexL)_+_IB1TipEM1(BldFlexL) EM1(BldFlexL) IB1TipEM1(BldFlexL)or:MHB1(0) =_BldFlexL0_MM1AeroB1(r) IM1(r) EM1(r) EM1(r) IM1(r)EM1(r)dr +_BldFlexL0rQS1(r) _FS1AeroB1(r) B1(r)gz2B1(r)EaS1(r)dr +rQS1(BldFlexL) _FS1TipDragB1mB1Tip_gz2 +EaS1(BldFlexL)_+_IB1TipEM1(BldFlexL) EM1(BldFlexL) IB1TipEM