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IntroductionLoads, dynamics and structural design2007-2008 4
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Harmonic loading Simple Harmonic Loading 1 F (t ) = F ⋅ sin (ω ⋅ t + ϕ ) ˆ Loading 0 1 0 1 2 3 4 5 6 Time- Gravity loads on blades- Mass imbalance rotor (1P)- Aerodynamic imbalance (1P) 27 RPM = 0.45 Hz- Small regular waves2007-2008 5
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Non-harmonic periodic loading F (t ) = Complex Cyclic Loading 0.5 1st Signal (0.5 Hz) 1 Loading 0 ∞ 0.5 0 1 2 3 4 5 6 a0 + ∑ ak F sin ( kωt + ϕ k ) Time ˆLoading 2nd Signal (1 Hz) 0.5 0 Loading 0 0.5 k 0 1 2 3 4 5 6 Time 3rd Signal (1.5 Hz) 1 0.5 Loading 0 1 2 3 4 5 6 0 Time 0.5 0 1 2 3 4 5 6 Time - Wind-shear - Yaw misalignment - Tower shadow 3P - Rotational sampling of turbulence (all 2P or 3P and multiples) 2007-2008 6
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Non-periodic random loading Random Load 1 Loading 0 1 0 1 2 3 4 5 6 Time- Turbulence (small scale)- Random waves2007-2008 7
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Other (non-periodic) loading Transients Short events Step Load Impulsive Load 1 1Loading Loading 0 0 1 1 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Time Time - Start/stop - Extreme gust - Turbine failures - Extreme waves - Storm front 2007-2008 8
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IntroductionLoads, dynamics and structural design2007-2008 9
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The effect of dynamics 2 Force [N] 1 0 F -1 1 k -2 m 2 1 Response [m] k = 1 [N/m] x 0 -1 -2 1 ?2007-2008 10
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The effect of dynamics 2 2 1 1 0 0 1 1 -1 -1 Force [N] m = 0.0005 [kg] -2 -2 2 2 1 (Static) 1 0 0 1 1 -1 -1 m = 0 [kg] m = 0.1 [kg] -2 -22007-2008 11
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The effect of dynamics 2 System F Force [N] 1 k 0 m 1 -1 -2 Fintern = k · x 2 Response [m]Internal forces ≠ external forces 1due to dynamics 0 1 -1Internal forces drive the design, m = 0.0005 [kg] -2not external forces!2007-2008 12
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Dynamic amplification factor 5 Dynamic Amplification Factor (DAF) 4 3 ResonanceDAF = Dynamic amplitude 2 Static deformation 1 0 0 1 2 3 4 5 Frequency Ratio fexcitation damping ratio = 0.1 damping ratio = 0.2 damping ratio = 0.5 fnatural Note: the DAF is defined for harmonic excitation2007-2008 13
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Character of resonance• Excitation frequency ≈ natural frequency• Large oscillations• Fatigue damage (due to severe cyclic loading)• Generally not destructive (anticipated in design) Natural frequencies of wind turbine (-components) are close to several excitation frequencies2007-2008 14
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Reduced response to loading Alleviation of (wind) loading byDAF shedding loads through motion Typical rotor loading frequencies 1 soft-soft structureTypical wave loading frequencies 1P 3P f02007-2008 17
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Increased response to loading Quasi-static or amplified response to wave loading 1-P *-P2007-2008 18
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Single degree of freedom system x k m c F F = m ⋅ && + k ⋅ x + c ⋅ x x &2007-2008 19
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Wind turbine characteristics • Stiffness • Material properties / soil properties • Buoyancy of a floating structure • Damping • Material properties / soil properties • Aerodynamic loading • Control • (Viscosity of water / radiation in soil) • Inertia • Material properties • Hydrodynamic loading (water added mass) • Entrained water mass2007-2008 20
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Linear / non-linear systemsLinear system: x(t) y(t) a·x1(t) + b·x2(t) a·y1(t) + b·y2(t) Initial condition x0: x(t) + x0 y(t) + y0Non-linear system: • No superposition possible • Possible dependency on initial conditions • Possible variation in output statistics for the same input (statistics) 2007-2008 21
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Non-linearities for wind turbines • Aerodynamic loading • Hydrodynamic loading • extreme waves • waves and currents • Speed and pitch control • some algorithms • settings for various wind speeds • Extreme deformations (2nd order effects)2007-2008 22
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IntroductionLoads, dynamics and structural design2007-2008 23
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Effects of loads and dynamics• Ultimate limit state (ULS) (maximum load carrying resistance) • Yield and buckling • Loss of bearing / overturning • Failure of critical components• Fatigue limit state (FLS) (effect of cyclic loading) • Repeated wind and wave loading • Repeated gravity loading on blade2007-2008 25
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Effects of loads and dynamics• Accidental limit state (ALS) (accidental event or operational failure, local damage or large displacements allowed) • Ship impact• Serviceability limit state (SLS) (deformations/motion, tolerance for normal use) • Blade tip tower clearance • Vibrations that may damage equipment • Tilt of turbine due to differential settlement2007-2008 26
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Design drivers of wind turbines Component Design drivers Ultimate Fatigue Tower top top mass - Tower - wind/wave Submerged wind/wave/current wind/wave tower Foundation wind/wave/current -2007-2008 27
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Importance of dynamics in design• Increase or decrease of maximum load • Affects Ultimate Limit State conditions• Increase or decrease of number of load cycles and their amplitudes • Affects Fatigue Limit State / Lifetime2007-2008 28
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Use of dynamic models 1. 2. 3.Analyse system Reduce internal Assess lifelongproperties loads and match loading resistanceAvoid resonance Validate reliabilityand instabilities Make lightest and technical and cheapest lifetime structural design 2007-2008 30
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Modelling of offshore wind turbinesStructural models of rotor, nacelle and support structure2007-2008 31
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Integrated dynamic model Offshore wind turbine ControllersWind Rotor Drive train Generator GridWave Support structure2007-2008 33
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Rotor model Distributed mass-stiffnessAerodynamic properties Beam theory FEM Tables µ, EIx,y,p Cl – α Cd – α2007-2008 34
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Drive train model Transmission ratio Igenerator Ihub+low speed shaftStiffness torsion in Damping transmissiontransmission and main shaft; suspension and generatormain shaft bending torque control2007-2008 35
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Generator model Tk,gsynchronous generator: generator 0 0 n0 n motor Tk,g induction generator: Slip generator 0 0 n0 2n0 n motor2007-2008 36
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Tower modelDistributed mass-stiffness Modal representation Beam theory FEM + = µ, EIx,y,p Deflection Deflection “Total” 1st mode 2nd mode deformation Effective reduction of DOFs 2007-2008 37
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Foundation model H kx x k xθ u M = k ⋅ kθθ θ θx 2007-2008 38
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Modelling of offshore wind turbinesDeriving parameters for foundation models2007-2008 39
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Importance of foundation model 3.5m Rotor/nacelle mass 130,000 kg 3.0m First natural frequency (Hz) 30 x 4.6m without foundation 0.34627 2800Ø x 20 6.0m 2800Ø x 25 with foundation 0.29055 40.0m 12.0m with scour 0.28219 2800Ø x 32 12.0m 2800Ø x 60 5.0m Second natural frequency (Hz) 12.0m Flange without foundation 2.2006 15.0m 2800Ø x 60 Tower 15.4m Boat Landing MSL with foundation 1.3328 2500Ø x 100 5.0m 6.0m with scour 1.2508 21.0m 100 x 3.0m Pile 3500Ø x 75 J-Tube 37.0m Scour Protection (25.0m Penetration)2007-2008 40
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Scour Pile 0 Vertical effective soil pressureSeabed General scour depth No scour condition Local scour depth Typically 1-1.5 times General scour only pile diameter Local scour condition Overburden reduction depth Typically 6 times pile diameter 2007-2008 42
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Effective fixity length Configuration Effective fixity Seabed length Effective Stiff clay 3.5 D – 4.5 D fixity Very soft silt 7D–8D length General calculations 6D Experience with 3.3 D – 3.7 D offshore turbines2007-2008 43
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Uncoupled springs Forced displacement/rotation u θ In exercise: Use M F ANSYS Macro’s and method B for Tower a monopile Method A Ignore M Ignore F Applied force/moment F MRotation u θSeabed Method B Translations Ignore θ Ignore u2007-2008 44
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Stiffness matrix Run two load cases with FEM Tower model with py-curves (See next slide) Seabed H kx x k xθ u M = k ⋅ kθθ θ θx Stiffness matrix2007-2008 45
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FEM-based pile-head stiffness1. Solve FEM for F1, M1 F1 = k xx ⋅ u1 + k xθ ⋅θ1 (F1, M1 near loading situation of interest) M 1 = kθx ⋅ u1 + kθθ ⋅θ12. Solve FEM for F2, M2 F2 = k xx ⋅ u 2 + k xθ ⋅θ 2 (F2, M2 near loading situation of interest) M 2 = kθx ⋅ u 2 + kθθ ⋅θ 23. Scratch one equation and solve kxx, kxθ, kθθ (kθx = kxθ, assume matrix equal for both loads)4. Check assumption with another FEM solution2007-2008 46
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Selection of pile foundation models• Foundation flexibility significant enough to require close consideration of modelling• Effective fixity length model dissuaded• Stiffness matrix much more favourable than uncoupled springs For exercise: Monopile in Bladed modeled with uncoupled springs (unfortunately)2007-2008 47
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Types of analysis and toolsNatural frequency and mode analysis2007-2008 49
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FEM modal analysis1 FEM analysis provides: • Natural frequencies • Mode shapes • (Pre-processed) Y Z X matrices of structural properties:Parametric support structure model generation • Mass • Stiffness • Damping 2007-2008 50
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Modes of the support structure Monopile 1st mode 2nd mode2007-2008 52
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Rayleighs method ˆ Z Velocity: SDOF : SDOF : k (x ) m((x )0 )2 1 1 Vmax = ˆ2 Tmax = & 2 2 = mω 2 (x ) 1 ˆ2 2 Maximum Maximumv(x, t ) = Ψ (x ) ⋅ Z sin (ω t ) ˆ strain energy = kinetic energy ˆ ~ Z2 ~ Z 2 ⋅ω 2 ˆ 2007-2008 53
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Rayleighs method• To estimate first natural frequency (lowest)• Based on energy conservation in undamped, free vibration: Exchange of energy between motion and strain• Mode shape must fit boundary conditions• Best estimate of mode shape results in lowest estimate of natural frequency• (Deflection under static top-load gives educated guess of mode shape)2007-2008 54
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Rayleigh’s method for stepped tower T = 2 4π 2 ≈ 4π 2 (m + meq L ) L 48 top 3 ⋅ 4 + C found ω 2 3EI eq π See document on Blackboard for: • Derivation of this equation • Explanation of EIeq, meq, Cfound2007-2008 55
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Free vibration of cylinder in water π π D 2 ⋅ a w, ⊥ − (C M − 1) ⋅ ρ ρD ⋅ v w, ⊥ − x c (v w,⊥ − x c ) 1f = CM ⋅ ρ D 2 ⋅ &&c + C D ⋅ x & & 4 4 2 Inertia force Drag force Inertia force due to moving cylinder • Still water remaining inertia term is called ‘water added mass’ • With CM ≈ 2 water added mass ≈ mass of replaced water But related to water surrounding the cylinder! • Use water added mass in analysis of natural frequency and modes 2007-2008 56
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Types of analysis and toolsResponse analysis2007-2008 57
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Types of response analysis • Static analysis with dynamic response factors • Time domain simulation • Frequency domain analysis • Mixtures All approaches can also be divided in:• Integrated combined loading• Superposition of effect of load components (wind, wave, current, gravity) 2007-2008 58
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Static + dynamic response factors• Calculate static response for several loading conditions (separate wind, wave, g)• Estimate a dynamic response factor per condition (comparison of characteristic frequencies) Typical 1.2-1.5• Superimpose results (including partial safety factors per loading type)2007-2008 59
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Superposition of forces Gravity Thrust Wind Waves + Superposition on tower current2007-2008 60
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Time domain simulation• Generate realisations of external conditions• Integrate equations of motion numerically• Analyse response (extremes, probability distribution, fatigue, …)• Repeat until statistically sound information is obtained The tool used in the exercise to do this is ‘Bladed’. See Blackboard item ‘Assignments’ for a tutorial and manuals.2007-2008 61
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Frequency domain analysis Fourier transforms and linear systems Time domain Frequency domain x(t ), y (t ), h(t ) X (ω ), Y (ω ), H (ω )y ( t ) = ∫ x (τ ) ⋅ h ( t − τ ) dτ = x ( t ) * h ( t ) Y (ω ) = X (ω ) ⋅ H (ω ) a ⋅ x1 (t ) + b ⋅ x2 (t ) → a ⋅ X 1 (ω ) + b ⋅ X 2 (ω ) → a ⋅ y1 (t ) + b ⋅ y 2 (t ) a ⋅ Y1 (ω ) + b ⋅ Y2 (ω ) 2007-2008 62
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Frequency domain analysis• Determine transfer function per load source Linearise system or use small harmonic loads• Multiply spectrum of load source with transfer function• Superimpose response spectra of different sources Due to non-linearity in the system, this procedure must be repeated for different average wind speeds2007-2008 63
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Time domain - frequency domain Time domain Frequency domain• Comprehensive non-linear • Simplified linear structuralstructural model model•Very time consuming • Very rapid calculation• Careful choice of time • Well documented windsignal turbulence spectra• Able to model control • Able only to model linearsystem dynamics control system• Established fatigue • Fatigue prediction toolsprediction tools relatively new 2007-2008 64
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Mixing time- and frequency domain + ≈TD simulation: FD analysis:- Transfer function tower top - Transfer function for wind loading loading (linearisation) - Aerodynamic damping as extra- Aerodynamic damping structural damping - Linear wave loading 2007-2008 65
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Aerodynamic dampingaerodyn. damping as frac. of crit. 6 5 soft-soft monopile 4 damp. [%] soft-stiff monotower 3 2 stiff-stiff lattice tower 1 Function of wind speed, turbine design (aerodynamic and control) and support structure! 0 5 10 15 20 25 wind speed at hub height [m/s] 2007-2008 67
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Some relevant analysis tools FEM Time Freq Rotor OffshoreANSYS XSesam X X X XAdams WT X X X XPhatas X X X XBladed X X X XFlex X X X XTurbu X X X X 2007-2008 68
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Loads and dynamics in designOverview of the process2007-2008 69
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Suggested steps• Choose a limited set of load cases• Make preliminary design based on static loads• Check for resonance*• Check extreme loads with time domain simulations*• Check fatigue damage** Adjust design when necessary2007-2008 70
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Partial safety factor method• Apply load and resistance factors to: • loads on the structure or load effects in the structure • resistance of the structure or strength of materials R• Fulfill design criterion: γ S ⋅ S ≤ γR• Combined loading with non-linear effects: • Apply one safety factor to combined load effect, determined from structural analysis of simultaneous loading2007-2008 71
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Values for safety factors• Importance of structural component w.r.t. consequence of failure considered• Typically between 0.7 and 1.35• ≤ 1.0 for favourable loads!• Load factor 1.0 for fatigue (safety in resistance)• See e.g. Offshore standard DNV-OS-J101 Design of offshore wind turbine structures2007-2008 72
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Loads and dynamics in designChoose load cases2007-2008 73
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Fundamental problems in evaluationResponse What is the true extreme?(loading + dynamics) More realisations at the same site Time Long time span (20 year) 2007-2008 74
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Load cases: Combine conditions external conditions normal extreme operational conditions normal conditions stand-by start-up The number of power production combinations that normal shut-down is required in the fault conditions standards is condition after occurrence of a fault enormous! erection2007-2008 75
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Reducing number of load cases(extremes)Select a few independent extreme conditions that might be design driving, e.g.:• Extreme loading during normal operation• Extreme loading during failure• Extreme wind loading above cut-out• Extreme wave loadingAnd combine these with reduced conditions for the other aspects (wind, wave, current)2007-2008 76
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Loads and dynamics in designMake preliminary design2007-2008 80
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Preliminary support structure design• Determine largest loads at several heights • Estimate wind, wave, current and gravity loads e.g. CD,AX = 8/9 (Betz) at Vrated & linear wave & DAF & safety • Superimpose and determine largest at each height• Dimension tower (moments / section modulus)• Rule of thump D/t • 200 tower section • ~60 driven foundation pile (see e.g. API on BB)• Estimate pile size with Blum’s method (See document on Blackboard!)2007-2008 81
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Loads and dynamics in designCheck for resonance2007-2008 82
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Campbell diagram margin 3-P ‘forbidden’ areaCharacteristic frequency Stiff- stiff 1e lead-lag Soft-stiff 1-P 1e flap Soft- 1st tower frequency stiff 1st tower frequency 1st tower frequency Soft- Soft-soft soft Wave-excitation Rotor speed2007-2008 83
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Design adaptations• Change diameters and/or wall thicknesses• Shift masses e.g. move transformer from nacelle to platform• Adjust rotor speed control e.g. skip resonance in partial load region• Change concept e.g. to braced tower / tripod2007-2008 84
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Loads and dynamics in designCheck lifetime fatigue2007-2008 85
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Fatigue Fatigue: after a number of cycles of a varying load below static strength failure occurs.F σ Fstatic failure fatigue test time number of cycles 2007-2008 86
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Variable amplitude loading ni n n n n n Miner’s Damage Rule: ∑ Ni = 1 + 2 + 3 + 4 + 5 N1 N 2 N 3 N 4 N 5 σamp N1 Damage < 1.0 n1 N2 n2 N3 n3 N3 n4 N4 n5 N5 log N2007-2008 88
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Stochastic loading Stress history can be converted to blocks of constant amplitude loadings (using counting method) Stress history stress histogramσ σamp log N Time Information about sequence lost 2007-2008 89
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Rainflow counting• Two parametric method: Range and mean• Display series of extremes with vertical time axis• Drip ‘rain’ from each extreme, stop at a larger extreme• Start and stop combine to one stress cycle2007-2008 90
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Rainflow counting• Established method• Several equivalent algorithms exist • Reservoir method • Intermediate extremes in groups of 4• Principle based on stress-strain hysteresis loops:2007-2008 91
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Frequency domain approachRayleigh: Theoretical, narrow band signals:Dirlik: Empirical, wide band signals:Used for spectra of random, Gaussian, stationary processes2007-2008 92
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Lifetime fatigue analysis HT 0-1s 1-2s 2-3s 3-4s 4-5s 5-6s s z 6-7s 7-8s 8-9s to l taDo the following for all load cases 5 -6m .5 5-5 m .5 4 -5m .5 Id gh hV > Vu ot lin, ig: w = ct_u 04 0 .0 08 .0 .3 0 .0 0 .1 0 .3(scatter diagram, operational and idle) 4-4 m .5 3 -4m .5 3-3 m .5 lu pdsas te me e ta 0 0 0 .3 0 8 .7 .7 .0 0 .4 0 .7 0 .7 2 -3m .5 0 .6 04 .0 0 .7 2-2 m .5 0 .2 0 .2 1 -2m .5 id glo :V <Vu in lin, w w ct_ 0 .0 1-1 m .5 3 .4 0 .4 3 .8 0 -1m .5 19 58 0 .7 7.7 7 0-0 m 0 8 .5 .6 1 .0 6 5 12 0 .1 01 .1 7.0 9 to l 0 ta .7 0 .0 1 .0 8.2 7.4 2 4 3 .0 1.9 0 .5 0 .0 14 6 σDetermine stress time series or PSD(PSD = Power Spectral Density) TimeDetermine stress histogram σamp(Rainflow counting – Dirlik) log N2007-2008 93
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Lifetime fatigue analysis ni Apply Miner’s rule to histogram n1 N1 D= ∑N i (damage per load case) σamp log N Apply Miner’s to all load cases: Damage of each load case (normalised to 1 unit of time) * Probability of load case * Total lifetime 0.14damage for load case with unity probability [-] 10. 1.0 Pile Load case type Pile Fatigue analysis type (absolute) damage per load case [-] 9. 5.5 m below mudline PP wind & waves 0.9 0.12 5.5 m below mudline No.1: wind & waves (full avail.) II, IP: waves 8. 0.8 probability of load case [-] No. 2: waves & no aerodyn. damping PP: wind 0.10 7. 0.7 No. 4: pure wind PP: waves probability No. 5: waves & aerodyn. damping 6. 0.6 0.08 5. 0.5 0.06 4. 0.4 3. 0.3 0.04 2. 0.2 0.02 1. 0.1 0. 0.0 0.00 prod. 8 m/s prod. 10 m/s prod. 14 m/s prod. 15 m/s prod. 17 m/s prod. 19 m/s prod. 20 m/s prod. 21 m/s prod. 8 m/s prod. 10 m/s prod. 14 m/s prod. 15 m/s prod. 17 m/s prod. 19 m/s prod. 20 m/s prod. 21 m/s Idle,low 1 Idle,low 2 Idle,low 3 Idle, high 1 Idle, high 2 Idle, high 3 Idle,low 1 Idle,low 2 Idle,low 3 Idle, high 1 Idle, high 2 Idle, high 3 2007-2008 94
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Integrated system dynamics integrated separate analyse analysis 50Equivalent bending moment 40 30 wind 20 wave 10 combined 0 wind wave super- combined loading loading position loading 2007-2008 95
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