Burton T. (et. al.) Wind energy Handbook

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(kNm)

Time (s)

-50

-100

100

150

200

250

300

0 20406080100120

Out-of-plane

In-plane

Figure 5.34 Blade Root Bending Moment in Turbulent Wind

Auto spectral density ((Nm)

/Hz)

Frequency (Hz)

1.0e+03

1.0e+04

1.0e+05

1.0e+06

1.0e+07

1.0e+08

1.0e+09

1.0e+10

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Hub thrust force

Blade root moment

Figure 5.35 Spectra of Out-of-plane Loads in Turbulent Wind

BLADE DYNAMIC RESPONSE 285

2P,4P etc. Only the peaks at multiples of 3P remain, since at these frequencies the

three blades act in phase with each other.

This effect is even more signiﬁcant in the in-plane load spectra (Figure 5.36). Of

the blade load peaks at multiples of 1P, only the relatively small peaks at 3P and 6P

come through to the hub torque. The 1P peak in the blade load, which is dominated

by gravity, is particularly large, but it is completely eliminated from the hub torque.

The to wer peak at 0.4 Hz is visible in both loads. A large blade load peak at the ﬁrst

in-plane blade vibrational mode at 4.4 Hz is also seen, but this is a mode which does

not include any rotation at the hub, and consequently is not seen in the hub torque.

Some higher frequency blade modes (not shown) will be coupled with hub rotation.

5.8.11 Ae roelastic stability

Aeroelastic instability can arise when the change in aerodynamic loads resulting

from a blade displacement is such as to exacerbate the displacement rather than

diminish it, as is normally the case. A theoretical example would be a teetering

rotor operating in stalled ﬂow, where the rate of change of lift coefﬁcient with angle

of attack is negative, so that the aerodynamic damping is negative likewise. In such

circumstances, teeter excursions would be expected to grow until the limits of the

negative damping band or of the teeter stops were reached. In practice, this

phenomenon can be avoided if the blade is designed so that the blade root ﬂapwise

bending moment increases monotonically with wind speed over the full wind

speed operational range (Armstrong and Hancock, 1991).

Auto spectral density (N

/Hz)

Frequency (Hz)

1.0e+03

1.0e+04

1.0e+05

1.0e+06

1.0e+07

1.0e+08

1.0e+09

1.0e+10

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Hub torque

Blade root moment

Figure 5.36 Spectra of In-plane Loads in Turbulent Wind

286

DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES

A real instance of incipient aeroelastic instability was the development of an

edgewise blade resonance under stalled conditions on some larger three-bladed

machines. A negative rate of change of lift coefﬁcient with angle of attack is

believed to have been the prime cause – see Section 7.1.9.

Another potential instance of aeroelastic instability is classical ﬂutter, encoun-

tered in the design of helicopter rotors, in which the blade structure is such that

out-of-plane ﬂexure in the downwind direction results in blade twisting, causing an

increase in the angle of attack. During the development of some of the early large

machines, the dangers of aeroelastic instability were considered to be a real concern,

and much analysis work was directed to demonstrating that individual turbine

designs would not be susceptible to it. However, partly no doubt because of the

high torsional rig idity of the closed cell hollow structure adopted for most wind

turbine blades, aeroelastic instability has not yet been found to be critical in

practice, and stability analyses are no longer regarded as an essential part of the

design process. This may change, however, if designs become more ﬂexible.

5.9 Blade Fatigue Stresses

5.9.1 Methodology for blade fatigue design

The v eriﬁcation of the adequacy of a blade design in fatigue requires knowledge of

the fatigue loading cycles expected over the lifetime of the machine at different

radii, derivation of the resultant stress cycles and calculation of the corresponding

fatigue damage number in relation to kno wn fatigue properties of the material. The

procedure is less or more complicated, depending upon whether blade loading in

one or two planes is taken into account. If bending about only the weaker principal

axis is taken into account, considering only aerodynamic lift forces, the steps

involved are as follows.

(1) Derive the individual fatigue load spectra for each mean wind speed and for

each radius. This is a non-trivial task because, unless wind simulation is used,

the information on the periodic and stochastic load components is available in

different forms, i.e., as a time history and a power spectrum respective ly.

Sections 5.9.2 and 5.9.3 consider methods of addressing this difﬁculty.

(2) Synthesize the complete fatigue load spectrum at each radius from the separate

load spectra for each mean wind speed, including start-ups and shutdowns (see

Section 5.5.1).

(3) Convert the fatigue load cycles (expressed as bending moments) to fatigue

stresses by dividing by the appropriate sec tion modulus. (The section modulus

with respect to a particular principal axis is deﬁned as Second Moment of Area

of the cross section about that axis divided by the distance of the point under

consideration from the axis.)

BLADE FATIGUE STRESSES 287

(4) Sum the fatigue damage numbers, n

, according to Miner’s rule, for each

moment range ‘bin’ in the fatigue load spectrum, according to the appropriate

SN curve for the material. SN curves for different blade materials are

considered in Section 7.1.6 and 7.1.7, together with the allowance necessary for

mean stress.

Sections 5.9.2 and 5.9.3 are concerned with the ﬁrst step of the sequence above. For

a given mean wind speed, the periodic component of blade loading will be invariant

over time, and the stochastic component will be stationary. As noted in Section 5.7.5,

the stochastic component can be analysed either in the frequency domain (provided

that a linear relationship between incident wind speed and blade loadings can be

assumed) or in the time domain, i.e., by using wind simulation. Section 5.9.2

considers how the deterministic and stochastic components may be combined if the

latter have been analysed in the frequency domai n, while Section 5.9.3 looks in detail

at the option of assessing fatigue damage completely in the frequency domain.

If the fatigue damage resulting from both in-plane and out-of-plane loading is to

be computed, it is necessary to revise the ordering of the steps above, in order to

derive the periodic and stochastic components of the stress variation for each point

under consideration and for each mean wind speed. For a chosen point, the

procedure becomes as shown below.

(A1) For a given mean wind speed, calculate the time histories of the bending

moments about the principal axes resulting from the periodic load compo-

nents over one blade rotation. The derivation of aerodynamic moments from

blade element loads is illustrated in Figure 5.37.

(A2) Convert these bending moment time histories to stress time histories by

dividing by the appropriate section modulus, and adding them together.

(B) For the same mean wind speed, convert the power spectrum of the stochastic

bending moment component (which, because of the linearity assumption,

arises from ﬂuctuating lift only) to a power spectrum of stress at the chosen

point.

stochastic stress components, using the methods of Sections 5.9.2 and 5.9.3.

(D) Repeat the above steps for the other mean wind speeds.

(E) Add together the fatigue damages arising at each mean wind speed to obtain

the total fatigue damage during normal running.

5.9.2 Combination of deterministic and stochastic components

Previous sections have shown how the deterministic (i.e., periodic) and stochastic

components of blade bending moments can be characterized in terms of time

288 DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES

histories and power spectra respectively. Unfortunately the spectral description of

the stochastic loading is not in a suitable form to be combined with the time history

of the periodic loading, but this difﬁculty can be resolved by one of two methods, as

follows.

(1) The power spectrum of the stochastic component can be transformed into a time

history by inverse Fourier transform, which can then be added directly to the

time history of the periodic component. Applications of this method have been

reported by Garrad and Hassan (1986) and Warren et al. (1988). With the

subsequent development of wind simulat ion techniques, this method is no

longer commonly used, because the use of transformations to generate time-

histories of wind speed rather than of wind loading avoids the need to assume

that wind speed and wind loading are linearly related when deriving the power

spectrum of the stochastic load component.

θ′

= M

cos θ' + M

sin θ'

= (C

cos φ + C

sin φ)

. c(r)

⫺F

= (C

sin φ + C

cos φ)

. c(v)

= M

sin θ' + M

cos θ'

. c(r)dr

c(r).dr

⫺F

.dr

.u M

σ′

= –

(

)

= ∫(

⫺

)(r-r*)dv

= ∫(F

)(r-r*)dv

Figure 5.37 Derivation of Blade Bending Stresses at Radius r



due to Aerodynamic Loads

BLADE FATIGUE STRESSES 289

(2) A probability density function for the load cycle ranges can be derived

empirically, based on the spectral properties of the power spectrum of the

stochastic and periodic components of loading co mbined.

The second approach is considered in the next Section.

5.9.3 Fat igue prediction in the frequency domain

The probability density function (p.d.f.) of peaks of a narrow band, Gaussian

process are given by the well-know n Rayle igh distribution. As each peak is

associated with a trough of similar mag nitude, the p.d.f. of cycle ranges is Rayleigh

likewise.

Wind turbine blade loading cannot be co nsidered as narrow band, despite the

concentration of energy at the rotational frequenc y by ‘gust slicing’ (Section 5.7.5),

and neither can it be considered as Gaussian because of the presence of periodic

components. Dirlik (1985) produced an empirical p.d.f. of cycle ranges applicable to

both wide and narrow band Gaussian processes, in terms of basic spectral proper-

ties determined from the power spectrum. This was done by considering 70 power

spectra of vari ous shapes, computing their rainﬂo w cycle range distributions and

ﬁtting a general expression for the cycle range p.d.f. in terms of the ﬁrst, second and

fourth spectral moments. Dirlik’s expression for the cycle range p.d.f. is:

p(S) ¼

Z=Q

( Z

=2R

)

þ D

( Z

=2)

ﬃﬃﬃﬃﬃﬃﬃ

(5:117)

where

Z ¼ S=

ﬃﬃﬃﬃﬃﬃﬃ

, D

2(x

 ª

)

1 þ ª

, D

(1  ª  D

þ D

)

1  R

, D

¼ 1  D

 D

Q ¼

1:25(ª  D

 D

, R ¼

ª  x

 D

(1  ª  D

þ D

)

, x

ﬃﬃﬃﬃﬃﬃﬃ

, ª ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ



(n)dnS



(n) is the power spectrum of stress,

and S is the cycle stress range:

Although the Dirlik cycle range p.d.f. was not intended to apply to signals

containing periodic components, several investigations (Hoskin et al. (1989), Mor-

gan and Tindal (1990), Bishop et al. (1991)) have been carried out to determine its

validity for wind turbine fatigue damage calculations, using monitored data for

ﬂapwise bending from the MS1 wind turbine on Orkney. Cycle range p.d.f.s were

calculated from power spectra of monitored strains using the Dirlik formula and

fatigue damage rates derived from these p.d.f.s compared with damage rates

290 DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES

derived directly from the monitored signal by rainﬂow cycle counting. The ratio of

damage calculated by the Dirlik method to damage calculated by the rainﬂow

method ranged from 0.84 to 1.46, from 1.01 to 2.48 and from 0.73 to 2.34 in the three

investigations listed above, using a S=N curve exponent of 5 in each case, as the

blade structure was of steel. In view of the fact that the calculated damage rates

vary as the ﬁfth power of the stress ranges, these results indicate that the Dirlik

method is capable of giving quite accurate results, despite the presence of the

periodic components.

There are two main drawbacks to the application of the Dirlik formula to power

spectra containing periodic components. First, the presence of large spikes in the

spectra due to the periodic components renders them very different from the

smooth distributions Di rlik originally considered, and second, information about

the relative phases of the periodic components is lost when they are transformed to

the frequency domain. Morgan and Tindal (1990) illustrate the effect of varying

phase angles by a comparison of plots of (cos øt þ 0:5 cos 3øt) and (cos øt

 0:5 cos 3øt) which is reproduced in Figure 5.38. For a material with a S=N curve

exponent of 5, stresses conforming to the ﬁrst time history would result in 5.25

times as much fatigue damage as stresses conforming to the second.

Bishop, Wang and Lack (1995) developed a modiﬁed form of the Dirlik formula

to include a single periodic component, using a neural network approach to

determine the different parameters in the formula from computer simulations.

Madsen et al. (1984) adopted a different approach to the problem of determini ng

fatigue damage resulting from combined stochastic and periodic loading, involving

the derivation of a single equivalent sinusoidal loading that would produce the same

fatigue damage as the actual loading. The method applies a reduction factor, g,

which is dependent on bandwidth, to account for the reduced cycle ranges implicit

in a wide band as opposed to a narrow band process, and utilizes Rice’s p.d.f. for the

peak value of a single sinusoid combined with a narrow band stochastic process,

-1.5

-1

-0.5

0.5

1.5

100

110

120

130

140

150

160

170

180

190

200

210

220

230

240

250

260

270

280

290

300

310

320

330

340

350

360

Blade azimuth (degrees)

Combined signal

Components in phase : (cosωt + 0.5cos3ωt)

Components out-of-phase : (cosωt - 0.5cos3ωt)

Figure 5.38 Effect of Variation of Phase Angle between Harmonics on Combined Signal

BLADE FATIGUE STRESSES 291

substituting half the maximum range of the periodic signal, including harmonics, for

the amplitude of the sinusoid. A fuller summary is given in Hoskin, Warren and

Draper (1989). They concluded, along with Morgan and Tindal (1990), that the

Madsen method yielded slightly less accurate fatigue damage values than the Dirlik

method for the MS1 monitored data for ﬂapwise bending referred to above.

5.9.4 Wind simulation

Wind simulation, which was introduced in Section 5.7.6, has two signiﬁcant

advantages over the methods described above for fatigue damage evaluation. First,

it can handle non-linear relationships between wind speed ﬂuctuations and blade

loadings in the calculation of stochastic loads, and second, it avoids the difﬁculty of

deriving the fatigue stress ranges arising from combined periodic and stochastic

load components. It is therefore currently the favoured method for detailed fatigue

design. The procedure is essentially as follows.

(1) Generate a three-dimensional ‘run of wind’ for the chosen mean wind speed,

with the desired shear proﬁle and tower shadow correction.

(2) Perform a step-by-step dynamic analysis on the turbine operating in this wind

ﬁeld, to obtain in-plane and out-of-plane bending moment time histories at

different radii.

(3) Convert these bending moment time histories to time histories of bending

moments about the principal axes.

(4) Compute stress–time histories at chosen points on each cross section.

(5) Derive the number of cycles in each stress range ‘bin’ by Rainﬂow Cycle

Counting (see Section 5.9.5 below).

(6) Scale up the cycle numbers in line with the predicted number of hours of

operation at the chosen mean wind speed.

(7) Calculate corresponding fatigue damage numbers based on the applicable S=N

curve.

(8) Repeat above steps for different mean wind speeds, and total the resulting

fatigue damages at each point.

A computationally simpler alternative is to generate a one-dimensional ‘run of

wind’ (in which only the longitudinal component of turbulence is modelled), and

run a number of simulations at different, ﬁxed yaw angles.

The duration of wind simulations is limited by available computer power, with a

time history length of 600 s being frequently chosen. A consequence of this is that a

single simulation will not provide an accurate picture of the infrequent high-stress

292 DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES

range fatigue cycles, which can have a disproportionate effect on fatigue damage

for materials with high m value such as those used for blades. However, this

inaccuracy can be reduced (and quantiﬁed) by running several simulations with

different random number seeds at each wind speed – see Thomsen (1998).

5.9.5 Fatigue cycle counting

As noted in Sectio n 5.9.4, the dynamic analysis of turbine behaviour in a simulated

wind ﬁeld yields time histories of loads or stresses which then need to be processed

to abstract details of the fatigue cycles. There are two established methods of fatigue

cycle counting: the reservoir method and the rainﬂow method, both of which yield

the same result.

In the reservoir method, the load or stress history (with time axis horizontal) is

imagined as the cross section of a reservoir, which is successively drained from

each low point, starting at the lowest and working up. Each draining operation then

yields a load or stress cycle (see BS 5400 (1980) for a full desc ription).

The rainﬂow method was ﬁrst proposed by Matsuishi and Endo in 1968, and its

title derives from the concept of water ﬂowing down the ‘rooves’ formed when the

time history is rotated so that the time axis is vertical. However, the following

description not involving the rainﬂow analogy may be easier to understand.

The ﬁrst step is to reduce the time history to a series of peaks and troughs, which

are then termed extremes. Then each group of four successive extremes is examined

in turn to determine whether the values of the two intermediate extremes lie between

the values of the initial and ﬁnal extremes. If so, the two intermediate extremes are

counted as deﬁning a stress cycle, which is then included in the cycle count, and the

two intermediate extremes are deleted from the time history. The process is contin-

ued until the complete series of extremes forming the time history has been

processed in this way. Then the sequence remaining will consist simply of a

diverging and a converging part from which the ﬁnal group of stress ranges can be

extracted (see ‘Fatigue Characteristics’ in the IEA series of Recommended Practices

for Wind Turbine Testin g and Evaluation (1984) for a full description of the method

and for details of algorithms that can be used for automating the process).

Although, in principle, the fatigue cycles obtained from, say, a 600 s time history

could be listed individually, it is normal to reduce the volume of data by allocating

individual cycles to a series of equal load or stress ranges known as ‘bins’ – e.g.,

0–2, 2–4, 4–6 N/mm

etc. The fatigue spectrum is then presented in terms of the

number of cycles falling into each ‘bin’.

5.10 Hub and Low-speed Shaft Loading

5.10.1 Introduction

The loadings on the hub consist of the aerodynamic, gravity and inertia loadings on

the blades and the equal and opposite (discounting hub self-weight) reactio n from

HUB AND LOW-SPEED SHAFT LOADING 293

the shaft. For ﬁxed hub machines, the loading on the shaft will include a signiﬁcant

moment arising from blade aerodynamic loads, but in the case of teetered two

bladed rotors this moment will be virtually eliminated. In either case, however, the

cantilevered low-speed shaft will experience large ﬂuctuating mome nts due to rotor

weight as it rotates. Figure 5.39 shows a low-speed shaft and front bearing in a

factory prior to assembly.

The shaft moments due to out-of-plane loads on the blades can be expressed as

moments about a pair of rotating axes, one perpendicular to blade 1 and the other

parallel to it. In the case of a three-bladed rotor, these moments are respectively as

follows:

¼ ˜M



(˜M

þ ˜M

) M

ﬃﬃﬃ

(˜M

 ˜M

)(5:118)

Here ˜ M

, ˜ M

and ˜ M

are the ﬂuctuations of the blade out-of-plane moments

about the hub centre (M

, M

and M

) about the mean value (see Figure 5.40).

5.10.2 Deterministic aerodynamic loads

The deterministic aerodynamic loads on the rotor may be split up into a steady

component, equal for each blade, and a periodic component, also equal for each

Figure 5.39 Low-speed Shaft and Front Bearing Before Assembly. The hub mounting ﬂange

at the right-hand end is bolted to a temporary support to allow the bearing to be threaded on

the shaft. (Reproduced by permission of NEG Micon)

294

DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES