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rotor thrust shows negligible azimuthal variation as a result of these effects. For

example, on two-bladed machines, a wind shear exponent of 0.2 results in a rotor

thrust variatio n of about þ/1 percent.

Tower shadow loading results in a sinusoidal tower top displacement at blade

passing frequency (see Figure 5.32). Figure 5.43 illustrates the variation of rotor

thrust with wind speed for stall and pitch regulated 40 m diameter three-bladed

machines.

5.12.4 Operational loads due to turbulence (stochastic compon ent)

Analysis in the frequency domain

Except near the top of the tower, the dominant source of fore-aft stochastic tower

bending moments is rotor thrust. The standard deviation of rotor thrust can be

expressed in terms of the turbulence intensity and the cross correlation function

between wind ﬂuctuations at different points on the rotor, following the method

used for deriving the standard deviation of stochastic blade root bending moment

in Section 5.7.5. As before, a linear relation between the wind ﬂuct uations and the

resultant load ﬂuctuations is assumed, so that the perturbation of loading per unit

length of blade, q, at radius r is given by

q ¼

rrc

dÆ

u (5:25)

100

0 5 10 15 20 25 30

Wind speed (m/s)

Rotor thrust (kN)

40 m diameter three-bladed rotor

with ‘TR’ blades

Rotational speed = 30 r.p.m.

500 kw Stall-regulated machine

Pitch-regulated machine

with 400 kW power limit

Figure 5.43 Rotor Thrust During Operation in Steady, Uniform Wind: Variation with Wind

Speed for Similar Stall-regulated and Pitch Regulated Machines

TOWER LOADING 305

and the perturbation of rotor thrust by

˜T ¼

r

dÆ



uc(r)r dr (5:128)

where the integral sign

signiﬁes that the integration is carried out over the wh ole

rotor. Hence the following expression for the variance of the rotor thrust is obtained:



r

dÆ





þþ

, r

,0)c(r

)c(r

(5:129)

where r

, r

, 0) is the normalized cross correlation function, k

, r

,0)=

for

points at radii r

and r

on the same or on different blades. k

, r

, 0) is given by

Equation (5.51), with  replaced by the phase angle between the two blades on

which r

and r

measured. For a three-bladed, 40 m diameter rotor and an integral

length scale of 73.5 m, the reduction in the standard deviation of the stochastic rotor

thrust ﬂuctuations is about 20 percent due to the lack of correlation of the wind

speed variations over the rotor. If the machine is rotating at 30 r.p.m. in an 8 m=s

wind and the turbulence intensity is 20 percent, the rotor thrust standard deviation

will be about 9 kN – i.e., 25 percent of the steady value

The derivation of the expression for the power spectrum of rotor thrust parallels

that for the pow er spectrum of blade root bending moment (Section 5.7.5), yielding:

(n) ¼

r

dÆ



þþ

uJ, K

, r

, n)c(r

)c(r

(5:130)

where S

uJ, K

, r

, n) is the rotationally sampled cross spectrum for points at radii

and r

on blades J and K respectively. Note that on a mach ine with three blades,

A, B and C, S

uJ,K

, r

, n) is complex when J and K are different, but

uA,B

, r

, n) and S

uA,C

, r

, n) are complex conjugates, so the double integral in

Equation (5.130) is still real. An example power spectrum of rotor thrust for a three

bladed machine is shown in Figure 5.44. It can be seen that there is some concentra-

tion of energy at the blade passing frequency of 1.5 Hz due to gust slicing, but that

the effect is not large. The concentration effect is signiﬁcantly greater for two-bladed

machines (see Figure 5.45). This shows the power spectrum of rotor thrust for a

two-bladed machine with the same blade planform, but rotating 22.5 percent faster

to give comparable performance.

In addition to thrust ﬂuctuations, longitudinal turbulence will also cause rotor

torque ﬂuctuations and in-plane rotor loads due to differential loads on different

blades, bot h of which will result in tower sideways bending moments. The expression

for the in-plane component of aerodynamic lift per unit length, F

(r) ¼

c(r) sin , can be differentiated with res pect to the wind ﬂuctuation as follows:



rc(r)

sin C

] ¼

rc(r)

[WfU

(1  a) þ ugC

]

ﬃ

rc(r)WC

þ sin 

dÆ



306 DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES

100

150

200

250

300

350

400

450

0.01 0.1 1 10

Frequency (Hz, logarithmic scale)

Power spectral density nS(n) (kN)

Wind speed = 8 m/s

Integral length scale = 73.5 m

Turbulence intensity = 20%

Von Karman spectrum

nS(n) for tower base fore-aft

bending moment divided by

hub height, H

nS(n) for Thrust x 10

Rotational speed = 0.5 Hz

Tower natural frequency = 1.16 Hz

Damping ratio = 0.022

Figure 5.44 Power Spectra of Rotor Thrust and Resultant Tower Base Fore–aft Bending

Moment for Three-bladed, 40 m Diameter Turbine

100

150

200

250

300

350

400

450

500

0.01 0.1 1 10

Frequency (Hz, logarithmic scale)

Power spectral density nS(n) (kN)

Wind speed = 8 m/s

Integral length scale = 73.5 m

Turbulence intensity = 20%

Von Karman spectrum

nS(n) for tower base fore-aft

bending moment divided by

hub height, H

nS(n) for Thrust x 10

Rotational speed = 0.5 x 1.225 Hz

Tower natural frequency = 1.16 x (1.225/1.5) Hz

Damping ratio = 0.022

Figure 5.45 Power Spectra of Rotor Thrust and Resultant Tower Base Fore–aft Bending

Moment for Two-bladed, 40 m Diameter Turbine

TOWER LOADING 307

so, approximately,



r

dÆ



c(r)r

=dÆ

þ sin 



(5:131a)

Thus the standard deviation of rotor torque is approximately given by



r

dÆ





c(r)

=dÆ

þ sin 





(5:131b)

(which parallels Equation (5.26)) provided the relationship between blade loading

and wind speed ﬂuctuation remains linear and the turbulence length scale is large

compared with rotor diameter. Equation (5.131b) can be used to derive an expres-

sion for the variance of the rotor torque in the same way as for rotor thrust above.

At the top of the tower the stochastic M

(i.e., side-to-side) moment due to rotor

torque ﬂuctuations is typically of the same order of magnitude as the stochastic M

(i.e., fore-aft) moment due to differential out-of-plane loads on the rotor, but at the

tower base the dominant effect of rotor thrust loading means that the stochastic

side-to-side moments are usually signiﬁcantly less than the stochastic fore-aft

moments before the excitation of tower resonance is taken into account.

Analysis in the time domain

As noted in Section 5.7.5, there are situations, such as operation in stalled ﬂow,

when the linear relationship between blade loading and wind speed ﬂuctuations

required for analysis in the frequency domain does not apply. In these cases,

recourse must be made to analysis in the time domain using wind simulation

techniques such as described in Section 5.7.6.

5.12.5 Dy namic response to operational loads

The power spectrum of rotor thrust will usually contain some energy at the tower

natural frequency, leading to dynamic magniﬁcation of deﬂections, and hence of

tower bending moments. The power spectrum of hub deﬂection, S

(n), resulting

from the excitation of the tower ﬁrst fore-aft ﬂexural mode, is related to the power

spectrum of rotor thrust by

(n) ¼

(n)

[(1  n

)

þ 4

]

(5:132)

This relation is analagous to Equation (5.90), and derived in the same way.

The amplitude of tower base fore-aft moment at resonance in the ﬁrst mode, M

can be derived from the corresponding amplitude of hub deﬂection, x

, as follows

308 DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES

¼ ø

m(z)(z)z dz ¼ ø

m(z)(z)z dz

m(z)

(z)dz

(5:133)

The quotient on the right-hand side is close to unity because of the dominance of

the tower head mas s, so, substituting k

for ø

, the equation reduces to

¼ x

H, which applies at any exciting frequency. Hence the power spect rum

for the tower base fore-aft bending moment due to rotor thrust loading is given by

My1

(n) ¼ S

(n)H

[(1  n

)

þ 4

]

(5:134)

The aerodynamic damping is almost entirely provided by the rotor, the dampi ng

ratio for the ﬁrst tower mode being approximately



¼ N

r

dÆ

rc(r)dr

(5:135)

where N is the number of blades (see Section 5.8.4). The overall damping ratio is

obtained by adding this to the structural damping ratio for the tower (see Table

5.4), and is generally low compared to the blade ﬁrst mode damping because of the

large tower head mass. The effect of a low damping ratio is illustrat ed by the power

spectrum of fore-aft tower bending moment shown in Figure 5.44, which has a very

high peak at the tower natural frequency of 1.16 Hz, despite this frequency being

somewhat remo ved from the blade passing frequency of 1.5 Hz. The damping ratio

is calculated as 0.022, consisting of 0.019 due to aerodynamic damping and 0.03 due

to structural damping (for a welded steel tower).

In the example shown in Figure 5.44, the tower dynamic response increases the

standard deviation of the tower base fore-aft bending moment by 15 percent.

However, the effect of tower dynamic response results in a larger increase of 25

percent in the case of the two-bladed machine featured in Figure 5.45 despite the

reduction in tower natural frequency in proportion to blade-passing frequency.

It is important to note that the rotor provides negligibl e aero dynamic damping in

the side-to-side direction, so that effectively the only damping present is the

structural damping. This means that, even though the side-to-side loadi ngs are

small in relation to the fore-aft loads, the side-to-side tower moment ﬂuctuations

can sometimes approach the fore-aft ones in magnitude.

5.12.6 Fatigue loads and stresses

The tower moments at height z are related to the hub-height loads as follows:

(z, t ) ¼ F

(H , t):(H  z) þ M

(H , t) M

(z, t ) ¼F

(H , t):(H  z) þ M

(H , t)

(z, t ) ¼ M

(H , t)(5:136)

TOWER LOADING 309

For three-bladed machines, the ﬁve hub-height fatigue loads are almo st entirely

stochastic, because the deterministic load component is either constant (for a given

mean wind speed) or negligible, and it is instructive to consider how they relate to

one another. Recognizing that the centre of any gust lying off the rotor centre will be

located at a random azimuth, then it is clear that the rotor out-of-plane loads, i.e., the

moment about the horizontal axis, M

(H , t), the hub moment about the vertical axis,

(H , t), and the rotor thrust, F

(H , t), will all be statistically independent of each

other. The same will apply to the rotor in-plane loads – the rotor torque, M

(H, t),

and the sideways load, F

(H , t). However, as the out-of-plane and in-plane loads on

a blade element are both assumed to be proportional to the local wind speed

ﬂuctuation, u, it follows that the rotor torque ﬂuctuations will be in phase with the

rotor thrust ﬂuctuations, and the rotor sideways load ﬂuctuations will be in phase

with the ﬂuctuations of the hub moment about the horizontal axis, M

(H , t).

The above relationships have implications for the combination of fatigue loads.

Clearly the power spect rum of the fore-aft tower moment at height z, S

(z, n), can

be obtained by simply adding the power spectrum of the hub moment about the

horizontal axis to (H  z)

times the power spectrum of the rotor thrust. Similarly

the power spectrum of the side-to-side tower moment at height z, S

(z, n), can be

obtained by adding the power spectrum of the rotor torque to ( H  z)

times the

power spectrum of the rotor sidew ays load.

Having obtained power spectra for the M

, M

and M

moments at height z, the

corresponding fatigue load spectra can be derived with reasonable accuracy by

means of the Dirlik method described in Section 5.9.3. As the tower stress ranges

will be enhanced by tower resonance, the input power spectra should incorporate

dynamic magniﬁcation, as outlined in Section 5.12.5.

Fatigue stress ranges due to bending about the two axes can easil y be calculated

separately from the M

(z) and M

(z) fatigue spectra, but the stress ranges due to

the two fatigue spect ra combined cannot be calculated precisely because of lack of

information about phase relati onships. However, as noted above, the M

(H )

component of the M

(z) ﬂuctuations is in phase with the F

(H ) component of the

(z) ﬂuctuations, and the F

(H ) component of the M

(z) ﬂuctuations is in phase

with the M

(H ) component of the M

(z) ﬂuctuations so the stress ranges due to the

(z) and M

(z) fatigue spectra combined can be conservatively calculated as if

they were in phase too. Theoretically this means pairing the largest M

(z) and

(z) loading cycles, the second largest, the third largest and so on, right through

the fatigue spectra, and calculating the stress range resulting from each pairing. In

practice, of course, the M

(z) and M

(z) load cycles are distributed between two

sets of equal size ‘bins’, so they have to be reallocated to bins in a two-dimensional

matrix of descending load ranges, as shown in the grossly simpliﬁed example given

in Tables 5.7 and 5.8 below:

Table 5.7 Example M

and M

Fatigue Spectra

˜M

(kNm) No. of ˜M

cycles ˜M

(kNm) No. of ˜ M

cycles

200–300 5 100–150 10

100–200 15 50–100 40

0–100 80 0–50 50

310

DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES

For a circular tower, the stress ranges would have to be computed at several

points around the circumference in order to identify the location (with resp ect to

the nacelle axis) where the fatigue damage was maximum.

A simpler but poten tially cruder approach to the combination of the two fatigue

spectra is to use the ‘Damage Equivalent Load’ method. This involves the calcula-

tion of constant ampl itude fatigue loadings, M

X:Del

and M

Y:Del

, of, say 10

cycles

each, that would respectively produce the same fatigue damages as the M

and M

spectra, using the S=N curve appropriate to the fatigue detail under consideration.

If the M

and M

ﬂuctuations are treated as being in-phase as before, the combined

‘Damage Equivalent Load’ moment is

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

X:Del

þ M

Y:Del

References

American Society of Civil Engineers, (1993). ASCE 7-93: Minimum design loads for buildings and

other structures.

Armstrong, J. R. C. and Hancock, M., (1991). ‘Feasibility study of teetered, stall-regulated

rotors’ ETSU Report No. WN 6022.

Batchelor, G. K., (1953). The theory of homogeneous turbulence. Cambridge University Press, UK.

Bishop, N. W. M., Zhihua, H. and Sheratt, F., (1991). ‘The analysis of non-gaussian loadings

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Bishop, N. W. M., Wang, R. and Lack, L., (1995). ‘A frequency domain fatigue predictor for

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Table 5.8 Joint M

and M

Cycle Distribution

˜M

(kNm)

˜M

(kNm) 200–300 100–200 0–100 Total No. of M

cycles

100–150 5 5 10

50–100 10 30 40

0–50 50 50

Total No. of M

cycles 5 15 80

REFERENCES 311

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1063. Riso National Laboratory, Roskilde, Denmark.

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312

DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES

Appendix: Dynamic Response of Stationary Blade in

Turbulent Wind

A5.1 Introduction

As described in Chapter 2, the turbulent wind contains wind speed ﬂuct uations

over a wide range of frequencies, as desc ribed by the power spectrum. Although

the bulk of the turbulent energy is normally at frequencies much lower than the

blade ﬁrst mode out-of-plane frequency, which is typically over 1 Hz, the fraction

close to the ﬁrst mode frequency will excite resonant blade oscillations. This

appendix describes the method by which the resonant response may be determined.

Working in the frequency domain, expression s for the standard deviations of bot h

the tip displacement and root bending moment respons es are derived, and then the

method of deriving the peak value in a given period is described. Initially the wind

is assumed to be perfectly correlated along the blade, but subsequently the

treatment is extended to include the effect of spatial variation.

A5.2 Frequency Response Function

A5.2.1 Equation of motion

The dynamic response of a cantilever blade to the ﬂuctuating aerodynam ic loads

upon it is most conveniently investigated by means of modal analysis, in which the

the excitatio ns of the vario us different natural modes of vibration are computed

separately and the results superposed. Thus the deﬂection x(r, t) at radius r is given

by:

x(r, t) ¼

i¼1

(t)

(r)

Normally, in the case of a stationary blade, the ﬁrst mode dominates and higher

modes do not need to be considered. The equation of motion for the ith mode,

which is derived in Section 5.8.1, is as follows:

€

(t) þ c

(t) þ m

(t) ¼



(r)q(r, t)dr (A5:1)

where q(r, t) is the applied loading, f

(t) is the tip displacement, 

(r) is the non-

dimensional mode shape of the ith mode, normalized to give a tip displacement of

unity, ø

is the natural frequency in radians per second, m

is the generalized mass,

m(r)

(r)dr, and c

is the generalized damping,

cc(r)

(r)dr.

FREQUENCY RESPONSE FUNCTION 313

A5.2.2 Frequency response function

If q(r, t) varies harmonically, with frequency ø and amplitude q

(r), then it can be

shown that:

(t) ¼



(r)q

(r)dr

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

(ø

 ø

)

þ (c

)

cos(øt þ 

)



(r)q

(r)dr

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

(1  ø

=ø

)

þ (c

)

cos(øt þ 

) (A5:2)

Deﬁning k

¼ m

, and noting that the damping ratio 

¼ c

=2m

, this becomes:

(t) ¼



(r)q

(r)dr

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

(1  ø

=ø

)

þ 4

=ø

cos(øt þ 

) ¼ A

cos(øt þ 

) (A5:3)

The numerator



(r)q

(r)dr: is the amplitude of the equivalent loading at the tip

of the cantilever that would result in the same tip displacement as the loading

q(r, t), and is known as the generalized load with respect to the ith mode , Q

(t).

Thus the ratio between the tip displacement amplitude and the amplitude of the

generalized load is



(r)q

(r)dr:

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

(1  ø

=ø

)

þ 4

=ø

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

(1  n

)

þ 4

¼jH

(n)j (A5:4)

The ratio jH

(n)j is the modulus of the complex frequency response function, H

(n),

and its square can be used to transform the power spectrum of the wind incident on

the blade into the power spectrum of the ith mode tip displacement. Thus, in the

case of the dominant ﬁrst mode, the tip displacement in response to a harmonic

generalized loading, Q

(t), of frequency n is given by

(R, t) ¼ f

(t) ¼ Q

(t)jH

(n)j

and the power spectrum of the ﬁrst mode tip deﬂection is S

(n) ¼ S

(n)jH

(n)j

In what follows, the simplifying assumption is made initially that the wind is

perfectly correlated along the blade.

314 DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES