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

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A5.3 Resonant Displacement Response Ignoring Wind

Variations along the Blade

A5.3.1 Linearization of wind loading

For a ﬂuctuating wind speed U(t) ¼ U þ u(t), the wind load per unit length on the

blade is

(t)c(r) ¼

r[U

þ 2Uu(t) þ u

(t)]c(r), where C

is the lift or drag

coefﬁcient, as appropriate, and c(r) is the local blade chord dimension. In order to

permit a linear treatment, the third term in the square brackets, which will normally

be small compared to the ﬁrst two, is ignored, so that the ﬂuctuating load q(r, t)

becomes C

rUu(t)c(r).

A5.3.2 First mode displacement response

Setting q(r, t) ¼ C

rUu(t)c(r), the ﬁrst mode tip displacement response to a sinusoi-

dal wind ﬂuctuation of frequency n (¼ ø=2) and amplitude u

(n) given by Equa-

tion (A5.3) becomes

(t) ¼



(r)C

rUc(r)dru

(n)jH

(n)jcos(2nt þ 

)

¼ C



(r)c(r)dru

(n)jH

(n)jcos(2nt þ 

) (A5:5)

Hence power spectrum of ﬁrst mode tip displacement is

(n) ¼ C



(r)c(r)dr

(n)jH

(n)j

(A5:6)

where S

(n) is the power spectrum for the along wind turbulence. Thus the

standard deviation of the ﬁrst mode tip displacement is given by



¼ [C



(r)c(r)dr]

(n)jH

(n)j

dn (A5: 7)

A5.3.3 Background and resonant respons e

Normally the bulk of the turbulent energy in the wind is at frequencies well below

the frequency of the ﬁrst out-of-plane blade mode. This is illustrated in Figure A5.1,

where a typical power spectrum for wind turbulence is compared with the square,

(n)j

, of an example frequency response function for a 1 Hz resonant frequency.

The power spectrum is that due to Kaimal (and adopted in Eurocode 1, 1997):

RESONANT DISPLACEMENT RESPONSE IGNORING WIND VARIATIONS 315

(n) ¼ 

0:1417nL

(0:098 þ nL

=U)

(A5:8)

and is plotted as the non-dimensional power-spectral density function, R

(n) ¼

(n)=

, against a logarithmic frequency scale. The time scale, L

=U, chosen is

5 s, based on a mean wind speed,

U,of45m=s and an integral length scale, L

,of

225 m.

In view of the fact that the resonant response usually occurs over a narrow band

of frequencies on the tail of the power spectrum, it is normal to treat it separately

from the quasistatic response at lower frequencies, and to ignore the variation in

(n) on either side of the resonant frequency, n

(see, for example, Wyatt (1980)).

The variance of total tip displacement then becomes:



¼ 

þ 

in which the variance of the ﬁrst mode resonant response, 

, is given by



¼ C



(r)c(r)dr

)

(n)j

dn (A5:9)

and the re sonant response of higher modes, 

, 

etc are ignored. The non-

resonant response, 

, is termed the background response, and can be derived from

simple static beam theory.

It has been shown by Newland (1984) that

(n)j

dn reduces to

(

=2)(n

), where  is the logarithmic decrement of damping. The logarithmic

decrement, ,is2 times the damping ratio, 

, deﬁned as 

¼ c

=2m

. Hence

Equation (A5.9) becomes

0.05

0.1

0.15

0.2

0.25

0.3

0.0001 0.001 0.01 0.1 1 10

Frequency (Hz, logarithmic scale)

Power spectral density,

nSu(n)/su

0.1417nT/(0.098 +

nT )

5/ 3

with T = 5 s

Square of frequency

response function,

x1/100, for damping ratio

of 0.1 and natural

frequency of 1 Hz

(stiffness, k, taken as

unity)

Figure A5.1 Power Spectrum and Frequency Response Function

316

DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES



¼ C



(r)c(r)dr

)



2

(A5:10)

For comparison, the ﬁrst mode component,

, of the steady response is obtained by

setting ø ¼ 0 and q

(r) ¼

c(r) in Equation (A5.3), yielding



(r)c(r)dr (A5:11)

Hence the ratio of the standard deviation of the ﬁrs t mode resonant response to the

ﬁrst mode component of the steady response is



¼ 2





ﬃﬃﬃﬃﬃﬃ

2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

)



¼ 2





ﬃﬃﬃﬃﬃﬃ

2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

)

(A5:12)

Note that towards the upper tail of the power spectrum of along wind turbulence,

where n

is likely to be located,

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

)

tends to

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

0:1417=(nL

)

A5.4 Effect of Across-wind Turbulence Distribution on

Resonant Displacement Response

In the foregoing treatment, the wind was assumed to be perfectly correlated along

the blade. The implications of removing this simplifying assumption will now be

examined.

The ﬂuctuating load on the blade, q(r, t), becomes C

rUu(u, r)c(r) per unit length,

and the generalized ﬂuctuating load with respect to the ﬁrst mode becomes

(t) ¼



(r)q(r, t)dr ¼ C

u(r, t)c(r)

(r)dr (A5:13)

The standard deviation, 

,ofQ(t) is given by



(t)dt ¼ (rUC

)

u(r, t)c(r)

(r)dr

u(r9, t)c(r9)

(r9)dr9

¼ (rUC

)

u(r, t)u(r9, t)dt

c(r)c(r9)

(r)

(r9)dr dr9 (A5:14)

Now the expression within the square brackets is the cross correlatio n function ,

(r, r9, ) ¼ Efu(r, t)u(r9, t þ )g with  set equal to zero. The cross correlation

function is related to the cross spectrum, S

(r, r9, n), as follows:

EFFECT OF ACROSS-WIND TURBULENCE DISTRIBUTION 317

(r, r9, ) ¼

1

(r, r9, n) exp(i2n)dn

giving

(r, r9,0)¼

u(r, t)u(r9, t)dt

(r, r9, n)dn for  ¼ 0 (A5:15)

Hence



¼ (rUC

)

(r, r9, n)dn



c(r)c (r9)

(r)

(r9)dr dr9 (A5:16)

The normalized cross spectrum is deﬁned as S

(r, r 9, n) ¼ S

(r, r9, n)=S

(n), and

like S

(r, r9, n), is in general a complex quantity, because of phase differences

between the wind speed ﬂuctuations at different heights. As only in-phase wind

speed ﬂuctuations will affect the response, we consider only the real part of the

normalized cross spectrum, known as the normalized co-spectrum, and denoted by

(r, r 9, n). Substituting in Equation (A5.16), we obtain:



¼ (rUC

)

(n)ł

(r, r9, n)dn



c(r)c(r9)

(r)

(r9)dr dr9 (A5:17)

From this, it can be deduced that the power spectrum of the gene ralized load with

respect to the ﬁrst mode is

(n) ¼ ( rUC

)

(n)ł

(r, r9, n)c(r)c(r9)

(r)

(r9)dr dr9 (A5:18)

Note that the power spectrum for the along wind turbulence shows some variation

with height, and so should strictly be written S

(n, z) instead of S

(n). However,

the variation along the length of a vertical blade is small, and is ignored here.

As for the initial case when wind loadings along the blade were assumed to be

perfectly correlated, the power spectrum for ﬁrst mode tip displacement is equal to

the product of the power spectrum of the generalized load (with respect to the ﬁrst

mode) and the square of the frequency response function, i.e.,

(n) ¼ S

(n)jH

(n)j

(A5:19)

As before, S

(n) is assumed constant over the narrow band of frequencies

straddling the resonant frequency, and the standard deviation of resonant tip

response becomes:



¼ S

)

(n)j

dn ¼ S

)



2

(A5:20)

318 DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES

A5.4.1 Formula for normalized co-spectrum

It remains to evaluate S

) ¼ (rUC

)

(r, r9, n) c(r)c(r9)

(r)

(r9)dr dr9. The normalized co-spectrum, ł

(r, r9, n), must decrease as the spacing

[r  r9] between the two points consid ered increases, and intuitively it is to be

expected that the decrease would be more rapid for the higher frequency compo-

nents of wind ﬂuctuation. On an empirical basis, Davenport (1962) has proposed an

exponential expression for the normalized co-spectrum as follows:

(r, r 9, n) ¼ exp[Cjr  r9jn=U] (A5:21)

where C is a non-dimensional decay constant. Davenport noted that measurements

by Cramer (1958) indicated values of C ranging from 7 in unstable conditions to 50

in stable conditions, but recommended the use of the lower ﬁgure as being the more

conservative despite the likeliho od of stable condi tions in high winds. Dyrbye and

Hansen (1997) quote Riso measurements reported by Mann (1994) which indicate a

value of C of 9.4, and they recomm end a value of 10 for use in design. A value of 9.2

is implicitly assumed in Eurocode 1 (1997).

There is an obvious inconsistency in the exponential expression for the normal-

ized co-spectrum – when it is integrated up over the plane perpendicular to the

wind direction, the result is positive instead of zero as it should be. This has led to

the development of more complex expressions by Harris (1971) and Krenk (1995).

However, the Davenport formulat ion will be used here, giving



¼ S

)



2

¼ (rUC

)

exp[Cjr  r9jn

=U]c(r)c(r9)

(r)

(r9)dr dr9



2

(A5:22)

The resonant response can be expressed in terms of the ﬁrst mode component,

,of

the steady response,



(r)c(r)dr

from Equation (A5.11) giving



¼ 4





2

)

exp [Cjr  r9jn

=U]c(r)c(r9)

(r)

(r9)dr dr9

c(r)

(r)dr

(A5:23)

Hence

EFFECT OF ACROSS-WIND TURBULENCE DISTRIBUTION 319



¼ 2





ﬃﬃﬃﬃﬃﬃ

2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

)

p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

)

(A5:24)

where

) ¼

exp [Cjr  r9jn

=U]c(r)c(r9)

(r)

(r9)dr dr9

c(r)

(r)dr

(A5:25)

is denoted the size reduction factor, which results from the lack of correlation of the

wind along the blade. As an example, the size reduction factor, K

), is plotted

out against frequency in Figure A5.2 for the case of a 20 m blade with chord

c(r) ¼ 0:0961R  0: 06467r (Blade TR), assuming a decay constant C of 9.2, and a

mean wind speed

U of 45 m=s. The mode shape taken is the same as for the

example in section 5.6.3 (see Figure 5.3). Also shown for comparison is the

corresponding parameter for a uniform cantilever.

A5.5 Resonant Root Bending Moment

For design purposes it is the augmentation of blade bending moments due to

dynamic effects that is of principal signiﬁcance. The ratio of the standard deviation

of the ﬁrst mode resonant root bending moment to the steady root bending moment

(allowing for the lack of correlation of wind ﬂuctuations along the blade) is derived

below.

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.5 1 1.5 2 2.5 3

First mode natural frequency (Hz)

Size redcution factor

Tapered blade ‘TR’

Blade of constant cross section

Figure A5.2 Size Reduction Factors for the First Mode Resonant Response due to Lack of

Correlation of Wind Loading along the Blade-Variation with Frequency for 20 m Blade

320

DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES

Deﬁning M

(t) as the ﬂuctuating root bending moment due to wind excitation of

the ﬁrst mode, we have

(t) ¼

m(r)

€

(t, r)r dr ¼

m(r)ø

(t, r)r dr ¼ ø

(t)

m(r)

(r)r dr

(A5:26)

Hence the standard deviation of M

(t),



¼ ø



m(r)

(r)r dr (A5:27)

The steady roo t bending moment,

M ¼

c(r)r dr ¼

c(r)r dr (A5:28)

Hence the ratio



m(r)

(r)r dr

c(r)r dr

(A5:29)

Substituting the expression for 

from Equation (A5.22), we obtain



rUC



ﬃﬃﬃﬃﬃﬃ

2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

)

m(r)

(r)r dr

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

exp[Cjr  r9jn

=U]c(r)c(r9)

(r)

(r9)dr dr9

c(r)r dr

(A5:30)

Noting that R

(n) ¼ nS

(n)=

, and that k

¼ m

, this simpliﬁes to



¼ 2





ﬃﬃﬃﬃﬃﬃ

2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

(n)

m(r)

(r)r dr

c(r)r dr

exp [Cjr  r9j n

=U]c(r)c(r9)

(r)

(r9)dr dr9

(A5:31)

RESONANT ROOT BENDING MOMENT 321

where m

m(r)

(r)dr is the generalized mass with respect to the ﬁrst

mode, and the exponential expression within the double integral allows for

the lack of correlation of wind ﬂuctuations along the blade. Substituting

(

c(r)

(r)dr):

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

)

for the square root of the double integral, using Equation

A5.25, leads to



¼ 2





ﬃﬃﬃﬃﬃﬃ

2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

(n)

m(r)

(r)r dr

c(r)r dr

c(r)

(r)dr

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

)

(A5:32)

Deﬁning the ratio of the integrals,

m(r)

(r)r dr

c(r)r dr

c(r)

(r)dr



(A5:33)

as º

we obtain



¼ 2





ﬃﬃﬃﬃﬃﬃ

2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

)

p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

)

(A5:34)

A5.6 Root Bending Moment Background Response

The root bending moment background response can be expressed in terms of the

standard deviation of the root bending moment excluding resonant effects. If the

wind is perfectl y correlated along the blade, this is given by



¼ C

rU

c(r)r dr (A5:35)

However, if the lack of correlation of wind ﬂuctuations along the blade is taken into

account,



¼ C

rU

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

(r  r9)c(r)c(r9)rr9 dr dr9

(A5:36)

where r

(r  r9) is the normalized cross correlation function between simultaneous

wind-speed ﬂuctuations at two different blade radii, and is deﬁned as

(r  r9) ¼



Efu(r, t)u(r9, t þ )g (A5:37)

322 DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES

with  set equal to zero. Measurements indicate that the normalized cross correla-

tion function decays exponentially, so it can be expressed as

(r  r9) ¼ exp[jr  r9j=L

] (A5:38)

where L

is the integral length scale for the longitudinal turbulence component

measured in the across wind direction along the blade, and is thus deﬁned as

(r  r)d(r  r9). As the integral length scale for longitudinal turbulence meas-

ured vertically in the across wind direction (L

) is, if anything, less than that

measured horizontally (L

), it is conservative to treat it as being equal to that

measured horizontally, with the result that L

can be taken as equal to L

also.

Typically L

is approximately equal to 30 percent of L

, the integral length scale for

longitudinal turbulence measured in the along wind direction. Obse rving that

M ¼

c(r)r dr

we can theref ore write



¼ 2



ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

SMB

(A5:39)

where K

SMB

, the size reduction factor for the root bending moment background

response, is deﬁned as

SMB

exp[jr  r9j=0:3L

]c(r)c(r9)rr9 dr dr9

c(r)r dr

(A5:40)

For a blade with a uniform chord, the integral is straightforward, giving

SMB

¼ 4

3





 exp()





where  ¼

0:3L

(A5:41)

As an example, K

SMB

comes to 0.927 for the case of R ¼ 20 m and L

¼ 230 m,

indicating that the effect of the lack of correlation of the wind ﬂuctuations is rather

small.

For blades with a normal tapering chord, K

SMB

can be evaluated numerically. In

the case of a blade with a tip chord equal to 25 percent of the maximum chord,

SMB

is 0.924 for the same value of  as before. It is seen that the taper has a

negligible effect on the end result.

ROOT BENDING MOMENT BACKGROUND RESPONSE 323

A5.7 Peak Response

One of the key parameters required in blade design is the extreme value of the out-

of-plane bending moment. The 50 year return moment is deﬁned as the expected

maximum moment occurring during the mean wind averaging period wh en the

mean takes the 50 year return value. Treating the moment as a Gaussian process,

Davenport (1964) has shown that the expected value of the maximum departure

from the mean is the standard deviation multiplied by the peak factor, g, where

g ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2ln(T)

0:5772

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2ln(T)

(A5:42)

In this formula,  is the mean zero-upcrossing frequency of the root moment

ﬂuctuations, and T is the mean wind speed averagi ng period. The variance of the

root bending moment is, in the same way as for the tip displacement, equal to the

sum of the variances of the background and resonant root bending moment

responses, i.e.,



¼ 

þ 

(A5:43)

Hence, from Equations (A5.39) and (A5.34), we obtain



¼ 

þ 

¼ 4M



SMB



2

)º



(A5:44)

Thus

max

¼ M þ g

¼ M 1 þ 2g



ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

SMB



2

)º

(A5:45)

The mean zero up-crossing frequency of the root moment ﬂuctuations, , is deﬁned

 ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

(n)dn

(A5:46)

where S

(n) is the power spectrum of the root moment ﬂuctuations. If we separate

the power spectrum of the background response from the ﬁrst mode resonant

response at frequency n

, then the above expression can be written

324 DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES