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

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machines of 40 m diameter and above, the fatigue equivalent load for in-plane

bending at the blade root is typically greater than that for out-of-plane bending.

Over most of the blade length, the chord dimension is much larger than the blade

thickness, so the section modulus for edgew ise bending will generally exce ed that

for ﬂapwise bending. However, for blades attached to the hub or pitch-bearing by a

circular ring of bolts, which is the normal arrangement, the blade structure adjacent

to the root is a cyl indrical shell, which will have the same section modulus about

both axes if the wall thickness is uniform. As a consequence, the blade root is the

ﬁrst area that should be checked for in-plane fatigue loading.

The procedure can be illustrated for a 20 m tip radius TR blade in GFRP, designed

for an extreme static root moment of 750 kNm at 0.5 m radius. Taking the gravity

moment at the root as 124 kNm, and assuming 2 :6 3 10

revolutions over the

machine lifetime, the notio nal one cycle in-plane bending equivalent load range,

eq(n¼1)

, becomes 248(2:6 3 10

)

0:1

¼ 1720 kNm and 

eq(n¼1)

=

ext

¼ 1720=750 ¼ 2:3.

This is less than the minimum value of 2.7 of the right-hand expression in the

inequality (7.11), so in-plane fatigue loading does not govern. In practice, the

cylinder wall thickness would have to be increased signiﬁcantly to pre vent buck-

ling, rendering fatigue less critical still at this diameter.

It may be concluded from the above example that in-plane fatigue loadings are

not a signiﬁcant consideration in the design of stall-regulated blades constructed in

GFRP or wood laminates, until much larger diameters are reached. They cannot be

ignored entirely, because of blade twist and because they add to the fatigue stress

ranges due to ﬂapwise bending at points on the cross section away from the neutral

axis for edgewise bending. In-plane fatigue loadings are of more signiﬁcance for

pitch-regulated machines because gravity loadings will contribute increasingly to

ﬂapwise loadings as the pitch angle increases.

Tip deﬂection

Under extreme operating conditions, tip deﬂections of up to about 10 percent of

blade radius can oc cur, so care is needed to avoid the risk of blade/tower collisions

in the case of upwind machines. GL specify that the quasi-static tip deﬂection under

the extreme unfactored operational loading is not to exceed 50 percent of the

clearance without blade deﬂection, which implies a safety factor of 2. IEC, on the

other hand, require no blade/tower contact when the extreme loads are multiplied

by the combined partial safety factors for loads and the blade material.

It is instructive to compare the tip deﬂections for similar blades designed in

different materials. If the skin thickness distributions are chosen so that the design

compression strength of each material is fully mobilized under the extreme load

case, then the tip deﬂection will be proportional to the design compression strength

to Young’s Modulus ratio, 

=E, of the blade material. These ratios are compared

for different materials in Table 7.4.

It is clear from the table that a GFRP blade will be about twice as ﬂexible as

blades in the other materials (apart from steel), provided that the spar is stocky

enough for buckling not to govern the design, e.g., as in Figure 7.4. In the case of

thin-walled cross sections, however, such as that in Figure 7.3, the GFRP compres-

BLADES 405

Table 7.4 Design Strength-to-stiffness Ratios for Different Wind Turbine Blade Materials

Material Ultimate

compression

strength, 

(MPa)

Partial safety factor

for material

strength, ª

Compression

design strength,



(MPa)

Young’s

Modulus, E

(GPa)

Strength-to-

stiffness ratio,

(

=E) 3 10

Glass/polyester laminate with 50% ﬁbre volume

fraction and 80% UD

580 2.94 197 (ignoring

buckling)

32.5 6.1

Carbon ﬁbre/epoxy ply with 60% ﬁbre volume

fraction and UD lay-up

1100 2.94 374 142 2.6

Khaya/epoxy laminate 50 1.5 33 10 3.3

Birch/epoxy laminate 81 1.5 54 15 3.6

Yield strength, 

High-yield steel (Grade Fe 510) 355 1.1 323 210 1.54

Weldable aluminium alloy AA6082 240 1.1 218 69 3.2

406 COMPONENT DESIGN

sive design stress has to be red uced signiﬁcantly to guard against buckling, with

the result that blade ﬂexibility is reduced. For example, in the case of the 19.5 m TR

blade, the compressive design stress is reduced to about 90 MPa, resulting in a tip

deﬂection of about 2 m under extreme loading, and a strength-to-stiffness ratio less

than that for wood/epoxy. In this connection, it is noteworthy that Hancock,

Sonderby and Schubert (1997) records a 130 percent proof load test on a 31.2 m

birch/epoxy blade which resulted in a tip deﬂection of 3.4 m.

7.1.9 Blade resonance

One of the most important objectives of blade design is the avoidance of resonant

oscillations, which, in a mild form, exacerbate fatigue damage and in an extreme

form can lead to rapid failure. The excitation of blade resonance can be minimized

by maximizing the damping and ensuring that the blade ﬂapwise and edgewise

natural frequencies are well separated from the exciting frequencies – i.e. the

rotational frequency and its harmonics, particularly the blade passing frequency –

and from the frequencies of other vibration modes with which there is an identiﬁ-

able risk of coupled oscillatio ns.

Vibrations in stall

On stall-regulated machines, the lift curve slope dC

=dÆ, goes negative when a

section of the blade goes into stall, resulting in local negative aerodynamic damping

of blade motion in the lift direction. If the overall aerodynamic damping for a

particular mode shape is negative, and exceeds the modal structural damping in

magnitude, then divergent oscillations can develop from any initial disturbance,

regardless of the relationship between the mode natural frequency and exciting

frequencies. The ﬁrst mode in each direction is most susceptible to such behaviour

because the structural damping increases with frequency while the aerodynamic

damping diminishes. If conditions favouring ﬁrst-mode oscillations are to be

avoided, the factors affecting the aerodynamic damping of both edgewise and

ﬂapwise oscillations need to be understood, so these are explored below.

Consider a turbine operating in steady co nditions in a perpendicular airﬂow. If a

blade cross section at radius r experiences out-of-plane and in-plane perturbations

with velocities

xx in the downwind direction and

yy in the direction opposite to that

of blade rotation (assumed clockwise), then the relative velocity triangle is as in

Figure 7.16(a). The lift and drag forces per unit length on a blade element, L and D,

can be resolved into out-of-plane and in-plane forces F

and F

(see Figure 7.16(b)),

leading to Equations (5.18) and (5.19). Ignoring the small rotational induction factor,

which is very small, these may be rewritten as:

¼ W[C

(1  a) 

xx) þ C

(r 

yy)]

rc (7:12)

¼ W[C

(r 

yy) þ C

(1  a) 

xx)]

rc (7:13)

BLADES 407

Here U

is the free stream wind speed and U

(1  a) the reduced wind speed at

the rotor plane as usual. The damping coefﬁcients per unit length for vibrations in

the in-plane and out-of-plane directions are then given by

(r) ¼

(7:14a)

(r) ¼

(7:14b)

Analagous ‘cross’ coefﬁcients relating the in-plane force to the out-of-plane velocity

and vice versa can also be deﬁn ed as:

(r) ¼

@ F

(7:15a)

(r) ¼

(7:15b)

(a)

(b)

(c)

θ*

Ωr –y

∞

(l –a) –x

(Down wind)

Plane of rotor

rotation

Figure 7.16 (a) Velocity Diagram for Vibrating Blade (Looking Towards Hub); (b) Out-

of-plane and In-plane Components of Lift and Drag Forces; (c) Directions of Vibration, x



and y



408

COMPONENT DESIGN

Substituting V for U

(1  a) for brevity, the in-plane damping coefﬁcient is derived

as follows:

(r) ¼

¼

[C

V þ C

r] þ W 

V þ

r  C



(7:16)

Noting that

¼

r

and

@Æ

@

@Æ

this equation becomes:

(r) ¼

r

VC

r

@Æ

2

þ V

r

 V

@Æ



(7:17)

The ‘cross’ coefﬁcients and the out-of-plane damping coefﬁcient and are derived by

a similar pro cedure:

(r) ¼

r





þ 2V

r

 V

@Æ

þ VC

þ r

@Æ



(7:18)

(r) ¼

r

2

þ V

r

 V

@Æ

þ VC



r

@Æ



(7:19)

(r) ¼

r

þVC

þ r

@Æ



þ 2V

r

þ V

@Æ



(7:20)

It is apparent from inspection of the expres sions for the two damping coefﬁcients,

and

, that the choice of an aerofoil with a gentler stall – i.e., with a smaller lift

curve slope after stall onset – will increase the damping coefﬁcient in both cases.

Note that the modal damping coefﬁcient is dominated by the damping per unit

length over the outboard part of the blade, so it is important to select an aerofoil

with a gentle stall in this a rea only.

The choice of aerofoil also affects performance, so there is merit in expressing the

damping coefﬁcients in terms of the power output in orde r to investigate possible

trade-offs between them. It transpires that the damping and ‘cross’ coefﬁ cients per

unit length can be formulated quite simply in terms of the power output per unit

length of blade, P9(r, V) ¼ r(F

), and the blade thrust per unit length, F

,as

follows:

BLADES 409

¼



P9 þ



@P9



2P9 þ V

@ P9



(7:21)

¼

@ F

r

(rF

) ¼

r

@P9

(7:22)

r

 V

@ F



(7:23)

¼

@ F

(7:24)

Equations (7.21) and (7.23) are derived from the equations

r

þ V

¼ 2F

¼2P9=r

and

r

þ V

¼ 2F

which may be veriﬁed using Equations (7.17) to (7.20) .

From Equation (7.21) it is clear that the damping coefﬁcient in the in-plane

direction,

, will always be negative when 2(P9=V) exceeds @ P9=@V, and that a

negative power curve slope should be avo ided if the size of the negative damping

is to be kept small.

Effect of blade twist

In the discussi on so far, damping of vibrations in the out-of-plane and in-plane

directions only has been considered. In practice, blade twist will result in the

ﬂapwise and edgewise vibrations taking place in directions rotated from the out-of-

plane and in-plane directions in the same sense as the blade twis t, but by a lesser

amount (see Section 5.8.1). If we deﬁne x



- and y



-axes in the directions of the

ﬂapwise and edgewise displacement s, each making an angle of Ł



to the x - and y-

axes respectively, as shown in Figure 7.16(c), then the edgewise damping coefﬁcient

per unit length is given by:



cos



 (

) sin Ł



cos Ł



sin



(7:25)

Substitution of Equations (7.21)–(7.24) in Equation (7.25) yields:



¼ cos





2P9 þ V

@ P9



þ cos Ł



sin Ł



r



@ P9

þ 2F

 V



þ sin



@ F



(7:26)

410 COMPONENT DESIGN

This expression also gives the ﬂapwise damping coefﬁcient per unit length if Ł



replaced by Ł



þ 908 through out.

The variation of the damping coefﬁcient



per unit length at 14 m radius with

vibration direction, Ł



, at three different wind speeds is illustrated in Figure 7.17

for a specimen aerofoil section on a 20 .5 m tip radius blade rotating at 29 r.p.m. The

data are taken from Petersen et al. (1998), and do not include allowance for the axial

induction factor. It can be seen that negative damping is worst at 20 m/s, and that

negative edgewise damping is ameliorated by increasing Ł



at the expense of

increasing negative ﬂapwise damping.

Although a plot of the local damping coefﬁcient at ca 70 percent radius can

provide a useful indication of trends, the best guide to the likel ihood of divergent

oscillations is provided by the modal damping coefﬁcient for the mode under

consideration. This is obtained by multiplying the right-hand side of Equation

(7.26) by the square of the local modal amplitude and integrating over the length of

the blade.

If comparison of the ﬁrst mode edgewise and ﬂapwise modal damping coefﬁ-

cients shows there is a beneﬁt to be gained from altering the direction of vibration,

small changes can be made by redistributing material within the blade cross section.

Alternatively the blade pitch could be altered in conjunction with a compensatory

change in aerofoil camber so that the aerodynamic properties for any given inﬂow

angle are unch anged.

The prediction of edgewise vibrations in stall is examined in detail by Petersen

et al. (1998), whose work provides the basis of the introductory survey given here.

-100

-50

100

150

-45 0 45 90 135

Direction of vibration relative to in-plane direction

Damping coefficient per unit length (Ns/m

)

8 m/s wind speed

20 m/s wind speed

25 m/s wind speed

Range of flapwise

vibration directions

Range of edgewise

vibration directions

Chord at 14 m radius = 1.06 m

Rotational speed = 29 r.p.m.

Tip radius = 20.5 m

Figure 7.17 Variation in Damping Coefﬁcient at 14 m Radius with Vibration Direction for

Example Aerofoil

BLADES 411

They concluded that the fundam ental cause of edgewise blade oscillations that had

been observed on some stall-regulated machines of 40 m diameter and over was

negative aerodynamic damping, but found that the use of dynamic stall models

improved the level of agreement with measurements.

Coupling of edgewise blade mode and rotor whirl modes

A further important ﬁnding was that, on one machine subject to stall-induced

vibrations which was investigated in detail, there was coupling between the blade

ﬁrst edgewise mode and one of the second rotor whirl modes. The rotor whirl

modes arise from the combination of simultaneous nodding and yawing oscillations

of the rotor shaft, which occur at the same frequency during operation due to

gyroscopic effects. As a result, the rotor hub traces out a circular or elliptical path,

running either in the same direction as rotor rotation or in reverse, which explains

the existence of two ﬁrst and second modes.

The explanation for the coupling was as follows. When a pair of blades vibrate in

the edgewise direction in anti-phase, they impart a sinusoidally varying in-plane

force to the rotor hub even though their edgewise root bending moments cancel

out. The direction of this oscillating force rotates with the rotor, so it has horizontal

and vertical components of the form sin(ø

t þ ):sin  t and sin(ø

t þ ):cos  t,

where ø

is the frequency of the blade ﬁrst edgewise mod, and  is the speed of

rotor rotation. With respect to stationary axes the in-plane loads on the hub

therefore act at two frequencies – namely ø

þ  and ø

 . In the case of the

machine investigated by Petersen et al., the upper frequency of 2:9 þ 0:5 ¼ 3:4Hz

coincided with the backwa rd second rotor whirl mode, allowing interaction be-

tween this mode and the blade ﬁrst edgewise mode.

Simulations were carried out on an aeroelast ic model of the turbine at various

wind speeds and satisfactory agreement obtained between simulated and measured

behaviour. In particular, the simulation at 23.2 m/s predicted the build up of large

blade root edgewise moment oscillations at the ﬁrst mode frequency, as observed

on the real machine at this wind speed. Signiﬁcantly, when the latter simulation

was repeated with the rotor shaft stiffness increased sufﬁciently to increase the

backward second rotor whirl mode frequency to 3.6 Hz, the predicted blade root

edgewise moment oscillations were negligible by comparison.

Mechanical damping

An alternative strategy for preventing damaging edgewise vibrations is the incor-

poration of a tuned mass damper inside the blade towards the tip. The performance

of such a damper on a 22 m tip radius blade is reported by Anderson, Heerkes and

Yemm (1998). It was found that the ﬁtting of a damper tuned to the ﬁrst mode

edgewise frequency, and weighing only 0.4 percent of the total blade weight,

effectively suppressed the edgewise vibrations which had previously been ob-

served during high wind speed operation.

412 COMPONENT DESIGN

7.1.10 Design against buckling

The stress at which a slender plate element without imperfections buckles under

compression loading is known as the critical buckling stress. The derivation of the

critical buckling stresses for thin-walled curved panels bounded by stiffeners,

which typically form the blade load-bearing structure, is relatively straightforward

when the panel mater ial is isotropic and solutions are provided in Timoshenko and

Gere (1961). These do not apply to composite materials such as the GFRP and wood

laminates commonly used in blade manufacture, however, as these are anisotropic,

but solutions can be derived for a symmetric laminate using the energy method, as

outlined below.

Consider a long cylindrical panel of length L and radius r, supported along two

generators and subtending an angle ł at the cylinder axi s, which is axially loaded

in compression. If it deﬂects to form n half-waves around the circumference

between supports and m half-waves along its length, then its out-of-pla ne deﬂection

can be written as:

w ¼ C sin

nŁ

sin

mx

(7:27)

where Ł and x are the co-ordinates of the deﬂected point with respect to one of the

long edges and one end respectively. In the absence of in-plane direct strains in the

plate, this out-of-plane deﬂected proﬁle will resu lt in circumferential deﬂections

Cł

n

cos

nŁ

sin

mx

(7:28)

These deﬂections will result in in-plane shear stresses, which reach a maximum at

the corners of each rectangular buckled panel. In practice, additional in-plane

deﬂections occur to moderate these shear stresses, as follows:

u ¼ A sin

nŁ

cos

mx

in the axial direction

v ¼ B cos

nŁ

sin

mx

in the circumferential direction (7:29)

The in-plane strain energy is calculated as

ðð

(



þ 



þ ª) r dŁ dx (7:30)

with the sufﬁces 1 and 2 denoting the axial and circumferential directions respec-

tively, so that



, 

r@Ł

, ª ¼

r@Ł

@(v

þ v)

(7:31)

BLADES 413

Substituting 

¼ E

(

þ ı



)=(1  ı

), 

¼ E

(

þ ı



)=(1  ı

) and  ¼

ª, where E

, E

and G

are the longitudinal, transverse and shear moduli of the

laminate respectively (obtained by averaging the corresponding moduli of the

individual plies) and ı

and ı

are the effective Poisson’s ratios, the in-plane strain

energy becomes:

2(1  ı

)

ðð



þ E



þ 2E



þ (1  ı

)ª

]r dŁ dx (7:32)

Substituting the expressions for 

, 

, and ª from Equation (7.31) and integrating

over the width of the panel, łr, and the length of one half wave, L=m, we obtain

1  ı

łr

m



þ 



þ2ı

Æ



þ (1  ı

)



þ  þ

n



(7:33)

where º ¼ młr =L and the ratios Æ ¼ A=C and  ¼ B=C are yet to be determined.

The expression for the strain energy of curvature is derived as follows. Replacing

the angular coordinate Ł by the linear coordinate y (¼ rŁ), the bending energy

absorbed in an area dx:d y is:

¼

þ M

@ y

dx:dy

where

¼D

 D

@ y

and

¼D

@ y

 D

for a specially orthotropic laminate, i.e. one in which the reinforcement in each layer

is either oriented at 08 or 908, or is bi-directional with the same amount of ﬁbres at

þŁ8 and Ł8. D

and D

are the ﬂexural rigidities of the laminate when ﬂat, for

bending about the y-axis and x-axis respectively, and D

is the ‘cross ﬂexural

rigidity’ i.e. the moment per unit width about one axis generated by unit curvature

about the other. Hence

414 COMPONENT DESIGN