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

Подождите немного. Документ загружается.



þ 2D

@ y

þ D

@ y

dx d y (7:34)

The twisting energy absorbed in an area dx dy is:

þ M

)

@x@ y

dx d y

where

¼ 2

h=2

h=2

(z):z

@x@ y

in which z is the distance measured from the mid-plane of the laminate, G

(z)is

the in-plane shear modulus at that distance and h is the laminate thickness. Denot-

ing the torsional rigidity,

h=2

h=2

(z):z

by D

, then

:4D

@x@ y

dx d y (7:35)

The total strain energy of curvat ure over the width of the panel and the leng th of

one half wave is found by substituting the out-of-plane deﬂection given by Equa-

tion (7.27) in Equations (7.34) and (7.35) and integrating over this area, which gives:

¼ U

þ U

łrL

m



1 þ



þ 4



(7:36)

The energy absorbed by the panel during buckling as a result of in-plane strains

and out-of-plane curvature is equal to the work done by the critical axial load as the

panel shortens. The shortening of the panel over one half wave length is given by

L=m



dx ¼



sin

nŁ

(7:37)

so the work done by the axi al force of N

per unit width over the panel width is



łrN

(7:38)

BLADES 415

The equality T

¼ U

þ U

yields the critical value of the axial force as follows:

)

¼ D

m



1 þ



þ 4



1  ı

þ 



þ2Æı

þ (1  ı

)

þ  þ

n



(7:39)

Noting that

m

młr

n

łr

n

łr

this equation becomes

(

)

n

łr



1 þ



þ 4



1  ı

þ 



þ2Æı

þ (1  ı

)

þ  þ

n



(7:40)

The right-hand side of Equation (7.40) contains four unknowns, the number of

transverse half waves, n, the ratio of longitudinal to transverse half wave length,

n=º, and the factors Æ and . Assuming that there is only one transverse half wave,

as is normally the case, the expression is minimized with respect to Æ, and  for

each value of n=º, and then with respec t to n=º to obtain the critical stress.

The results of this exercise are illustrated for a particular curved laminate panel

in Figure 7.18. The radius of curvature, r, of 1150 mm and thickness, h,of15mm

are chosen to be representative of the values likely to obtain at 70 percent radius on

a blade with 20 m tip radius. The laminate has 80 percent of its plies with ﬁbres

oriented axially and 20 percent with ﬁbres at  458 to resist shear loads. In each case,

the ﬁbre volume fraction is 50 percent. The 458 plies are concentrated about the

laminate mid-plane so that they do not detract signiﬁcantly from the longit udinal

ﬂexural rigidity. Thus, if the longitudinal modulus of the UD plies is denoted by E

and the Poisso n’s ratios by ı

and ı

, the longitudinal ﬂexural rigidity is given

approximately by:

12(1  ı

)

(7:41)

The other ply and laminate properties required for evaluation of the critical stress

are detailed on the ﬁgure. Note that in the derivation of the laminate in-plane

stiffness properties it is necessary to transform the in-plane stiffness properties of

416 COMPONENT DESIGN

the 45 8 plies obtained initially in relation to the ply axes (which are parallel to the

ﬁbre directions) to the set of properties in relation to the global x- and y-axes of the

laminate as a whole (see, for example, Barbero (1998) for the requisite formulae).

The heavy curve in Figure 7.18 shows the variation in axial critical stress with

panel width (in terms of subtended angle) when the buckled shape has only a single

half wave in the transverse direction and the ﬁne lines below show the separate in-

plane and ﬂexural contributions. The minimum stress of 110 MPa occurs when the

angle subtended by the panel is about 208, but there is only a gradual increase in

critical stress as the angle increases above this. When the subtended angle exceeds

about 358, buckling with two half waves in the transverse direction takes over as the

critical mode – see dashed line. Also shown for comparison is the critical buckling

stress variation for an isotropic plate with Young’s modulus equal to the long-

itudinal modulus of the UD plies of the laminate – see dotted line. The minimum

critical stress in this case is 298 MPa, about 2.7 times as big.

7.1.11 Blade root ﬁxings

The ﬁxing of the blade root to the hub is one of the most critical areas of blade

design, because the order of magnitude difference between the relative stiffnesses

of the steel hub and the blade material – usually GFRP or wood – militates against

a smooth load transfer. The co nnection is usually made by steel bolts, which can

either be embedded in the blade material in the axial direc tion or aligned radially to

pass through the blade skin, but in either case stress concentrations are inevitable.

Figure 7.19 illustrates four different types of blade root ﬁxings in section. The

blade structure is usually a cylindrical shell at the root, in which case the stud or

bolt ﬁxings are arranged in a circle. Figure 7.19(a) shows the carrot connector,

100

150

200

250

300

350

400

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Angle subtended by panel (radians)

Critical stress (MPa)

Laminate critical stress with one transverse half wave (n = 1)

Laminate critical stress

with two transverse

half waves (n = 2)

Critical stress for

isotropic panel with E = 37.7 GPa

Laminate properties:

= 33.6 GPa, E

= 12.4 Gpa

vx = 0.38, vy = 0.14, G

= 4.8 GPa

= 0.30, D

= 0.087, D

= 0.086

Laminate structure:

80% of plies with axial fibres (outer plies);

20% of plies with fibres at ⫾ 45⬚ (inner plies);

50% fibre volume fraction

Ply properties:

= 37.7 GPa, E

= 11.3 GPa,

= 0.29, ν

= 0.087, G

= 3.3 GPa

In-plane contribution to critical stress (n = 1)

Flexural contribn to critical stress (n = 1)

Figure 7.18 Variation of Axial Critical Buckling Stress with Panel Width for Specimen

Curved Anisotropic Panel with Radius 1150 mm and Thickness 15 mm

BLADES 417

which is the standard ﬁxing for laminated wood blades. The connector consists of a

tapered portion carbon-epoxy grouted into a stepped hole drilled into the end of

the blade, together with a projecting threaded stud for attachment to the hub or

pitch bearing. Conn ectors are either machined from high strength steel or cast from

spheroidal graphite iron (SGI). They are normally preloaded to reduce fatigue

loading. A similar connector, in which the embedded portion is cylindrical rather

than tapered, is in common use on GFRP blades.

Figures 7.19(b)–(d) show three further ﬁxing arrangeme nts used on GFRP blades.

The T-bolt connector, shown in Figure 7.19(b), consists of a steel stud inserted into a

longitudinal hole in the blade skin, which engages with a cylindrical nut held in a

transverse hole. The stud is preloaded to reduce fatigue loading. The ‘pin-hole

ﬂange’ arrangement in Figure 7.19(c) uses the same method of load transfer

between the GRP and the steel – i.e., bearing on a transverse rod – but the interface

does not lend itself to preloading. Moreover the bolts attaching the ﬂange to the

hub are eccentric to the blade skin, so the ﬂange has to resist the resultant bending

moment as well.

In the trumpet ﬂange detailed in Figure 7.19(d), the blade root is splayed out in

the form of a trumpet mouth and clamped between inner and outer ﬂanges by the

ring of bolts which attach the ﬂange to the hub. These bolts also pass through the

GFRP skin to provide positive anchorage. Again the ﬂange has to resist bending

moments arising from the eccentricity of the ﬁxing bolts to the blade skin where it

emerges from the ﬂange. The pin-hole and trumpet ﬂange arrangements are rarely

used for larger blades.

(a)

(c)

(b)

(d)

Blade Blade

Blade

Carbon epoxy grout

Cylindrical nut

Figure 7.19 (a) Carrot Connector; (b) T-bolt Connector; (c) Pin-hole Flange; (d) Trumpet

Flange

418

COMPONENT DESIGN

The stress distributions calculated for blade root ﬁxings are subject to signiﬁcant

levels of uncertainty, so it is normal to conduct both static and fatigue tests on them

to verify the suitability of the design. Static pull-out failures of carrot connectors

occur as a result of shearing of the wood surrounding the grout, but fatigue failures

can also occur in the connector itself or the grout. However, SGI studs subjected to

R ¼ 0:1 fatigue loading at over 60 percent of the UTS have survived for approxi-

mately 10

cycles.

Mayer (1996) records the results of fatigue tests on the other blade root ﬁxings

featured in Figure 7.19, but in no case did failure occur as a result of fatigue of the

GFRP in the region of the root ﬁxing. In the case of the T-bolt ﬁxing arrangement,

failure occurred in the studs rather than in the GRP. The pin-hole ﬂange specimens

developed fatigue cracks in the GFRP in areas remote from the root ﬁxings and the

trumpet ﬂange specimens developed cracks in the ﬂanges themselves.

7.2 Pitch Bearings

On pitch-regulat ed machines a bearing similar to a crane slewing ring is interposed

between each blade and the hub to allow the blade to be rotated or ‘pitched’ about

its axis. A typic al arrangement is as shown in Figure 7.20, in which the inner and

outer rings of the bearing are bolted to the blade and hub respectively. The different

types of bearings available can be classiﬁed according to the rolling elements used

and their arr angement, in order of increasing moment capacity:

(a) single-row roller bearings, with alternat e rollers inclined at þ458 and 458 to the

plane of the bearing;

Pitch bearing

outer ring

Blade

Pitch bearing

inner ring

Linkage arm

for pitch actuation

Hub wall–

cylindrical

geometry

Hub wall

–

spherical

geometry

Blade wall

Figure 7.20 Typical Pitch-bearing Arrangement

PITCH BEARINGS 419

(b) single-row ball bearings,

(d) three-row roller bearings.

These are shown in cross section in Figure 7.21. The single-row ball bearing slewing

rings are normally designed to transmit axial loads in both directions and are

therefore known as four point contact bearings. Low contact stresses are achieved

by making the radii on each side of the grooves only slightly larger than that of the

balls.

At low wind speeds, the cyclic in-plane bending moment at the blade root due to

gravity is of similar magnitude to the out-of-plane moment due to blade thrust, so

bearing loads will alternate in direction over portions of the bearing circumference.

Accordingly it is desirable to avoid the risk of play by preloading the bearing. This

can be achie ved relatively easily on bearings in which one of the rings is split on a

plane normal to the axis, suc h as typ es (c) and (d), but is more difﬁcult when both

rings are solid. In this case it is necessary to force the rolling elements into the races

one by one during manufacture.

The bearing selected for a particular application needs to have suf ﬁcient moment

capacity to both resist the extreme blade root bending moments and provide

adequate fatigue life. Manufacturers’ catalogues typically specify both the extrem e

moment capacity and the steady moment loading that will give a life of, say, 30 000

bearing revolutions, so the wind turbine designer’s chief task is to convert the

(a) (b)

(c)

(d)

Figure 7.21 (a) Single-row Crossed Roller Bearings; (b) Single-row Ball Bearings; (c) Double-

row Ball Bearings; (d) Three-row Roller Bearings

420

COMPONENT DESIGN

anticipated pitch bearing duty into the equivalent constant loading at the appro-

priate number of revolut ions. If the rolling elements are ball bearings, the bearing

life is inversely proportional to the cube of the bearing loading, so the equivalent

loading at N revolutions of the pitch bearing can be calculated according to the

formula:

eqt

1=3

(7:42)

where n

is the total pitch bearing movement anticipated over the design life at

moment loading M

, expressed as a number of revolutions. In the case of roller

bearings, the index of the S–N curve is 10/3 instead of 3, so the above formula

should be modiﬁed accordingly. As the blade root out-of-plane moment drops as

the wind speed increases above rated, the fatigue damage will be concentrated at

wind speeds near rated.

The total pitc h bearing movement over a period of operation at a particular wind

speed is a function of the turbulence intensity and the pitch control algorithm, and

is best predicted by means of a wind simulation. The mean blade pitching rate

during operation above rated wind speed is found to be of the order of 1 degree/s,

assuming the pitch system only responds to wind speed ﬂuctuations at a frequency

less than the speed of rotation.

The performance of slewing ring bearings such as those employed as pitch

bearings is critically dependent on the extent of bear ing distortio n under load, so

manufacturers normally specify a maximum axial deﬂection and tilt of the bolted

contact surfaces. For example, the limiting values given by Rothe-Erde for a single-

row ball bearing slewing ring with a 1000 mm track diameter are 0.6 mm and 0.178

respectively. Local tilting of the bearing rings could clearly be minimized if the

blade wall, bearing trac k and hub wall were all positioned in the same plane.

However, this would necessitate the provision of ﬂanges, so the simpler arrange-

ment shown is Figure 7.20, in which the ﬁxing bolts are inserted central ly into the

blade and hub walls, is generally preferred. The designer must then ensure that the

blade and hub structures are of sufﬁcient stiffness to limit the bearing distortion

due to the eccentric loading to acceptable values.

It is standard practice to preload the bearing ﬁxing bolts in order to minimize bolt

fatigue loading. Grade 10.9 bolts are commonly used so that the preload can be

maximized.

7.3 Rotor Hub

The rela tively complex three-dimensional geometry of rotor hubs favours the use of

casting in their manufacture, with spheroidal graphite iron being the material

generally chosen.

Two distinct shapes of hub for three-bladed machines can be identiﬁed: tri-

cylindrical or spherical. The former consists of three cylindrical shells concen tric

ROTOR HUB 421

with the blade axes, which ﬂare into each other where they meet, while the latter

consists simply of a spherical shell with cut-outs at the three-blade mounting

positions. Diagrams of both types are shown in Figure 7.22, while an actual

spherical hub is illustrated in Figure 7.23. The structural action of the hub in

resisting three loadings is discussed in the following paragraphs.

(1) Symmetric rotor thrust loading. The blade root bending moments due to sym-

metric rotor thrust loading put the front of the hub in bi-axial tension near the

rotor axis and the rear in bi-axial compression, while the thrust itself generates

out-of-plane bending stresses in the hub shell adjacent to the low speed shaft

ﬂange connection. The load paths are easy to visualize in this case.

(2) Thrust loading on a single blade. This generates out-of-plane bending stresses in

the hub shell at the rear, and in-plane tensile stresses around a curved load path

between the upwind side of the blade bearing and the portion of the low-speed

shaft ﬂange connection remote from the blade (see dashed line in Figure 7.22(b).

The resultant lateral loads will result in out-of-plane bending.

(3) Blade gravity moments. On the tri-cylindrical hub, equal and opposite blade

gravity moments are co mmunicated via the cylindrical shells to areas near the

rotor axis at front and rear where they cancel each other out. It is less

straightforward to visualize the corresponding load paths on the spherical hub,

as out-of-plane bending is likely to be mobilized.

Side view

Front view

(a)

(b)

Figure 7.22 (a) Tri-cylindrical Hub; (b) Spherical Hub

422

COMPONENT DESIGN

The complexity of the stress states arising from the latter two types of loading

renders ﬁnite-element analysi s of rotor hubs more or less mandatory. At the most,

six load cases need to be analysed, corresponding to the separate application of

moments about the three axes and forces along the three axes at a single hub/blade

interface. Then the distribution of hub stresses due to combinations of loadings on

different blades can be obtaine d by superposition. Similarly the ﬂuctuation of hub

stresses over time can be derived by inputting the time histories of the blade loads

obtained from a wind simulation.

The critical stresses for hub design are the in-plane stresses at the inner or

outer surface, where they reach a maximum because of shell bending. For any

one location on the hub, these are deﬁned by three quantities at each surface: the

in-plane direct stresses in two directions at right angles, and the in-plane shear

stress. In general, these stresses will not vary in-phase with each other over time,

so the principal stress directions will change, complicating the fatigue assess-

ment.

There is, as yet, no generally recognized procedure for calculating the fatigue

damage accumulation due to multi-axial stress ﬂuctuations, although the following

methods have been used, despite their acknowledged imperfections. They all cater

for one or more series of repeated stress cycles rather than the random stress

ﬂuctuations resulting from turbulent loading.

Figure 7.23 Rotor Hub. View of Spherical-shaped Rotor Hub for the 1.5 MW NEG Micon

Turbine Awaiting Installation. The Hub and Spinner are Temporarily Oriented with the

Rotor Shaft Axis Vertical. The Turbine is Stall-regulated, so Slotted Blade Fixing Holes are

Provided to Allow for Fine Adjustment of Blade Pitch to Suit the Site. (Reproduced by

permission of NEG-Micon)

ROTOR HUB 423

(1) Maximum shear method. Here the fatigue evaluation is based on the maximum

shear stress ranges, calculated from either the (

 

)=2, 

=2or

=2 time

histories. The effect of mean stress is allowed for using the Goodman relation-

ship:



(7:43)

where 

is the alternating shear stress, 

is the mean shear stress, S

is the

alternating shear stress for N loading cycles from the material S–N curve, S

the ultimate shear strength, and ª is the safety factor.

Having used Equation (7.43) to determine S

, the permitted number of

cycles for this loading range can be derived from the S–N curve, enabl ing the

corresponding fatigue damage to be calculated.

(2) ASME Boiler and pressure vessel code method. This is similar to the maximum

shear method, but the shear stres s ranges are based on notional principal

stresses calculated from the changes in the values of 

, 

, 

and 

from datum values occurring at one of the extremes of the stress cycle. Mean

stress effects are not included.

(3) Distortion energy method. In this method, the fatigue evaluation is based on the

ﬂuctuations of the effective or Von Mises stress. In the case of the hub shell, the

stress perpen dicular to the hub surface (and hence the third principal stress) is

zero, so the effective stress is given by:

 9 ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

(

 

)

þ 

(7:44)

As the effective stress is based on the distortion energy, it is a scalar quantity, so

it need s to be assigned a sign corresponding to that of the dominant principal

stress. The effect of mean stress is allowed for in the same way as for the

maximum shear method, except that the stresses in Equation (7.43) are now

direct stresses instead of shear stresses. S–N curves for spheroidal graphite iron

are given in Hu

ck (1983).

7.4 Gearbox

7.4.1 Introduction

The function of the gearbox is to step up the speed of rotor rotation to a value

suitable for standard induction generators, which, in the case of ﬁxed-speed

machines or two-speed machines operating at the higher speed, is usually

1500 r.p.m. plus the requisite slip. For machines rated between 300 kW and

2000 kW, with upper rotational speeds between 48 and 17 r.p.m., overall gear ratios

424 COMPONENT DESIGN