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

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of between about 1:31 and 1:88 are therefore required. Normally these large step-

ups are achieved by three separate stages with ratios of between 1:3 and 1:5 each.

The design of industrial ﬁxed ratio gearboxes is a large subject in itself and well

beyond the scope of the present work. However, it is important to recognize that

the use of such gearboxes in wind turbines is a special application, because of the

unusual environment and load characteristics, and the sections which follow focus

on these aspects. Sections 7.4.2 to 7.4.6 consider variable loading, including drive

train dynamics and the impact of emergency braking loads, and examine how gear

fatigue design is adapted to take accou nt of it. The relative beneﬁts of parallel and

epicyclic shaft arrangements are discussed in Section 7.4.7, while subsequent

sections deal with noise reduction measures, and lubrication and cooling. A useful

reference is the American Gear Manufacturers Association Information Sheet (1996)

which covers the special requirements of wind turbine gearboxes in some detail.

7.4.2 Variable loads during operation

The torque level in a wind turbine gearbox will vary betwe en zero and rated torque

according to the wind speed, with excursions above rated on ﬁxed-speed pitch-

regulated mach ines due to slow pitch response. The short-term torque ﬂuctuations

will be subject to dynamic magniﬁcation to the extent that they excite drive train

resonances (see Section 7.4.3 below). In addition there will be occasional much

larger torques of short duration due to braking events, unless the brake is ﬁtted to

200

400

600

800

1000

1200

0 100 200 300 400 500 600 700 800 900 1000

Power output (kW)

Duration (h/annum/20 kW bin)

Two-bladed 500 kW, 40.4

r.p.m.

pitch-regulated machine

500 kW,

stall-

regulated

machine

Curves based on 8.5 m/s annual mean wind speed

and IEC Class A turbulence intensity

Figure 7.24 Load Duration Curves for 500 kW, Two-bladed Pitch-regulated and 500 kW,

Stall-regulated Fixed-speed Machines

GEARBOX 425

the low-speed shaft. Figure 7.24 shows example load–duration curves (excludi ng

dynamic effects and braking) for two 500 kW, ﬁxed-speed machines – one stall-

and the other pitch-regulated. The curve for the former is calculated by simply

combining the power curve with the distribution of instantaneous wind speeds,

which is obtained by superposing the turbulent variations about each mean wind

speed on the Weibull distribution of hourly means. Excursions above rated power

are not included.

In the case of a pitch-regulated machine, the pitch control system is not normally

designed to respond to wind speed ﬂuctuations at blade-passing frequenc y or

above, as this would impose excessive loads on the control mechanism. Thus there

is no attenuation of the signiﬁcant power ﬂuctuations that occur at blade-passing

frequency due to turbulence, which are illustrated for the example two-bladed,

500 kW machine operating in a 20 m/s mean wind with 16.5 percent turbulence

intensity in Figure 7.25.

The load duration curve for a ﬁxed-speed pitch-regulated machine can be derived

approximately from the distribution of instan taneous wind speeds below rated

wind speed, and the distribution of short-term mean wind speeds (i.e. those to

which the pitch system can respond) above. The former can be combined with the

power curve to give the power distribution due to instantaneous winds below rated

directly, while the winds above rated are assumed to produce Gaussian spreads of

power outputs about the rated value, with the standard deviation depending on the

short-term mean wind. The standard deviation of power ﬂuctuations when the

pitch control system is operational can be related to that portion of the wind

ﬂuctuations above the pitch control system cut off frequency:



, r

, n)dn



d p



d p



(7:45)

100

200

300

400

500

600

700

800

900

0 1020304050

Time (s)

Power output (kW)

Mean wind = 20 m/s, Turbulence intensity = 16.5%, Integral length scale = 73.5 m, Rotational speed = 40 r.p.m.

Rated power = 500 kW, Standard deviation of power fluctuations = 91 kW

Figure 7.25 Simulated Power Output for Two-bladed, 40 m Diameter Pitch-regulated

Machine Operating in Above-rated Wind Speed

426

COMPONENT DESIGN

where S

, r

, n) is the rotationally sampled cross spectrum of the wind speed

ﬂuctuations at a pair of points, j and k , on the rotor (see Section 5.7.5) and (d p=du)

is the rate of change with wind speed of the power generated by the blade elements

at r

on all N blades if the pitch does not change. The summations are carried out

over the whole rotor, and give 

¼ 0:213(dP=du)

¼ 91 kW for the example two-

bladed machine operating at 40.4 r.p.m. in a 20 m/s mean wind with 16.5 percent

turbulence intensity. Here dP=du is the rate of change of turbine power with wind

if the pitch does not change. The standard deviation of the power ﬂuctuations for a

three-bladed machine of similar size would be ab out one third less.

7.4.3 Drive-train dynamics

All wind turbines experience aerodynamic torque ﬂuctuations at blade-passing

frequency and multiples thereof because of the ‘gust slicing’ phenomenon, and

these ﬂuctuations will inevitabl y interact with the dynamics of the drive train,

modifying the torques transmitted. The resulting drive train torque ﬂuctuations can

be assessed by dynamic analysis of a drive train model consisting of the following

elements connected in series:

• a body with rotational inertia and damping (representing the turbine rotor),

• a torsional spring (representing the gearbox),

• a body with rotational inertia (representing the generator rotor) ,

• a torsional damper (modelling the resistance produced by slip on an the

induction generator), and

• a body of inﬁnite rotational inertia rotating at constant speed (the mechanical

equivalent of the electrical grid).

The inertias, spring stiffness and damping must all be referred to the same shaft.

7.4.4 Braking loads

Most turbines have the mechanical brake locate d on the high-speed shaft, with the

result that braking loads are transmitted through the gearbox. If, as is usually the

case, the mechanical brake is one of the two independent braking systems required,

then it must be capable of decelerating the rotor to a standstill from an overspeed,

e.g., after a grid loss. This typically requires a torque of about three times rated

torque.

The mechanical brake is only required to act alone during emergency shut-

downs, which are comparatively rare. During normal shut-downs the rotor is

decelerated to a much lower speed by aerodynamic braking, so the duration of

mechanical braking is much less, but the braking torque is the same, unless there is

provision for two different braking torque level s.

Figure 7.26 is a typical record of low-speed shaft torque during a normal shut-

GEARBOX 427

down, in which the mechanical brake is applied as soon as the generator has been

taken off-line. It is apparent that the braking torque is far from constant, taking

about 2 s to reach its ﬁrst maximum and then falling off slightly before reaching a

higher maximum just before the high-speed shaft stops. Following this, there are

signiﬁcant torque oscillations due to the release of wind-up in the drive train. These

result in torque reversals accompanied by tooth impacts and take some time to

decay.

Although braking loads are infrequent and of short duration, their magnitude

means that they can have a decisive effect on fatigu e damage. The AGMA/AWEA

document (1996) recommends that the time histories of braking and other transient

events are simulated with the aid of a dynamic model of the drive train for input

into both the gear extreme load design calculations and the fatigue load spectrum.

7.4.5 Effect of variable loading on fatigue design of gear teeth

Gear teeth must be designed in fatigue to achieve both acceptable contact stresses

on the ﬂanks and acceptable bending stresses at the roots. In non-wind turbine

0 5 10 15 20

Time (s)

⫺20

Shaft torsion (kNm)

Aero tips

deploy

Brake torque

rising

Generator

off-line

Normal

operation

HSS

stops

Torque reversals

and teeth impacts

Figure 7.26 Low-speed Shaft Torque During Braking at Normal Shut-down. Extracted from

AGMA/AWEA 921–A97, Recommended practices for design and speciﬁcation of gearboxes

for wind-turbine generator systems, with permission of the publisher, the American Gear

Manufacurers Association, 1500 King Street, Suite 201, Alexandria, Virginia 22314, USA.

428

COMPONENT DESIGN

applications, gearboxes typically operate at rated torque throughout their lives, so

the gear strengths are traditionally modiﬁed by ‘life factors’ which are derived from

the material S–N curves on the basis of the predicted number of tooth load cycles

for the gear in question. The Briti sh code for gear design, BS436 (British Standards

Institution, 1986), recognizes an endurance limit for both contact stress and bending

stress, so that the life factors are unity when the number of tooth load cycl es exceeds

and 3 3 10

respectively, but increase for lesser numbers of cycles.

The Hertzian compression stress between a pair of spur gear teeth in contact at

the pitch point (i.e., at the point on the line joining the gear centres) is given by



ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

(1  ı

)

u þ 1

sin Æ cos Æ

(7:46)

where F

is the force between the gear teeth at right angles to the line joining the

gear centres, b is the gear face width, d

is the pinion pitch diameter, u is the gear

ratio (greater than unity), and Æ is the pre ssure angle, i.e., the angle at which the

force acts between the gears – usuall y 208–258.

Note that the contact stress increases only as the square root of the force between

the teeth because the area in contact increases with the force as well.

The maximum bending stress at the tooth root is given by



(7:47)

where h is the maximum height of single tooth contact above the critical root

section, t is the tooth thickness at the critical root section, K

is a factor to allow for

stress concentration at the root.

For gearing operating at rated torque only, the designer needs to show that the

resultant bending stress multiplied by an appropriate safety factor is less than the

endurance limit multiplied by the life factor, Y

, and a number of stress modifying

factors, as follows



:ª < 

B lim

......... (7:48)

A similar calculation is requir ed in relation to the contact stress.

Given the predicted turbine load spectrum (Section 7.4.2), which should include

dynamic effects (see Section 7.4.3), it is then necessary to establish the required

design torque at the endurance limit. Normally this is done by invoking Miner’s

rule and determining the inﬁnite life torque for which the design torque spect rum

yields unity fatigue damage in conjunction with the prescribed S–N curve. Y

Equation (7.48) can then be set to unity, as the life factor has been accounted for in

the derivation of the required inﬁnite life torque.

Figure 7.27 shows specimen torque–endurance curves laid down by BS 436 for

case-hardened gears for tooth bending and tooth contact stress (with no pitting

allowed) plotted in terms of the torque at the endurance limit. Hence in each case

the design inﬁ nite life torque, T

, is calculated according to:

GEARBOX 429



1=m

(7:49)

where N

is the number of cycles at torque level T

, and torques less than T

are

omitted from the summation. The number of cycles at the lower knee of the

torque–endurance curve, N

, is always 3 3 10

cycles for tooth bending but is

generally higher for contact stress, varying according to the material. Note that the

slope index, m, of the torque–endurance curve for contact stress is half that of the

contact stress–endurance curve because contact stress only increases as the square

root of torque (Equation 7.46).

Leaving braking loads out of consideration to begin with, the design inﬁnite life

torque will be equal to the rated torque if there are no power ﬂuctuations above

rated, because the number of gear tooth loading cycles at rated torque will be well

above N

. For example, in the case of the 500 kW stall-regulated machine featured

in Figure 7.24, the teeth on the critical pinion driven by the 30 r.p.m. low-speed

shaft will experience 3 3 30 3 60 3 1050 3 20 ¼ 1:13 3 10

load cycles at rated

torque over 20 years, assuming a ﬁrst stage gear ratio of 3. On the other hand, for

the 500 kW, two- bladed pitch-regulated machine, the power ﬂuctuations above

rated detailed in the Figure 7.24 load–durati on curve result in a design inﬁnite life

torque for the ﬁrst stage pinion tooth bending stress of 1.36 times the rated torque,

with most of the damage coming from torques just above this value. (The ﬁrst stage

gear ratio is assumed to be three as before and the turbi ne rotational speed is taken

as 40.4 r.p.m.) The design inﬁnite life torque for tooth contact stress is only 1:17 3

0.1

1 10 100 1000 10000 100000 1000000 10000000 100000000 1000000000

Endurance, N (cycles)

Torque/Endurance limit torque (BS436) or Torque/Torque at

cycles (ANSI/AGMA)

BS 436 (Appendix A) T–N curve

for case hardened gear tooth

bending (heavy line)

ANSI/AGMA T–N curve

for gear tooth bending

(light line)

BS 436 (Appendix A) T–N curve

for case hardened gear tooth contact

stress with no pitting allowed

(heavy dashed line)

ANSI/AGMA T–N curve

for gear tooth contact stress

(light dashed line)

Figure 7.27 Specimen Torque-endurance Curves for Gear Tooth Design under Variable

Amplitude Loading

430

COMPONENT DESIGN

rated torque – signiﬁcantly less than for bending, as expected from comparison of

the BS 436 torque–endurance curves in Figure 7.27.

Figure 7.27 also shows specimen torque–endurance curves derived from S – N

curves in the American National Standards Institute/AGMA 2001-C95 plotted in

terms of the torque at 10

cycles. The torque–endurance curve for tooth bending

stress, which is based on a middle of the range Brinell Hardness value of 250 HB,

closely parallels the selected BS 436 curve, except that the curve continues with a

very shallow slope beyond 3 3 10

cycles instead of displaying an endurance limit.

The design torques at 10

cycles for tooth bending for the example 500 kW machines

featured in Figure 7.24 are similar to the design inﬁnite life torques obt ained using

the BS 436 torque–endurance curves.

The ANSI/AGMA (1995) torque–endurance curves for tooth contact stress are

signiﬁcantly more conservative than the selected BS 436 curve. This is particularly

so in the case of the ANSI curve selected, which is the one recommended for wind

turbine applications, in view of the elimination of the lower knee. The absence of

the lower knee increases the design torque at 10

cycles for tooth contact to 1.4 times

the rated torque for the stall regulated machine, but the ﬁgure for the pitch

regulated machine is only about 10 percent higher.

From the above discussion, the general conclusion can be drawn that tooth

bending fatigue usually governs the increased gearbox rating required to take care

of load excursions above rated.

The effect of braking loads on the design inﬁnite life torque according to BS 436

can be illustrated with respect to the example machines discussed in Section 7.4.2.

Although the mechanical brake must be capable of decelerating an overspeeding

rotor unassisted, a shut-down under these conditions will be a very rare event.

Accordingly the typical emergency shut-down considered for fatigue design pur-

poses is deceleration from normal rotational speed under the action of mechanical

and aerodynamic braking combined, with an assumed stopping time of 3 s. An

emergency shut-down frequency of 20 per annum is assumed. Normal shut-downs

are assumed to occur on average twice a day, with a stopping time of 1.5 s, because

of the reduced rotational speed at which mechanical braking is initiated for parking.

In each case the braking torque is assumed to remain constant at three times rated

torque throughout the brake application for simplici ty. Based on these ass umptions,

the percentage increases in design inﬁnite life torque for gear tooth bending in

fatigue, due to the inclusion of braking loads in the load spectrum, are shown in

Table 7.5 for emergency braking alone on the one hand and normal plus emergency

shut-downs on the other.

Also shown in the table are the percentage increases in the ANSI/AGMA design

life for gear tooth bending at 10

cycles due to the inclusion of braking loads. It is

seen that the inclusion of emergency braking loads alone makes very little differ-

ence to design torques in the case of the pitch-regulated machine, but is signiﬁcant

in the case of the stall-regulated machine. The addition of braking loads at normal

shut-downs incurs a much greater penalty in both cases because of the large

number of stops involved, indicating that provision for brake application at

reduced torque on these occasions would probably be worthwhile. Note that the

larger percentage increases in design torques due to braking indicated by BS 436

are a conseque nce of the assumption that there is an endurance limit.

GEARBOX 431

7.4.6 Effect of variable loading on fatigue design of bearings and

shafts

Bearing lives are approximately inversely proportional to the cub e of the bearing

loading. Applying Miner’s rule, the equivalent steady bearing loading over the

gearbox design life can thus be calculated from the load duration spectrum

according to the formula

eqt

1=3

(7:50)

where N

is the number of revolutions at bearing load level F

. Gravity often

dominates the loading on the low-spe ed shaft bearings, but on the other shafts the

bearing loads result from drive torque only, so the bearing load duration spectrum

can be scaled directly from the torque duration spectrum. Note that the S– N curve

for bearings is much steeper than those for gear tooth design, so that occasional

large braking loads will be of less signiﬁcance.

The nature of the fatigue loading of intermediate shafts is essentially different

from that of gear teeth, as the former is governed by the torque ﬂuctuations as

opposed to the absolute torque magnitude. Consequently the fatigue load spectrum

for shaft design should be derived from rainﬂow cycle counts on simulated torque

time histories rather than on the load duration curve used for gear tooth design.

Table 7.5 Illustrative Increases in Design Torques for Gear Tooth Bending due to Inclusion

of Braking Loads in Fatigue Load Spectrum, According to BS 436 and ANSI/AGMA Rules

500 kW Stall-regulated

machine

500 kW Two-bladed pitch-regulated

machine

Percentage

increase in BS

436 design

inﬁnite life

torque for tooth

bending

Percentage

increase in

ANSI/AGMA

250 HB design

torque at 10

cycles for tooth

bending

Percentage

increase in

BS 436 design

inﬁnite life

torque for

tooth bending

Percentage

increase in

ANSI/AGMA

250 HB design

torque at 10

cycles for tooth

bending

Emergency braking

at 3 3 FLT

30% 16% 4% 3%

Emergency plus

normal braking,

each at 3 3 FLT

65% 47% 25% 21%

432

COMPONENT DESIGN

7.4.7 Gear arrangements

Parallel axis gear s may be arranged in one of two ways in eac h gear stage. The

simplest arrangement within a stage consists of two external gears meshing with

each other and is commonly referred to as ‘parallel shaft’. The alternative ‘epicyclic’

arrangement consists of a ring of planet gears mounted on a planet carrier and

meshing with a sun gear on the inside and an annulus gear on the outside. The sun

and planets are external gears and the annulus is an internal gear as its teeth are on

the inside. Usually eithe r the annulus or planet carrier are held ﬁxed, but the gear

ratio is larger if the annulus is ﬁxed.

The epicyclic arrangement allows the load to be shared out between the planets,

reducing the load at any one gear interface. Consequently the gears and gearbox

can be made smaller and lighter, at the cost of increased complexity. The scope for

material savings are greatest in the input stages of the gear train, so it is common to

use the epicyclic arrangement for the ﬁrst two stages and the parallel shaft

arrangement for the output stage. A further advantage of epicyclic gearboxes is

greater efﬁciency as a result of the reduced sliding that takes place between the

annulus and planet teeth.

The derivation of the optimum gear ratio in a series of parallel shaft stages is

fairly straightforward and is desc ribed below. Equation (7.47) for tooth bending

stress can be modiﬁed as follows



¼ F

6(h=m)

bm(t=m)

¼ F

(h=m)

(t=m)

(7:47a)

where m is the module, deﬁned as d

for spur gears and z

is the number of

pinion teeth. If the ratios h=m and t=m are treated as constants, then the bending

stress is proportional to the number of teeth for a given size of gear. Hence the

design of the gears is governed by contact stress because, in principle, the bending

stress can always be reduced by reducing the number of pinion teeth. Thus, based

on Equa tion (7.46), the permitted tangential force, F

, is proportional to bd

u=(u þ 1)

so that the permitted low-speed shaft torque, T

LSS

¼ F

=2 is given by

LSS

/ d

u=(u þ 1) ¼ bd

=(u þ 1) (7:51)

Hence the volumes of the low-speed shaft gear wheel and the meshing pinion can

be expressed as V

¼ kT

LSS

(u þ 1), where k is a constant, and V

¼ V

respec-

tively. These can be used to derive an expression for the volume of gears in a drive

train with an inﬁnite number of stages each with the same ratio. It is found that the

total gear volume is a minimum for a gear stage ratio of 2.9, but increases by only

10 percent when the ratio drops to 2.1 or rises to 4.3.

The gear teeth of parallel shaft gear stages are only loaded in one direction, so the

permitted alternating bending stress in fatigue, 

alt

, is modiﬁed to account for the

non-zero mean value in accordance with the Goodman relation:

GEARBOX 433



alt



lim

¼ 1 



ult

(7:52)

where 

lim

is the permitted alternating bending stress with zero mean,  is the

mean bending stress and 

ult

is the ultimate tensile strength. Settin g  ¼ 

alt

results in:



alt



lim



ult

(

ult

þ 

lim

)

(7:53)

If the 

lim

=

ult

ratio is 0.2, then 

alt

¼ 0:833

lim

and the permitted peak bending

stress at the endurance limit is 1:667

lim

. In epicyclic gearboxes, by contrast, the

gear teeth on the planet wheels are loaded in both directions, so the permitted peak

bending stress at the endurance limit is only 0:5

lim

. As the number of teeth on the

smallest gear cannot be reduced indeﬁnitely, this means that tooth bending is more

likely to govern in the case of epicyclic gearing.

The minimum total gear volume for an inﬁnite series of epicyclic gear stages with

ﬁxed annuli is obtained for a gear stage ratio of two, which implies that the radius

of the sun gear is the same as that of the annulus gear and that there are an inﬁnite

number of planets! This is not realistic, and the annulus radius is in practice

typically double the sun radius, giving a gear ratio of three. It is instructive to

compare the volume of gears for an epicyclic and parallel gear stages with this ratio,

assuming that tooth bending stress governs in each case.

For the parallel stage, it can be shown using Equation (7.47a) that the volume of

the pinion is



¼ k

=

alt

¼ k



alt

¼ k

LSS

0:833

lim

(7:54)

where k

is a constant. This gives a volume for gear wheel and pinion of

2:4k

LSS

(1 þ 1=u

)u=

lim

¼ 8k

LSS

=

lim

for u ¼ 3. For the epicyclic stage, the

volume of the planet, which is assumed to have the same number of teeth as the

pinion of the parallel stage, i.e., the minimum permissible, is



¼ k

=(0:5

lim

)(7:55)

If the low-speed shaft drives the planet carrier and the N planets are spaced at 1.15

diameters, then the low speed shaft torque is

LSS

¼ F

N(r

þ r

)

where

N ¼

(r

þ r

)

1:15(r

 r

)

434 COMPONENT DESIGN