Tong W. Wind Power Generation and Wind Turbine Design

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Optimized Gearbox Design 515

fdm

(3 )

where m = module = d / N . Then,

fd =

(4 )

where f , d and T are as before and N is the number of teeth.

By equating the surface and bending volumes derived as above, it is possible to

obtain a non-dimensional “optimum” number of pinion teeth based on the balance

of bending to surface criteria and the gear ratio viz. By comparing eqns (1) and (4),

it yields,

N= +

⎛⎞

⎜⎟

⎝⎠

(5)

Since tooth number is unaffected by face width to diameter ratio or torque, it is

only necessary to choose a rounded down number compatible with the nearest

standard pitch and the required face width, diameter and ratio. Thus, root ﬁ llet

stress is not usually a limiting criterion because the pitch is easily increased by

reducing the number of teeth. Nonetheless, there are big incentives for making

pitch as ﬁ ne as possible.

A smaller pitch with bigger tooth numbers has a somewhat greater contact ratio 1.

but a shorter path of contact and commensurately lower tooth sliding veloci-

ties. This improves efﬁ ciency and reduces sliding losses and associated sur-

face related problems such as scufﬁ ng. It also reduces surface stress slightly by

increasing the relative radius at the chosen load point.

For a given load, the reduced bending moment on the root of a shorter tooth 2.

means a thinner rim is required for its support. This is very important in epicyclic

gears as described later.

For simplicity, the foregoing consideration of gear geometry is conﬁ ned to spur

rather than helical gears. The latter also embody tip relief to mitigate pitch error

problems as the teeth enter and leave the contact zone. However, experience sug-

gests that although they are generally quieter than spurs, they are both subject to

the same problems associated with the effects of parasitic loads and deﬂ ections

and the helix corrections required to compensate for them. Unfortunately, such

corrections only work for one condition. Attempts to cater for varying conditions

by crowning the teeth, inevitably lead to higher stresses.

3 G eartrains

It is not practical to have a single stage gearbox to provide a step-up ratio of 100:1.

In practice, such an overall ratio invariably requires three stages. To minimise size

516 Wind Power Generation and Wind Turbine Design

and weight, particularly in the ﬁ rst two, high torque low speed stages, it is usual to

employ epicyclic gearing in which load is shared via three or more parallel load paths.

As shown in Fig. 5 , such gears have the further advantage of having co-axial input and

output shafts rather than the offset parallel axes of a simple wheel and pinion.

The simplest form of epicyclic gear comprises three co-axial elements; a sun

wheel, a planet carrier, which provides a straddle mounting for a number of equi-

spaced planet wheels and an internally toothed ring gear or annulus. The ﬁ gure

shows that the planet wheels serve as idlers (no residual torque) between the sun and

annulus wheels. If the planet carrier is ﬁ xed, the sun and annulus rotate in opposite

directions, with the sun rotating at − R times the speed of the annulus where

R= =

(6 )

where N

and N

are the teeth numbers, and d

and d

are the pitch diameters of the

annulus (ring) and sun wheel, respectively.

It can be seen that the carrier has a torque reaction equal and opposite to the sum

of the sun and annulus torques. From this, it can be inferred that if the annulus is

ﬁ xed then the sun will rotate at +( R + 1) times the speed of the carrier and in the

same direction .

Conventionally, most simple epicyclic gears have three planets and to ensure

equal load sharing the sun is allowed to ﬂ oat so that it can ﬁ nd an axis which ensures

its equilibrium and compensates for the collective errors in the concentricity of the

respective axes of the sun wheel, planet carrier and annulus. This therefore requires

a suitable ﬂ exible coupling to transmit the sun wheel torque.

Planet

bearing

load

Sun

wheel

Planet

wheel

Annulus

wheel

Figure 5: Half section epicyclic gear.

Optimized Gearbox Design 517

Various solutions have been used to provide load sharing for epicyclic gears

having more than three planets. The most widely used have employed a ﬂ exible

annulus ring which subject to its tooth forces deﬂ ects as shown in Fig. 6 . However,

the maximum number of planets is usually limited to 6, because with greater num-

bers, load sharing becomes less effective as the deﬂ ections decrease. Even though

more planets enable the ring thickness and weight to be appreciably reduced, it is

not enough to give the required deﬂ ections without excessive stresses. In addition,

the planet spindles are straddle mounted in a carrier which requires rigid webs

between the planets to try and minimise its torsional wind up and the mal-distribution

across the meshing faces of the planet wheel teeth.

The main problem associated with ﬂ exible annulus rings is that even with con-

stant torque, they are subject to fully reversed cyclic bending stresses due to the

outward and inward deﬂ ections, with the passage of each planet (see Fig. 7 ).

The most logical location for ﬂ exibility is in fact the planet spindle. Because it

serves as a mounting for an idler with zero torque, the relative load on the spindle

is always in a constant direction, whether or not the carrier is rotating. It follows

therefore that subject to constant torque, deﬂ ection is static, and not subject to a

Figure 6 : Annulus ring bending deﬂ ections. The deﬂ ection curves should not be

offset laterally but located symmetrically so that they show the radial

inward and outward distortions of the respective 3 and 8 planet annulus

rings from their circular shapes.

Bending

stress

ular distance between planets

Figure 7: Cyclic stress reversals.

518 Wind Power Generation and Wind Turbine Design

primary fatigue condition. However, when torque is variable, there is clearly an

associated secondary unidirectional fatigue condition in the planet spindles as well

as the gear teeth. In any case, even with constant torque, sun and annulus gear teeth

are subject to unidirectional fatigue as they rotate, in and out of mesh with the

planet wheel whose teeth are subject to full fatigue load reversals as they alter-

nately mesh on opposite ﬂ anks with the sun and annulus. Therefore, all gears

whether epicyclic or otherwise, have to be designed to accept primary fatigue

loads as well as the secondary effects of torque ﬂ uctuation.

As stated previously, conventional planet carriers cannot be made completely

rigid so that inevitably, the webs joining the two ﬂ anges which support either end

of the planet spindle are subject to shear and bending deﬂ ections that create a tor-

sional deﬂ ection of one ﬂ ange with respect to the other to misalign the planet

wheel. While it is feasible to calculate this deﬂ ection and compensate for it either

by boring the carrier skewed or by helix corrections on the mating gears this only

helps at one nominal torque. It is therefore, usual to crown the face widths of the

gear teeth to avoid edge contacts on either end which would otherwise occur at

different loads. This reduces the contact area and increases the local stresses. Fur-

thermore, the planet bearing load is no longer on the centre of its spindle which

can also be a source of bearing problems.

Figure 8 shows the principles of the compound cantilever ﬂ exible planet spindle

comprising a ﬂ exible inner member and a comparatively rigid co-axial outer sleeve.

Central tooth loads at the planets sun and annulus mesh points create equal and

opposite moments at either end of the inner pin with a point of inﬂ ection at the cen-

troid of load, where the bending moment is zero. The spindle is very soft in an angu-

lar sense to such an effect that it cannot sustain any unequal loading across either

gear face, e.g. if a planet wheel has a helix error which could lead to heavier loads

at opposite ends of its respective face contacts with the sun and annulus, then the

Planet carrier

Flexible pin

Spindle

Planet

Tooth load

Point of inflexion

Figure 8: Flexible planet spindle.

Optimized Gearbox Design 519

effective centroid of its tangential loads would still be at the midpoint of the spindle.

However, the associated radial components of tooth load due to pressure angle create

a tilt in the radial plane, which reduces the tipping couple and the mal-distribution of

load by which it is generated, so that the planet adopts a skewed equilibrium attitude

with a low mal-distribution commensurate with its very low angular rigidity. The

crossed helix effect created by having non-parallel axes leads to a notional point

contact rather than line contact on the tooth faces. Since the crossed helix angle is

very small it relieves the tendency for edge contacts in a manner analogous to crown-

ing. The ﬂ exible spindle has proved conclusively that it can compensate for helix

errors of different magnitude and hand in sun, planet and annulus by tilting in a

complex way to a position of minimum strain energy to enable the planet wheel to

avoid the load mal-distributions that are imposed by a more rigid support. In simple

terms, the planet dictates where it wants the spindle to be rather than vice versa.

Unlike a ﬂ exible annulus, the planet spindles are all independent of one another so

they are all free to do their own thing and because they are cantilevers, the only limit

on the number of planets is the clearance of their adjacent tip diameters and the

annulus to sun ratio. As this ratio varies from 2.15 to 5.2 the number of planets

reduces from 8 to 4. For bigger ratios than 5.2 only 3 can be accommodated.

A larger number of equally loaded planets directly reduce the overall volume of

an epicyclic geartrain. This is shown by deriving a similar volumetric expression

as that shown above for a simple parallel shaft pinion and wheel viz.

It can be seen in Fig. 5 that the relationship of three pitch diameters can be

expressed as

−

(7 )

Epicyclic analogy gives

n= =

−

(8 )

Noting that

12 1

R+

+=+ =

nRR−−

(9 )

Thus,

R+ C

⎛⎞

⎜⎟

⎝⎠

−

(10 )

R+ C

N=NR=R

⎛⎞

⎜⎟

⎝⎠

−

(11 )

1(1)

R+

fd = =

QK R QK R

⎛⎞

⎜⎟

⎝⎠

−−

(12 )

520 Wind Power Generation and Wind Turbine Design

and

222

(1)

fd = fd R =

QK R −

(13 )

where d

is the planet wheel pitch diameter, d

the sun wheel pitch diameter, d

the

annulus pitch diameter, f the sun wheel face width, N

the sun wheel tooth number,

C the planet tooth bending criterion, K the sun wheel surface criterion, T

the sun

wheel torque, T

the planet carrier torque and Q is the number of planets.

From Fig. 5 it can be seen that in effect, an annulus has a negative diameter

exempliﬁ ed by the concave ﬂ anks on internal teeth. This means that given the

same pressure angles, the product of the annulus and planet base tangent lengths is − R

times that of the planet and sun whereas the sum of the respective base tangent

lengths are equal and opposite so that algebraically, the relative radius of curvature

at the planet/annulus mesh point is precisely R times that of the planet/sun. Given

the same face widths its K value is reduced accordingly by the reciprocal of R . The

internal tooth root thickness is also somewhat thicker due to its concavity so that

lower grade material and/or a smaller face width may be used.

The signiﬁ cance of the above is illustrated by comparing the annulus volumes

of two planetary gears having the same carrier torque and sun wheel surface stress

but with R equal to either 2 or 3, i.e. planetary ratios of 3 and 4 having either 8 or

5 planets respectively viz.

ac c

(4 / 8) 0.5fd = T = T

(14a )

ac c

(9 /10) 0.9fd = T = T

(14b )

The larger ratio annulus is therefore, 1.8 times the volume!

The comparable volumes of a simple wheel subject to the same torques, surface

stress and ratios are viz.

wc c

(3 1) 4fd = T + = T

(15a )

wc c

(4 1) 5fd = T + = T

(15b )

Without considering the pinion offset, the ﬁ rst is 8 times and the second 5.56 times

the volumes of the annuli of the alternative planetary gears.

Even with only three planet wheels, the volume of the annuli is always 30% of

an equivalent parallel shaft wheel for any ratio from 3 to 12.

4 B earings

Rolling element bearings are the type most commonly used in wind turbines for both

parallel shaft and epicyclic gears. Generally, the design criteria for such bearings

Optimized Gearbox Design 521

leads to a ﬁ nite life which takes account of the total number of hours at varying

loads. The most heavily loaded are in the high torque low speed primary trains and

in particular the planet spindles which sustain the double tooth loads on the planet

meshes with the sun and annulus. The most successful arrangement has been a

pair of preloaded taper roller bearings which ensure that at light loads there is no

risk of skidding.

To maximise the bearing space available between the bore of the planet and the

spindle especially for low annulus/sun ratios it helps to have ﬁ ne pitch teeth to

increase the root diameter, reduce rim thickness and increase the bore. It also helps

if roller outer races are embodied in the planet bores. Timken have gone further by

also integrating the inner races in the planet spindle and using full complement

preloaded tapered rollers. All planet bearings together with all other lower loaded

higher speed bearings in the secondary trains require a pressurised supply of lubri-

cant. No bearing should be subjected to misalignment and self-aligning bearings

should be avoided. They cannot be effectively preloaded because they have clear-

ances which may lead to skidding on low loads.

In this context, the ﬂ exible planet spindle ensures that however much the torque

may transiently vary, the bearing load always stays in the same place, i.e. the plane

of the face width centres so that it is equally shared when two or more bearings are

required to carry the load.

For smaller gears it is quite possible to have fully ﬂ oating suns and annuli whose

dead weight can, without detriment, be supported on their gear tooth meshes but

generally not for planet carriers. As power increases, the tooth force to component

weight diminishes and there comes a point where annulus rings and even sun

wheels have a signiﬁ cant effect on load sharing and need support.

5 Gear arrangements

As shown in Fig. 9 the most commonly used arrangement employs two plan-

etary step-up gear stages (with ﬁ xed annuli) coupled in series with the second-

ary sun wheel driving a parallel shaft wheel via a double tooth type coupling.

This wheel meshes with a pinion having a parallel offset determined by the

required location of the generator which it drives via a proprietary spacer type

coupling. The primary reason for the offset is to provide a co-axial access to

the turbine rotor from the rear of the gearbox for pitch control purposes, e.g.

electrical slip rings.

Figure 10 shows an arrangement of the epicyclic stages featuring a star/plane-

tary differential with its input torque divided between the annulus of a primary star

stage and the planet carrier of a secondary differential stage whose annulus is

coupled to the primary sun wheel. Thus the primary planet carrier is the sole static

torque reaction member of the combined trains, while the secondary differential

sun wheel is the output coupled to the parallel shaft wheel. The signiﬁ cance of this

is that the torque reaction is no longer transmitted to the gear case via a live gear

such as an annulus. This reduces structure-borne vibrations particularly when

ﬂ exible planet spindles are used.

522 Wind Power Generation and Wind Turbine Design

It is usual to mount a brake disc on the output shaft of the gear. This has two

functions, ﬁ rst as a parking brake and secondly to stop the turbine in an emergency.

The second function generally imposes up to three times the nominal full power

torque on the drive train. However, in the light of earlier comments, this is quite

probably no worse than the torque ﬂ uctuations it experiences in normal operation.

Figure 9 : Conventional 3 train arrangement.

Fully floating torque reaction arm

Figure 10: Star/planetary differential arrangement.

Optimized Gearbox Design 523

The most critical aspects affecting reliability are the mounting of the gearbox

and the coupling of the turbine shaft to its input. Hitherto, a majority have had the

turbine shaft supported at its front by a single bearing mounted on the nacelle bed

plate while its rear end has been supported by the gearbox input shaft to which it

is rigidly coupled via a shrink ﬁ t coupling. The rear end of the turbine shaft is

therefore, supported by the gear case and its resilient mounts via the input shaft

bearings. This has created detrimental parasitic loads on the gearbox due to the

pitching and yawing couples and associated shaft bending deﬂ ections plus deﬂ ec-

tions in the mounts due to torque ﬂ uctuations. In the light of the problems that

have arisen from this situation, as powers have increased, most recent designs have

featured large back to back taper roller bearings in TDO conﬁ guration to indepen-

dently support the turbine rotor in a mounting frame. The gearbox input shaft is

rigidly coupled to the rotor hub while the gear case torque reaction is supported by

a suitable mechanism designed to impose only pure torque (see Fig. 11 ). In effect

the rotor hub supports the gearbox, not vice versa.

6 T orque limitation

In its simplest form, the differential properties of an epicyclic gear can be exploited

by allowing what would otherwise be its ﬁ xed reaction member to rotate in the

direction imposed by its torque. This is effected by gearing it to a ﬁ xed stroke posi-

tive displacement pump with its delivery bypassed to its inlet via a pressure relief

valve to give a limited slip at a controlled torque. Such a gear is best used on the

ﬁ nal low torque high speed stage where the component sizes are much smaller and

more manageable. This has been used very successfully by the Windﬂ ow company

in New Zealand for 500 kW turbines driving synchronous generators. They have

Rotor

TDO brg

Nacelle

Fully floating

torque reaction

Figure 11: Independent rotor support arrangement.

524 Wind Power Generation and Wind Turbine Design

found that at this size, it has worked quite successfully without heat dissipation

problems with a limited slip of up to 5%.

For larger powers, to provide better control and a bigger speed range with greater

energy capture without excessive losses, it is necessary to control the reaction with

a closed loop bypass branch comprising either a hydraulic pump and motor or an

electrical equivalent to recover the power which would otherwise be lost. With

such a system, torque may be monitored to enable transient referred inertia effects

to be eliminated. Variable ratio gears using this principle have been successfully

developed for powers up to 3.6 MW with synchronous generators driven by tur-

bines with speeds ranging from 60 to 100%. In effect, such gears allow turbine

speeds to increase when subject to a transient torque increase so that the excess

torque is absorbed by the increased kinetic energy in the rotor while the excess

speed is absorbed by the reaction member. Conversely, when the turbine torque

has a transient decrease its speed can be reduced by a ratio change to recover the

kinetic energy. For more sustained changes the gear ratio is changed accordingly

(see Fig. 12 ).

7 C onclusions

The purpose of this chapter is to show how the transient torque/speed charac-

teristics of a wind turbine affects the volume/weight of the drive train and the

beneﬁ ts that accrue due to the use of epicyclic gears not only for reducing weight

and increasing compliance but also for their differential torque limiting properties.

Generator

Reaction

M/C2

Reaction

M/C1

Output gears

Epicyclic

gears

Figure 12 : Variable ratio gear arrangement.