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CHAPTER 15

Optimized gearbox design

Ray Hicks Ray Hicks Limited, UK.

Superfi cially, gearboxes for wind turbines are required for a low technology, low speed and relatively low power application. However, their very high torque and speed increasing ratio requirements coupled with the capricious nature of the power source have created many problems which have had a detrimental effect on reliability. In reality therefore, they have had to be manufactured to the highest possible quality with corrections to gear teeth, etc. to compensate for the parasitic loads and defl ections to which they are subjected. This chapter explains the basics of gear design criteria and offers solutions to the various problems.

1 Introduction

Wind turbines in common with virtually all other rotary machinery are subject to speed limits such that the product of rotor diameter and rotational speed is a con-stant, i.e. blade tip diameter is inversely proportional to speed. Since the propor-tions of the blade length and chord section tend to be constant, then given similar materials, wind speeds, etc., the rotor weight and torque are proportional to the linear dimension cubed.

Because of the inverse relationship of diameter and speed, the product of torque and speed, i.e. power, is directly proportional to the rotor diameter squared and there-fore, the swept area. Thus, the power to weight ratio diminishes as power increases.

For example, if power is increased from 750 to 3000 kW, i.e. by a factor of 4, the rotor diameter is doubled and its rotational speed is halved. It follows that the weight and torque of the rotor are increased by a factor of 8. Incidentally the moment of inertia (related to the 5th power of the diameter) is increased by a factor of 32.

Unlike other methods of power generation such as gas turbines, the input energy source of wind turbines is of an uncontrolled stochastic nature. Its velocity, direc-tion and pressure distribution over the swept area are all subject to sudden changes

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510 Wind Power Generation and Wind Turbine Design

which require complementary changes in the rotor speed, blade pitch and nacelle orientation. However, because of inertia effects such changes cannot be made within a compatible time scale during which, the rotor hub is transiently required to sustain whatever loads this might entail.

Since power is proportional to wind speed cubed, a transient speed increase of only 50% will more than double the torque and treble the power. Even if the wind speed remains constant, a change of its direction with respect to the axis of rota-tion means that the rotor will run yawed such that the angle of attack on the indi-vidual blades will vary continuously as they rotate. Since it is virtually impossible to keep moving the nacelle in step with every transient it is only practicable to respond to a sustained change of direction. Thus, the turbine could spend a signifi cant amount of its time running yawed.

In any case, most wind turbines face upwind with their rotor axes tilted some 5° up at the front to reduce overhang from the nacelle and the danger of blades col-liding with the tower. The rotor therefore, will always be yawed even if in other respects it is perfectly aligned to the wind. Over the large swept area of a turbine there are signifi cant variations in wind speeds, angles of attack and blade defl ec-tions which inevitably promote angular fl uctuations at the rotor hub and conse-quentially, large cyclic torque and electrical power variations in the generator. While the electrical fl uctuations may be dealt with electronically, the associated mechanical torque fl uctuations due to the referred inertia of the generator rotor can only be absorbed by strain energy defl ections in the drive train and/or an active form of torque control.

Assuming similar density materials, a direct drive generator will have 100 times the torque and weight of with a step up ratio of 100/1 the geared version and if their respective generator rotor lengths are approximately the same, it would have the same polar moment of inertia as that of the high speed generator whose inertia is multiplied by gear ratio squared when referred to the turbine rotor. The power to weight ratio of the direct drive generator, like the turbine will be subject to the same disproportionate decrease in its power to weight ratio, whereas that of a geared generator is constant.

Due to the universal application of constant frequency grid systems and cheap standardised high speed generators produced in large numbers, the cost of direct drive generators produced in relatively small numbers is inevitably much greater.

As power increases, the input shaft of a gearbox is subject not only to the same disproportionate increase in turbine torque but also a bigger step up ratio. How-ever, this incurs a much smaller increase in overall weight and cost of the nacelle/tower assembly compared with the direct drive alternative. Thus, despite their reli-ability problems, geared generators have generally been the preferred option for the vast majority of wind turbines.

2 Basic gear tooth design

Toothed gearing is historically the most effective and effi cient mechanism for cou-pling machines having different optimum speeds. Its development has therefore been driven by purely economic considerations.


Optimized Gearbox Design 511

In its simplest form, a fi xed ratio gear comprises a pinion with a smaller number of teeth meshing with a wheel having a larger number of teeth whose respective axes are parallel.

The difference in tooth numbers then determines the ratio between the respec-tive speeds of pinion and wheel; e.g. a 100-tooth wheel will drive a 20-tooth pinion at fi ve times its own speed. As shown in Fig. 1 , the wheel and pinion rotate in opposite directions.

To provide a constant velocity ratio, the respective teeth must have the same precise circular pitch and a geometric shape which enables the torque to be transmit-ted from one tooth to the next by a slide/roll mechanism which ensures a constant circumferential velocity. The universally chosen tooth form is an involute whose properties are clearly described in any gearing text book. While toothed gearing is very simple in principle, it is very diffi cult to implement in practice. Torque is trans-mitted as a normal load between the mating teeth .but even if they are geometrically perfect, this load creates surface and bending defl ections which in effect create pitch errors that vary with torque. In addition, misalignments occur due to associated defl ections in the shafts, bearings, mountings, casings, etc. which support the gears. It becomes even more diffi cult when the gearbox is subjected to externally generated forces due to the variable nature of the wind. All these effects create unacceptable mal-distribution of tooth load across the face width of the gears.

Figure 2 shows the pitch circles of a pinion and wheel which contact one another at a pitch point on the line joining their respective centres. The pitch line passing through this point is tangential to the pitch circles and therefore, crosses the centre line at right angles. The circumference of the respective pitch circles is equal to their tooth numbers multiplied by the common circular pitch. As shown, the path of contact between the mating gears is a straight line common tangent to their respective base circles from which the involute tooth fl anks are generated. This passes through the pitch point at an angle to the pitch line known as the pressure angle (usually 20°). Its length is determined by the distance between the two points where the respective tooth tip diameters cut across the common tangent. For con-tinuity of transmission the normal distance between successive tooth fl anks (the base pitch) has to be less than this length by a factor known as the contact ratio. For most standard gears this varies between 1.4 and 1.7. Thus, at the beginning and

Figure 1: Simple wheel and pinion.



end of the contact path there are two pairs of teeth engaged, whereas in the centre, there is only one. Considering the meshing sequence: just as an unloaded pair of teeth are about to enter the double tooth contact zone at the beginning of contact there is only one pair of teeth transmitting the load at one base pitch from the beginning of the contact path. These loaded teeth will therefore, have a combined defl ection which creates a relative pitch error with respect to the unloaded teeth. It is standard practise to modify the involute profi les of mating gears by tip relief designed to ensure that the tooth load increases progressively from zero to its nominal value as it passes through the double tooth contact zone at the beginning of the contact path into the single tooth contact zone, with a complementary decrease as it subsequently passes through the double contact zone at the exit.

Gear tooth design is required to satisfy two basic fatigue stress criteria, i.e. tooth root tensile bending and surface compressive stresses. The critical area therefore, for both is in this single tooth contact zone.

Surface contact stress is the criterion which effectively determines the pitch cylinder volumes of a pair of gears, i.e. their respective diameters squared multi-plied by their face width. The compressive stress generated by the normal force between the teeth is determined by dividing this force by the meshing face width and the relative radius of curvature at the contact point which varies as it pro-gresses from the beginning to the end of the path of contact. This is because rela-tive radius is the product of the respective tangent lengths to the contact point divided by their sum, i.e. the constant length of the common tangent. For a given common tangent length, the product of respective pinion and wheel tangent lengths would be a maximum if they were equal. Clearly, this would only happen if the

Base circle

Tip circle

Pitch circle

Pressure angle

Common tangent of pitch circles

Pitch point

Base pitch

Figure 2 : Base tangent contact path.



pinion and wheel were of the same size. It follows that relative radius of curvature is minimum at the lowest point of contact between the wheel tip diameter in the root of the pinion. However, this is in the double tooth contact zone and thus the chosen load point for calculating the highest surface stress is at the lowest point of single tooth contact on the pinion fl ank.

The criterion calculated as above is known as the Sc factor whose value is directly proportional to torque. It is therefore, valid for directly comparing load capacity taking into account any linear application and service factors (factors of ignorance!). While superfi cially, it has the dimensions of stress, in fact it is neces-sary to take the square root of Sc (after it has been multiplied by the various fac-tors) then further multiplying this by a constant (190 for N/mm 2 ) or (2290 for lb/in 2 ) to get the “actual” compressive stress. The reason for the non-linear relation-ship between load and stress is that the contact area increases as it fl attens so that if load is increased by a factor of 4, stress is only doubled. Most international design standards use this as their surface stress criterion. This leads to the anomaly that an acceptable surface safety factor based on stress is the square root of the associated Sc and bending safety factors directly related to load.

Historically, a simplifi ed surface criterion known as the “ K ” factor, has been universally used for gear design. In effect, it is similar to Sc but as an approxima-tion, it takes the pitch point as the chosen load point and further simplifi es calcula-tion by treating the sine and cosine of pressure angle as constants. Arbitrary limits for K may then be used as appropriate, for different applications, gear materials, pressure angles, etc. Using this approach, it is much easier to relate the volume of gears directly to the torque and ratio in a particular application viz.

112 T

fd = +n K

⎛ ⎞⎜ ⎟⎝ ⎠

(1 )

ww ( 1)2 T

fd = nK

+ (2 )

where K is the surface criterion, n the wheel/pinion ratio, f the face width, d the pinion pitch diameter, d w the wheel pitch diameter, T the pinion torque and T w is the wheel torque.

The chosen load point for calculating bending stress in both pinion and wheel, is their respective highest point of single tooth contact, i.e. one base pitch from either end of the contact path as appropriate. Figure 3 shows the angle at which the normal tooth load at this point crosses the centre line of the tooth.

This load is then resolved into its tangential and radial components which respectively, create bending and direct compressive stresses in the tooth root. The resultant maximum tensile and compressive root fi llet stresses, in particular the tensile, may be determined for the actual pitch, face width, etc. by an iteration process which takes account of the precise geometric shape of the fi llet and the associated stress concentration factor. The results are then compared with the per-missible tensile fatigue limit suggested by the Goodman diagram as shown in Fig. 4 . This shows that when the load is unidirectional the mean stress is half the tensile



maximum with an alternating range from zero to the maximum. In the case of an idler such as an epicyclic planet as shown later, tooth load reverses as it alternately meshes with the sun and annulus. The mean root stress obtained by taking the alge-braic sum of the tensile and compressive stresses divided by 2 is therefore, negative and while this leads to a greater permissible alternating range about the mean, the allowable tensile maximum stress is only some 70% of that of the limit for unidirec-tional application. As a fi rst approximation, for a given gear volume, fi llet stresses are inversely proportional to pitch, i.e. if pitch is doubled, stress is halved.

Again, historically, this has led to a simple criterion for tooth bending known as the “ C ” factor. As for the “ K ” factor this can be arranged as a volumetric expression viz.

Figure 3: Highest point of single tooth contact.

cycles

σUTS

mean allowable tensile stress

max allowable stress – 70% of allowable for unidirectional application

Figure 4 : Goodman fatigue stress diagram.



TC =

fdm (3 )

where m = module = d / N . Then,

2 TNfd =

C (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,

11

CN = +

n K⎛ ⎞⎜ ⎟⎝ ⎠

(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



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

a a

s s

N dR = =

N d (6 )

where N a and N s are the teeth numbers, and d a and d s 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.

Tf

Tf

Planet bearing load

ds

da

dp

Sun wheel

Planet wheel

Annulus wheel

Figure 5: Half section epicyclic gear.



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.

Bendingstress

Angular distance between planets

Figure 7: Cyclic stress reversals.



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.



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

a sp 2

d dd =

−

(7 )

Epicyclic analogy gives

p

s

1

2

d Rn = =

d

−

(8 )

Noting that

1 2 11 1

1 1

R++ = + =

n R R− − (9 )

Thus,

s

1

1

R+ CN =

R K⎛ ⎞⎜ ⎟⎝ ⎠−

(10 )

a s

1

1

R+ CN = N R = R

R K⎛ ⎞⎜ ⎟⎝ ⎠−

(11 )

2 s cs

1

1 ( 1)

T TR+fd = =

QK R QK R⎛ ⎞⎜ ⎟⎝ ⎠− −

(12 )



and

22 2 2 ca s ( 1)

T Rfd = fd R =

QK R − (13 )

where d p is the planet wheel pitch diameter, d s the sun wheel pitch diameter, d a the annulus pitch diameter, f the sun wheel face width, N s the sun wheel tooth number, C the planet tooth bending criterion, K the sun wheel surface criterion, T s the sun wheel torque, T c 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.

2a c c(4 / 8) 0.5fd = T = T

(14a )

or

2a c 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.

2w c c(3 1) 4fd = T + = T

(15a )

or

2w c 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



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.



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.



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 brgNacelle

Fully floatingtorque reaction

Figure 11: Independent rotor support arrangement.



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

ReactionM/C2

ReactionM/C1

Output gears

Epicyclicgears

Figure 12 : Variable ratio gear arrangement.



It also emphasizes the importance of isolating the gearbox from the parasitic forces imposed by the turbine on its rotor support.

The volumetric concept facilitates the synthesis of the initial design of gears rather than using an analytical/iterative approach. It helps to optimise the overall size and weight of gears by showing the value of using lower ratios in the high torque low speed stages of high ratio applications, particularly when epicyclic trains are involved. Ultimately, all stress criteria are subject to arbitrary limits embodying a string of "factors of ignorance" which tend to be treated as virtual constants. Ten such factors, with a 5% increase in each, would reduce permissible load by 40%!


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