Tong W. Wind Power Generation and Wind Turbine Design

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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

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.

512 Wind Power Generation and Wind Turbine Design

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.

Optimized Gearbox Design 513

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

) or (2290 for lb/

) 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.

fd = +

⎛⎞

⎜⎟

⎝⎠

(1 )

(1)

fd = n

(2 )

where K is the surface criterion, n the wheel/pinion ratio, f the face width, d the

pinion pitch diameter, d

the wheel pitch diameter, T the pinion torque and T

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

514 Wind Power Generation and Wind Turbine Design

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

UTS

mean allowable

tensile stress

max allowable

stress – 70% of

allowable for

unidirectional

application

Figure 4 : Goodman fatigue stress diagram.