Effect of Geometrical Imperfections of Gears in Large...

Effect of Geometrical Imperfections of Gears in Large Offshore Wind Turbine Gear Trains: 0.6–10 MW Case

Studies Amir Rasekhi Nejad Torgeir Moan IMT, CeSOS, NTNU Trondheim, Norway

CeSOS, NTNU Trondheim, Norway

[email protected] [email protected] EWEA 2012, Copenhagen, Denmark

Keywords: Gear geometrical imperfections, large wind turbine gear trains, offshore drivetrains

Abstract Current wind turbine gear design standards,

such as ISO/IEC 61400-4 [1] and ANSI/AGMA/ AWEA 6006-A03 [2] cover turbines with capacity up to 2 MW. According to these design codes, gear geometrical tolerances shall be taken from ISO 1328-1 & 2 [3,4] where gear quality is classified in grades. For instance based on AWEA wind turbine design code [2], the maximum quality grade for carburized external gears shall not be more than 6 in ISO 1328-1 [3] ranking level. However, in absence of design guidelines for above 2 MW turbines, selecting right gear quality grade is a challenge for designers.

In this paper, the effect of gear geometrical variation in large wind turbines is investigated. In general gear geometrical imperfections are classified in two groups of assembly dependent and assembly independent variations. They influence gear load sharing, vibration, contact pattern, contact stress and finally reaction loads imposed on bearings. Both two groups of variations are considered in this study and their effect on contact stress, vibration and bearing load variation is evaluated through case studies of 0.6 to 10 MW. The outcome presents the effect of gear quality changes in large wind turbine gear trains and sensitivity to each category of imperfection.

1. Introduction Wind is taking the industry further offshore and

deeper water exposing wind turbine machineries to extreme loads and higher design uncertainties. Without doubt, current challenges with land based and fixed offshore wind turbines needs to be well understood in order to limit uncertainties for future floating fleets. Particularly for large multi mega watts offshore wind turbines as blade diameter increases, the rotational speed decreases, thus, drive trains with higher ratio are needed.

In this paper, the effect of gear geometrical quality in large wind turbines is investigated. Manufacturing deviations create nonlinearity in the dynamic behaviour of the gears as indicated by gear researchers [5,6,7]. According to Musial W. et al [8] some common issues have been observed in wind turbine drivetrain failures: “most of the problems with the current fleet of wind turbine

gearboxes are generic in nature, meaning that the problems are not specific to a single gear manufacturer or turbine model. Most gearbox failures do not begin as gear failures or gear-tooth design deficiencies”. One of possible failure reasons highlighted by Musial W. et al [8] is that the transfer of loads from gears and bearings to shafts is occurring in non-linear or unpredicted manner meaning that observed experimental loads are higher than expected values obtained from simulations.

Although research works directly addressing gear trains in wind turbines, to the author’s best knowledge, are rather limited [9-11], nonlinear dynamic behaviour of gear trains has been studied for both spur and helical gears and manufacturing imperfections are claimed to be important players in this nonlinear performance [12-17].

Wind turbine geometrical manufacturing imperfections of gears can be classified in four general categories: Tooth profile deviations (assembly

independent) Misalignment (assembly dependent) Backlash (assembly dependent) Mesh phasing (assembly dependent)

The influence of each category is investigated in reference [18] but in the current study, first two groups are assessed.

2. Study method & case studies There are well established standard

calculation methods for gear contact stress, transmission error and bearing reaction specified in design codes like ISO 6336 [19] which are reflected in gear design tools. In this paper, imperfection effects are investigated through study of gear trains in various sizes by means of design and analysis software, KISSsoft [20]. The gear parameters are calculated by this program in rated wind speed.

In accordance with ANSI/AGMA/AWEA 6006-A03 [2] the gear quality shall follow grading level listed in table 1 in which the higher grade number means the larger tolerances and lower quality. Gear quality grade limits gear tooth tolerances including profile deviations which are assembly independent. There are other manufacturing limits

like axial misalignment that are not dictated by the gear quality grade.

Table 1: required gear accuracy [2]

Gear type Heat treatment Max. Accuracy per

ISO 1328-1 External Carburized 6 Internal Carburized 7

Internal Nitrided 7 (with 8 for runout

and total cumulative pitch deviation)

Internal Through hardened

8

In this study both groups are considered by

case studies listed in table 2 and 3. Gear quality investigation is carried through R1 and C1 cases. R1 reflects the lowest permitted quality in accordance with wind turbine design codes [1,2] while gears in C1 are one grade lower than permissible level. It is worthwhile to indicate that lower quality than C1 case is not possible because the tooth thickness tolerances are out of the acceptable standard range.

Table 2: Assembly independent study cases

Case # Gear quality grade

(ISO1328-1) Wind speed

External Internal R1 6 7 Rated C1 7 8 Rated Case R2 and C2 in table 3 cover the axial

misalignment – shown in Fig. 1 – which is an assembly dependent parameter. The misalignment values selected for C2 and R2 cases are based on experimental observations [11] and standard values of total helix deviations [3]. The axial misalignment is then applied only on planets as the floating sun concept is assumed for all case studies.

Fig. 1: Axial misalignment

Table 3: Assembly dependent study cases

Case #

Axial misalignment ( f ) Wind speed

R2 50 Rated C2 200 Rated

It is known that gear geometrical imperfections

influence contact and root stresses, contact pattern, support reactions and vibration throughout the system. Therefore in order to capture their effects, following parameters are calculated for each case:

Bearing reactions: by measuring planet bearing force

Vibration: by measuring transmission error (TE)

Contact stress : by measuring contact stress along line of action

Variation of force along the face width: by

measuring HK ; face load factor

Bearing reaction varies in each gear rotation cycle. Geometrical imperfections influence load distribution on the bearing which is captured by

recording the reaction and HK in each case study.

Transmission Error (TE) is another important factor which is affected by the manufacturing imperfections. Transmission error is the single most important factor in the generation of gear vibration and is defined as “the difference between the actual position of the output gear and the position it would occupy if the gear were perfectly conjugate” [21]. Transmission error is the combination of gear pitch, profile and helix errors together with tooth bending, gear body deformation and support deflections which give an overall relative deflection at the meshing point between the gears and the deviation from the true involute profile. The mean value of TE is not important in vibration generation as it is due to elastic tooth deflection but the varying part is causing the oscillating acceleration and vibration through the system.

Contact stress is also measured for each case through the line of action for planet in the middle section of face width.

The design concept of case studies covers high ratio gear trains suitable for high speed generators with specification listed in table 4 and 5. Besides that, since scope of this study is limited to the gear quality, shaft and gear train support deflections are excluded.

Table 4: Rotor speed and generated power

of study cases (rpm/MW) Capacity(MW) Cut in Rated

0.6 12/0.05 24/0.6 2 9/0.18 15/2 5 7/0.35 12/5

10 5/0.70 12/10

Table 5: Gear trains of study cases Capacity (MW)

1st

stage 2nd

stage 3rd

stage 4th

stage

0.6 Type P H H Ratio 1:4.07 1:4.00 1:3.77

2 Type P P H Ratio 1:4.03 1:5.06 1:5.09

5 Type P P P H Ratio 1:4.00 1:4.00 1:4.00 1:1.95

10 Type P P P H Ratio 1:4.00 1:4.00 1:4.00 1:1.97

P: Planetary, H: Parallel Helical

2.1. 0.6 MW case study Fig. 2 presents the schematic of a 0.6 MW

model consisting of one planetary and two helical stages.

Fig. 2: 0.6 MW, 3 stage gear train

In Fig. 3 planet bearing, contact stress and TE

for cases R1 and C1 is presented. The noticeable observation is the increase of TE for gear quality of 7.

-10 -5 0 5 10 15 207.5

8

8.5

9

9.5x 10

4 Bearing Force (N)

Pla

net,

1st s

tag

e

-10 -5 0 5 10 15 20 257

7.5

8

8.5x 10

4

Gea

r, 2

nd

sta

ge

-10 -5 0 5 10 15 20 252.8

3

3.2

3.4

3.6x 10

4

Rotation Angle(degree)

Gea

r, 3

rd s

tage

Blue: Gear Quality Grade 6Green: Gear Quality Grade 7

Fig. 3-1: 0.6 MW, effect of gear quality on bearing

force

-10 -8 -6 -4 -2 0 2 40

500

1000

1500Contact Stress (Mpa)

1st

sta

ge

-20 -15 -10 -5 0 5 10 15600

800

1000

1200

1400

2nd

sta

ge

-20 -15 -10 -5 0 5 10400

600

800

1000

1200

1400


3rd

stag

e


Fig. 3-2: 0.6 MW, effect of gear quality on contact

stress

-10 -5 0 5 10 15 20-230

-225

-220

-215

-210

-205Transmission Error (micron)

Sun

/Pla

net

TE

1st

sta

ge

-10 -5 0 5 10 15 20 25-220

-218

-216

-214

-212

-210

Pin

ion/

Gea

r T

E 2

nd s

tag

e

-10 -5 0 5 10 15 20 25-180

-175

-170

-165


Pin

ion/

Ge

ar T

E 3

rd s

tag

e


Fig. 3-3: 0.6 MW, effect of gear quality on TE

In this case as it shown in above figures, the

effect of gear quality is limited to TE variation. The misalignment effect (R2 and C2 cases) is shown in

Fig. 4-1 to 4-3 with HK in table 6.

Table 6: 0.6 MW, HK

CASE # 1st stage 2nd stage 3rd stageR2 1.42 1.34 1.51 C2 2.56 2.29 2.83

-10 -5 0 5 10 15 206

6.5

7

7.5

8

8.5x 10

4 Bearing Force (N)

Pla

net

, 1

st s

tag

e

-10 -5 0 5 10 15 20 258

8.5

9

9.5

10

10.5x 10

4

Gea

r, 2

nd

sta

ge

-10 -5 0 5 10 15 20 253.5

4

4.5

5x 10

4


Gea

r, 3

rd s

tage

Blue: misalignment 50 micronGreen: misalignment 200 micron

Fig. 4-1: 0.6 MW, effect of gear misalignment on

bearing force

-10 -8 -6 -4 -2 0 2 40

500

1000


1st

sta

ge

-20 -15 -10 -5 0 5 10 150

500

1000

1500

2nd

sta

ge

-20 -15 -10 -5 0 5 100

500

1000

1500


3rd

stag

e


Fig. 4-2: 0.6 MW, effect of gear misalignment on

contact stress

-10 -5 0 5 10 15 20-230

-220

-210

-200


Sun

/Pla

net

TE

1st

sta

ge

-10 -5 0 5 10 15 20 25-230

-220

-210

-200

-190

Pin

ion/

Gea

r T

E 2

nd s

tage

-10 -5 0 5 10 15 20 25-180

-170

-160

-150

-140

-130


Pin

ion/

Ge

ar T

E 3

rd s

tage


Fig. 4-3: 0.6 MW, effect of gear misalignment on TE

Misalignment of 200 m appears too large for

this gear train as the face load distribution factor is too high. Because of large face load factor, the load is not distributed equally along the face. Thus, the bearing force and contact stress obtained from middle of the gear is less than the 50 m case.

2.2. 2 MW case study

The 2 MW gear train includes two planetary stages and one parallel helical as shown in Fig. 5.

Fig. 5: 2 MW, 3 stage gear train

-10 -5 0 5 10 15 201.9

2

2.1

2.2

2.3x 10

5 Bearing Force (N)

Pla

ne

t, 1

st s

tag

e

-10 -5 0 5 10 15 207.5

8

8.5

9

9.5x 10

4

Pla

net,

2nd

stag

e

-10 -5 0 5 10 15 20 256

6.5

7

7.5x 10

4


Gea

r, 3

rd s

tage


Fig. 6-1: 2 MW, effect of gear quality on bearing

force

-14 -12 -10 -8 -6 -4 -2 0 2 4400

600

800

1000


1st

sta

ge

-10 -5 0 50

500

1000

1500

2nd

sta

ge

-15 -10 -5 0 5 10 150

500

1000

1500


3rd

sta

ge


Fig. 6-2: 2 MW, effect of gear quality on contact

stress

-10 -5 0 5 10 15 20-250

-245

-240

-235

-230


Sun

/Pla

net

TE

1st

sta

ge

-10 -5 0 5 10 15 20-206

-204

-202

-200

-198

-196

Sun

/Pla

net

TE

2n

d st

age

-10 -5 0 5 10 15 20 25-275

-270

-265

-260

-255

-250


Pin

ion/

Ge

ar T

E 3

rd s

tag

e


Fig. 6-3: 2 MW, effect of gear quality on TE

Fig.6 presents the effect of gear quality and

Fig. 7 the misalignment influence.

-10 -5 0 5 10 15 201.4

1.6

1.8

2

2.2x 10

5 Bearing Force (N)

Pla

net,

1st

sta

ge

-10 -5 0 5 10 15 205

6

7

8x 10

4

Pla

net,

2nd

stag

e

-10 -5 0 5 10 15 20 256

7

8

9

10x 10

4


Gea

r, 3

rd s

tage


Fig. 7-1: 2 MW, effect of gear misalignment on

bearing force

-14 -12 -10 -8 -6 -4 -2 0 2 40

500

1000


1st

sta

ge

-10 -5 0 50

500

1000

1500

2nd

sta

ge

-15 -10 -5 0 5 100

500

1000

1500


3rd

sta

ge



contact stress

-10 -5 0 5 10 15 20-260

-240

-220


Sun

/Pla

net

TE

1st

sta

ge

-10 -5 0 5 10 15 20-210

-200

-190

-180

-170

-160

Pin

ion/

Gea

r T

E 2

nd s

tag

e

-10 -5 0 5 10 15 20 25-280

-270

-260

-250

-240

-230


Pin

ion/

Ge

ar T

E 3

rd s

tag

e


Fig. 7-3: 2 MW, effect of gear misalignment on TE

Table 7: 2 MW, HK

CASE # 1st stage 2nd stage 3rd stageR2 1.30 1.52 1.35 C2 2.19 2.86 2.34

The change of gear quality does not have a

considerable impact on measured parameters shown in fig. 6, while changes in planet axial misalignment has increased TE variation, causing loss of contact. The effect of misalignment appears to be not the same for each stage. The first stage is less sensitive to the misalignment than others. The load reduction observed in bearing reactions, confirms the unequal load distribution along the face width.


The 5 MW example gear train consists of three planetary stages (Fig. 8) and one helical stage.


-5 0 5 10 15 203

3.5

4x 10

5 Bearing Force (N)

Pla

net,

1st s

tag

e

-10 -5 0 5 10 15 201.4

1.5

1.6

1.7x 10

5

Pla

net

, 2n

d st

age

-10 -5 0 5 10 15 206

6.5

7x 10

4

Pla

net,

3rd

stag

e

-6 -4 -2 0 2 4 6 8 10 12 145.5

6

6.5x 10

4


Gea

r, 4

th s

tage



force

-12 -10 -8 -6 -4 -2 0 2 4 60

500

1000


1st

sta

ge

-14 -12 -10 -8 -6 -4 -2 0 2 4 60

500

1000

1500

2nd

sta

ge

-12 -10 -8 -6 -4 -2 0 2 4 60

500

1000

1500

3rd

sta

ge

-10 -8 -6 -4 -2 0 2 4 6 8400

600

800

1000


Gea

r, 4

th s

tage



stress

-5 0 5 10 15 20-330

-320

-310


S/P

TE

1st

sta

ge

-10 -5 0 5 10 15 20-250

-240

-230

-220

S/P

TE

2nd

sta

ge

-10 -5 0 5 10 15 20-210

-205

-200

S/P

TE

3rd

sta

ge

-6 -4 -2 0 2 4 6 8 10 12 14-240

-235

-230

-225


TE

4th

sta

ge



Table 8: 5 MW, HK

CASE # 1st stage 2nd stage 3rd stage 4th

stage R2 1.14 1.19 1.52 1.38 C2 1.78 2.04 2.85 2.44

-5 0 5 10 15 202.5

3

3.5

4x 10

5 Bearing Force (N)

Pla

net,

1st s

tag

e

-10 -5 0 5 10 15 201

1.2

1.4

1.6x 10

5

Pla

net

, 2n

d st

age

-10 -5 0 5 10 15 204

5

6

7x 10

4

Pla

net,

3rd

stag

e

-6 -4 -2 0 2 4 6 8 10 12 146

7

8x 10

4


Gea

r, 4

th s

tage



bearing force

-10 -8 -6 -4 -2 0 2 4 60

500

1000


1st

stag

e

-14 -12 -10 -8 -6 -4 -2 0 2 4 60

500

1000

1500

2nd

sta

ge

-12 -10 -8 -6 -4 -2 0 2 4 60

500

1000

1500

3rd

stag

e

-10 -8 -6 -4 -2 0 2 4 6 80

500

1000


4th

sta

ge



contact stress

-5 0 5 10 15 20-340

-320

-300


S/P

TE

1st

sta

ge

-10 -5 0 5 10 15 20-260

-240

-220

-200

S/P

TE

2nd

sta

ge

-10 -5 0 5 10 15 20-220

-200

-180

-160

S/P

TE

3rd

sta

ge

-6 -4 -2 0 2 4 6 8 10 12 14-240

-230

-220

-210


TE

4th

sta

ge


Fig. 10-3: 5 MW, effect of gear misalignment on TE According to figures 9-1 to 9-3, low gear

quality of 7 has minor influence on the bearing reaction, vibration and planet contact stress for 5 MW gear train. In axial misalignment cases, the 1st and 2nd stage accept more misalignment than the last two stages. Table 8 shows a decline in face load factor comparative with the 0.6 and 2 MW gear trains.


The 10 MW gear train includes three stage planetary with one stage parallel helical gear as illustrated in Fig. 11. The overall gear ratio of this gear train is 1:126 which is suitable for high speed generators.


-10 -5 0 5 10 15 205.5

6

6.5

7x 10

5 Bearing Force (N)

Pla

net,

1st s

tag

e

-10 -5 0 5 10 15 202.2

2.4

2.6

2.8x 10

5

Pla

net

, 2n

d st

age

-10 -5 0 5 10 15 208

9

10

11x 10

4

Pla

net,

3rd

stag

e

-5 0 5 10 15 209

9.5

10

10.5x 10

4


Gea

r, 4

th s

tage



force

-12 -10 -8 -6 -4 -2 0 2 4 60

500

1000


1st

sta

ge

-14 -12 -10 -8 -6 -4 -2 0 2 40

500

1000

1500

2nd

stag

e

-14 -12 -10 -8 -6 -4 -2 0 2 4 60

500

1000

1500

3rd

stag

e

-12 -10 -8 -6 -4 -2 0 2 4 6 80

500

1000

1500


Gea

r, 4

th s

tage



stress

-10 -5 0 5 10 15 20-325

-320

-315


S/P

TE

1st

sta

ge

-10 -5 0 5 10 15 20-295

-290

-285

-280

S/P

TE

2nd

sta

ge

-10 -5 0 5 10 15 20-215

-210

-205

-200

S/P

TE

3rd

sta

ge

-5 0 5 10 15 20-280

-270

-260

-250

-240


TE

4th

sta

ge



Table 9: 10 MW, HK

CASE # 1st stage 2nd stage 3rd stage 4th

stage R2 1.20 1.30 1.46 1.27 C2 1.78 2.19 2.70 2.07

-10 -5 0 5 10 15 204

5

6

7x 10

5 Bearing Force (N)

Pla

net,

1st

stag

e

-10 -5 0 5 10 15 201.5

2

2.5

3x 10

5

Pla

net,

2nd

sta

ge

-10 -5 0 5 10 15 206

8

10x 10

4

Pla

net,

3rd

stag

e

-5 0 5 10 15 200.8

1

1.2

1.4x 10

5


Gea

r, 4

th s

tage



bearing force

-12 -10 -8 -6 -4 -2 0 2 4 60

500

1000


1st

sta

ge

-14 -12 -10 -8 -6 -4 -2 0 2 4 60

500

1000

1500

2nd

sta

ge

-14 -12 -10 -8 -6 -4 -2 0 2 4 60

500

1000

1500

3rd

sta

ge

-12 -10 -8 -6 -4 -2 0 2 4 6 80

500

1000

1500


4th

stag

e



contact stress

-10 -5 0 5 10 15 20-330

-320

-310


S/P

TE

1st

sta

ge

-10 -5 0 5 10 15 20-300

-290

-280

-270

S/P

TE

2nd

sta

ge

-10 -5 0 5 10 15 20-220

-200

-180

S/P

TE

3rd

sta

ge

-5 0 5 10 15 20-280

-260

-240

-220


TE

4th

sta

ge


Fig. 13-3: 10 MW, effect of gear misalignment on TE From Fig. 12 it is observed that change of gear

quality, does not affect the 10 MW gear train considerably. Similar to 5 MW case, 1st and 2nd stages are less sensitive to the misalignment shown in Fig. 13.

3. Comparison In above case studies, planet bearing force, transmission error, face load factor and contact stress were measured for two groups of gear grades and axial misalignment. In Fig. 14 and 15, the mean value of these parameters among the stages are drawn and compared. For bearing force, relative maximum variations are considered in comparison while for TE, standard deviations are compared. In contact stress, the maximum values are considered.

From Fig. 14 and 15 it is observed that misalignment holds stronger influence than gear quality on gear load sharing, vibration and contact stress variation. As the gear train capacity goes

higher to 10 MW, the face load factor ( HK )

declines, but it still remains in the range above 2 for 200 m misalignment which is not an

acceptable value. The same trend is observed for transmission error and contact stress in 200 m

for 10 MW gear train.

Fig. 14: Effect of gear quality grade, 0.6 to 10 MW

Fig. 15: Effect of misalignment, 0.6 to 10 MW

4. Conclusion The effect of gear quality grade and axial

misalignment for a range of medium to large wind turbine gear trains are investigated. Bearing reaction force, face load distribution factor, transmission error and maximum contact stress

are measured for each case study with varying gear quality and planet axial misalignment. In large gear trains such as 5 and 10 MW it is found that the gear quality of 7 for external gear and 8 for internal do not affect the measured parameters considerably even though they are one grade lower than permitted level. Transmission Error is the only parameter changes but within a small

range. However, larger TE variation can influence the load sharing behaviour of planets especially under low speed.

For assembly dependent imperfections like axial misalignment although a negative trend is observed for face load factor and contact stress toward larger wind turbines for large misalignment, they are found still not within acceptable range. Therefore, the assembly dependent tolerances still remain crucial even for large gear trains. This shows that the special consideration shall be taken in design of large turbines to accommodate the assembly imperfections or load dependent deformations because they are not less sensitive than small turbines to the misalignments.

It is also observed that each stage behave different than others to the misalignment. For instance the first two stages in 5 and 10 MW gear trains can hold larger misalignment than the other stages.

This study is conducted in rated wind speed for all the cases. Since the transmission error variation is higher in low wind speed, it is required to evaluate all cases in both low and rated wind speeds to confirm the results which are carried further in reference [18].

Acknowledgement The first author would like to thank KISSsoft

AG, Switzerland and Dr. Stefan Beermann for providing KISSsoft and KISSsys programs.

References [1] ISO/IEC 61400-4, Design and Specification of Wind Turbine Gearboxes, 2012 [2] ANSI/AGMA/AWEA 6006-A03, Standard for Design and Specifications of Gearboxes for Wind Turbines, 2010 [3] ISO 1328‐1, Cylindrical gears – ISO system of accuracy – part 1: definitions and allowable values of deviations relevant to corresponding flanks of gear teeth, first edition, 1995 [4] ISO 1328‐2, Cylindrical gears – ISO system of accuracy – part 2: definitions and allowable values of deviations relevant to radial composite deviations and runout information, first edition, 1997 [5] Kahraman A., Singh R., Non-linear dynamics of spur gear pair, Journal of sound and vibration; 1990: 142(1), 49-75 [6] Litvin F. L., Fuentes A., Gear geometry and applied theory, second edition, 2004, Cambridge Press [7] Smith J. D., helical gear vibration excitation with misalignment, Proceeding of Institute of Mechanical Engineers; 1994: 208, 71-79 [8] Musial W. et al, Improving wind turbine gear train reliability, proceeding of European Wind Energy Annual Conference, EWEA 7-10 May 2007, Milan, Italy

[9] Peeters J., Vandepitte D., Sas P., Analysis of internal drive train dynamics in a wind turbine, Wind Energy; 2006: 9, 141-161 [10] Heege A., Betran J., Radovcic Y., Fatigue load computation of wind turbine gearboxes by coupled finite element, multi-body system and aerodynamic analysis, Wind Energy; 2007: 10, 395-413 [11] Crowther A. et al, Sources of time-varying contact stress and misalignments in wind turbine planetary sets, Wind Energy; 2011: 14, 637-651 [12] Parker R. G. et al, Non-linear dynamic response of a spur gear pair: modelling and experimental comparisons, Journal of Sound and Vibration; 2000: 237(3), 435-455 [13] Comparin R. J., Singh R., Non-linear frequency response characteristics of an impact pair, Journal of sound and vibration; 1989: 134(2), 259-290 [14] Singh R., Xie H., Comparin R. J., Analysis of automotive neutral gear rattle, Journal of sound and vibration; 1989: 131 (2), 177-196 [15] Guo Y., Parker R. G., Purely rotational model and vibration modes of compound planetary gears, Mechanism and Machine Theory; 2010: 45, 365–377 [16] Parker R. G., Agashe V., Vijayakar S. M., Dynamic response of a planetary gear system using a finite element / contact mechanics model, ASME Journal of Mechanical Design; 2000: 122, 304-310 [17] Litak G., Friswell M. , Vibration in gear systems, Chaos Solutions & Fractals; 2003: 16, 795-800 [18] Rasekhi Nejad A, Moan T., Gear geometrical imperfections in large wind turbine drivetrains, Proc. IMechE, Part B: J. Engineering Manufacture, submitted 2012 [19] ISO 6336-1, Calculation of load capacity of spur and helical gears – part 1: basic principles, introduction and general influence factors, 2006 [20] KISSsoft version 03/2011, KISSsoft AG, Switzerland [21] Smith J. D., Gear noise and vibration, second edition, 2003, Marcel Dekker Inc.

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