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This report is available at no cost from the National Renewable Energy

Laboratory (NREL) at www.nrel.gov/publications.

Contract No. DE-AC36-08GO28308

Gearbox Reliability 

Collaborative Analytic 

Formulation for the Evaluation 

of Spline Couplings 

Yi Guo and Jonathan Keller

National Renewable Energy Laboratory 

Robert Errichello

GEARTECH 

Chris Halse

Romax Technology

Technical Report

NREL/TP-5000-60637

December 2013
NREL is a national laboratory of the U.S. Department of Energy

Office of Energy Efficiency & Renewable Energy

Operated by the Alliance for Sustainable Energy, LLC

This report is available at no cost from the National Renewable Energy

Laboratory (NREL) at www.nrel.gov/publications.

Contract No. DE-AC36-08GO28308

National Renewable Energy Laboratory

15013 Denver West Parkway

Golden, CO 80401

303-275-3000 • www.nrel.gov

Gearbox Reliability 

Collaborative Analytic 

Formulation for the Evaluation 

of Spline Couplings 

Yi Guo and Jonathan Keller

National Renewable Energy Laboratory

Robert Errichello 

GEARTECH 

Chris Halse 

Romax Technology

Prepared under Task No. WE11.0301

Technical Report

NREL/TP-5000-60637

December 2013
NOTICE

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Cover Photos: (left to right) photo by Pat Corkery, NREL 16416, photo from SunEdison, NREL 17423, photo by Pat Corkery, NREL 

16560, photo by Dennis Schroeder, NREL 17613, photo by Dean Armstrong, NREL 17436, photo by Pat Corkery, NREL 17721.

Printed on paper containing at least 50% wastepaper, including 10% post consumer waste. 
iii

Acknowledgments

This work was supported by the U.S. Department of Energy under contract number DE-AC36-

08GO28308 with the National Renewable Energy Laboratory. The authors would also like to 

thank Jan Helsen of the Katholieke Universiteit Leuven for her advice in developing the research 

topic, Jason Austin and Donald Houser from The Ohio State University for providing the 

Transmission 3D model of the spline, Sandeep Vijayakar from Advanced Numerical Solutions 

for developing the conformal contact functionality in Transmission 3D, and Sharad Jain from 

Romax Technology for valuable discussions regarding spline contact properties. 
iv

List of Acronyms

2D two-dimensional 

3D three-dimensional 

AGMA American Gear Manufacturers Association

FE finite element

GRC Gearbox Reliability Collaborative

GUI graphical user interface

IEC International Electrotechnical Commission

NREL National Renewable Energy Laboratory

SCouP Spline Coupling Program 
v 

Nomenclature

C Bending stiffness of a single pair of spline teeth

Cψ Helical overlap ratio

Do Hub major diameter

Di Sleeve minor diameter

E Young’s modulus 

F Facewidth

cf Half tooth facewidth in contact 

h Hub tooth tip chamfer 

h1 Active tooth flank height 

hF Height of Lewis parabola 

max , , d ii i Misalignment angle, design misalignment angle, and jam angle

J Geometry factor

K Accuracy factor

a K Surface factor

b K Size factor

c K Reliability factor

d K Temperature factor

e K Stress concentration factor

f K Hub tooth AGMA stress correction factor

Km Miscellaneous effects factor

R k Rotational stiffness of a single pair of spline teeth
vi 

v K Velocity factor

mN Coupling load sharing ratio 

n Safety factor

N Number of spline teeth

, , nnn bf cf sf Bending, contact, and shear fatigue safety factors

, , nnn by cy sy Bending, contact, and shear yield safety factors

P Diametral pitch

Pd Normal tooth load carried by the teeth adjacent to the highest-loaded tooth 

Pi Normal tooth load carried by the highest-loaded tooth 

′ Pi Normal tooth load carried by the highest-loaded tooth, per unit length 

Pt Total normal tooth load 

q Number of teeth in contact 

R Pitch radius 

Rc Hub root crown radius at pitch diameter

F R Hub face crown radius (normal plane) 

, , bf cf sf SSS Bending, contact, and shear endurance strength

, , by cy sy SSS Bending, contact. and shear yielding strength

c2 S Design sleeve circular space width

' , e e S S Modified and absolute bending endurance strength

F S Tooth thickness at the critical section

u S Tensile strength

y S Yielding endurance strength

T Transmitted torque 
vii 

1 2 , c c t t Hub and sleeve tooth circular thickness 

*

c1 t Hub tooth circular thickness at the contact point 

Wt Tangential tooth load 

1 2 Y Y, Hub and sleeve tooth AGMA form factor 

1 2 Z Z, Hub and sleeve elasticity factor

z Elastic tooth deformation of the teeth adjacent to the highest-loaded tooth 

o z Maximum tooth separation

e z Maximum elastic tooth deformation of the highest-loaded tooth 

φ Normal pressure angle

σ a Alternating stress

σ b Bending stress

σ c Contact (Hertzian) stress

σ e Effective distortion energy stress

σ m Mean stress

τ Shear stress

ρF Minimum radius of curvature of the fillet curve
viii

Table of Contents

1 Introduction........................................................................................................................................... 1

1.1 GRC Overview ................................................................................................................................ 1

1.2 Study Motivation .............................................................................................................................. 1

1.1.1 GRC Survey of Designers ................................................................................................... 2

1.1.2 GRC Damage Analysis ....................................................................................................... 2

1.1.3 Other Reliability Analysis and Prediction Results .............................................................. 3

1.3 Objective.......................................................................................................................................... 4

2 GRC Test Article and Instrumentation ............................................................................................... 5

3 Modeling Approaches .......................................................................................................................... 9

3.1 Analytic Formulation ....................................................................................................................... 9

3.1.1 Assumptions........................................................................................................................ 9

3.1.2 Stiffness of a Single Tooth Pair ........................................................................................ 10

3.1.3 Hertzian Stress .................................................................................................................. 10

3.1.4 Bending and Shear Stress ................................................................................................. 11

3.1.5 Tooth Load Distribution ................................................................................................... 12

3.1.6 Jam Angle ......................................................................................................................... 12

3.1.7 Strength Analysis .............................................................................................................. 14

3.1.8 Safety Factors ................................................................................................................... 16

3.1.9 Solution Methodology ...................................................................................................... 16

3.2 Conformal Contact and Thin Strip Model: RomaxWind ............................................................... 17

3.3 Line Contact and Three-Dimensional Finite Element Model: Transmission3D............................ 18

4 Results and Discussion ..................................................................................................................... 19

4.1 Spline Design Parameters .............................................................................................................. 19

4.2 Spline Loads .................................................................................................................................. 19

4.2.1 Effects of Shaft Misalignment .......................................................................................... 20

4.2.2 Effects of Transmitted Torque .......................................................................................... 23

4.3 Spline Safety Factors ..................................................................................................................... 24

4.4 Gear Spline Coupling Program (Gear SCouP) .............................................................................. 26

5 Conclusions ........................................................................................................................................ 27

5.1 Spline Modeling and Behavior ...................................................................................................... 27

5.2 Future Work ................................................................................................................................... 27

5.3 Recommendations for Current Standards ...................................................................................... 28

6 References .......................................................................................................................................... 29
ix

List of Figures

Figure 1. Gearbox configuration. .............................................................................................................. 2

Figure 2. Fretting corrosion of the sun spline. ........................................................................................ 3

Figure 3. Close-up of fretting corrosion of the sun spline. .................................................................... 3

Figure 4. Drivetrain configuration ............................................................................................................. 5

Figure 5. Sun proximity sensor schematic............................................................................................... 6

Figure 6. Sun proximity sensor (one of two sensors)............................................................................. 7

Figure 7. Measured sun gear orbit (top) and frequency spectrum (bottom) ........................................ 8

Figure 8. Misalignment spline contact on both sides of the tooth flank............................................. 13

Figure 9. Flow chart of the spline stress, deformation, and safety factor calculations .................... 17

Figure 10. Tooth load distribution across spline teeth with (left) and without (right) misalignment20

Figure 11. Effect of misalignment on the spline tooth loads................................................................ 20

Figure 12. Spline tooth load distributions from the analytic model .................................................... 21

Figure 13. Spline tooth load distributions from RomaxWind............................................................... 22

Figure 14. Spline tooth load distributions from Transmission3D........................................................ 23

Figure 15. Effect of torque on the spline tooth loads............................................................................ 24

Figure 16. Effect of misalignment and torque on the spline safety factors........................................ 25

Figure 17. Gear SCouP graphical user interface ................................................................................... 26

List of Tables

Table 1. Estimated Failure Rate of the R80 Turbine Planetary System................................................. 4

Table 2. Sun Spline Coupling Specified Geometry ................................................................................. 6

Table 3. Fatigue and Yielding Strength of Splines with Different Heat Treatment Processes.......... 16

Table 4. Rotational Stiffness of a Pair of Spline Teeth.......................................................................... 19
1 

1 Introduction

1.1 GRC Overview

Many gearboxes in wind turbines have not been achieving their expected design life [1]; 

however, they commonly meet or exceed the design criteria specified in current standards in the 

gear, bearing, and wind turbine industry as well as third-party certification criteria. The cost of 

gearbox replacements and rebuilds, as well as the downtime associated with these failures, has 

elevated the cost of wind energy. The National Renewable Energy Laboratory (NREL) Gearbox 

Reliability Collaborative (GRC) was established by the U.S. Department of Energy in 2006; its 

key goal is to understand the root causes of premature gearbox failures and improve their

reliability using a combined approach of dynamometer testing, field testing, and modeling [2]. A 

hypothesis being examined by GRC is that the gap between design-estimated and actual wind 

turbine gearbox reliability is caused by an underestimation of loads, inaccurate design tools, the 

absence of critical elements in the design process, or insufficient testing. One of the overall goals 

of the GRC is to facilitate an increase in the accuracy of existing gearbox design and modeling 

tools, or to produce these tools when none are available. Another goal is to make 

recommendations to improve gearbox design standards. 

1.2 Study Motivation

As part of the GRC program, this paper investigates the design of the spline coupling often used 

in modern wind turbine gearboxes to connect the planetary and helical gear stages. The GRC 

gearbox configuration with the sun spline is shown in Figure 1. Aside from transmitting the 

driving torque, another common function of the spline coupling is to allow the sun to float 

between the planets. The floating principle equalizes the load distribution between planets as 

much as possible, given the realities of imperfections such as machining errors and misalignment 

and loads imparted to the planetary system other than torque alone. A freely floating sun 

minimizes the negative effect of these realities on the planet, sun, and ring gear mesh contact 

patterns. Conversely, without the floating sun, gearbox misalignment and unequally shared loads 

can occur. As a result, edge loading of the gears and planet-bearing forces increase, leading to 

reduced gear and bearing life and potential premature failure [3].

The amount the sun can float is determined by the spline design and the sun shaft flexibility 

subject to the operational loads. American Gear Manufacturers Association (AGMA) 6006-A03

[4], AGMA 6123-B06 [5], and IEC 61400-4 [6] address spline coupling design requirements in 

varying detail, with the most detailed guidance provided in AGMA 6123-B06 for single 

articulation couplings. This report provides additional insight beyond these current standards to 

help quickly evaluate spline coupling designs. 
2 

Figure 1. Gearbox configuration. Illustration by Powertrain Engineers Inc.

The following sections provide additional motivation for investigating the design and modeling 

of the sun spline. 

1.1.1 GRC Survey of Designers

One of the overall goals of the GRC is to facilitate an increase in the accuracy of existing 

gearbox design and modeling tools, or to produce these tools when none are available. To collect 

objective insights into the important issues that wind turbine gearbox designers are facing, a 

blind survey was distributed by the GRC in late 2011 to the designers of leading wind turbine 

gearbox firms. A portion of the survey was dedicated to gathering opinions regarding the 

gearbox components for which design methodologies are either not mature enough to yield

sufficient reliability or have limited availability. The design of the sun spline was commonly 

mentioned as one of these components. Therefore, the GRC program began to collect expert 

opinions on the features of a tool that would be most valuable to the industry. 

1.1.2 GRC Damage Analysis

One of the GRC gearboxes was installed in a turbine at the Ponnequin wind farm in September 

2009 to gather field loading data. After approximately 1 month, the test was stopped because of

bearing temperature exceedances and reports of oil loss from the gearbox. An inspection 

revealed that the high-speed stage gear teeth showed signs of significant overheating. As a result, 

testing was suspended to avoid the potential for catastrophic gearbox failure. Subsequently, the 

gearbox was removed from the turbine and shipped back to NREL. After conducting a limited 

set of condition monitoring tests in the NREL dynamometer, the gearbox was disassembled and 

inspected [7]. 

Some of the noted damage to the gearbox was related to the sun spline connection. The primary 

damage to the sun spline was the severe fretting corrosion shown in Figure 2 and Figure 3. The 

root cause of this damage was probably poor load sharing and lubricant starvation. The fretting 

corrosion was concentrated on about half of the spline teeth, indicating that about half of the 

teeth were carrying the torque load. It was recommended by the GRC gearbox designer to 

increase the accuracy of the spline teeth to improve the tooth-to-tooth load distribution. 

Planetary 

gear stage

Sun gear

Sun spline

Hollow shaft
3 

Figure 2. Fretting corrosion of the sun spline. Photo by Robert Errichello, GEARTECH, NREL

19852

Adhesion between the external sun spline and the hollow shaft internal spline transferred lumps 

of material from the hollow shaft as shown in Figure 3. Axial sliding broke the welds and 

deformed the material in the axial direction. The red debris is hematite (Fe2O3), a polishing agent 

that created polishing wear that surrounds the fretting damage.

Figure 3. Close-up of fretting corrosion of the sun spline. Photo by Robert Errichello, GEARTECH, 

NREL 19853

1.1.3 Other Reliability Analysis and Prediction Results

The estimated failure rates of different gearbox subcomponents for several representative wind 

turbines were reported in the framework of the Reliawind project [8] [9] and results on R80 

gearboxes are listed in Table 1. The original failure data on gearbox systems are from the WMEP 

scientific measurement and evaluation program and Landwirtschaftskammer (LWK) database

[10, 11]. The sun spline failure rate in this analysis was similar to other planetary system gears 

and bearings. In addition to the outright failure of the sun spline itself, poorly functioning splines 

could also be a contributor to failures in the rest of the planetary stage because of their influence 

on load-sharing characteristics. According to NREL’s gearbox failure database [12] that is based 

on gearbox rebuilt information over 10 years, the average gearbox failure rate was 5%, with the 

highest rate of 10% occurring after 5 years in service. Therefore, the actual sun spline failures 

and associated planetary gear failures could be higher than the estimations in [8].
4 

Table 1. Estimated Failure Rate of the R80 Turbine Planetary System [8] 

Part Failures over 13 years

Low Speed Shaft 0.0325

Low Speed Shaft Bearing 0.0012

Planet Shaft 0.0325

Planet Bearing 0.0029

Planet Gear 0.0022

Ring Gear 0. 0022

Sun Pinion 0. 0022

Sun Shaft 0.0325

Sun Spline 0.0033

Planetary gear section 0.1835

1.3 Objective

Based on the GRC survey and spline failure rate of the R80 turbine, it appears that the current 

spline design methods are not sufficient for wind turbine gearbox systems. As a result, there is an

opportunity for the GRC program to develop a greater understanding of spline modeling, 

contribute to design methodologies, and positively influence spline design standards. 

The objective of this report is to describe a simple, analytical formulation for quick spline 

coupling design and spline coupling rating analysis. The formulation has been coded into 

software and is compared to existing higher fidelity spline coupling modeling tools. This analytic 

formulation provides essential design information and could be easily integrated into gearbox 

design and simulation tools. The formulation also provides insight into the effect of spline design 

parameters on the spline behavior, some of which are not discussed in the current gearbox design 

standards. 
5 

2 GRC Test Article and Instrumentation

The GRC test article drivetrain originally was designed for a two-speed, stall-controlled, threebladed upwind turbine with a rated power of 750 kilowatts (kW) [1]. The design follows a 

conventional configuration wherein all the drivetrain components are mounted onto the bed 

plate. These components include the hub, main bearing, main shaft, gearbox, brake, generator 

shaft, and generator as shown in Figure 4. Everything but the hub is included in the 

dynamometer tests. The drivetrain follows a common configuration of megawatt-scale turbines 

used in the industry today. The gearbox is mounted with a three-point configuration in which 

torsional loads are transferred to the main frame through two torque arms, and forces are reacted 

mostly at the main bearing.

Figure 4. Drivetrain configuration

The gearbox is composed of one low-speed planetary stage, accommodating three planet gears, 

and two parallel shaft stages as shown in Figure 1. The gears and bearings were redesigned and 

have been modified from the original gearbox configuration used in the commercial versions of 

this wind turbine. This redesigned gearbox is hereafter termed the “the GRC gearbox.” 

In the GRC gearbox, the sun gear is in a floating configuration to equalize the load distribution 

among the planets. The sun gear is integral to a long shaft with an external spline (hub) on its 

downwind end as shown in Figure 1. The hub is connected to the hollow low-speed shaft 

(sleeve) through an internal spline. The spline coupling is a single articulation design with 

significant crowning. Key parameters of the external spline (hub) are listed in Table 2. The 

spline carries an operational torque of 57 kiloNewton meters (kNm) (504,000 lb-in) that is

transferred to the parallel-shaft stages.

Generator

Main 

Bearing Main Shaft

Brake

Generator Shaft

Gearbox

Bed Plate

Hub

Torque Arms
6 

Table 2. Sun Spline Coupling Specified Geometry

Parameter Symbol Value

Hub major diameter, inch Do 8.42 

Sleeve minor diameter, inch Di 7.80

Effective facewidth, inch F 4.61 

Number of teeth N 48

Diametral pitch, 1/inch P 6 

Pitch radius, inch R 4.00

Hub face crown radius, inch F R 444.97

Hub circular tooth thickness, inch c1 t 0.285–0.286

Sleeve circular tooth thickness, inch c2 t 0.225–0.226

Normal pressure angle, degree φ 20.00

Lead modification type Lead full crown

Heat treatment technique for hub Carburized

Heat treatment technique for sleeve Nitrided

The GRC gearbox is highly instrumented both internally and externally [13]. Of special interest 

in this work is the measurement of the main shaft torque and bending, plus the radial position of 

the sun gear relative to the planet carrier. This radial motion is measured using two orthogonal 

proximity sensors mounted on the planet carrier as shown in Figure 5 and Figure 6. The sensors 

discern the upwind end of the sun shaft in the area of the shaft that extends about 50 mm beyond 

the end of the sun pinion. There is no direct measurement acquired on the spline itself because of 

the challenges of instrumentation and wiring. 

Figure 5. Sun proximity sensor schematic. Illustration by McNiff Light Industry
7 

Figure 6. Sun proximity sensor (one of two sensors). Photo by Edward Overly, NREL 26666

The sun gear orbiting motion measured during the dynamometer testing is shown in Figure 7. 

The measurement was taken at rated torque (325 kNm) and two amplitudes of main shaft 

pitching moments (also called non-torque loads). Aerodynamics loads and turbine gravity forces 

cause non-torque loads transmitted into the main shaft. Pitching moments tilt the planetary gears 

and cause unequal planetary load sharing [14]. In order to accommodate this tilt, the sun spline 

allows the sun shaft to self-adjust and orbit near freely in the planetary system plane. As a result, 

the sun orbiting motion increases with pitching moment significantly. The sun gear has up to 

0.25 mm of orbiting motion about the spline tooth center. Assuming this motion results only in 

angular misalignment, the sun shaft misalignment is up to 0.023° as shown in Figure 7. AGMA 

6123-B06 recommends a reasonable limit of 0.057° (0.001 radian) for a single articulation 

coupling, which is slightly over double the measured misalignment. 

The pitching moments measured in field tests were much higher (275 kNm) than the maximum 

applied during the dynamometer tests (125 kNm) [13]. Therefore, it is expected that sun orbiting 

motion occurring in the field is even higher than measured during the dynamometer test. The 

frequency spectrum of the sun orbit, shown in Figure 7, consists of higher-energy resonances at 

the planetary rotation frequencies than the energy at the planet-sun mesh frequency, which 

suggests potential manufacturing deviations and misalignments might exist.
8 

Figure 7. Measured sun gear orbit (top) and frequency spectrum (bottom)
9 

3 Modeling Approaches

Three modeling approaches of varying fidelity are explored in this work: 1) an analytic model 

formulated in Section 3.1, 2) a hybrid, two-dimensional (2D) finite element (FE) and an analytic

model described in Section 3.2, and 3) a fully three-dimensional (3D) FE model described in 

Section 3.3. The hybrid model (model 2) is RomaxWind 14.5.0 and the 3D FE model (model 3) 

is Calyx: Transmission3D version 2.2700|0.1195. These FE models of the drivetrain have 

previously been validated against relevant GRC experimental data for bearing and gear loading 

[2, 14-16].

3.1 Analytic Formulation

The analytic formulation described herein provides much of the same information as modern 

gearbox design software, plus estimation of spline safety factors. By its very nature, the 

formulation (model 1) provides greater insight into the effect of the spline coupling design

parameters upon the spline performance and resulting safety factors than the other two 

approaches. Solutions can be calculated two orders of magnitude faster than higher fidelity 

models, making it very useful for early design stage parametric studies.

The analytic formulation was developed over many years by GEARTECH and was recently 

updated in a collaborative effort between GEARTECH and NREL. The formulation was also 

recently coded into MATLAB stand-alone software by NREL and then correlated by NREL and 

Romax Technology using the existing GRC models. 

3.1.1 Assumptions

The major assumptions of the analytic formulation makes are that: 

• The hub and sleeve shafts are considered rigid. This assumption is valid most of the time 

because these shafts typically have large wall thicknesses to resist shaft twist. However, 

the analytical model could have conservative results when compared to a model that

accounts for compliance of the connecting shafts. 

• Only angular misalignment is considered. This assumption is valid, at least for the GRC 

gearbox, because there was no strong evidence that the spline teeth were moving radially 

and contacting the roots during its inspection. 

• Only torsional loading is considered. Radial gear loads are not included because, at least 

for the GRC gearbox and other common wind turbine gearboxes, they are reacted by 

bearings supporting the hollow shaft. 

• The crowning of the spline teeth leads to constant base pitch spacing in all planes. 

• The bending and contact stiffnesses are derived at a specified, steady torque, and tooth 

contact occurs only on the drive-side.

• The spline teeth are equally spaced circumferentially.
10 

3.1.2 Stiffness of a Single Tooth Pair

Spline tooth bending stiffness is estimated using an analytical approach [17], considering 

combined tooth deformations caused by bending and compression. Line contact at the pitch 

diameter is assumed. The actual bending stiffness may be somewhat different because the 

contact is conformal between two identical involute curves. The empirical values are based on 

measured deformation of gear teeth, rather than gear coupling teeth, under known test loads in 

which the tooth contact is localized near the pitch line. The spline tooth bending stiffness is: 

= +

1 2

1 2

Z Z

C FE

Z Z (1) 

where the elasticity form factors are [17]: 

= = + +

1 2

1 2

1 2 0.76 7.25 0.76 7.25

Y Y

Z Z

Y Y (2) 

The FE models typically express stiffness in terms of the rotational stiffness of a single tooth pair 

rather than bending stiffness. For the purpose of comparing the analytical formulation to the FE 

models, this rotational stiffness of a single tooth pair can be approximated as: 

= φ 2 2 cos ( ) R k CR (3) 

3.1.3 Hertzian Stress

Given that the curvature in the tooth profile direction is the same on both the hub and the spline, 

the Hertzian contact is modeled as a cylinder with radius F R and length h1 contacting on a semiinfinite plane. The contact stress is [19]: 

σ

  =    

1/2 '

0.418 i

c

F

P E

R (4) 

where the load per unit length is: 

'

1

i

i

P P

h = (5) 

and where the active tooth flank height is: 

− 1 = − 2

D D o i h h (6) 

The total normal tooth load is: 

cos( ) t

T P

R φ = (7) 
11 

where the pitch radius equals: 

2

N R

P = (8) 

3.1.4 Bending and Shear Stress

The difference in bending and shear stress between a line contact model with a conformal 

contact model is expected to be insignificant because the total load carried by the tooth is the 

same. Assuming line contact at the pitch diameter, the gear coupling bending stress is: 

σ = t

b

v

W P

K FJ (9) 

The velocity factor is assumed to be equal to one in this study. The tangential force is: 

= cos( ) φ W P t i (10) 

The geometry factor equals [18]: 

1

f N

Y C

J

K m

ψ = (11) 

Where the hub tooth AGMA stress correction factor is [18]: 

L M

F F

f

F F

S S K H

ρ h

  

= +      

(12) 

The empirical factors H , L , and M are: 

HLM =− =− =+ 0.331 0.436 0.324 0.492 0.261 0.545 φφ φ (13) 

Spline connections typically use spur teeth, therefore, mN and Cψ equal one. The final form of 

the bending stress is: 

( ) φ

σ =

1

cos i f

b

P PK

FY (14) 

At the pitch line, the spline shear stress is: 

( ) φ

τ =

1

cos i

c

P

t F (15) 
12 

3.1.5 Tooth Load Distribution

The normal tooth load carried by the teeth adjacent to the highest loaded tooth is assumed to be 

proportional to the difference z z e − [20]: 

e

d i

e

z z P P

z

  − =    

(16) 

The elastic tooth deformation of the highest loaded tooth equals: 

i

e

P

z

C = (17) 

The maximum tooth separation occurs at the teeth located 90° from the highest loaded tooth 

[20]: 

( )

( ) φ =

2 tan

2tan

c

o

R i

z (18) 

Where the hub root crown radius is: 

RR i cF d = cos sin ( ) ( ) φ (19) 

Therefore, the total load (or load capacity) of the spline is: 

P KqNP t i = (20) 

For very accurate splines, the accuracy factor equals one as used in this study. For commercially 

accurate splines, the accuracy factor is less than one. 

3.1.6 Jam Angle

The jam angle, or maximum misalignment angle, defines the upper boundary of the spline shaft 

tilt. Couplings must be designed so that their jam angle is larger than the maximum misalignment 

expected during operation as stated in AGMA 6123-B06 [5]; however, no simple guidance is 

given in the standard to calculate the jam angle. The circumferential space between two adjacent

sleeve teeth is limited by the amount of c2 S , that is, 

( ) ( ) *

1 2 2 sin cos cc c f it i S + = (21) 

where: 

fR i c F = sin( ) (22) 

( ) ∗ = − −  1 1   2 1 cos cc F tt R i (23) 
13 

where the sleeve circular space width is given by: 

c c 2 2 S t

P

π

= − (24) 

Inserting equations (22) through (24) into equation (21) yields: 

( ) ( ) 1 2 2 cos 2 0 c F F c t R i RS − + −= (25) 

Therefore, the jam angle equals: 

2

max

1

2

acos

2

F c

F c

R S i

R t

  − =     − (26) 

Figure 8. Misalignment spline contact on both sides of the tooth flank

However, equation (25) is only valid when f F c ≤ 2 . If the hub face crown radius is very large, it 

is possible that equation (22) will calculate that f F c > 2 . In this case, the contact point simply 

occurs at the edge of the spline and the jam angle is: 

  =     max asin

2 F

F i

R (27) 

Therefore, the final form of jam angle is the smaller of the quantities: 

      −   =             −  

2

max

1

2

min acos , asin

2 2

F c

F c F

R S F i

Rt R (28) 

Note that the jam angle is entirely determined by spline geometric parameters. It can be 

increased primarily by reducing the hub face crown radius or hub tooth thickness. 
14 

3.1.7 Strength Analysis

3.1.7.1 Fatigue Strength

For steel, the endurance limit of a beam specimen is: 

' S S e u = 0.50 (29) 

Applying factors for the spline geometry and operating conditions, the bending endurance limit 

is [21]: 

= '

e a b c d eme S KKKKKK S (30) 

Assuming the gear coupling is not subjected to torque reversals that are high enough to load the 

teeth in reverse bending, the fluctuating bending stress varies sinsusoidally from zero to the 

maximum bending stress:

σ

σ σ= = ,max

2

b

a m (31) 

from the modified Goodman’s fatigue failure criterion: 

σ σ

+ = a m 1

e u SSn (32) 

If the safety factor is assumed to be one and the maximum bending stress is assumed to equal the 

bending endurance strength, the result is:

=

+

2

1

e

bf

e

u

S

S

S

S

(33) 

For pure rolling, fatigue cracks initiate below the surface of Hertzian contact in the area of 

maximum shear stress. For rolling and sliding, the maximum shear stress moves to the surface. 

Juvinall [22] suggests the maximum shear stress exists at the surface when the coefficient of 

friction is greater than 1/9 and under the surface when it is less than 1/9. Spline connections 

typically have a coefficient of friction greater than 1/9 (AGMA 6123-B06 recommends 0.25); 

therefore, the surface-initiated fatigue case is considered here. It is assumed the coefficient of 

friction is 1/3 and Poisson’s ratio is 1/4. The principal compressive stresses at the surface are 

then: 

σ σσ σσ σ 123 =1.39 0.72 0.53 ccc = = (34) 

The effective distortion energy stress is: 

( ) ( ) ( ) 222

12 23 31

2

2

σ σσ σσ σσ e = − +− +− (35) 
15 

Substituting equation (34) into equation (35), the final form of the effective distortion energy 

stress becomes: 

σ σ e c = 0.782 (36) 

The fluctuating contact stress varies sinsusoidally from zero to the maximum effective distortion 

energy stress, thus: 

σ σ

σ σ= = = 0.782

2 2

e c

a m (37) 

Again using the Goodman fatigue failure criterion and assuming the safety factor is one and the 

maximum contact stress is equal to the contact endurance strength, the result is:

=

+

2.56

1

e

cf

e

u

S

S

S

S

(38) 

According to the maximum shear stress theory, the shear endurance strength is: 

= 1

2 sf bf S S (39) 

3.1.7.2 Yielding Strength

Assuming Hertzian yielding occurs when the distortion energy stress equals the yielding 

endurance strength: 

e y σ = S (40) 

and letting the contact stress equal the contact yield strength, the bending, contact, and shear 

yield strengths equal: 

= = = 1 1.28

2 by y cy y sy y S SS SS S (41) 

Spline connections can be made using different heat treatment processes. Table 3 summarizes the 

endurance and yield strengths for splines made from steel and heat treated by carburizing and 

nitriding and through hardening and induction hardening. 
16 

Table 3. Fatigue and Yielding Strength of Splines with Different Heat Treatment Processes

Strength

(ksi)

Through 

hardened,

HB = 300

Carburized Nitrided

Induction 

hardened, 

flank and root

Induction 

hardened, 

flank only

bf S 62 83 65 81 33

cf S 79 106 106 103 103

sf S 31 42 33 41 17

by S 130 140 140 120 120

cy S 166 300 179 290 290

sy S 65 70 70 60 60

Safety Factors 

The safety factors based on fatigue are: 

σσ τ = = = bf cf sf

bf cf sf

b c

SSS

n nn (42) 

The safety factors based on yielding are: 

σσ τ = = = by cy sy

by cy sy

b c

SSS

n nn (43) 

3.1.8 Solution Methodology

The model is solved using numerical iterations and assuming an initial guess for q . The new 

value of q is calculated using the relationship curves [20]: 

 

= +   ≤ <

 

    

=+ − − <          

= − ≥      

1

2 2

0.36 0.14 0.4 1.0

1.195 1.75 1.033 0.4

1 1 1.0

2

e e

o o

e e e

o o o

e

e o

o

z z

q

z z

z z z

q

z z z

z

q

z z

z

(44) 

where the quantity e o z z is a function of q and is calculated from equation (17). The preceding 

process iterates until the assumed q equals the calculated q within the specified tolerance (10-3 

used in this study). The calculation process is described in Figure 9. 
17 

Figure 9. Flow chart of the spline stress, deformation, and safety factor calculations

3.2 Conformal Contact and Thin Strip Model: RomaxWind

The approach used in the commercial software package RomaxWind builds on an analytical 

foundation and then combines it with a numerical “thin-strip” model to capture the facewidth in 

contact. A key difference between the RomaxWind approach and the analytical and 

Transmission3D approaches described in this paper is that the contact is assumed to be 

conformal between the hub external teeth and sleeve internal teeth. The contact is assumed to 

occur at all points up the tooth height simultaneously. Local contact deformation is assumed to 

be negligible.

The first step in the approach is to determine the tooth stiffness per unit length assuming an 

unmodified, perfectly aligned spline. This stiffness is determined with a 2D finite element model 

of a single pair of teeth. The FE mesh is generated based on the parametrically defined tooth 

geometry. The rim thickness of the spline and hub meshes are used as specified for the true 

system. A unit load is applied to the inner rim of the hub, and the outer diameter of the sleeve is 

fixed. The contact is assumed to occur over the complete overlapping tooth profiles of the 

internal and external splines.

The spline is then broken into a finite number of lamina. Each lamina has a stiffness proportional 

to its width. The number of lamina in contact depends on the reduction in tooth thickness caused 

by crowning; the deflection caused by torque, radial forces, and tilting moment; and the 
18 

misalignment. Pitch errors can also be included but are not provided in this analysis. The 

deflections are iterated as part of a system nonlinear static analysis to find the position where the 

forces, moments, and deflections balance.

3.3 Line Contact and Three-Dimensional Finite Element Model: 

Transmission3D

The 3D finite element model in Transmission3D includes the flexibility of the spline teeth. For 

the purpose of comparing the 3D FE model to the analytical model, it can be configured to 

exclude the flexibility of the connection shafts. For the purpose of comparing it to the 2D FE 

model, it can be configured to include the flexibility of the connecting shafts. The model 

includes profile modification, lead modification, tooth spacing error, tooth flank flexibility, and 

rim flexibility. Assuming that the tooth contact is a line contact along the pitch line, the surface 

contact is solved by a surface integral approach that analyzes the near-field contact mechanics by 

integrating, in the style of a Green’s function, the solution for a point load on a half space over 

the contact area [23]. FE analysis calculates far-field elastic deformations starting a small 

distance away from the contact area. Matching of the contact deflections and FE solutions at the 

matching surface yields a combined contact solution for near-field surface deformations [24, 25]. 

This FE model calculates tooth load share, load distribution, tooth deformation, contact stiffness, 

and contact stress.
19 

4 Results and Discussion

The following sections evaluate the GRC sun shaft spline coupling using the developed 

analytical model, RomaxWind model, and the Transmission3D model considering both rigid and 

3D FE shaft models. Because of the lack of experimental data, results of the analytic model 

formulation are simply compared to these outputs from the high-fidelity finite element models.

4.1 Spline Design Parameters

Two important spline design parameters are the rotational stiffness of a single tooth pair and the 

jam angle, which are compared for each model in Table 4. For the rotational stiffness, the 

analytical model predicts a significantly higher stiffness than the FE models that account for 

shaft flexibility and conformal contact. However, for the purpose of simply correlating the 

analytical model, the best comparison is with the Transmission3D rigid shaft model, because 

each model assumes line contact and a rigid shaft. In this case, there is only a 3% difference

between the two models.

For the jam angle, the analytical model predicts an angle that is 50% higher than all of the FE 

models, which are in agreement with each other. The analytical formulation assumes a slightly 

different tooth profile shape than the FE models, which assume pure involutes. This different 

profile causes the difference in jam angles. The calculated jam angle for the GRC design of 0.20° 

is almost nine times the measured misalignment angle of 0.023° as shown in Figure 7. The GRC 

sun shaft 25.9-inch length tends to reduce the effect of the sun orbit; that is, for a given 

deflection of the sun, the longer the sun shaft, the lower the misalignment angle.

Table 4. Rotational Stiffness of a Pair of Spline Teeth

Analytical RomaxWind Transmission3D: 

Rigid Shaft

Transmission3D: 

3D FE Shaft

Rotational stiffness, Nm/rad 13.4 ×106 8.62×106 13.8×106 12.5×106

Jam angle, degree 0.30 0.20 0.20 0.20

4.2 Spline Loads

When the spline shaft is perfectly aligned with the sleeve shaft (hollow shaft), all of the teeth are 

in contact and the torque is transmitted evenly for ideally manufactured splines as shown in 

Figure 10 (left). That is, the loads are centered on the facewidth and the maximum load on a 

single tooth equals the average load per tooth. However, when the sun shaft is misaligned, the 

number of teeth in contact is reduced, the loads migrate toward the edges, and the maximum 

tooth load increases significantly as shown in Figure 10 (right). The variation of the loads is 

explored in the following sections. 
20 

Figure 10. Tooth load distribution across spline teeth with (left) and without (right) misalignment

4.2.1 Effects of Shaft Misalignment

Figure 11 shows the effects of shaft misalignment angle on the number of teeth in contact and 

the maximum load on a single tooth at rated torque. In general, the analytical model agrees 

reasonably well with all of the FE models. 

Figure 11. Effect of misalignment on the spline tooth loads

For the analytical model, an increase in the shaft misalignment from 0° to 0.2° (i.e., the jam 

angle) results in an immediate, but gradual decrease in the dimensionless number of teeth in 

contact from 1 to 0.36. At the jam angle, only 16 out of 48 teeth are in contact. The maximum 

load that a single tooth carries increases more than three times within the range of misalignment. 

The elevated tooth load increases the bending, contact, and shear stresses that could exceed their 

strength limits, leading to reduced service life.
21 

The number of teeth in contact for the FE models does not immediately decrease with 

misalignment. This is a result of the inclusion of spline teeth and shaft elasticity and the 

subsequent additional compliance in the system. The misalignment angle at which the number of 

teeth in contact finally begins to decrease correlates with the amount of flexibility in the system. 

The fully flexible shaft models have the highest number of teeth in contact. However, at the jam 

angle the number of teeth in contact agrees reasonably well for all four models. Despite the fact 

that the number of teeth in contact is generally higher for the FE models than for the analytical 

model, the maximum tooth load for the FE models is also generally higher than the analytical 

model. This is especially evident at higher misalignment angles.

Figure 12 to Figure 14 show the tooth load distribution at rated torque at selected misalignment 

angles. Figure 12 is for the analytic formulation, Figure 13 is for the RomaxWind model, and

Figure 14 is for the Transmission3D FE shaft model. The models are in good agreement, each

capturing the influence of misalignment on the tooth load distribution. 

Figure 12. Spline tooth load distributions from the analytic model

lbf

lbf

lbf

lbf
22 

When perfectly aligned, the teeth share loads equally, the tooth loads are centralized around the 

tooth facewidth center, and the load profile shape is parabolic. For small misalignment angles, 

the center of the contact area deviates from the tooth geometric center in a sinusoidal pattern 

around the circumference of the spline. The further the center of the contact is from the tooth 

geometric center (center-to-center distance), the larger the tooth load is. As the misalignment 

angle grows, the center-to-center distance increases and so does the maximum tooth load. Near a 

misalignment angle of half the jam angle, some teeth are entirely unloaded. The remaining teeth 

in contact carry the load and their contact area migrates even closer to the tooth edge. At the jam 

angle itself, only about one-third of the teeth are carrying any load at all. In this situation, the 

maximum tooth load is approximately quadruple the nominal load and is located very close to 

the edge of the teeth. These edge-loaded teeth are at risk of pitting.

Figure 13. Spline tooth load distributions from RomaxWind

lbf

lbf

lbf

lbf
23 

Figure 14. Spline tooth load distributions from Transmission3D

4.2.2 Effects of Transmitted Torque

The measured torque spectrum from field testing of the GRC gearbox in one turbine ranged from 

negative torque to two times the rated torque. The effects of torque on the number of teeth in 

contact and the maximum tooth load are shown in Figure 15. In these figures, the spline shaft is 

assumed to be misaligned by 0.1° (i.e., half the jam angle). The overall agreement among the 

models is acceptable.

The analytical model predicts lower maximum tooth loads than the high-fidelity models. This 

might be because of the simplification of the contact stiffness calculation of the proposed model. 

The proposed approach does not include the influence of torque on contact stiffness in equation 

(7) while the RomaxWind and Transmission3D models do include it. A safety factor of 1.5 is 

recommended for the analytical model for applications where the torque is less than 75% rated or 

greater than 125% rated. When torque increases from 50% to 150% of rated, the maximum tooth 

load for the analytical model approximately doubles as shown in Figure 15. The number of teeth 

in contact increases approximately 50% over this range. Therefore, spline loading and contact 

situations are clearly torque dependent. It is important to verify the spline design within the 

entire torque spectrum. 

lbf

lbf

lbf

lbf
24 

All of the FE results show similar trends with varying torque. The number of teeth in contact for 

the RomaxWind model is unaffected by torque over this range because of the high level of 

compliance in the system; however, the maximum tooth load correlates with the other models. 

With a rigid shaft, the Transmission3D model predicts the highest loads of all. 

Figure 15. Effect of torque on the spline tooth loads 

4.3 Spline Safety Factors

Safety factors compare the rated strengths of the spline against its bending, contact, and shear 

stresses in both fatigue and yielding. When the safety factor is less than one, the spline stress 

exceeds its rated strength. When the safety factor is larger than one, the spline design satisfies the 

strength requirement. The spline design is acceptable only when all of the safety factors are 

larger than one. 

The effects of shaft misalignment and transmitted torque on fatigue and yield safety factors are 

shown in Figure 16. The operating torque and misalignment values measured during the GRC 

dynamometer and field tests are highlighted in each figure. In general, the spline safety factors 

are well above one. The bending safety factor based on yielding is low compared to the other 

yielding safety factors. However, even in the most extreme field conditions, this safety factor is 

still above three. The bending safety factor based on fatigue can go below one; however, this is 

only in high-torque and large misalignment conditions that would very rarely occur in the field. 
25 

Figure 16. Effect of misalignment and torque on the spline safety factors

The GRC spline design meets all of the fatigue and yielding strength requirements; therefore, the 

spline damage shown in Figure 2 and Figure 3 is unlikely to have been caused by excessive loads 

or stresses. Although the spline meets all the strength requirements, the planetary section has 

poor load-sharing characteristics [2]. It could be possible to alter the design of the spline to both 

meet strength requirements and improve planetary section load sharing, essentially optimizing 

the spline design. This proposed model, together with the sun motion and torque measurements, 

can evaluate the spline’s performance, possibly for use in real-time condition monitoring. 
26 

4.4 Gear Spline Coupling Program (Gear SCouP)

The analytic formulation described in this report was coded into MATLAB software and is

available as a stand-alone program (i.e., a .exe) on an NREL-secure FTP website [26]. The 

program itself is called the Gear Spline Coupling Program (Gear SCouP) and the program 

graphical user interface (GUI) is shown in Figure 17. The input to the program includes spline 

dimensions, microgeometry, material properties, heat treatment, misalignment angle, and 

transmitted torque as described in Figure 9. These parameters are specified in an input data file 

called “input.txt” and can be read by SCouP by clicking the “Load Input File” button in the GUI 

interface. The “Review Input Parameter” button allows the user to review and edit these input 

parameters. Clicking the “Start Analysis” button initiates the analytic evaluation, normally 

requiring just a few seconds to complete the calculations. The program output consists of the 

spline stiffness, jam angle, spline load, plus contact, bending, and yielding stresses and their 

corresponding safety factors. The results can be plotted in the GUI screen by clicking the “Result 

Review” button or saved as an image file by clicking the “Capture Image” button. All of the 

input and output data can also be saved in a spreadsheet by clicking the “Save Result” button. 

Figure 17. Gear SCouP graphical user interface
27 

5 Conclusions

Splines are a critical but often overlooked part of the gearbox. In addition to the outright failure 

of the sun spline itself, poorly functioning splines could also be a contributor to failures in the 

rest of the planetary stage because of their influence on load-sharing characteristics. In this 

report, a new analytical model is described to evaluate spline designs commonly used in geared 

systems. Through this model, a greater understanding of the behavior of spline connections has 

been achieved and recommendations to improve design standards are summarized below. 

5.1 Spline Modeling and Behavior

Evaluation of the spline design through direct measurements is extremely difficult; however, 

measurement of the sun orbit and resulting shaft misalignment is feasible. When combined with 

the developed model or high-fidelity FE models, the sun orbit measurement facilitates the best 

available evaluation of the spline design. The analytical model results compare favorably with 

high-fidelity FE analyses and quickly yield insights into relationships between the spline design 

parameters and resulting loads and stresses. Simulation time with the proposed analytical model 

is two orders of magnitude lower than high-fidelity FE analyses. 

Given the spline properties, torque and shaft misalignment, the new analytic model calculates 

essential spline design information, including the jam angle, load share, maximum tooth loads, 

stresses, and safety factors. The sensitivity of spline loads, stresses, and safety factors to 

misalignment, transmitted torque, tooth crowning, and heat treatment was examined. Major 

findings from this study include the following: 

• The spline jam angle is solely determined by geometric parameters and is independent of 

the modeling approach. Reducing the hub face crown radius increases the jam angle.

• When the spline is in perfect alignment, the load is shared equally across all spline teeth 

and the tooth load distribution has a parabolic shape. When the spline is misaligned, the 

number of teeth in contact decreases and the maximum tooth load increases sharply. The 

contact area deviates from the tooth center and moves toward the tooth ends. In the 

extreme, the spline teeth are edge loaded and at risk of failure. 

• Torque affects the spline load share, maximum tooth load, and safety factors. It is 

important to evaluate the spline design within the entire torque spectrum. 

5.2 Future Work

Future development of the analytic formulation and Gear SCouP will account for: 1) influences 

of the geometric spacing error of the hub and sleeve teeth on spline loads and stresses, 2) axial 

friction force that can cause unexpected forces and excitations, and 3) spline teeth life 

calculation, which is missing in current standards and literature. 
28 

5.3 Recommendations for Current Standards 

Existing wind turbine gearbox design standards do not address spline design in great depth. In 

AGMA 6123, spline design is treated in terms of an allowable torque value based on semiempirical models of shaft failure in shear and bursting (a failure mode in which the shell of the 

gear teeth ruptures), and spline tooth failure in shear at the pitch-line and wear [5]. These

methods originate from the studies by Dudley [27] and Drago [28]. Assumptions are made 

regarding the number of teeth in contact, the engaged spline length, and the load distribution 

factor. All of these parameters are essential for evaluating spline loads and stresses. The GearSCouP program calculates the values of these crucial parameters and their effect on safety 

factors, which can significantly increase the confidence in the spline design.

As described in Section 10.4.3 of AGMA 6123, the load distribution factor is 2 for misaligned 

splines with tooth crowning. That is, the allowable torque for a misaligned, crowned spline is 

reduced by half. However, this is an oversimplification. Misalignment and crowning have a 

nonlinear effect on the tooth load distribution and maximum tooth load. The Gear-SCouP 

program calculates the effect of misalignment and crowning on the resulting load distribution

and maximum tooth load. 

Additionally, other effects are mentioned in AGMA 6123 but are not explored. For example, 

crowning is mentioned in Section 10.4.4.3, which states that “external teeth should be crowned 

to avoid high end loading and to improve load distribution at the ends of teeth.” However, the 

section also states “Crowning may reduce the load capacity of the coupling at zero degree 

misalignment, but improve the coupling life.” It is clear that optimal tooth crowning is important 

for spline reliability, but no method is given to evaluate the effects of crowning. The GearSCouP 

program is suitable for performing a fast crowning optimization study during the gearbox design 

stage.

As stated in Section 10.4.8.2: “Couplings which will be misaligned must be designed with 

adequate radial and tangential clearance so that they will not jam at the maximum misalignment 

permitted by the design.” However, a method to calculate the jam angle is not given. This study 

derives the jam angle and provides insight upon the relationship between the tooth geometry and 

jam angle. This formulation can reduce the number of iterations required to determine the spline 

geometry.
29 

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