1. DriveSE: An Analytical Formulation for Sizing and Estimating
Cost of Wind Turbine Hub and Drivetrain Components
Taylor Parsons, Yi Guo, Ryan King, Katherine Dykes, and Paul Veers
Saturday 15th November, 2014

2. Abstract
This report summarizes the theory, verification and validation of a new set of sizing models for wind turbine hub and
drivetrain components. The Drivetrain Systems Engineering (DriveSE) model provides a set of modules to determine
the dimensions and mass properties of a wind turbine hub, low speed shaft, main bearing(s), gearbox, bedplate and
yaw system. The levels of fidelity for each module range from semi-empirical parametric to full physics-based
models with internal iteration schemes for sizing components based on different system constraints and design
criteria. This report documents the details on the model assumptions, theories and formulations. Every component
model is validated against available industry data on component sizes. In addition, physics-based models are verified
against finite-element models. The verification and validation results show that the models to a reasonable job of
capturing first-order drivers for the sizing and design of major drivetrain components. However, due to the simple
nature of the model and their underlying physics, there is still significant deviation in the resulting component sizes
from DriveSE and actual industry data. Still, the resulting DriveSE model can be quite useful as a simple drivetrain
design tool in a larger wind turbine system design or analysis and will provide a good first-order approximation of
key nacelle and subcomponent attributes as a function changes to the rest of the system design.
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3. Table of Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Drivetrain Configurations and DriveSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Drivetrain Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1 Design Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Component Model Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.1 Hub . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.1.1 Hub Mass Properties Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.2 Main Shaft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.2.1 Shaft Load Analysis: Three-Point Suspension Drivetrain . . . . . . . . . . . . . . 9
2.2.2.2 Shaft Load Analysis: Four-Point Suspension Drivetrain . . . . . . . . . . . . . . . 10
2.2.2.3 Shaft Dimension Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.2.4 Low Speed Shaft Mass Properties Calculations . . . . . . . . . . . . . . . . . . . . 14
2.2.2.5 Shaft Design with Parameterized Fatigue Spectrum . . . . . . . . . . . . . . . . . 14
2.2.2.6 Shaft Design under User-Defined Fatigue Spectra . . . . . . . . . . . . . . . . . . 21
2.2.3 Main Bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.3.1 Bearing Sizing under Extreme Loads . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.3.2 Bearing Sizing under Parameterized Fatigue Loads . . . . . . . . . . . . . . . . . . 25
2.2.3.3 Bearing Sizing under user-defined fatigue loads . . . . . . . . . . . . . . . . . . . 27
2.2.3.4 Bearing Mass Properties Calculations . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.4 Gearbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2.4.1 Single external gear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2.4.2 An external gear pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2.4.3 Planetary gear stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2.4.4 Gearbox Weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2.4.5 Determination of gearbox speed ratio per stage . . . . . . . . . . . . . . . . . . . . 36
2.2.4.6 Gearbox Mass Properties Calculations . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2.5 Yaw System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2.5.1 Yaw System Mass Properties Calculations . . . . . . . . . . . . . . . . . . . . . . 38
2.2.6 Additional Nacelle Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.2.6.1 Other Nacelle Component Mass Properties Calculations . . . . . . . . . . . . . . . 39
2.2.7 Bedplate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.2.7.1 Load Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.2.7.2 Deflection Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.2.7.3 Bending Stress Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.2.7.4 I-beam Sizing and Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.2.7.5 Mass and Geometric Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3 Model Verification Against Higher Fidelity Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.1 Description of Modeling Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2 Results, Analysis, and Discussion of Model Verification . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.1 Low Speed Shaft Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.2 Bedplate Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
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4. 4 Model Validation and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.1 Hub . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 Main Shaft and Bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3 Bedplate Weight and Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.4 Gearbox Stage Ratio, Weight, Cost, Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.5 Nacelle Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.0.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6 Appendix B: Parameterized Fatigue Loads Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
7 Appendix C: Simplified Four-Point Suspension Main Shaft Model . . . . . . . . . . . . . . . . . . . . 64
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
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5. List of Figures
Figure 1. Comparison of three-point (above) and four-point (below) drivetrain configurations . . . . . . . . 2
Figure 2. DrivePY calculation flow chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Figure 3. Nacelle layout assumed in component center of mass definitions for three-point and four . . . . . 6
Figure 4. Flow chart of main shaft/bearing sizing tool. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Figure 5. Force diagram of a main shaft in three-point suspension drivetrain. . . . . . . . . . . . . . . . . . 9
Figure 6. Force diagram of a main shaft in four-point suspension drivetrain. . . . . . . . . . . . . . . . . . 11
Figure 7. Flowchart of main shaft and bearing fatigue sizing tool . . . . . . . . . . . . . . . . . . . . . . . 16
Figure 8. Stochastic stress distribution with generic S-N curve of high-strength steel . . . . . . . . . . . . . 19
Figure 9. Generic S-N relationship of materials without an endurance limit, taken from Norton (2014) . . . 20
Figure 10. Flowchart of user-defined main shaft and bearing fatigue sizing tool . . . . . . . . . . . . . . . . 22
Figure 11. CARB mass and facewidth interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Figure 12. SRB mass and facewidth interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Figure 13. TRB1 mass and facewidth interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Figure 14. CRB mass and facewidth interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Figure 15. TRB2 mass and facewidth interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Figure 16. RB mass and facewidth interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Figure 17. Gearbox weight vs. rated input torque computed by the Sunderland model (Harrison and Jenkins,
1993a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Figure 18. Flow chart of the gearbox sizing tool. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Figure 19. Scaling relationship for ABB 24 kV dry transformers. . . . . . . . . . . . . . . . . . . . . . . . . 40
Figure 20. Loads and constraints applied to the 750 kW LSS (A), the mesh used in analysis (B), and plots of
Von Mises stress (C), shear stress (D), and deformation (E). . . . . . . . . . . . . . . . . . . . . . . . . 44
Figure 21. Low speed shaft and hub solid models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Figure 22. DriveSE basic I-beam assembly with large front cast iron beam and long rear steel piece. Note
split lines across top faces of I-beams at component locations. . . . . . . . . . . . . . . . . . . . . . . . 46
Figure 23. Von Mises stress plot of rear I-beam with loads, mesh, and emphasized deflection. . . . . . . . . 47
Figure 24. Von Mises stress and deflection results across the length of the rear beam. . . . . . . . . . . . . . 47
Figure 25. Deformation plot of front I-beam after application of all DriveSE loads. . . . . . . . . . . . . . . 48
Figure 26. Industry trend and model results: hub mass vs. turbine nameplate rating . . . . . . . . . . . . . . 49
Figure 27. Industry trend and model results: main shaft mass vs. turbine rating . . . . . . . . . . . . . . . . 51
Figure 28. Industry trend and model results: bedplate mass vs. rotor diameter . . . . . . . . . . . . . . . . . 52
Figure 29. Industry trend and model results: Gearbox weight vs. rated torque. . . . . . . . . . . . . . . . . . 53
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6. Figure 30. Industry trend and model results: nacelle mass vs rotor diameter . . . . . . . . . . . . . . . . . . 54
Figure 31. Force and Moment Spectra Defined by DS472 Using Inputs From a 750kW Rotor . . . . . . . . . 62
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7. List of Tables
Table 1. Nomenclature used during shaft design: main variables . . . . . . . . . . . . . . . . . . . . . . . ix
Table 2. Nomenclature used during shaft design: sub- and superscripts . . . . . . . . . . . . . . . . . . . . x
Table 3. Nomenclature used during gearbox design: main variables . . . . . . . . . . . . . . . . . . . . . . x
Table 4. Nomenclature used during gearbox design: sub- and superscripts . . . . . . . . . . . . . . . . . . xi
Table 5. Input Summary for DriveSE Component Sizing . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Table 6. Maximum Ranges for Slopes and Transverse Deflections (Shigley et al., 2003) . . . . . . . . . . . 13
Table 7. Additional Inputs Required for Parameterized Fatigue Analysis . . . . . . . . . . . . . . . . . . . 15
Table 8. Bearing calculation factors used by bearing type . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Table 9. I-beam Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Table 10. DLC loads applied to LSS components, in global nacelle coordinate system . . . . . . . . . . . . . 45
Table 11. Resultant maximum stress concentrations and deflections at the location of the bearings . . . . . . 45
Table 12. Maximum bending stress and tip deflection compared for rear beams . . . . . . . . . . . . . . . . 47
Table 13. Maximum bending stress and tip deflection compared for front beams . . . . . . . . . . . . . . . . 48
Table 14. Comparison of hub model outputs to solid model masses . . . . . . . . . . . . . . . . . . . . . . . 49
Table 15. Comparison of no-fatigue main shaft model output and solid model dimensions . . . . . . . . . . 50
Table 16. Comparison of fatigue main shaft model output and solid model dimensions . . . . . . . . . . . . 50
Table 17. Comparison of Fatigue and non-fatigue models to solid model mass . . . . . . . . . . . . . . . . . 50
Table 18. Comparison of bedplate model outputs to solid model masses . . . . . . . . . . . . . . . . . . . . 51
Table 19. Effects of transformer location on bedplate dimensions and mass . . . . . . . . . . . . . . . . . . 52
Table 20. Gearbox speed ratio per stage of the GRC 750kW gearbox . . . . . . . . . . . . . . . . . . . . . . 52
Table 21. 750 kW reference inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Table 22. 750 kW reference fatigue inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Table 23. 1.5 MW reference inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Table 24. 1.5 MW reference fatigue inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Table 25. 5 MW reference inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Table 26. 5 MW reference fatigue inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Table 27. Description of three stress sources considered in main shaft fatigue analysis . . . . . . . . . . . . 60
Table 28. Relationship Between IEC Class and Turbulence Intensity Factor . . . . . . . . . . . . . . . . . . 62
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8. List of Terms
Table 1. Nomenclature used during shaft design: main variables
Symbol Meaning
Aw Weibull scale parameter
CL Coefficient of lift
CM Center of mass
B Blade number
d Diameter
D Damage due to fatigue
r Radius
P Power
p0 Aerodynamic line-load
E Young’s Modulus
ρ Density
f Frequency
F Force or Load
FW Face width
F∆∗() Stochastic load range at given load count
H Height
I Second moment of area
It Turbulence intensity factor
k Safety factor
kw Weibull shape parameter
L Length
m Mass
M Moment
n(),N() Number of stress cycles, number of cycles to failure at given amplitude
Nf Maximum number of load cycles experienced by components
p Diametral pitch
SF Stress Range due to specified Force (fatigue)
SM Stress Range due to specified Moment (fatigue)
Sy Yielding strength
Sf Fatigue strength
Sm Fatigue failure point at 103 cycles
t Time
th Thickness
T Torque
TD Tip Deflection
TL Turbine Life
v Transverse deflection
V Volume
Vmin,Vmax Cut-in, Cut-out windspeeds
V0 Nominal windspeed
W Weight
X Tipspeed ratio
ηd Drivetrain efficiency
γ Tilt angle
ω Lengthless weight
σ Normal stress
τ Shear stress
σv Von Mises stress
σ1 Maximum principal stress
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9. Table 2. Nomenclature used during shaft design: sub- and superscripts
Subscript Meaning
b Blade
c Characteristic
hb Hub
r Rotor
rb Rotor-main bearing
ms Main shaft
ms,i Inner main shaft
f Main shaft flange
mb1,mb2 Upwind and downwind main bearing for four-point suspension
mb Main bearing for three-point suspension
gb Bearing-gear coupling
gc Gear coupling
gbx Gearbox
hs High speed shaft / coupling
gen Generator
t Transformer
RNA Rotor-Nacelle Assembly
norm Normal stress range
bend Bending stress range
eq equivalent stress range (Goodman)
Superscript Meaning
x, y, z Coordinates
determ, mean, ult, max Loads type (deterministic, mean, ultimate, maximum)
Table 3. Nomenclature used during gearbox design: main variables
Symbol Meaning
B Number of planet gears
dp Gear diameter
KAG Application factor
Kγb Load sharing factor between rows
Kγ p Load sharing factor among planets
Kr Scaling factor
Ksh Shaft factor
Kv Dynamic factor
np Speed
P Power
Qo Input torque to the main shaft
Qp Input torque to the pinion
U Speed ratio
WGBPN Gear pair weight
Wt Tangential driving force
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10. Table 4. Nomenclature used during gearbox design: sub- and superscripts
Subscript Meaning
GB Gearbox
s Sun
p Planet
rg Ring
xi
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11. 1 Introduction
The Drivetrain Systems Engineering (DriveSE) model is a set of models for sizing wind turbine drivetrain com-
ponents that is designed for use as part of a larger wind turbine design and analysis tool. A previous similar tool
was developed in the late 1990’s (Harrison and Jenkins, 1993b). This model, known as the Sunderland model, used
semi-empirical formulations were developed for all major wind turbine components in order to provide the mass
of each which could be then converted into cost for an overall turbine capital cost estimate. These semi-empirical
formulations were based on a collected industry database that is not representative of today’s multi-megawatt (MW)
size wind turbines. Thus, a need exists to develop an accurate drivetrain sizing tool that can be used for dimension-
ing drivetrain components as part of a case study on drivetrain design or, more importantly, as part of a larger wind
turbine and system study.
Wind turbine drivetrains physically connect the rotor to the tower and serve as a load-path from one to the other.
The drivetrain is also responsible for converting the aerodynamic torque of the rotor into electrical power that can
be fed to the grid. Therefore, the drivetrain model interacts with the rotor and tower designs and it is important
in looking at the overall design of a wind turbine to consider the coupling that exists between these three primary
subsystem. DriveSE provides the capability to take in the aerodynamic loads and rotor properties and to estimate
the mass properties and dimensions for all major components; the overall nacelle properties can then be used in
subsequent tower design and analysis or as part of a system-level optimization of the wind turbine. In addition, the
resulting mass and dimension estimates can then be used to feed into a turbine capital cost model as well as a balance
of station cost model that considers cost of assembly and installation of a wind turbine so that a full wind plant
system level cost analysis could be performed. Thus, while DriveSE can be used to do drivetrain specific analysis as
illustrated in (King et al., 2014), the model set can also be used as part of larger wind turbine and plant system level
studies (Dykes et al., 2014).
DriveSE uses a more rigorous set of physics-based analyses than was used in the Sunderland model to estimate the
size of a subset of the major load-bearing components (the low speed shaft, main bearing(s), gearbox and bedplate)
and parametric formulations representative of current wind turbine technology for the remaining components (the
hub and yaw system). The high-speed side of the drivetrain including the high speed shaft and coupling, mechanical
brake, generator and other auxiliary components are not modeled and an existing set of models based on the Sunder-
land Model and other more recently developed models (Dykes, 2013). The only exception is the transformer which
was not included in previous models. A simple model of the transformer based on industry data is included since
up-tower transformers are relatively common to modern wind turbines and, as a heavy component, have a significant
impact on the overall mass properties of the nacelle assembly.
1.1 Drivetrain Configurations and DriveSE
Geared drivetrains, the most prevalent design for land-based wind turbines, consist of a main shaft, main bearing(s),
gearbox, generator coupling, and generator. Different rotor supports and bearing configurations are used across
various manufacturers, which can be grouped into four categories: 1) three-point suspension, 2) two-main-bearing
suspension (four-point suspension), 3) integrated drivetrain, and 4) hub supported drivetrain. In the three-point
suspension, the rear main bearing is integrated into the gearbox at the planetary stage as the planetary carrier bearing.
The two-main-bearing suspension uses two separate main bearings that ideally carry all the nontorque loads from
the rotor and transmit them into the tower through the bedplate. The integrated drivetrain has the main bearings
integrated into the gearbox. The nontorque loads are transmitted through the gearbox housing. The Pure Torque hub
support is distinct from the others as it uses a set of circumferential flexible couplings to connect the rotor with the
main shaft and thus isolates any nontorque loads from the drivetrain. Among all different drivetrain configurations,
the three-point suspension and two-main-bearing suspension (four-point suspension) are most common. Thus, these
are the configurations that are modeled in DriveSE. Figure ?? shows a graphical comparison of three-point and
four-point suspension configurations. Note the location of the main bearing(s) and the sizing of the main shaft.
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12. Figure 1. Comparison of three-point (above) and four-point (below) drivetrain configurations
DriveSE considers three-point and four-point drivetrain configurations with gearboxes and high speed generators.
Medium-speed and direct-drive configurations are outside the scope of the work presented here and will be addressed
in future model versions. Furthermore, the gearbox model of DriveSE includes several layouts of parallel and plane-
tary gear stages. In all cases, the loads are fed into the models from the rotor and at this time, extreme loads are used
to size all components with the exception on the shaft sizing model. The constraints on system design are as close
as possible to those used in practice for designing commercial components. The particular design methodology and
limitations of that methodology for each component are discussed in the model description section of the report.
The drivetrain designs calculated using the developed models are compared against actual industry data or higher
fidelity finite element analysis. For the physics-based models, verification is performed against higher fidelity finite-
element models for key design criteria. This involved creating representations of each component for different sizes -
using data on real turbine components where possible. An iterative process was used to evaluate the DriveSE model
in comparison to the higher fidelity model and corrections to the DriveSE models were made as necessary. For both
the physics-based and parametric models, validation of each model is performed against available industry data on
component sizes. Given the simplifications of the DriveSE models, final scaling factors were included with some the
DriveSE models in order to calibrate them with the industry data and to account for design factors that were outside
of the current scope.
In Chapter 2 of the report, the theory for the overall model and each component model is described. In Chapter 3,
the physics based models are verified against higher fidelity finite element models. In Chapter 4, all models are
compared against industry data as a validation step. Finally, the conclusion identifies the strength and weaknesses of
each of the models. The resulting model set may then be used for standalone drivetrain analysis and design or as part
of a larger wind turbine or plant study.
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13. 2 Drivetrain Model Description
DriveSE consists of a series of coupled mathematical models of drivetrain subcomponents as shown in Figure 2.
At this time, DriveSE contains models for the hub, low speed shaft, main bearings, gearbox, bedplate and yaw
system. The remainder of the components in the hub and nacelle systems are sized using DriveWPACT (Dykes,
2013) that are primarily based on empirical data. Master routines DriveWPACT interface with other wind turbine
components, namely, the rotor and tower. At this top level, design criteria on allowable stress and deflection are
inherently included for individual drivetrain subcomponents. These design criteria, together with the minimum
weight objective for sub-optimizations, are used to determine the subcomponent dimensions.
Figure 2. DrivePY calculation flow chart
Key model inputs include the extreme aerodynamic rotor loads (both torque and non-torque), gravity loads, gearbox
configuration parameters, and overall turbine design parameters such as rotor overhang and gearbox location. An op-
tional fatigue analysis of the low speed shaft and main bearing(s) is included that requires several additional inputs.
The outputs of DriveSE fall into two categories: subcomponent outputs and system outputs. Subcomponent outputs
include the dimensions and mass properties of individual subcomponents that are preliminary design parameters
for these subcomponents. The current model implementation calculates the size, mass, center-of-mass (CM), and
moment of inertia for the hub, low speed shaft, upwind main bearing, downwind main bearing (if used), gearbox,
bedplate, and yaw system. For the gearbox, individual stage ratios, volumes and masses are also computed. The
mass outputs for all the individual components are then used in wind turbine capital cost and balance of station cost
models. The system outputs are the cumulative weight, moments of inertia, and center of gravity of the entire hub
and nacelle assemblies, which are used as inputs at the tower design level and also for wind turbine and plant cost
models.
Each component model takes a unique design approach. Firstly, the hub is modeled entirely off of scaling arguments
calibrated to industry data due to its geometric complexity. The current implementation treats the hub as a thin
walled ductile cast iron cylinder with circular holes for blade root openings and lowspeed shaft flange. The hub outer
dimensions and thickness scale with the rotor diameter and blade root thickness.
The main shaft design is sized first by determining the length from deflection limitations imposed by main bearings,
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14. with a maximum length constrainted by the overhang distance. Distortion energy failure theory is used to determine
the outer diameters at main bearings, with the final design consisting of a hollow shaft with a taper in between
bearing locations. The bearing selection criteria are based on shaft geometry and load capacity. After main bearings
are selected, shaft dimensions will be updated to match the bearing bore diameters. Thus, there is a sub-iteration in
the model between the main shaft and bearing modules.
DriveSE designs the gearbox for the minimum weight by optimizing the speed ratio of each stage. The model re-
quires as a minimum input set: the transmitted torque, overall speed ratio, stage number, and gearbox configuration.
The gearbox model outputs the weight, volume, and speed ratio of each gearbox stage as well as the overall gearbox
weight.
The bedplate size is approximated by modeling the bedplate as two parallel I-beams and separately treating the
upwind and downwind sections. The upwind section is assumed to be made of ductile cast iron, while the downwind
section is steel. Static point loads from the nacelle components are superimposed on the bedplate structure at the
center of mass of each component, and rotor aerodynamic loads are superimposed on the upwind bedplate section
as well. The upwind and downwind bedplate sections are individually sized to meet deflection and bending stress
constraints. A scaling factor is applied to the bedplate at the end to model mass which is not incorporated into the
I-beam structure.
The yaw system is composed of a friction plate yaw bearing at the nacelle tower interface and also includes several
yaw motors. The friction plate bearing is treated as a steel annulus and is sized according to the tower top diame-
ter and rotor diameter. The motors are assumed to be a common Bonfiglioli design and the number of motors is a
function of the rotor diameter if not specified by the user.
The rest of the drivetrain, hub system and nacelle components are included in analysis for the purpose of bedplate
sizing and determination of overall system-level outputs. Masses and sizes of these components are currently calcu-
lated using a modified version of the DriveWPACT (Dykes, 2013), with updated center of mass calculations based
on component sizes and CM optimization. The DriveWPACT models for the high-speed shaft, coupling, brake and
generator are highly simplified parametric models. A physics-based generator and generator-coupling model are not
included at this stage, though the addition of a parametric model of the transformer has been made.
2.1 Design Loads
The design loads used in the model are based on those that would be obtained via wind turbine dynamic simulations
such as those from an aeroelastic code or similar model. The extreme loads are the primary design loads used in
DriveSE. These include IEC 61400-1 standard loads obtained during normal power production, when a wind gust
occurs, during an electrical fault, etc. ??. The required input loads along with other key inputs for each drivetrain
component are summarized in Table 5. External loads are specified using the load analysis results while the internal
loads are calculated inside of DriveSE and drive the design of components downstream of one another.
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15. Table 5. Input Summary for DriveSE Component Sizing
Input Parameter Hub Low Speed Shaft Main Bearing Gearbox Bedplate Yaw
Power Rating (External) - -
Rotor Diameter (External) - - - - -
Blade Root Diameter (External) - - - - -
Drivetrain Topology (External) - -
Rotor Weight (External) - - -
Rotor Overhung (External) - - -
Rotor Loads (External) - - - -
Rotor Torque (External) - - -
Main Shaft Dimensions (Internal) - - - -
Bearing Type (External) - - - - -
Bearing Weight (Internal) - - - - -
Gearbox Ratio (External) - - - - -
Gearbox Topology (External) - - - - -
Gearbox Weight (Internal) - - - - -
Generator Type (External) - - - - - -
Generator Weight (Internal) - - - - -
Converter Weight (Internal) - - - - -
Transformer Weight (Internal) - - - - -
2.2 Component Model Formulations
A number of general assumptions have been made to facilitate the drivetrain model development. The current effort
focuses on component designs driven by maximum stress analyses and deflection criteria. By using industry standard
safety margins this approach sizes components to avoid catastrophic failure. The deflection constraints ensure com-
ponents are properly aligned and within geometrical limits for bearing alignment, gear tooth meshing, etc. As will be
discussed later, these geometrical requirements are an important part of the integrated design work-flow connecting
main bearings, low speed shaft, and gearbox components. The optional fatigue analyses for the main shaft and bear-
ings are based on a linear damage assumption, and rely on a Miner’s Rule summation of damage across the design
life of a turbine. Other general assumptions include homogeneously distributed component masses for calculating
moment of inertia (MOI) and center of mass (CM). Where models rely on scaling arguments, such as for the hub
or transformer mass in sizing the bedplate, the departure from physics-based analysis is noted. Under the typical
3-pt or 4-pt suspension drivetrain, CM calculations are made under the assumption that drivetrain components are
distributed according to figure 3. Coordinate directions follow the IEC standard wind turbine coordinate system with
the origin shifted to the yaw axis at the top of the bedplate as shown. For individual CM calculations, see the CM
documentation at the end of each section.
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16. Figure 3. Nacelle layout assumed in component center of mass definitions for three-point and four
2.2.1 Hub
DriveSE models the hub entirely off scaling arguments due to its geometric complexity, and will be refined in future
studies. The current implementation treats the hub as a thin walled ductile cast iron cylinder with holes for blade root
openings and low speed shaft flanges. The hub outer dimensions and thickness scale with the blade root thickness.
Pitch motors and bearings are not considered at this point and are modeled using the DriveWPACT model.
The hub radius is assumed to be 1.1× the blade root radius, the hub height is assumed to be 2.8 × the hub radius,
and the hub thickness is assumed to be 1/10th the hub radius. These relationships are based off existing hub designs
and will be updated as new data becomes available.
2.2.1.1 Hub Mass Properties Calculations
The total hub material volume Vhub is then given by:
Vhb = 2πrhbHhbthhb −(1+B)πr2
bthhb (2.1)
where material has been removed for each blade root and the lowspeed shaft flange opening is assumed to be the
same size as a blade root. The hub mass mhb, is simply calculated by mhb = ρVhb where ρ is the hub cast iron density
which is 7200 kg/m3 in this module.
In the case that the blade root diameter is unknown to the user, a default is set using the scaling relationship in
equation 2.2. This relationship was found using industry data gathered from a wide range of turbines, and scales
much better with the machine’s power rating, P[MW], than with its rotor diameter.
db = 2.659×P.3254
(2.2)
The CM of the hub is determined by the location of the upwind main bearing, and the variable Lrb (shown in figure
5) as such:


CMx
mb2 = CMmb1 −Lrb
CMy
mb2 = 0
CMz
mb2 = CMmb1 −Lrb sin(γ)

 (2.3)
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17. 2.2.2 Main Shaft
The inputs for main shaft and main bearing design are the static extreme rotor aerodynamic and gravity loads simu-
lated by wind turbine load analyses or measured by experiments. The main shaft dimensions are designed based on
the distortion energy failure theory. A peak load safety factor of 2.5 [(AGMA, 2008)] is applied. Deflection limita-
tions imposed by main bearing(s) are then applied to calculate the shaft length. In addition to shaft dimensions, the
shaft program also calculates main bearing loads that are used to select the main bearings. Main bearings are picked
out of the SKF bearing database. The type of bearings used is used along with criteria based on shaft geometry and
load capacity to select the final bearing size(s). Once the main bearing(s) are selected, the shaft dimensions will be
updated to match the bearing bore diameter(s). The design process of main shaft is illustrated in Figure 4. This chap-
ter covers two models for the main shaft design depending on whether the drivetrain configuration is a three-point or
four-point suspension. An alternative and simplified four-point suspension design model is also included in DriveSE
and described in Appendix C.
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18. Torque, rotor aerodynamic forces moments,
rotor drivetrain weight, bedplate tilt angle
Calculate von Mises stress of main shaft
Calculte shaft diameters based on shaft
stress and allowable safety factor
Allowable safety factor
for shaft material
Meet bearing deflection
requirements?
Bearing types
Calculate shaft deflection
Yes
No
Shaft geometry finalized
Assume shaft length
bearing locations
Calcuate bearing loads
moments
Select bearings from
database based on shaft
geometry carried loads
Allowable safety
factor for bearings
Update shaft
length
Bearing geometry matches
shaft geometry?
No
Yes
Update shaft
geometry
Shaft/bearing unit design complete
Figure 4. Flow chart of main shaft/bearing sizing tool.
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19. 2.2.2.1 Shaft Load Analysis: Three-Point Suspension Drivetrain
Shaft geometry depends on the highest stresses experienced at stress concentration locations, typically at the main
bearing location. As the first step of shaft design, the shaft stress in the longitudinal direction is determined given
the transmitted torque and bending moments. Figure 5 shows the force diagram of a main shaft in a three-point
suspension drivetrain. The nomenclature is shown in Table 1. The following equations are derived based on the force
and moment balance of the system.
Blade
Main
Bearing Main Shaft
Wms
Wgb
Fmb
Fgb
Las
Lbg
Wa
Lms
Lgc
z
z
z
x
y
Mmb
y
x Fmb
y
Mmb
z
x
Fgb
y
Lgb
COG High Speed
Shaft
Gearbox
x
Generator
Coupling
Fgc
Fgc
y
z
Mgc
y
Mgc
z
Mgb
y
Mgb
z
Fmb
x
Fgb
x
Fgc
x
Gearbox
Trunnion
Wr
Mr
y
Mr
z
Fr
x
Lrb
x
Hgb
Hgc
Figure 5. Force diagram of a main shaft in three-point suspension drivetrain.
The force balance along x axis ∑Fx = 0 leads to:
Fx
r +Fx
mb +Fx
gb +Fx
gc +(Wr +Wms +Wa +Wgb)sin(γ) = 0 (2.4)
The force balance along y axis ∑Fy = 0 leads to:
Fy
r +Fy
mb +Fy
gb +Fy
gc = 0 (2.5)
The force balance along z axis ∑Fz = 0 leads to:
Fz
r +Fz
mb +Fz
gb +Fz
gc −(Wa +Wr +Wms +Wgb)cos(γ) = 0 (2.6)
The balance of pitching moments ∑My = 0 around gearbox trunnions leads to:
My
r +My
mb +Fx
r Hgb −Wr cos(γ)(Lrb +Lbg)+Fz
r cos(γ)(Lrb +Lbg)
+Fz
mbLbg +Fx
mbHgb −Wms(Lbg −Las)cos(γ)+Wgb cos(γ)Lgb +My
gb +My
gc
−Wa cos(γ)(Lbg −Lms)+Fx
gcHgc −Fz
gc(Lgc −Lbg) = 0
(2.7)
The balance of yaw moments ∑Mz = 0 around gearbox trunnions leads to:
Mz
r +Mz
mb −Fy
r (Lbg +Lrb)−Fy
mbLbg +Mz
gb +Mz
gc +Fy
gc(Lgc −Lbg) = 0 (2.8)
Model assumptions are made based on systems stiffness properties and common design criteria of individual compo-
nents. For example, the radial stiffnesses of main bearings are orders of magnitude higher than the tilting stiffnesses,
resulting in higher radial loads than moments. Main bearings have large load capacity in the axial direction com-
pared gearbox bearings and support arms. They react the rotor thrust and transfer it to the tower.
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20. 1. The main bearing does not carry moments, such that:
My
mb = Mz
mb = 0 (2.9)
2. The generator coupling does not carry radial loads or moments, therefore:
Fx
gc = Fy
gc = Fz
gc = 0
My
gc = Mz
gc = 0
(2.10)
3. Main shaft bending moments are carried by the main bearing and bearings of the gearbox low-speed stage,
resulting in zero gearbox trunnion moments:
My
gb = Mz
gb = 0 (2.11)
4. The rotor axial loads are carried by the main bearing. The drivetrain axial loads caused by self-weight are
carried by gearbox trunnions.
Fx
r +Fx
mb +Wr sin(γ) = 0
Fx
gc +Fx
gb +(Wms +Wa +Wgb)sin(γ) = 0
(2.12)
By substituting Eq. 2.9-Eq. 2.12 into Eq. 2.4-Eq. 2.8, the solutions of Eq. 2.4-Eq. 2.8 can be derived. The loads at
the main bearing and gearbox trunnions are












Fx
mb = −Fx
r (t)−Wr sin(γ)
Fy
mb = Mz
r (t)
Lbg
−
F
y
r (t)(Lbg+Lrb)
Lbg
Fz
mb = 1
Lbg
{−My
r (t)+Wr[cos(γ)(Lrb +Lbg)+sin(γ)Hgb]−Fz
r (t)cos(γ)(Lrb +Lbg)
Wms(Lgb −Las)cos(γ)+Wa cos(γ)(Lbg −Lms)−Wgb cos(γ)Lgb
Fx
gb = −(Wms +Wa +Wgb)sin(γ)}
Fy
gb = −Fy
mb −Fy
r
Fz
gb = −Fz
mb +(Wa +Wr +Wms+Wgb
)cos(γ)−Fz
r (t)
(2.13)
The bending moments along the main shaft in the pitching and yaw directions are
My(x) =
−Fz
r (t)x+Wrx−My
r (t)+ x
0 ωms(x)xdx, 0 x ≤ Lrb
−Fz
r (t)x+Wrx−My
r (t)+ x
0 ωms(x)xdx−Fz
mb(x−Lrb), Lrb x ≤ Lrb +Lms
(2.14)
where ωms = Wms
Lms
.
Mz(x) =
−Mz
r (t)−Fy
r (t)x, 0 x ≤ Lrb
−Mz
r (t)−Fy
r (t)x−Fy
mb(x−Lrb), Lrb x ≤ Lrb +Lms
(2.15)
2.2.2.2 Shaft Load Analysis: Four-Point Suspension Drivetrain
Figure 6 shows the force diagram of a main shaft in four-point suspension drivetrain. The nomenclature is shown in
Table 1. The following equations are derived based on the force and moment balance of the system.
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21. Blade
Upwind
Bearing Main Shaft
Wms
Wgb
Fmb1
Fgb
Las
Lbg
Wa
Lms
Lgc
z
z
z
x
y
Mmb1
y
x Fmb1
y
Mmb1
z
x
Fgb
y
Lgb
COG High Speed
Shaft
Gearbox
x
Generator
Coupling
Fgc
Fgc
y
z
Mgc
y
Mgc
z
Mgb
y
Mgb
z
Fmb1
x
Fgb
x
Fgc
x
Gearbox
Trunnion
Wr
Mr
y
Mr
z
Fr
x
Lrb
x
Hgb
Hgc
x
Dnwind
Bearing
x
Lmb
Fmb2
z
Mmb2
y
Fmb2
y
Mmb2
z
Fmb2
x
Figure 6. Force diagram of a main shaft in four-point suspension drivetrain.
The force balance along x axis ∑Fx = 0 leads to:
Fx
r +Fx
mb1 +Fx
mb2 +Fx
gb +Fx
gc +(Wr +Wms +Wa +Wgb)sin(γ) = 0 (2.16)
The force balance along y axis ∑Fy = 0 leads to:
Fy
r +Fy
mb1 +Fy
mb2 +Fy
gb +Fy
gc = 0 (2.17)
The force balance along z axis ∑Fz = 0 leads to:
Fz
r +Fz
mb1 +Fz
mb2 +Fz
gb +Fz
gc −(Wa +Wr +Wms +Wgb)cos(γ) = 0 (2.18)
The balance of pitching moments ∑My = 0 around the upwind main bearing leads to:
My
r +My
mb1 +My
mb2 −Wr cos(γ)Lrb +Fz
r cos(γ)Lrb
−WmsLas cos(γ)+Wgb cos(γ)Lgb +My
gb +My
gc −Fz
mb2Lmb
−Wa cos(γ)Lms −Fz
gcLgc = 0
(2.19)
The balance of yaw moments ∑Mz = 0 around the upwind main bearing leads to:
Mz
r +Mz
mb1 +Mz
mb2 −Fy
r Lrb −Fy
mb2Lmb +Mz
gb +Mz
gc +Fy
gcLgc = 0 (2.20)
Additional assumptions are made based on system stiffness properties besides those used for three-point drivetrains
as follows. The gearbox supports, flexible in tilting and axial directions, are designed to allow gearbox compliance in
the associated directions.
1. Gearbox weight is carried by the trunnions:
Fz
gb +Wgb cos(γ) = 0 (2.21)
2. Gearbox trunnion force in the y direction is negligible compared to main bearing forces:
Fz
gb +Wgb cos(γ) = 0
Fy
gb Fy
mb1, Fy
mb2
(2.22)
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22. By substituting these assumption into Eq. 2.16-Eq. 2.20, the solutions of Eq. 2.16-Eq. 2.20 can be derived. The
loads at the main bearings and gearbox trunnions are:









Fx
mb2 = −Fx
r (t)−Wr sin(γ)
Fy
mb2 = −Mz
r (t)+F
y
r (t)Lrb
Lmb
Fz
mb2 = 1
Lmb
{My
r (t)−Wr cos(γ)Lrb −Wmscos(γ)LasWgb cos(γ)Lgb −Wa cos(γ)Lms
Fx
mb1 = 0
Fy
mb1 = −Fy
mb2 −Fy
r
Fz
mb1 = −Fz
mb2 +(Wa +Wr +Wms)cos(γ)−Fz
r (t)
(2.23)
The bending moments along the main shaft in the pitching and yaw directions are:
My(x) =


−Fz
r (t)x+Wrx−My
r (t)+ x
0 ωms(x)xdx, 0 x ≤ Lrb
−Fz
r (t)x+Wrx−My
r (t)+ x
0 ωms(x)xdx−Fz
mb1(x−Lrb), Lrb x ≤ Lrb +Lmb
−Fz
r (t)x+Wrx−My
r (t)+ x
0 ωms(x)xdx−Fz
mb1(x−Lrb)−Fz
mb2(x−Lrb −Lmb), Lrb +Lmb x ≤ Lrb +Lms
(2.24)
Mz(x) =


−Mz
r (t)−Fy
r (t)x, 0 x ≤ Lrb
−Mz
r (t)−Fy
r (t)x−Fy
mb1(x−Lrb), Lrb x ≤ Lrb +Lmb
−Mz
r (t)−Fy
r (t)x−Fy
mb1(x−Lrb)−Fy
mb2(x−Lrb −Lmb), Lrb +Lmb x ≤ Lrb +Lms
(2.25)
2.2.2.3 Shaft Dimension Determination
The stresses at a point on the surface of a solid round shaft of diameter d subject to bending, axial loading, and
twisting are:
σx =
32(M2
y +M2
z )1
2
πd3
+
4F
πd2
τxy =
16T
πd3
(2.26)
By use of a Mohr’s circle it can be shown that the two nonzero principal stresses are:
σa, σb =
σx
2
±[(
σx
2
)2
+τ2
xy]
1
2 (2.27)
These principle stresses can be combined to obtain the maximum shear stress τmax and the von Mises stress σv
τmax = [(σx
2 )2 +τxy
2]
1
2
σv = (σx
2 +3τ2
xy)
1
2
(2.28)
The shaft design is based on the distortion-energy theory of failure, the allowable von Mises stress is :
σv
all =
Sy
k
(2.29)
Under most conditions, the axial component F is either zero or so small that it can be neglected. With F = 0 and Eq.
2.26 substituted into Eq. 2.28, Eq. 2.28 becomes:
τmax = 16
πd3 (M2
y +M2
z +T2)
1
2
σv = 16
πd3 (4M2
y +4M2
z +3T2)
1
2
(2.30)
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23. Using the distortion-energy theory, the shaft diameter is solved as:
d(x) =
16k
πSy
[4My(x)2
+4Mz(x)2
+3T2
]
1
2
1
3
(2.31)
where My and Mz are calculated from Eq. 2.14 and Eq. 2.15 or Eq. 2.24 and Eq. 2.25 . The shaft inner diameter is
assumed as 10% of the outer diameter. This value is selected based on the collected industry data.
Shafts are designed to meet the requirements on deflections and rigidity. When shaft preliminary design is complete,
the shaft deflections and misalignments are calculated and checked for whether they meet the deflection requirements
at critical locations, including at the interfaces with main bearing(s) and the low speed stage of the gearbox. The
typical maximum ranges for misalignment and transverse deflections for bearings and gears are shown in Table 6.
Table 6. Maximum Ranges for Slopes and Transverse Deflections (Shigley et al., 2003)
Misalignment
Tapered roller 0.0005-0.0012 rad
Cylinder roller 0.0008-0.0012 rad
Deep-groove ball 0.001-0.003 rad
Spherical ball 0.026-0.052 rad
Self-align ball 0.026-0.052 rad
Uncrowned spur gear 0.0005 rad
Transverse deflections
Spur gears with P 10 0.010 inch
Spur gears with 11 P 19 0.005 inch
Spur gears with 20 P 50 0.003 inch
Shaft deflection for the three-point suspension drivetrain is derived as:



My(x) = −Fz
r (t)x+Wrx−My
r (t)−Fz
mb(x−Lrb)+ x
0 ωms(x)xdx, Lrb x ≤ Lrb +Lms
EI
dvy
dx = −Fz
r (t)x2
2 +Wr
x2
2 −My
r (t)x−Fz
mb
(x−Lrb)2
2 + y xωms(x)xdxdy+C1, Lrb x ≤ Lrb +Lms
EIvy = −Fz
r (t)x3
6 +Wr
x3
6 −My
r (t)x2
2 −Fz
mb
(x−Lrb)3
6 + z y xωms(x)xdxdy, dz+C1x+C2, Lrb x ≤ Lrb +Lms
(2.32)
Where the boundary conditions include:
vy = 0, x = Lrb
vy = 0, x+Lrb +Lbg
(2.33)
Shaft deflection for four-point suspension drivetrain is derived as:



M1
y (x) = −Fz
r (t)x+Wrx−My
r (t)+ x
0 ωms(x)xdx−Fz
mb1(x−Lrb), Lrb x ≤ Lrb +Lmb
EI
dv1
y
dx = −Fz
r (t)x2
2 +Wr
x2
2 −My
r (t)x−Fz
mb1
(x−Lrb)2
2 + y xωms(x)xdxdy+D1
EIv1
y = −Fz
r (t)x3
6 +Wr
x3
6 −My
r (t)x2
2 −Fz
mb1
(x−Lrb)3
6 + z y xωms(x)xdxdy, dz+D1x+D2
(2.34)
Where the boundary conditions include:
v1
y = 0, x = Lrb
v1
y = 0, x+Lrb +Lmb
(2.35)
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24. And after the second bearing as:



M2
y (x) = −Fz
r (t)x+Wrx−My
r (t)+ x
0 ωms(x)xdx−Fz
mb1(x−Lrb)−Fz
mb2(x−Lrb −Lmb), Lrb +Lmb x ≤ Lrb +Lms
EI
dv2
y
dx = −Fz
r (t)x2
2 +Wr
x2
2 −My
r (t)x−Fz
mb1
(x−Lrb)2
2 + y xωms(x)xdxdy−Fz
mb2
(x−Lrb−Lmb)2
2 +D3
EIv2
y = −Fz
r (t)x3
6 +Wr
x3
6 −My
r (t)x2
2 −Fz
mb1
(x−Lrb)3
6 + z y xωms(x)xdxdy, dz−Fz
mb2
(x−Lrb−Lmb)3
6 +D3x+D4
(2.36)
Where the boundary conditions include:
v2
y = 0, x = Lrb +Lmb
dv1
y
dx =
dv2
y
dx , x+Lrb +Lmb
(2.37)
2.2.2.4 Low Speed Shaft Mass Properties Calculations
The volume of the main shaft between the bearings is calculated using the following formula:
Vms = π
12 dmb1
2
+dmb2
2
+dmb2dmb1 Lmb − FWmb1+FWmb2
2 Volume of solid taper
+π
4 dmb1
2
FWmb1 + π
4 dmb1
2
FWmb1 Volume of shaft contained by main bearings
−π
4 dms,i
2
Lmb + FWmb1+FWmb2
2 Volume of hole
(2.38)
The mass of the main shaft is taken to be the density multiplied by the volume, where density of the shaft material
is specified as 7800/frackgm3. The mass of the flange is then accounted for by multiplying this total mass by 1.33
(found from an average of solid model flange masses percentages). The total length of the main shaft takes into
account the flange length, distance between main bearing centers, and facewidths in the following way:
Lms = Lmb +
FWmb1 +FWmb2
2
+Lf (2.39)
where Lf is either specified by the user or, if unspecified, found from an approximate scaling argument with rotor
diameter: Lf = 0.9918×e0.0068dr .
The available solid models for both 3-point and 4-point main shafts all have a center of mass location which is very
close to 65% of their length upwind of the gearbox connection. Using this data, the shaft CM is therefore modeled in
the following way:



CMx
ms = CMx
gbx −
Lgbx
2 −0.65Lms cos(γ)
CMy
ms = CMy
gbx
CMz
ms = CMz
gbx +0.65Lms sin(γ)


 (2.40)
2.2.2.5 Shaft Design with Parameterized Fatigue Spectrum
If desired, a fatigue analysis can be included in the main shaft and bearing sizing analysis. Fatigue analysis is a
user-specified option in DriveSE which may take the form of user-defined lifetime loads spectra or, if this data is not
available, the parameterized loads spectrum defined in Appendix B. This model uses cyclically-varying stochastic
loads, as well as deterministic loads from the rotor and component masses to size the main shaft and bearings. Using
a simplified representation of cyclic and mean loads experienced by the rotor, forces and moments are resolved into
stresses at the location of the main bearings. Beginning with the shaft diameters calculated in the above extreme-
loads shaft model, stresses and damage-equivalent-loads (DELs) are calculated and diameters are increased until the
resultant damage does not result in failure over a specified component lifetime. Additional inputs for this portion of
the DriveSE model are shown in Table 7.
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25. Table 7. Additional Inputs Required for Parameterized Fatigue Analysis
Input Variable Units
Cut-in wind speed m/s
Rated wind speed m/s
Cut-out wind speed m/s
IEC Class letter A,B,C
Availability (optional) %
Blade number -
Weibull shape parameter -
Weibull scale parameter m/s
Design life yrs
Shaft fatigue exponent (optional) -
If users chose to implement this fatigue analysis, DriveSE also uses the loads generated in this section to select
bearings which satisfy a calculated dynamic loads criterion. The design process for this sizing tool is illustrated in
figure 7. Bearing locations and component lengths are taken directly from the extreme loads model and assumed
to be suitable. When calculating the total number of cycles experienced by the shaft during the design life of the
turbine, it is assumed that the rated frequency, design life, and probability of operation (taken from Weibull param-
eters an cut-in/cut-out wind speed) can be multiplied to give an approximate lifetime number of shaft rotations. The
equation using this assumption can be found in the loads documentation. Damage resultant from each load cycle is
assumed to be linear, and wake effects from neighboring turbines are assumed to be nonexistent in the calculation of
aerodynamic rotor load cycles.
The loads definition used in this section is detailed in Appendix B. After all rotor loads are defined, the model re-
solves them into mean and alternating forces and moments at the location of the main bearings. In this step, care is
taken not to combine the stochastic-alternating, deterministic-alternating, and mean sources of stress on the main
shaft. Note that this model uses a simplified, parameterized loads spectra to calculate DELs from rotor loads oc-
curring during turbine life. The option to utilize a user-defined loads spectrum was added so that one might specify
the loads from a rotor model as part of a larger wind turbine design process where the rotor design is constantly
changing.
15
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26. Turbine inputs and
Shaft Size from
main shaft model
Define Stochastic
loading cycles across
turbine lifetime
Calculate
deterministic mean
loads
Calculate
deterministic
alternating loads
Resolve into
deterministic
alternating stress at
bearing locations
Resolve into
deterministic mean
stress at bearing
locations
Resolve into
stochastic
alternating stress at
beating locations
Define equivalent
zero-mean
deterministic stress
Define equivalent
zero-mean stochastic
stress range
S-N relationship of
high-strength
steel
Sum fatigue effects
across turbine life
(Miner’s Rule)
Damage results
in failure?
YES
NO
Increase shaft
diameter
Calculate axial and radial
forces experienced during
lifetime
Resolve into
equivalent loads
Integrate bearing life
consumed across
revolution lifetime
Bearing Data Table
Calculate required
dynamic load rating
Select smallest bearing
subject to load rating and
bore diameter constraints
Update Shaft size to
match bearing bore and
face width
Fatigue-driven design of
Shaft and Bearing(s)
complete
Bearing routine
Bearing types
Figure 7. Flowchart of main shaft and bearing fatigue sizing tool
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27. Three Point Stress Calculation
Using the same component location assumptions as the extreme load sizing tool, the fatigue model calculates the
stresses at the bottom of the shaft, above the single main bearing of a 3-pt drivetrain, where stresses are the highest
magnitude. The stochastic stresses are found considering the bending, torsional shear, and normal stresses, then
combined into an equivalent stress value using the following equations:




bending stress: σstoch
bend = − Mstoch
y
2
+Mstoch
z
2 dms
2I
normal stress: σstoch
norm = −Fstoch
x cosγ
A
torsional shear stress: τstoch = Mstoch
x dms
2J



 (2.41)
σstoch
v = σstoch
bend +σstoch
norm
2
+3τstoch2 (2.42)
where the geometrical constants are calculated as:
I = π
64 (d4
ms −d4
ms,i)
J = π
32 (d4
ms −d4
ms,i)
A = π
4 (d2
ms −d2
ms,i)
(2.43)
Note that these calculations are performed for every stochastic force and moment in the array of values found in the
Force-N and Moment-N spectrum, and assumes that maximum loads can all be combined to form a maximum stress
instance, while minimum loads can be combined to form a minimum stress instance. This approach is considered
valid because all damage from the resulting stresses is ultimately summed into the same DEL figure.
The deterministic alternating stress on the main bearing is defined as the stress which occurs every time the main
shaft makes one rotation. This stress takes into account the rotor loads and weights of drivetrain components. In this
case, the moment about the y-axis caused by the rotor overhang and mean x-force is the source of the deterministic
alternating stress at the main bearing:
σdeterm
v =
Mydms
2I
=
[(Wr cosγLrb)−(Fmean
x sinγLrb)]dms
2I
(2.44)
The mean stress is assumed to be constant, and exist during both deterministic and stochastic stress cycles. This
mean stress is a consequence of a mean torque value, mean compressive force on the rotor, and the compressive
effects of component weights. The following equations encompass the mean stress calculations used in this model:
normal stress: σmean
norm = −Fmean
x cosγ+(Wr+Wms)sinγ
A
torsional shear stress: τmean = Mmean
x dms
2J
(2.45)
σmean
v = (σmean
norm )2
+3(τmean)2
(2.46)
Four Point Stress Calculation
In the four-point suspension drivetrain, stress calculations are performed at both upwind and downwind bearings,
and a suitable shaft diameter is selected at both locations. Because the load path up to the first main bearing is the
same as that of the three-point suspension configuration, stress calculations at the upwind bearing are the same as
those in the three-point drivetrain, and follow equations 2.41 to 2.45. Stresses at the downwind bearing are again
separated into a single mean stress value, a distribution of stochastic stresses, and a deterministic alternating stress.
Geometrical constants found in equation 2.43 are found from the diameters at the downwind bearing.
17
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28. In order to determine the stresses at the downwind bearing, the model first calculates the forces at the upwind bear-
ing. Again, it uses the same assumptions, force balances, and moment diagrams as the extreme loading model. The
forces on the upwind bearing due to the stochastic forces and moments are:
Fy
mb1
stoch
=
−Mstoch
z
Lmb
Fz
mb1
stoch
=
Mstoch
y
Lmb
Fx
mb1
stoch
= 0
(2.47)
Stochastic moments at the downwind bearing are then calculated to be:
My
mb2
stoch
= Mstoch
y +Fz
mb1
stoch
Lmb = 0
Mz
mb2
stoch
= Mstoch
z −Fy
mb1
stoch
Lmb = 0
(2.48)
Because the moments at the downwind bearing due to stochastic forces is zero, there is no bending stress. This leads
to a stochastic stress at the downwind bearing due only to the torsion on the shaft and the axial load:
normal stress: σstoch
norm = −Fstoch
x cosγ
A
torsional shear stress: τstoch = Mstoch
x dms
2J
(2.49)
σstoch
v = (σstoch
norm )
2
+3(τstoch)
2
(2.50)
Mean Stresses are calculated in the same way, taking into account the deterministic axial and torsional stresses which
are present at the downwind bearing. Because all axial forces on the shaft are assumed to be held by the downwind
bearing, the mean stress at the second bearing is found to be:
normal stress: σmean
norm = −Fmean
x cosγ+(Wr+Wms)sinγ
A
torsional shear stress: τmean = Mmean
x dms
2J
(2.51)
σmean
v = (σmean
norm )2
+3(τmean)2
(2.52)
Deterministic alternating stresses which occur once every shaft rotation are found from the resultant bending stresses
at the second bearing location. First, the deterministic forces at the upwind bearing are found to be
Fz
mb1
determ
=
−Wr(Lmb+Lrb)−
Lmb
0
ωms(x)dx(Lms)+Wgb(Lgb)
Lmb
Fy
mb1
determ
= 0
(2.53)
where ωms = Wms
Lms
. It is assumed that Lms is half of the distance between main bearings.
The moment at the downwind bearing due to component weights and the reaction force calculated above is
My
mb2
determ
= −Wr(Lrb +Lmb)+Fz
mb1
determ
−
Lmb
0
ωms(x)dx(Lms)+WgbLgb = WgbLgb
Mz
mb2
determ
= 0
(2.54)
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29. From this, the deterministic alternating stress at the downwind bearing is found to be
σdeterm
v =
My
mb2
determ
dms
2I
(2.55)
DEL Summation and Sizing
From the stresses at each bearing location, the model calculates the damage at each bearing and ensures that damage
does not result in failure. Using the assumption that the mean stress is relatively constant over the operation of the
turbine, the stochastic stresses and alternating deterministic stresses are converted into stresses with zero mean using
a Goodman correction. Under Goodman, cyclic stresses with a non-zero mean contains a failure envelope described
by equation 2.56:
σa
σv
+
σm
SUT
= 1 (2.56)
, where σa is the alternating stress amplitude, σm is the mean stress, SUT is the ultimate strength of the material, and
σe is the effective alternating stress at failure. This is solved for the effective zero-mean alternating stress in equation
2.57. Figure 8 is an example of the stochastic alternating stress distribution plotted against the equivalent zero-mean
distribution, with a generated S-N curve for reference. Note that because the mean stress is compressive, the effective
alternating stress is lower than the unadjusted stress distribution.
σeq =
σa
1− σm
SUT
(2.57)
Figure 8. Stochastic stress distribution with generic S-N curve of high-strength steel
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30. After using this correction to convert both stochastic and deterministic stresses, the resultant damage from each
stress cycle is summed using the Palmgren-Miner Linear Damage Rule. The general form of Miner’s rule is shown in
equation 2.58. Here, n(Si) is the number of cycles at a given stress amplitude that the material experiences and N(Si)
is the number of cycles at the stress amplitude which are needed to fail the material. The part does not fail as long as
the accumulated damage, D does not exceed 1.
D =
n
∑
i=1
n(Si)
N(Si)
(2.58)
The value N(Si) is taken from the estimated S-N relationship of high-strength steel. According to Norton (2014),
the following procedure is a reasonable estimate for creating an approximated S-N diagram of the main shaft. The
high-strength steels used in main shaft manufacture typically do not exhibit an endurance limit, so we define the S-N
diagram of the material to be similar to the one shown in figure 9.
Figure 9. Generic S-N relationship of materials without an endurance limit, taken from Norton (2014)
With an ultimate strength of 700MPA, we assume that the failure point at 103 cycles, Sm is 90% of the ultimate
strength. The fatigue strength Sf of the material is calculated from the unadjusted fatigue strength Sf and a variety of
correction factors, as shown below (Norton (2014)). This value is taken to be the point at which the component will
fail at 5×108 cycles.
variable value comment
Se = 0.5SUT
Csize = 0.6 Diameter 250mm
Csur f = 4.51S−.265
UT Machined Surface
Ctemp = 1.0 Normal Operating Temperatures
Creliab = 0.814 99% reliability
Cenvir = 1.0 Enclosed environment
Sf = SeCsizeCsur fCtempCreliabCenvir
(2.59)
The equation for the line which connects Sm and Sf can be defined by the equation:
S(N) = aNb
(2.60)
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31. (Norton (2014)), where
b =
log Sm
Se
log(N1)−logN2
(2.61)
a =
Sm
Nb
1
(2.62)
This creates an S-N curve with a fatigue exponent, m = −1
b , of approximately 8.55, which reflects material data for
several high-strength steel alloys.
The damage summation formula using Miner’s Rule then becomes:
D =
n
∑
i=1
n(Si)
Si
a
1/b
(2.63)
In summing the damage at these bearings, we have a smooth function defining the stochastic spectrum of stresses,
and a single point defining the deterministic stress at a cycle number equal to the number of rotor rotations Nr.
Therefore, the final damage summation is defined by equation 2.64.
D =



Nf
Ni
n(Si)
Si
a
1/b
dN


+
Nr
σdeterm
a
1/b
1.0 (2.64)
As the flowchart in figure 7 shows, the model iterates the stress and damage calculations at the bearing location(s),
and increases the shaft diameter until the total damage due to fatigue does not result in failure. Once the fatigue-
driven design of the shaft has been complete, the model uses the forces and moments from this model to calculate
fatigue-driven design in the bearing routine.
2.2.2.6 Shaft Design under User-Defined Fatigue Spectra
If users have their own fatigue range spectra and would like to use them in the main shaft and bearing sizing tool, the
model performs the calculations using the same methods as the parameterized model above, but with several marked
differences in the way damage is calculated.
Because this data will be coming from physical test data, the deterministic effects of component weights are in-
cluded in the spectra rather than considered separately. Secondly, because vectors defining load count in each load
range may not be the same as they are in the case of the parameterized loads spectrum, the damage due to each force
and moment spectrum must be integrated separately and added into a damage figure rather than performing a sin-
gle integration with the combined stress effects. Note that this model ignores the effects of the shaft angle on the
forces experienced by the bearings, as this angle would contribute to unnecessarily large complexity and marginally
increased accuracy.
Additional inputs to this model include vectors defining the load ranges at the location of the rotor Fx
r ,Fy
r ,Fz
r ,Mx
r ,My
r ,
and Mz
r , as well as their corresponding cycle counts NFx , NFy ,NFz ,NMx ,NMy , and NMz . All load spectra are assumed
to be zero-mean. As in the parameterized loads spectrum, this model uses the shaft lengths defined from the main
shaft deflection and stress analysis, and increases the diameter at the bearing location(s) until the fatigue loads do
not result in failure. Bearings are again selected based off of a calculated dynamic load rating and shaft diameter as
detailed below. Material properties of the shaft are taken to be the same as above. A flowchart detailing the design
process for this model is included in figure 10.
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32. Figure 10. Flowchart of user-defined main shaft and bearing fatigue sizing tool
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33. DEL Summation and Sizing Approach
The Damage due to each load range is calculated using Miner’s Rule. The damage due to a particular load and load
count is found using the following equation:
D = ∑
i
ni
Ni(Li)
(2.65)
Where ni denotes the cycle count, and Ni() the number of cycles to failure at the load cycle Li. The relationship
between load range and cycles to failure using the S-N curve is modeled by:
Ni =
Lult
1
2 Li
m
(2.66)
Where the load at failure, Lult, is calculated using the S-N curve and the geometry of the shaft, as discussed in the
following section.
The damage due to a single loads spectrum is then found by integrating the damage across turbine life:
D =
i
ni
Ni(Li)
(2.67)
This damage figure is calculated for each loads spectrum, and summed in the following way:
Dtotal =
i
ni
Ni(Fx
r )
+
i
ni
Ni(Fy
r )
+
i
ni
Ni(Fz
r )
+
i
ni
Ni(Mx
r )
+
i
ni
Ni(My
r )
+
i
ni
Ni(Mz
r )
1.0 (2.68)
If the total damage exceeds unity, the shaft diameter is increased until the shaft does not fail, as the flowchart in
Figure 10 shows.
Three Point Load Calculation
The ultimate loads Lult in equation 2.66 are found using the S-N curve and the geometry of the shaft.
For a three-point suspension drivetrain with a single main bearing, the thrust loads Fx
r contribute to normal stress on
the shaft, so that the stresses due to the axial force ranges are modeled as:
SFx
r
=
Fx
r
π
4 d2
ms −d2
ms,i
(2.69)
The load Fy
r creates a moment at the main bearing which contributes a bending stress to the damage figure.
MF
y
r
= Fy
r Lrb (2.70)
SF
y
r
=
MF
y
r
dms
π
32 d4
ms −d4
ms,i
=
Fy
r Lrbdms
π
32 d4
ms −d4
ms,i
(2.71)
The load Fz
r also contributes to a bending stress in the same manner as above, so that the stress due to Fz
r is:
23
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34. SFz
r
=
MFz
r
dms
π
32 d4
ms −d4
ms,i
=
Fz
r Lrbdms
π
32 d4
ms −d4
ms,i
(2.72)
The torque, Mx
r , contributes to a shear stress along the length of the shaft:
SMx
r
=
Mx
r dms
π
16 d4
ms −d4
ms,i
(2.73)
Note that we assume the shear yield strength is 1/
√
3 × Sy, so a factor of
√
3 is added in the final ultimate loads
calculations below.
The moments My
r and Mz
r contribute to bending stresses:
SM
y
r
=
My
r dms
π
32 d4
ms −d4
ms,i
(2.74)
SMz
r
=
Mz
r dms
π
32 d4
ms −d4
ms,i
(2.75)
Ultimate loads are found using the S-N relationship of the material, defined in section 2.2.2.5, by setting the above
stresses equal to a, the material constant defining failure at a single cycle count. After solving for Lult, the resultant
ultimate loads are summarized below.





























Lult
Fx
= a π
4 d2
ms −d2
ms,i
Lult
Fy
= a
π
32 (d4
ms−d4
ms,i)
Lrbdms
Lult
Fz
= a
π
32 (d4
ms−d4
ms,i)
Lrbdms
Lult
Mx
= a
π
16 (d4
ms−d4
ms,i)√
3dms
Lult
My
= a
π
32 (d4
ms−d4
ms,i)
dms
Lult
Mz
= a
π
32 (d4
ms−d4
ms,i)
dms
(2.76)
Four Point Load Calculation
The upwind main bearing of a four-point configuration drivetrain is modeled in the same way as the main bearing in
the three-point analysis. Under the assumption that the bearings handle bending moments from the rotor, the shaft
at the downwind location sees stresses from only the axial and torque loads. Because we assume that the axial loads
are carried by the downwind bearing and torque is constant throughout the shaft, these ultimate loads are the same as
found above.
24
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35. 




Lult
Fx
= a π
4 d2
ms −d2
ms,i
Lult
Mx
= a
π
16 (d4
ms−d4
ms,i)√
3dms
(2.77)
Only these loads are considered in the final damage total.
2.2.3 Main Bearings
DriveSE lets users select between six different bearing types: CARB toroidal roller bearings, spherical roller bear-
ings (SRB), single-row tapered roller bearings (TRB1), double-row tapered roller bearings (TRB2), cylindrical roller
bearings (CRB), and single-row deep-groove radial ball bearings (RB). It is assumed that the user is able to make
their own judgments with respect to the bearing selection and configuration. Often, it is recommended that three-
point suspension turbines are configured using a TRB bearing and four-point suspensions using a CARB and SRB in
the standard configurations.
2.2.3.1 Bearing Sizing under Extreme Loads
If users decide to use the extreme loads analysis alone to size the main shaft, bearings are selected whose bore
diameters are greater than the shaft diameter while minimizing bearing and shaft size. The selected bearing geometry
is then used to resize the shaft. Note that this step is documented in section 2.2.2 under figure 4.
2.2.3.2 Bearing Sizing under Parameterized Fatigue Loads
If additional fatigue analysis is performed, bearings are selected to satisfy a calculated dynamic load rating in ad-
dition to the criterion listed above. Information on fatigue analysis and the steps used to arrive at this value can be
found in sections 2.2.2.5 and 2.2.3.3.The selected bearing geometry is then used to resize the shaft.
Bearing fatigue is analyzed from the calculated axial and radial loads experienced by the main bearings across the
life of the turbine. The model calculates a dynamic load rating from the summation of equivalent forces during each
shaft rotation.
It is assumed that when integrating across turbine life, the domain of the integration is taken to be up to Nr (found
from
Nf
B , where B is blade number), so summation across the domain representing each shaft rotation is achieved
by dividing all values in the shaft N vector by B. We also assume that mean forces (both axial and radial) can be
added to the stochastic force amplitudes to give a spectrum of dynamic loads with a maximum load value. These
assumptions effectively convert the forces used in shaft analysis into ones which are usable in the bearing routine.
At the upwind bearing, the shaft model already defines the stochastic force distribution and the deterministic forces
resultant from component weights. In finding the force spectrum at this bearing, the radial and axial forces here are
taken to be:
Fr
mb1 = Fz
mb1
stoch
+Fz
mb1
determ
2
+Fy
mb1
stoch2
(2.78)
and because we assume the upwind bearing carries all axial load,
Fa
mb1 = Fstoch
x cosγ +(Wr +Wms)sinγ (2.79)
Keep in mind that all stochastic forces are defined as a vector of values, and that adding stochastic and deterministic
forces gives the total bearing force during each revolution.
25
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36. For a four-point drivetrain configuration, the force balance on the shaft system yields downwind bearing forces of:
Fy
mb2 = −Fy
mb2
determ
=
Mstoch
z
Lmb
Fz
mb2 = Fz
mb2
determ
+ Fz
mb2
stoch
= Wr +Wms −Fz
mb1
determ
+ −Fz
mb1
stoch
Fx
mb2 = 0
(2.80)
giving a radial load distribution of:
Fr
mb2 = Fy
mb2
2
+Fz
mb2
2
(2.81)
and axial force of:
Fa
mb2 = 0 (2.82)
For each bearing, the model then calculates an equivalent load, P, using the conditional equation:
P = Fr +Y1Fa ,Fa
Fr ≤ e
P = XFr +Y2Fa ,Fa
Fr e
(2.83)
where the variables Y1, Y2, X, and e are the calculation factors specific to each bearing type. Table 8 gives the ap-
proximate calculation factors for each bearing type, found from characteristic values in the (SKF, 2014a) bearing
catalogs.
Table 8. Bearing calculation factors used by bearing type
Bearing Type e Y1 Y2 X
CARB 1.0 0 0 1.0
SRB 0.32 2.1 3.1 0.67
TRB1 0.37 0 1.6 0.4
TRB2 0.4 2.5 1.75 0.4
CRB 0.2 0 0.6 0.92
RB 0.4 1.6 2.15 0.75
As an added requirement on cylindrical roller bearings, the ratio of axial to radial loads may not exceed 0.5, as
specified by (SKF, 2014a). In addition, because CARB bearings are not designed to carry axial load, an error is
returned if significant axial loads are present. P = Fr for all CARB bearings.
After converting all axial and radial loads into a single load array, the damage consumed by each load is summed
into a single equivalent load. To do this, we consider the total life, L1 (revolutions), of a bearing under load P1 and
dynamic load rating C, to be
L1 =
C
P1
3
×106
cycles (2.84)
so that the life consumed in one revolution at P1 is:
1
L1
=
P3
1
C3
×
1
106
(2.85)
If the bearing makes n1 revolutions at load P1, then:
n1
L1
=
P3
1
C3
×
1
106
×n1 (2.86)
26
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37. adding the effects of multiple loads and revolution counts, we get:
n1P3
1
106C3
+
n2P3
2
106C3
+...+
nnP3
n
106C3
=
(n1 +n2 +...nn)Pe
106C3
(2.87)
Where the equivalent load Pe can be solved as:
Pe = 3 ∑nP3
∑n
(2.88)
The relationship between loads and rotations is a smooth function, so this equivalent load is found by compressing
the summation into an integration.
Pe = 3 P3dn
dn
(2.89)
Equations up to this point were derived in (Jindal, 2010). From the equivalent load, the necessary dynamic load
rating of the bearing can be found from solving equation 2.84 for C, with P = Pe and L = Lbearing = Nr. The required
dynamic load rating of the bearing is found to be:
C = Pe
3 Lbearing
106
= 3 P3dn
dn
Lbearing
106
(2.90)
Note that equations 2.84-2.90 use an exponent of 3, as is specified for a typical ball bearing. For CARB, SRB,
TRB1, TRB2, and CRB bearings, The exponent used is 10/3. These values come directly from (SKF, 2014b), which
uses ISO 281:2007-02 as the basis for their calculations.
The model then selects the most suitable bearing from a lookup table of bearing sizes. The selected bearing is of
the type specified by the user, has a bore diameter greater than or equal to the shaft diameter, and has a dynamic
load rating greater than or equal to the calculated rating. When multiple bearings satisfy these constraints, the model
selects the one with the lowest bore diameter and face width, to minimize the mass of the main shaft and bearings.
After the final bearing has been selected, the model resizes the main shaft diameter(s) and bearing length(s) to fit the
bore diameter and face-widths of the bearings. This completes the design of the shaft and bearing system.
2.2.3.3 Bearing Sizing under user-defined fatigue loads
The model calculates the dynamic load rating of the bearing(s) using the same methods as in the parameterized loads
model, but with the damage due to each loads spectrum again added separately. We assume that the elements in the
user-specified cycle count arrays can be divided by the blade number to give an approximation of the rotation count
seen by the bearings, as is done in the previous section.
Dynamic Load Rating Calculation
Bearing life consumed by each load is integrated across the spectra separately, and summed to calculate a single
equivalent load and load rating. Because the timing of each load is not known from a simple load-cycle-count spec-
trum, the assumption is made that the axial and radial load ratios Fa
Fr can be calculated using the maximum values
from the load spectra, and this relationship is relatively constant during operation.
Forces are calculated at the locations of the main bearings, as detailed in the following section, and converted into an
equivalent bearing load as specified by (SKF, 2014b). Recalling the bearing calculation factors found in table 8, the
impact that each load spectrum has on the equivalent load can be found using the following conditional equation:
27
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38. P = Fr +Y1Fa ,Fa
Fr ≤ e
P = XFr +Y2Fa ,Fa
Fr e
(2.91)
The axial force on a bearing due to Fx, for example, is either multiplied by Y1 or Y2 depending on the ratio Fa
Fr , while
the loads contributing to radial loads are multiplied by 1 or X conditionally. This factor gives the equivalent load
spectrum PL due to each load type.
Once the equivalent load spectra is found, equation 2.88 is used to integrate the individual spectra across the turbine
life. The final equivalent bearing load is then found from equation 2.92.
Pe =
3 P3
fx
dn
dn
+
3 P3
fy
dn
dn
+
3 P3
fz
dn
dn
+
3 P3
My
dn
dn
+
3 P3
Mz
dn
dn
(2.92)
After the equivalent load is found, the dynamic load rating is calculated as:
C = Pe
3 Lbearing
106
(2.93)
Where Lbearing is the bearing life in revolutions. See section 2.2.3.2 for this derivation. Note again that these equa-
tions use exponents and roots of power 3 to calculate the equivalent bearing load. For CARB, SRB, TRB1, TRB2,
and CRB bearings, this power is in fact 10/3, as specified by (SKF, 2014b).
Once the force contributions are calculated and integrated into a dynamic load rating figure, the model selects a
bearing which satisfies the required rating, whose diameter is at least the diameter of the shaft, and whose mass and
size is a minimum.
Force Calculations
The above calculations are performed for each bearing after the effects of individual loads are isolated into radial or
axial forces held by the bearings. The Fx spectrum, for example, contributes the axial loads to the main bearing of a
3-point machine and the downwind bearing of a four-point machine.
The spectra of rotor loads, Fy
r ,Fz
r ,My
r , and Mz
r all contribute to radial loads acting on the bearings. At the upwind
bearing location, force balances due to the isolated loads Fz
r and My
r contribute a radial load in the z-direction as
follows
Fz
mb1 due to Fz
r = Fz
r
Lmb+Lrb
Lmb
Fz
mb1 due to My
r = M
y
r
Lmb
(2.94)
Force balances due to isolated loads Fy
r and Mz
r contribute a radial load in the y-direction as follows
Fy
mb1 due to Fy
r = −Fy
r
Lmb+Lrb
Lmb
Fy
mb1 due to Mz
r = Mz
r
Lmb
(2.95)
In a downwind bearing of a 4-point machine, force balances yield the following
Fz
mb1 due to Fz
r = Fz
r
Lrb
Lmb
Fz
mb1 due to My
r = M
y
r
Lmb
(2.96)
28
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39. Fy
mb1 due to Fy
r = Fy
r
Lrb
Lmb
Fy
mb1 due to Mz
r = Mz
r
Lmb
(2.97)
Each of these individual contributions to the radial load are input into the bearing routine and integrated into a
final damage figure. Note that normally, in the case where the time-history of all loads is known, the radial load Pr
would be Pr = Fy2 +Fz2. Here, because the times at which each load in the spectrum occurs are not known, the
calculation of maximum Fr for the Fa
Fr comparison assumes maximum values from each load spectrum.
2.2.3.4 Bearing Mass Properties Calculations
Individual bearing information is stored in DriveSE for bearings of all six bearing types. Information on inner and
outer diameters, face-widths, load ratings, and masses are used to define the bearing selection in the model, which is
collected from SKF bearing database. The maximum bore diameters for this database can be as low as 1.25m, so an
interpolation of data past the largest known bearings is needed. If a suitable bearing is not found within the database,
the model rounds up the bore diameter to the nearest 20 mm, which is a standard step size between commercial
bearing diameters, and determines facewidths and masses from an interpolation of the known data.
Figures 11-16 show the curve-fits for facewidth and mass data if the bearing sizing exceeds that of known data. Note
that this approach does not take into account load rating for fatigue, but assumes an average size from the observed
trend of bearing size.
Figure 11. CARB mass and facewidth interpolation
29
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40. Figure 12. SRB mass and facewidth interpolation
Figure 13. TRB1 mass and facewidth interpolation
30
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41. Figure 14. CRB mass and facewidth interpolation
Figure 15. TRB2 mass and facewidth interpolation
31
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42. Figure 16. RB mass and facewidth interpolation
The bearing housing mass is calculated as 2.92× the bearing mass, then added to the bearings to get entire bearing
assembly mass. This is based on Sunderland model scaling relationships, but uses current industry data on contem-
porary bearing sizes.
Main bearing CM location(s) are defined by the distance between main bearings, and their relation to gearbox loca-
tion in the following way:




CMx
mb1 = CMx
gbx −
Lgbx
2 − Lmb + FWmb2
2 cos(γ)
CMy
mb1 = CMy
gbx
CMz
mb1 = CMz
gbx + Lmb + FWmb2
2 sin(γ)



 (2.98)
If a downwind main bearing exists, its CM is calculated in a similar way:



CMx
mb2 = CMx
gbx −
Lgbx
2 − FWmb2
2 cos(γ)
CMy
mb2 = CMy
gbx
CMz
mb2 = CMz
gbx + FWmb2
2 sin(γ)


 (2.99)
2.2.4 Gearbox
Gearboxes are expensive components in wind turbine drivetrains. Gearbox weight estimates are important for calcu-
lating overall drivetrain capital, operation, and maintenance costs. A gearbox sizing model was previously developed
by the University of Sunderland in 1993 detailed in (Harrison and Jenkins, 1993a). This model was developed based
on the industrial data of wind turbines with rotor torque less than 1MegaNewon−meter (power rating 2MW). Fig-
ure 17 shows the comparison between the actual and calculated gearbox weight using the Sunderland model. While
32
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43. the model matches well for small size machines, significant deviation of the model output from the actual gearbox
weights are present for larger machines. Over 100% difference is observed between the model and manufacture data
for a 5MW wind turbine.
0
10000
20000
30000
40000
50000
60000
70000
0.5 1.25 2 2.75 3.5 4.25 5
Mass[kg]
Rated Power [MW]
Gearbox Weights
Model Output - Sunderland NREL 5 MW Ref GRC drivetrain GE 1.5sleFigure 17. Gearbox weight vs. rated input torque computed
by the Sunderland model (Harrison and Jenkins, 1993a).
In this study, a new design code is developed that uses turbine torque and overall speed ratio as input parameters.
This model gives gear/bearing weight and housing weight and stage ratios per stage as well as overall gearbox
weight. The size of the gearbox is determined for different gearbox configurations for minimizing the gearbox
weight. Figure 18 shows major steps for the developed sizing model.
The gearbox model includes an internal design optimizer to reduce its weight. It selects the best combination of
speed ratios for each gear stage to achieve the lowest weight, given the user specified overall speed ratio of the entire
gearbox. This sizing model is also suitable for a sensitivity study of gearbox weight to various design parameters.
The parameters of interest include the number of stages, number of planets in planetary gears, gearbox configuration,
and overall speed ratio.
Input torque drives wind turbine gearbox design. Influences of non-torque loads caused by rotor overhung weight
and aerodynamic forces on gearbox weight are considered in this work. For three-point suspension gearboxes, a
factor of 1.25 is multiplied to the gearbox weight in order to take into account the nontorque loads applied on the
gearbox. This factor is selected based on GRC measured load sharing factor in the upwind planetary gear section.
For other drivetrain configurations, nontorque loads are small and therefore not considered in the model. The design
criteria is the surface-durability recommended in ISO/AGMA gearbox design standards (AGMA, 2000, 2010; ISO,
1996a,b, 2012). The gearbox rating (bending and pitting resistance) analysis is not the focus of this approach so that
the resulting changes to the gearbox that would stem from these design drivers are not included.
This model focuses on the design of three-stage gearboxes with common configurations: one planetary gear and two
parallel stages, and two planetary gears and one parallel stage.
2.2.4.1 Single external gear
The relationship between the overall gear dimensions, the speed ratio Us, and power P for external gears is discussed
33
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44. Initial Dimension Selection of Gears (Diameter, Aspect Ratio)
Bearing Selection
Calculate Gearbox Stress Tolerance and Load Capacity
Satisfy Current Standards?
Yes
No
Gearbox Size/Weight Determined
Input torque, overall speed ratio, number of stages,
gearbox configuration, main shaft configuration
Stage Speed Ratio Iteration Program
Optimize Number of Planets in
Planetary Gears for Minimizing Weight
Estimate Stage Weight
Using Application Factor
Speed Ratio
Current
Approach
Advanced
Approach
Detailed Gear
Bearing
Information
Figure 18. Flow chart of the gearbox sizing tool.
in (Dudley, 1984):
C2
F =
31,500P(Us +1)3
KnpUs
(2.100)
where C = 0.5dp(Us + 1) is the center distance. dp and F are the gear diameter and facewidth. The transmitted
power P =
Tnp
63,000 is linearly correlated to the gearbox torque T and speed np. K factor is an index for measuring
the intensity of tooth loads (Dudley, 1984). There are different ways to calculate K factor: 1). it can be estimated
from the empirical table in (Dudley, 1984)(2.45); 2). it can be calculated by the formula below when the gearbox
component dimensions are designed:
K =
Wt
FWd
Us +1
Us
(2.101)
where Wt =
2Qp
dp
is the tangential driving force. Qp is the input torque to the pinion.
In the study, the first approach is used.
Rewriting Eq. 2.100, results in:
Fd2
p =
2Qp
K
Us +1
Us
(2.102)
The gearbox stage weight is estimated by WGB = KAGFd2
p, where KAG is the application factors for weight estima-
tions (Willis, 1963). The final form is:
WGB = KAG
2Qp
K
Us +1
Us
(2.103)
34
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45. 2.2.4.2 An external gear pair
The driving volume equals FWd2
p. The driven gear volume is FWd2
pU2
s . Therefore, the total size of the gear pair
equals:
∑FWd2
= FWd2
p +FWd2
g = FWd2
p +FWd2
pU2
s (2.104)
The total weight of the gear pair equals:
WGBPN = KAG
2Qp
K (Us+1
Us
)+KAG
2Qp
K (Us+1
Us
)U2
s
= KAG
2Qp
K (1+ 1
Us
+Us +U2
s )
(2.105)
2.2.4.3 Planetary gear stage
The volume of a planetary gear consists of the sun, ring, and B planet gears. The sun gear volume is:
FWd2
s =
2Qs
BK
(
USN +1
USN
) (2.106)
where uSN = 0.5Us −1 is the speed ratio between the sun and planet. Qs is the input torque to the sun gear.
The volume of a planet is :
FWd2
p = FWd2
s U2
SN =
2Qs
BK
(
USN +1
USN
)U2
SN (2.107)
The volume of the ring gear depends on both its diameter and thickness. AGMA 6123 (AGMA, 2006) defines the
ring thickness no less than 3 times module. The ring gear volume is approximated empirically without designing
individual gear dimensions. The ring volume considers the weight of the housing and carrier.
Vrg = KrFWd2
s (
drg
ds
)2
= Kr
2Qs
BK
(
USN +1
USN
)(
drg
ds
)2
(2.108)
where Kr = 0.4 is the scaling factor, selected from (Willis, 1963).
Therefore, the overall planetary gear volume is:
FWd2
s +BFWd2
p +Vr = 2Qs
BK (USN+1
USN
)+B2Qs
BK (USN+1
USN
)U2
SN +Kr
2Qs
BK (USN+1
USN
)(drg
ds
)2
= 2Qs
BK [ 1
B + 1
BUSN
+USN +U2
SN +Kr
(Us−1)2
B +Kr
(Us−1)2
BUs
]
(2.109)
The planetary gear weight equals:
WGSEN = KAG
2Qs
K
[
1
B
+
1
BUSN
+USN +U2
SN +Kr
(Us −1)2
B
+Kr
(Us −1)2
BUSN
] (2.110)
2.2.4.4 Gearbox Weight
The gearbox weight is the summation of individual stage weight, which depends on the input torque Q1, Q1, Q3
and speed ratio U1, U2, U3. For instance, the 750kW gearbox (NREL GRC gearbox (Link et al., 2011)) utilizes a
planetary-parallel-parallel configuration. The total weight of this gearbox is
WGB = W1
GSEN +W2
GBPN +W3
GBPN
= KAG
2Q1
K [ 1
B + 1
BUSN
+USN +U2
SN +Kr
(U1−1)2
B +Kr
(U1−1)2
BU1
]
+KAG
2Q2
K (1+ 1
U2
+U2 +U2
2 )+KAG
2Q3
K (1+ 1
U3
+U3 +U2
3 )
(2.111)
where Q0 is main shaft input torque. Q0 = Q1U1. Q1 = Q2U2. Q2 = Q3U3.
35
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46. The final gearbox design takes into account the gear dynamic effects on loads, overload, unequal load sharing for
planetary gears, and the main shaft configuration. Therefore, the gearbox weight considers the overload factor K0
(Avallone et al., 2006), dynamic factor Kv (AGMA, 2010), load sharing factor among planets Kγ p (ISO, 2012),
load sharing factors between rows Kγb (based on GRC test data which has results in a new design parameter being
proposed to AGMA standard committee), and a new factor that captures the effects of main shaft configurations on
gearbox loads KSH. In the model, KSH = 1 is used as default.
W0
GB = K0KvKγ pKγbKSHWGB (2.112)
2.2.4.5 Determination of gearbox speed ratio per stage
This method selects the optimal speed ratios of individual gear stages for minimizing gearbox weight.
Gearboxes with two parallel stages
The total volume of the gearbox is proportional to:
∑Fd2 = FW1d2
p1 +FW1d2
g1 +FW2d2
p2 +FW2d2
g2
= KAG
2Q1
K (1+ 1
U1
+U1 +U2
1 )+KAG
2Q2
K (1+ 1
U2
+U2 +U2
2 )
= KAG
2Q1
K (1+ 1
U1
+U1 +U2
1 )+KAG
2Q1U1
K (1+ U1
M0
+ M0
U1
+ M0
U1
2
)
(2.113)
For minimum volume, set the derivative to zero. That is:
d(∑FWd2)
dU1
= −
1
U2
1
+2+2U1 +2
U1
M0
−
M2
0
U2
1
= 0 (2.114)
An iteration program is needed to find the roots of Eq. 2.114:
Gearboxes with three stages
There are two primary configurations for three-stage gearboxes included in the current version of the model: parallel-
parallel-parallel and planetary-parallel-parallel.
Planetary-Parallel-Parallel Configuration
The total volume of the gearbox equals:
V = 2Q0
K
1
U1
1
B1
+ 1
B1(
U1
2 −1)
+(U1
2 −1)+(U1
2 −1)2 +Kr
(U1−1)2
B1
+Kr
(U1−1)2
B1(U
2 −1)
+2Q0
K
1
U1U2
1+ 1
U2
+U2 +U2
2 + 2Q0
K
1
U1U2U3
1+ 1
U3
+U3 +U2
3
(2.115)
Let M1 = U1U2, M2 = U2U3, and M0 = U1U2U3 and rewrite Eq. 2.115 as:
V(M1,U1) = 2Q0
K
1
U1
1
B1
+ 1
B1(
U1
2 −1)
+(U1
2 −1)+(U1
2 −1)2 +Kr
(U1−1)2
B1
+Kr
(U1−1)2
B1(U
2 −1)
+2Q0
K
1
M1
1+ U1
M1
+ M1
U1
+(M1
U1
)2 + 2Q0
K
1
M0
1+ M1
M0
+ M0
M1
+(M0
M1
)2
(2.116)
V(M2,U2) = 2Q0
K
M2
M0
1
B1
+ 1
B1(
M0
2M2
−1)
+( M0
2M2
−1)+( M0
2M2
−1)2 +Kr
(
M0
M2
−1)2
B1
+Kr
(
M0
M2
−1)2
B1(
M0
2M2
−1)
+2Q0
K
M2
M0U2
1+ 1
U2
+U2 +U2
2 + 2Q0
K
1
M0
1+ U2
M2
+ M2
U2
+(M2
U2
)2
(2.117)
The gearbox volume reaches the minimum when dV(M1,U1)
dU1
= 0 and dV(M2,U2)
dU2
= 0.
36
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47. dV(M1,U1)
dU1
= 1
M2
1
+ (Np+4Kr)
4B U3
1 (U
2 −1)2 − (Kr+B+1)
B U1(U1
2 −1)2
−2M1(U1
2 −1)2 +
2Kr(U1−1)U2
1 (U
2 −1)
B − (Kr(U1−1)2+1)U1(U1−1)
B
(2.118)
dV(M2,U2)
dU2
=
(M2
2 +1)
M0M2U3
2
− 2M2
M0U2
− 2M2
M0(M2+1) (2.119)
Planetary-Planetary-Parallel Configuration
The total volume of the gearbox equals:
V = 2Q0
K
1
U1
1
B1
+ 1
B1(
U1
2 −1)
+(U1
2 −1)+(U1
2 −1)2 +Kr1
(U1−1)2
B1
+Kr1
(U1−1)2
B1(
U1
2 −1)
+2Q0
K
1
U1U2
1
B2
+ 1
B2(
U2
2 −1)
+(U2
2 −1)+(U2
2 −1)2 +Kr2
(U2−1)2
B2
+Kr2
(U2−1)2
B2(
U2
2 −1)
+2Q0
K
1
U1U2U3
1+ 1
U3
+U3 +U2
3
(2.120)
Let M1 = U1U2, M2 = U2U3, and M0 = U1U2U3 and rewrite Eq. 2.120 as:
V(M1,U1) = 2Q0
K
1
U1
1
B1
+ 1
B1(
U1
2 −1)
+(U1
2 −1)+(U1
2 −1)2 +Kr1
(U1−1)2
B1
+Kr1
(U1−1)2
B1(
U1
2 −1)
+2Q0
K
1
M1
1
B2
+ 1
B2(
M1
2U1
−1)
+( M1
2U1
−1)+( M1
2U1
−1)2 +Kr2
(
M1
U1
−1)2
B2
+Kr2
(
M1
U1
−1)2
B2(
M1
2U1
−1)
+2Q0
K
1
M0
1+ M1
M0
+ M0
M1
+(M0
M1
)2
(2.121)
V(M2,U2) = 2Q0
K
M2
M0
1
B1
+ 1
B1(
M0
2M2
−1)
+( M0
2M2
−1)+( M0
2M2
−1)2 +Kr1
(
M0
M2
−1)2
B1
+Kr1
(
M0
M2
−1)2
B1(
M0
2M2
−1)
+2Q0
K
M2
M0U2
1
B2
+ 1
B2(
U2
2 −1)
+(U2
2 −1)+(U2
2 −1)2 +Kr2
(U2−1)2
B2
+Kr2
(U2−1)2
B2(
U2
2 −1)
+2Q0
K
1
M0
1+ U2
M2
+ M2
U2
+(M2
U2
)2
(2.122)
The derivatives are then:
dV(M1,U1)
dU1
= −(U1−1)(1+Kr(U1−1)2)
B1U2
1 (
U1
2 −1)2
+ 2Kr(U1−1)
B1U1(
U1
2 −1)
− (1+Kr)
B1U2
1
+ 1
4 + Kr
B1
+ 1
2B2(
M1
2 −U1)2
− 2Kr(M1−U1)
B2U2
1 (
M1
2 −U1)
+ Kr(M1−U1)2
2B2U2
1 (
M1
2 −U1)2
+(1
2 + 2Kr
B2
)( 1
U2
1
− M1
U3
1
)
(2.123)
dV(M2,U2)
dU2
= −M2(U2−1)(1+KrM2(U2−1)2)
M0B2U2
2 (
U2
2 −1)2
+ 2KrM2(U2−1)
M0B2U2(
U2
2 −1)
+( M2
4M0
+ KrM
M0B2
+ 1
M2M0
)+
(−
M2
M0B2
− KrM
M0B2
−
M2
M0
)
U2
2
− 2M2
M0U3
2
(2.124)
dV(M1,U1)
dU1
, dV(M1,U1)
dU1
can also be calculated numerically by using finite differencing. Central difference formulation
dV(U+dU)−dV(U−dU)
2dU (second order accuracy) is used here to check the accuracy of the analytical formulations in Eqs
2.118, 2.119, 2.123, and 2.124. dU = 2 × 10−5 is selected through a sensitivity test to find the optimal step size to
avoid machine round off.
Optimizer Solving Approach Global Newton iteration is used to find the roots of Eq. 2.123 and Eq. 2.124: U1 and
U2 for a given M1 = U1U2. The iteration procedure is based on the (Errichello). The general procedure is described
as below:
37
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Energy Laboratory at www.nrel.gov/publications

48. 1. Select an initial value for M1 = U1U2
2. Set the derivative of the gearbox volume to zero. dVGB
dU1
= 0
3. Solve for the root U1
4. U2 = M1/U1
5. U3 = M0/M1, where M0 is total gear ratio
6. M2 = U2U3
7. Solve for U2 from dVGB
dU2
= 0
8. Iterate until U2 from step 7 equals U2 from step 4.
Convergence tolerance used in the iteration is 5×10−3.
2.2.4.6 Gearbox Mass Properties Calculations
Having found the mass via the approach corresponding to the selected gearbox configuration as described above,
the next step is to calculate the mass properties for the gearbox. The Gearbox CM is found using the user-input
x-location (defaulted above tower top center), and the height of the gearbox, as shown below:



CMx
gbx = 0 or user input
CMy
gbx = 0
CMz
gbx =
Hgbx
2


 (2.125)
Where the gearbox dimensions still rely on scaling arguments based on NacelleSE, as follows:
Lgbx = 0.012×Dr
Hgbx = 0.015×Dr
(2.126)
2.2.5 Yaw System
DriveSE assumes the yaw system is composed of a friction plate yaw bearing at the nacelle tower interface and
several yaw motors. The friction plate bearing is treated as a steel annulus and is sized according to the tower top
diameter and rotor diameter. The motors are assumed to be a common Bonfiglioli design from the 700T series used
in the mid-2000’s on 2 MW size turbines. They are 690V electric motors with a hybrid planetary and worm gear
design resulting in a 1:1100 gear ratio and weighing 190 kg according to manufacturer specifications. The number of
motors is a found as a function of the rotor diameter if not specified by the user.
2.2.5.1 Yaw System Mass Properties Calculations
The friction plate surface width is assumed to be 1/10th the tower top diameter and the friction plate thickness
is assumed to be 1/1000th the rotor diameter. These ratios resulted in reasonable agreement with known turbine
specifications, however the variability in tower top diameter, mass, and friction or rolling element bearings results in
a range of observed masses for this component. The mass and geometrical properties are calculated according to the
following equation
Vyaw =
πD2
tower
10
Dr
1000
(2.127)
38
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Energy Laboratory at www.nrel.gov/publications