LOAD DATA ANALYSIS FOR WIND TURBINE GEARBOXES Bernd ...

LOAD DATA ANALYSIS FOR WIND TURBINE GEARBOXES
Bernd Niederstucke, Andreas Anders, Peter Dalhoff, Rainer Grzybowski
Germanischer Lloyd WindEnergie GmbH
Johannisbollwerk 6-8, 20459 Hamburg
ABSTRACT: Gearboxes for wind turbines have to ensure highest reliability over a period of approximately 20 years,
withstanding high dynamic loads. At the same time lightweight design and cost minimization are required.
These demands can only be met by a thought-out design, high-quality materials, high production quality and maintenance.
In order to design a reliable and lightweight gearbox it is necessary to describe the loads acting on the gearbox as
exact as possible. For fatigue this can be done by using the load-duration-distribution (LDD) of the torque at the
input shaft.
In the following the fatigue resistance of a gearbox will be analysed using the torque-LDD. Methods of calculating the
life time of gearings and bearings with a given LDD will be described. The influence of the mean wind speed on the life
time of teeth and bearings will be pointed out.
1 GEARBOXES IN WIND-TURBINES
A common gearbox for wind turbines is shown in figure
1. A planetary stage with 3-4 planets is followed by two
spur gear stages. The output shaft has an offset from the
input shaft in order to lay wiring and piping through the
hollow input shaft.
In the 1 – 2 MW class of wind turbines the gear ratio
goes from about 80 to 100. In order to reduce noiseemission,
all stages can be designed with helical gearing.
The shafts are carried radial with spherical or cylindrical
roller bearings, axial with four-point-contact ball
bearings or tapered roller bearings.
Torque support
Input shaft
Torque
Output shaft
3 rd stage (helical spur gear)
1 st stage (planetary) 2 nd stage (helical spur gear)
Fig. 1: Gearbox for wind turbines (by METSO DRIVES
OY)
2 LOAD DURATION DISTRIBUTION
2.1 Determination of a LDD
The fatigue analysis for gearing and bearings is partly
different to that of other components, e.g. a rotor hub or a
rotor shaft. Especially derivation of the load spectra
differs. This is due to the fact that a stress cycle for a
tooth emerges with every tooth flank contact (e.g. with
every revolution for a spur gear stage), whereas a stress
cycle for e.g. a rotor hub occurs in correspondence with
fluctuations (cycles) of its external loads.
For the determination of load spectra for the gear box a
load duration distribution (LDD) of the input torque is
needed. The main advantage of the LDD is that it can
easily be transferred into local stress spectra. By using the
rotational speed of pinion or wheel the duration of torque
levels can be transferred into a number of load cycles.
The LDD are derived from the load time series of the
torque of the main shaft (see fig. 2) by level distribution
counting [6]: The load time series is divided into
equidistant time intervals. At each time interval the level
of the load time series is read and counted into the
respective bin.
Fig. 2: Level distribution counting of load time series
The number of countings per bin (1 to 8 in fig. 2) is a
measure for the load duration in a bin. The sampling
frequency should be at least ten times the highest
frequency of the load time signal in order to obtain an
accurate result. Information on the number of load cycles
is lost.
The load levels (bins) may be shown above their
duration or above their accumulated duration (see fig. 3).
The accumulated duration distribution is used for the
further calculations.
Fig. 3: LDD as a result of level distribution counting

2.2 Description of a LDD
Fig. 4 shows typical LDDs for various yearly average
wind speeds, i. e. 8,5m/s, 10 m/s and 15 m/s with 20%
constant turbulence according to [1] for a 80m diameter
rotor.
T/Nm
2,5E+06
2,0E+06
1,5E+06
1,0E+06
5,0E+05
0,0E+00
8,5m/s 15m/s
10m/s
1,0E+00 1,0E+02 1,0E+04 1,0E+06
accumulated duration/hours
Fig. 4: LDDs for different yearly average wind speeds
The vertical axis shows the input torque in linear scale,
the horizontal axis the accumulated duration of the
torque levels in logarithmic scale. The duration sums up
to about 170000 hours, which equals with a life-time of
20 years.
The first two LDDs (8,5 m/s and 10 m/s) are those of the
GL-type-classes II and I (see [1]), the third one (15 m/s)
is a theoretical one in order to bring the examined
gearbox to its limits. The LDDs are derived by an aeroelastic
simulation of the dynamic wind turbine
behaviour. Since the wind turbine is regularly stopped by
application of aerodynamic braking (pitch) the influence
of these stop sequences can be neglected.
Because of the logarithmic display the differences seem
indistinct. For example: The accumulated duration of
torque levels above the rated torque of 1200 kNm for the
examined gearbox is 1,5 times higher for 10m/s and 2,5
times higher for 15m/s than for 8,5m/s. This leads to
significantly higher fatigue loads on the gearing and
bearings.
3 CALCULATION PROCEDURE
3.1 Calculation of gearing durability
The given LDD is transformed into the local stress
spectra σ F/N and σ H/N for tooth root and flank (see
fig.5). σ F is the root stress, σ H the hertzian contact stress
at the tooth flanks, t the accumulated duration and N the
number of load cycles.
To establish the stress spectra three transfer-functions are
used:
σ F = f(T), σ H = f(T) and N = f(t)
The transfer-function between the torque T and the root
stress σF according to ISO 6336, method B is as follows:
2 ⋅ Ti
σF
= ⋅ YF ⋅ YS ⋅ Yβ ⋅ KA ⋅ KV ⋅ KFα ⋅ KFβ ⋅ SF ⋅ SRFF d ⋅ b ⋅ m
1
n
with
d 1 = standard pitch diameter of pinion,
T i = pinion torque
b = face width
m n = normal module
S F = required safety factor for root stress
SRF F = stress reserve factor for root stress.
The pinion torque is the input shaft torque divided by the
gear ratio up to the stage.
The Y- and K-factors take into account the form and
stiffness of the teeth and their dynamic behavior . They
are according to ISO 6336.
In the present examination the factors are calculated with
the gear program ST-plus [5].
The application factor K A is set to 1 as the dynamic
loading of the gearing is embodied in the load spectrum
resembled by the LDD (see [1], chapter 6). The safety
factor is 1,5 according to [1].
N
N
Fig. 5: Transformation of LDD into local stress spectra
Furthermore a S/N-curve is needed with which the stress
spectra can be compared. The S/N-curve depends on the
material, heat treatment, surface roughness and the size
of the gearing. The examined gearbox is equipped with
gears of case hardening steel for which S/N-curves with
the following characteristics were applied (following
[4]):
Slope in the limited life region (10 3 to 3 ⋅ 10 6 load
cycles): 8,7, slope in the long life region (3 ⋅ 10 6 to 10 10
load cycles): 50. To simplify the calculation, the slope
was extended beyond 10 10 load cycles.
After transformation of the LDD into a σ F/N-spectra and
having determined the S/N-curve, a damage
accumulation according to Palmgren-Miner-rule is
carried out. This leads to a damage D. If the damage is
greater 1 the gearing is liable to fail within the claimed
life-span, if it is lesser 1 the gearing has reserves against
fatigue failure. The stress reserve factor SRF F is varied
until the damage equals 1 so that the gearing lasts until
the end of the claimed life-span.
The determined stress reserve factor together with the
safety factor S F shows the margin of the chosen gearing
against fatigue fracture at the tooth root.
The transfer-function between torque T and tooth flank
stress σ H is:
σ
H
with
o F
=
2⋅
T ⋅(
u + 1)
⋅
d ⋅b
⋅u
i
2
1
⋅Z
⋅S
⋅SRF
β
H
H
T
K
A
⋅K
t
S/N-curve
V
⋅K
o H
Hβ
⋅K
Hα
⋅Z
H
⋅ Z ⋅Z
⋅ Z
E
B
ε

u = gear ratio = z2/z1, positive for external gears,
negative for internal gears
SH = required safety factor for contact stress
SRFH = stress reserve factor for contact stress
Ti, d1, b same as with root stress.
The Z- and K-factors are according to ISO 6336 and take
into account the geometry, elasticity and dynamic
behavior of the gearing.
As with the root stress the application factor KA is set to
1 and the safety factor SH to 1,2 according to [1].
The S/N-curve for the contact stress is determined by the
material, heat treatment, lubrication, surface roughness
and circumferential velocity.
As with the S/N-curve for root stress it was determined
following [4] with the characteristics:
Slope in the limited life region (10 5 to 5 ⋅ 10 7 load
cycles): 13,22
Slope in the long life region (5 ⋅ 10 7 to 10 10 load cycles):
32,6. To simplify the calculation, the slope was extended
beyond 10 10 load cycles.
After calculation of the accumulated damage according
to Palmgren-Miner the stress reserve factor SRF is varied
until the damage equals 1 thus giving the margin of the
examined gearing against pitting.
The number of load cycles follows the equation
N = t ⋅ n ⋅ p ⋅60
with
n = rounds per minute
p = number of meshing per round
t = duration of input-torque-levels in hours.
3.2 Calculation of bearing durability
The given LDD is also used to determine the life-time of
the bearings. The given input-torque-LDD is transferred
into LDDs for axial load on bearing F A and radial load
F R. The transfer–functions TF between torque of input
shaft T and loads F A or F R depend on the geometric
arrangement of bearings and meshing to each other, i. e.
pitch diameter and distance of bearings. The forces are
proportional to the input torque:
F A = T ⋅ TF A and F R = T ⋅ TF R
With the F/t distribution the mean axial and radial loads
FA and F R are:
p
p
p ∑ FAi
⋅ t i
p ∑ FRi
⋅ ti
FA
= and FR
=
t
t
with FAi and FRi = axial and radial load level during ti, t = accumulated time,
p = 3 for ball bearings and p = 10/3 for roller bearings.
With the mean loads and the axial and radial factors X
and Y depending on the bearing type the equivalent
dynamic load P acting on the bearing is:
P = X ⋅ F + Y ⋅ F
R
A
The modified life span of the bearings L10ah in hours
follows with
p
6
⎛ C ⎞ 10
L10ah = ⎜ ⎟ ⋅ ⋅ a1
⋅ a 23
⎝ P ⎠ 60 ⋅ n
with C = basic load rating,
p = 3 for ball bearings and p = 10/3 for roller bearings,
n = rounds per minute,
a1 = coefficient for survival probability, here set to 1 for
a probability of 90 %
a23 = coefficient for material and operating conditions
(see [7]).
4 EXAMPLE
The calculation procedures described above are applied
on a newly designed gearbox with the following
characteristics:
ratio ≈ 100
rated torque ≈ 1200 kNm
rated rounds per minute, slow speed shaft = 18
material of all gears: case hardening steel
The LDDs are the ones shown in fig. 4.
4.1 Gearing
The results for the gearing can be seen in fig. 6 and 7.
SRF F
SRF H
1,6
1,4
1,2
1
0,8
1,3
1,2
1,1
1
0,9
8 10 12 14
Annual average wind speed (m/s)
Planet W heel
Sun Gear
W heel Interm ed. Stage
Pinion Interm ed. Stage
Fig. 6: Stress reserve factor at tooth root for different
gears and wind speeds
8 10 12 14
Annual average wind speed (m/s)
Planet Wheel
Sun Gear
Wheel Intermed. Stage
Pinion Intermed. Stage
Fig. 7: Stress reserve factor at tooth flank for different
gears and wind speeds
The stress reserve factors for tooth root and tooth flank
are examined for the four gears with the highest load, i.
e. planet gear, sun gear and wheel and pinion of the
intermediate stage.
It can be seen that all gears have sufficient reserves for
the 8,5m/s-wind-class for which the gearbox is intended.
For the tooth root the pinion of the intermediate stage has
the least reserves followed by the planet gear.
Concerning the tooth flank wheel and pinion of the
intermediate stage show the least stress reserves.
The influence of the wind speed on the SRF is made
more clearly in table I: Taking the stress reserve at
8,5 m/s wind speed for 100% the reserve decreases to
91% at the root and to 93% at the flank for the wind

speed of 15 m/s. For being the gear with the least stress
reserve, the pinion of the intermediate stage was picked
out as an example Compared with the bearings, this
decrease is relatively small (see table II).
Annual average
wind speed
L 10ah (hours)
3 ,5 E + 0 5
2 ,5 E + 0 5
1 ,5 E + 0 5
5 ,0 E + 0 4
SRF F in % SRF H in %
8,5 m/s 100 100
10 m/s 95 97
15 m/s 91 93
Tab. I: SRF in % for pinion of intermediate stage
4.2 Bearings
All bearings of the gearbox show sufficient durability for
the intended wind class, i. e. more than 130000 hours
life-span (see fig. 8).
The ones with the least life-span are the roller bearings
of the planet wheel, the roller bearing directly at the
torque output and the four-point-axial bearings of the
intermediate and high-speed-shaft. All other bearings
have significantly higher life-times.
8 1 0 1 2 1 4
A n n u al av erag e w in d sp eed (m /s)
P lan et w h eel b earin g
In term ed iate sh aft, ax ial b earin g
O u tp u t sh aft, righ t rad ial b earin g
O u tp u t sh aft, axial b earin g
Fig. 8: Modified life span of bearings for different wind
speeds
With the mean axial and radial forces FA and F R being
proportional to the input torque T it is possible to
calculate an equivalent torque
_
T :
p
p ∑ Ti
⋅ t i
T =
t
with
Ti = torque level during ti and t = accumulated time and
p = 3 for ball bearings and p = 10/3 for roller bearings.
The results of the bearing calculation are shown in tab.
_
II: Column 2 shows the equivalent torque T in relation
to the rated torque. Column 3 shows the life span of the
axial bearing of the intermediate shaft in % and column 4
the relative load reserve. The relative load reserve is the
ratio of the values of the equivalent torque
_
T towards
each other with the one for 8,5 m/s wind speed set to
100%. This gives a comparable value to the stress
reserve factor of the bearings: It is apparent, that the
bearings react much stronger on an increase of the
average wind speed than the gears.
Annual
average wind
speed (m/s)
Equivalent
torque in %,
T rated =100%
Life span in
%
Load reserve
factor in %
8,5 68 100 100
10 74 75 92
15 84 50 81
Tab. II: Results of bearing calculation for the axial
bearing of the intermediate shaft.
5 SUMMARY
The aim of the present work was to show how the load
duration distribution of the input torque may be used for
the design of gearboxes of wind turbines. Therefore
calculation procedures for gearing and bearings were
described.
These procedures were then applied on a real gearbox.
It’s durability was examined by increasing the average
yearly wind speed i. e. the load set on the gearbox.
This examination should be verified by a long-time
survey of the gearbox in service.
6 ACKNOWLEDGEMENT
The work described in this paper is part of the research
project ELA “Enhanced Life Time Analysis of Wind
Turbine Structures” supported by the Federal German
Ministry for Economy and Technology BMWi.
7 REFERENCES
[1] Germanischer Lloyd, Regulations for the
Certification of Wind Energy Conversion Systems,
Germanischer Lloyd, Hamburg, 1999
[2] DIN 3990, Tragfähigkeitsberechnung von
Stirnrädern, Beuth Verlag, Berlin, 1987
[3] Niemann/Winter, Maschinenelemente, Bd. 2,
Springer Verlag, Berlin, 1985
[4] ISO 6336, Calculation of load capacity of spur and
helical gears, 1996
[5] NN., Spur gear program ST Plus,
Forschungsvereinigung Antriebstechnik, Frankfurt, 1997
[6] Westermann-Friedrich, A., Zenner, H., FVA-
Merkblatt 0/14, Zählverfahren zur Bildung von
Kollektiven aus Zeitfunktionen, Frankfurt, 1999
[7] NN., FAG rolling bearing cataloge, FAG,
Schweinfurt, 1999

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