28th International Cong ress of Conditio n Monitoring
and Diagnostic Engineerin g
10th Regional Congress on Non Destructive a nd Structural Tes ting
Modelling and measurements of dynamic ring gear strains at 1st stage
of main gearboxes in wind turbines
Martin-Christopher Nolla *, Dominik Radnera, Dennis Wittera and Ralf Schelenza
aCenter for Wind Power Drives, RWTH Aachen University, Campus Boulevard 61, 52074 Aachen, Germany
Keywords: Wind Turbine, Planetary Gearbox, Surface Strain, Ring Gear, Modelling, Measurement
Along with the continuing increase in rated power of modern WT comes an increase in
rotor diameter and hence an increase in torque and a decrease in revolution speeds at
gearbox input. Furthermore gearboxes, which are employed to raise revolution speeds to
a grid compatible level, highly dynamic loads resulting from the stochastic nature of the
turbulent wind field acting on the rotor.
While main gearboxes are crucial components of WT they can also suffer from various
damages especially in bearings. To avoid long downtimes with high cost modern WT
gearboxes are equipped with CM systems, which rely on vibration measurements (0.1 Hz
– 20 kHz) and oil particle data(1, 2). Low revolution speeds with resulting low gear and
damage cycling frequencies and weak excitations when damages of roller bearings are
cycled lead to problems in damage detection: While low frequencies require long
recording periods (> 100 s), which are necessary to average the frequency-domain data
over several revolutions for signal quality improvements, highly dynamic loads require
short recording periods (< 10 s) to minimize the influence of load conditions on damage
detection. As a result other measurement parameters, such as acoustic emissions are
investigated. In spite of promising results, this technology is only applied in small scale
due to expensive measurement hardware for high sampling frequencies up to several
Another measurement parameter, which is investigated in this work, is the surface strain
on main gearbox components. It was shown by NASA and the University of Maryland
that surface strain on the ring gear of planetary gearboxes in helicopters measured with
Fiber-Bragg-Sensors can be used for damage detection (planetary, sun and ring gear as
Corresponding author. Tel.: +49-241-80-95925; fax: +49-241-80-92885; e-mail: firstname.lastname@example.org
Condition monitoring (CM) of main gearboxes in wind turbines (WT) mainly relies on mechanical vibration(1, 2). However,
low revolution speeds, low damage cycling frequencies, long recording periods and highly dynamic wind loads lead to difficult
damage detections in first planetary stages using vibration. Signals of ring gear strains present another parameter for the CM
of planetary gearboxes(3).
In a recent publication signals of piezoelectric strain sensors and strain gauges mounted on the outer ring gear surface of the
first stage of a 1 MW WT-gearbox were analyzed regarding signal quality and inferable information. Both sensors showed
very good signal to noise ratios and high reproducibilities(4).
To further investigate these signals, dynamic ring gear surface strains of WT-gearboxes in undamaged condition are modelled,
measured and evaluated in this paper. First the complex strain condition on ring gears is investigated with analytical and
numerical models. In a second step strain measurements are conducted on a 3 MW gearbox, which is equipped with extensive
measurement hardware. Measured signals compare well to modelled results. The comparison of tooth root and ring gear strains
shows that the planetary load sharing, which is influenced by many relevant gearbox parameters, can be inferred from the
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well as planetary bearings)(3). It was shown that usage of the surface strain signal has
various advantages compared to vibration measurements:
• Due to a very high reproducibility the signal does not need to be averaged over
• the signal shows a very high signal-to-noise ratio and
• the low frequency signal component correlates with the driving torque.
In previous work it was shown that the advantageous characteristics of the signal could
also be observed with piezoelectric strain sensors as well as with strain gauges (SG)
mounted on the ring gear of the planetary stage of a 1 MW class WT-gearbox on a nacelle
test rig(4). However, no gearbox internal measurements could be carried out.
In the work on hand dynamic ring gear surface strains of the planetary stages of WT-
gearboxes in undamaged condition are modelled in detail with analytical and numerical
approaches as well as measured and evaluated on a 3 MW class gearbox with internal
measurement equipment in order to gain a deeper understanding of the acquired signals.
By identifying advantages of the signal and it’s related influence factors, this deeper
understanding can be seen as a first step towards future applications of strain
measurements in the CM of slowly spinning planetary gearboxes.
2.1 Analytical Approach
As a first step towards a further understanding of the strain condition on ring gears of
planetary gearboxes, an analytical model is developed. The ring gear is modeled as a non-
rotating freely deformable two-dimensional circular bending beam neglecting any further
connections to a supporting structure. The axial force resulting from helical gears is
neglected. At first the forces are considered to be transmitted at only one tooth at each
meshing. The tooth forces are applied at the maximum tooth height. In Figure 1 the
derivation of the analytical approach is shown for a planetary stage with four planets.
Figure 1: Derivation of an analytical model for the analysis of tangential surface strains on ring
The torque transmission results in two forces at each tooth meshing: A radial and a
tangential component. For the further calculations both components are considered in two
different load cases, which are superposed in the end . In a symmetric load case (left), the
radial forces cause a local expansion of the ring gear at the meshing point, which results
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in a polygonization of the ring gear. In the anti-symmetrical load case (right) the
tangential forces cause a torsion of the ring gear as well as a bending moment at the tooth
Symmetrical and anti-symmetrical load case lead to respective bearing reactions at the
lines of symmetry. The two-dimensional modelling results in two displacement functions:
The radial displacement v and the tangential displacement u. Considering the bearing
reactions at the lines of symmetry, boundary conditions as shown in figure are applied.
The resulting qualitative influence lines around one tooth meshing (between – π / p and
+ π / p, where p is the number of planets) of tangential force and bending moment for an
exemplary gearbox are shown in the following figures.
Figure 2: Normal forces (above) and bending moments (bottom) in ring gear (left: symm. load case
/ radial force, right: anti-symm. load case / tangential force)
A resulting tangential surface strain can be calculated by using the linear dependencies of
a straight beam, which is possible due to the only small curvature of the ring gear:
With the surface strain ε, Young’s-Modulus E, total bending moment Mb, section
modulus W, total tangential force T and ring gear cross section A.
All normal stresses due to tangential force and bending moments of both load cases are
superposed. A strain in time-domain as a function of the carrier rotation angle, which
would be acquired by a real sensor, as it measures the integral of surface strain over a
finite length, is calculated by employing the following formula.
with the sensor length ls and the radius of the ring gear’s neutral fiber R. Here the rotation
angle of the sensor-signal is inverted as calculated strains in front of the planet’s position
are measured before the planet passes the sensor position. Hence, these strains occur at a
lower rotation angle of the planet carrier.
The resulting total bending moment and the strain, which would be measured by a sensor
mounted to the ring gear surface are shown in Figure 3.
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Figure 3: Resulting total bending moment and strain measured by a sensor in time-domain
Until now all tooth forces are transferred at only one tooth meshing at phi = 0°. For the
right graph in Figure 3, the total tooth forces at one meshing are distributed over three
teeth according to the overlap ratio (here represented by a 25 %, 50 % and 25 % load
distribution acting on three adjacent teeth). Steps in the strain curve represent various
tooth meshes around a sensor location at phi = 0°, while the relations between the strain-
deltas at the different steps symbolize the overlap ratio.
2.2 Numerical Approach
For the analytical approach various influences are neglected: Axial tooth forces, the
stiffness of the supporting structure, which surrounds the ring gear, as well as a realistic
tooth contact. To model these effects a numerical approach is chosen. For this purpose a
generic gearbox model with a rated power of 6 MW is used. The gearbox consists of three
stages, a first planetary stage with four planets, a second planetary stage with three planets
and a spur gear stage. As FE-solver ABAQUS is used.
The FE-model consists of 936.043 nodes forming 641.316 elements (mostly linear
hexahedrons at the contacting tooth flanks as well as quadratic tetrahedrons at planetary
carriers). The torque arms are fixed in all degrees of freedom. A pure torque load is
applied at gearbox output (high speed shaft), while the rotation angle is set at gearbox
input (low speed shaft). The bearings are modelled with multi-point-constraints while the
coupling between inner and outer ring is realized with radial and axial spring elements
having non-linear stiffnesses. All teeth contacts are modeled with frictional (µ = 0.05)
ABAQUS-standard contacts. Parts which have no relative motion are connected using
tie-constraints. Tooth flank corrections are not implemented.
In the following figures the mises stresses are shown for a static bracing of the gearbox.
Local mises stresses at the tooth contacts can be made visible. By concentrating on the
ring gear, the complex deformations resulting from tangential, radial and axial tooth
forces can be made visible: around tooth meshes the ring gear is expanded, polygonalized
as well as twisted locally by the resulting bending moment.
Figure 4: Mises stresses in static restraint, deformed ring gears and sign of the tangential strain
component (blue = negative, red = positive strains)
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Tangential and axial surface strain on the ring gear are evaluated in static restraint using
a circumferential path in the middle of the ring gear width. To emulate a sensor signal the
angle axis is inverted (see chapter 2.1 Analytical Approach).
Figure 5: FE-simulated tangential and axial ring gear surface strain along a circumferential path
It can be seen that both strain functions are not symmetrical around the circumference.
This is due to the influence of the stiffness of the surrounding support structure, for
example the torque arms.
3. Experimental methods and Measurements
For this study measurements are carried out on a 3 MW gearbox deployed on a 4 MW
WT-system test rig at RWTH Aachen University. The test rig is described in detail in a
publication by Franzen et al.(6).
The test rig consists of a direct drive motor connected to a non-torque load application
system, which is capable of dynamically applying realistic wind loads to the specimen.
The WT is electrically connected to a grid simulator, which is capable of recuperating the
electrical power, generated by the WT-Generator to the motor.
The tested gearbox has two planetary stages (4 planets and 3 planets) at first and second
stages due to very high input torques. The ring gears are fixed to the supporting structure
while the input torque is applied to the planet carrier and the output torque is transmitted
to the sun gear. For all measurements SG are applied, which are connected to a HBM
QuantumX MX1615. Different load conditions are imposed by the system test rig.
The gearbox is equipped with extensive measurement technology. Internal loads are
measured at various tooth roots on the ring gear. For these measuring positions SG with
a length of 2.29 mm are employed. These gauge applications are mounted at four
positions in orbit. Each application consists of six gauges applied over the tooth width at
maximum strain position (60° tangent to the tooth root, see Figure 6).
For the external measurements SG with a length of 3 mm are used. At each location two
gauges (tangential and axial) are attached to the grinded gearbox surface in the middle of
the ring gear width by using cyanoacrylate. All measurements are taken on the first
planetary stage with four planets. During all measurements only pure torque is
transmitted, while the load application system is set to zero forces and bending moments
at the input flange of the gearbox in force controlled mode.
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Figure 6: External and gearbox internal SG positions
In this work, strain measurements on the outer ring gear surface are directly compared to
gearbox internal strain measurements at a tooth root. In the following figures exemplary
measured tangential and axial time-domain strain signals at the 12 o’clock outer ring gear
position are shown over one carrier revolution. A qualitative comparison of the
numerically simulated (see chapter 2.2. Numerical Approach) and measured signals
shows a very good consistency.
Figure 7: Time-domain signals of the surface strain in tangential (left) and axial (right) direction
In Figure 8 acquired internal ring gear strain signals in time-domain are shown for two
different SG in one tooth root. Due to the passing planet first a negative (compressive
stress resulting from previous mesh) and afterwards a positive strain peak occurs. The
difference in both strain signals is due to load distribution across the tooth width.
Figure 8: Time-domain signals of the tooth root strain for two different locations across the tooth
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In the following, the planetary load sharing factor (PLSF) Kv Kγ, which describes
deviations in force transmission between the planets, is used as an example for the
comparison of internal and external measurements.
The PLSF are calculated the same way for both signals:
Hence, for each revolution, the maximum positive strain peaks (representing tensile
stress) of all four planets passing the sensor position are divided by the mean value of all
four peaks. Only the 12’o clock measurement positions are evaluated here. Since there is
no change in turning direction it is clear that every n-th peak with n = 1 .. 4 corresponds
with the same passing planet. Furthermore one revolution always consists of four peaks.
In the following, PLSF are determined dynamically for each revolution during a torque
ramp from zero to about 100 % of nominal load and back to zero at a constant speed of
12 revolutions per minute. Figure 9 shows, how the planetary load sharing factor is
derived from strain data. The PLSF is calculated for every external and internal SG. All
internal SG over the tooth width are averaged for the calculation of the red curve. SG 4,
mounted in the middle of the tooth width, is also compared to the external signal.
Figure 9: Description of measurement procedure and derivation of the planetary load sharing
factor from strain measurements (vertical lines dividing carrier revolutions)
Figure 10: Derived PLSF (averaged over 10 revolutions = 50 s)
While it can be seen that the PLSF decreases with increasing load it can also be shown
that the internal averaged and external measurements have only very small deviations at
nominal load. At lower loads higher deviations arise. The PLSF generated from SG 4
shows lower values. Hence, it can be seen that with only one external sensor position an
averaged value can be generated. This is due to the force flux between inner tooth flank
and outer ring gear surface. To make these measurements more comparable, further
measurements are carried out at three different load levels (50 %, 75 %, 100 % of nominal
torque) for 100 s each, which equals 20 revolutions at a constant speed of 12 revolutions
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per minute, with following calculated deviations (the internal tooth-averaged value at
100% nominal load is set to 0%):
Table 1: Calculated deviations in load sharing factors for different load levels
Kv Kγ external
Kv Kγ internal measurement
averaged over tooth
Kv Kγ derived from
SG position 4
+ 0,68 %
+ 1,26 %
- 1,06 %
- 0,19 %
+ 0,58 %
- 1,45 %
- 1,84 %
It was shown that acquired measurement signals compare well to the results of the
numerical FE-approach and less to the results of the analytical approach. The main reason
for that is the assumption of a freely deformable ring gear underlying the analytical model.
This assumption leads to great bending deflections at the bearing reactions for the anti-
symmetrical load case resulting from tangential forces. These deflections are reduced
significantly in the real as well as the FE-system by the clamping of the ring gear, which
is caused by the supporting gearbox structure i.e. the casing. Hence, it can be stated, that
the supporting gearbox structure has significant influence on the strain graph.
Further investigations show, that by applying external strain measurements, the PLSF, an
important figure symbolizing internal gearbox parameters can be inferred. This is shown
by comparison with gearbox internal strain measurements. Planetary load sharing is
influenced by various factors, for example: Load, bearing stiffnesses of planetary and
carrier bearings, tooth flank corrections, misalignments, manufacturing quality (lower
manufacturing tolerances lead to a better planetary load sharing factor) and many more.
In this paper, by deriving an analytical and a numerical FE-approach the surface strain
condition on planetary ring gears was studied. It could be shown that the loads transmitted
by single planets as well as the geometry of ring gear itself and the structure surrounding
it have great influence on the ring gear surface strain.
Measurements are taken on a 3 MW class WT main-gearbox equipped with extensive
measurement technology and deployed on a WT nacelle test rig. It was shown that the
PLSF could be inferred by analyzing surface strain peaks in time-domain. The determined
factors were compared to gearbox internal measurements and a very good accordance
While the PLSF is susceptible to changes in many gearbox internal parameters such as
bearing stiffnesses, misalignments and tooth flank corrections further investigations
should be undertaken to analyze a potential applicability for the condition monitoring of
slowly spinning planetary gearboxes. Furthermore by reducing various influences to one
single factor a significant data reduction is possible.
This work is supported by the European Regional Development Fund (ERDF) and the
Ziel2.NRW Program of the Ministry for Innovation, Science and Research of the state
North Rhine-Westphalia, Germany.
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