3.1. Predicted Film Thickness and Structure Deformation
The first-stage bearing is utilized as an example to compare the film thickness under part-load
20% and nominal load
100% conditions. The dimensionless axial coordinates
0 and
1 in
Figure 5 represent the generator and rotor side of the bearing. As shown by the black point in
Figure 5, the minimum film thickness is smaller for the nominal load
100%, with a value of 8.9% of absolute radial clearance. In the area localized by the red line, the film thickness is below 21.1% of absolute radial clearance
. A comparison of the results in
Figure 5a,b indicates that this area is much larger for the nominal load
100%, as expected. The film thickness in this load zone remains on a level slightly above the minimum film thickness, as shown by the blue line in an angular span ranging from
214° to 338°.
The dimensionless total radial deformations on the sliding surface of the planet and pin for loads
20% and 100% are shown in
Figure 6a,b, respectively. The region with the larger deformation is closer to the generator side (GS,
0) for the relative load
20% in
Figure 6a due to the mesh force offset position, while an axially more homogenous deformation can be observed for the nominal load case in
Figure 6b. Additionally, both deformation fields feature significant local maximum values in the load zone between
240° and 310° as well as in the area 180 degrees offset from it, which exhibits a characteristic oval shape. This property can also be observed in
Figure 7 and is caused by the highly flexible planet deformed by the combination of fluid film and mesh forces. From the left to the right, the axial position
in
Figure 7 shifts from the generator side (GS) to the center (Mid) and the rotor side (RS) of the bearing individually. The radial bearing clearance is expressed in a dimensionless form relative to the absolute radial clearance
, where
is contour of pin, planet or resultant gap.
Figure 7a displays the magnitude of the oval shape for load
20% is decreasing from GS to RS, since local pressure loads concentrate on the GS. In contrast, the resultant contour for load
100% shows the oval shape in
Figure 7b over the entire bearing width due to the broader pressure distribution in the axial direction.
3.2. Validation of Pressure Distribution
Figure 8 presents the predicted rising trend of the dimensionless maximum hydrodynamic pressure in the planet gear bearings of the three different gear stages for increasing relative loads, where
means the maximum of all pressures. The maximum hydrodynamic pressure of third-stage bearing under relative load
40–110% is higher than that of the other two stages, although its nominal specific bearing load is lower.
Figure 9 shows the predicted pressure distribution in the lubricant gap for relative loads of
20%, 60%, and 100%, which is expressed in dimensionless form relative to the maximum pressure at
100% load in each stage. The locations of the three pressure sensors for each pin in the experiment are marked with red points to allow comparison between measured and simulated results. As already discussed for the predicted deformation and film thickness distribution, the pressure in the first-stage bearing for
20% load in
Figure 9a is mainly concentrated on the GS due to the additional moment about the
y-axis. With a rising load on the bearing, the range of high pressure becomes broader both in axial and circumferential directions. This tendency can be observed for all three stages. However, the circumferential growth of the load section is significantly lower for the third stage. In addition, the planet bearings in the first and second stages exhibit two peak pressure sections with local maxima in a circumferential direction at
100% load, while the third stage has a more homogeneous pressure distribution.
Table 3 includes the dimensionless axial offset positions of the mesh forces on tooth flanks relative to the bearing center for loads
20%, 60%, and 100%, i.e., the lever arms for the moment are represented by
and
. Positive and negative values of
and
represent the offset position in the GS and RS direction, respectively. The resulting pressure distribution closer to the generator side for load
20% results from the fact that the values of
and
in
Table 3 are both positive and comparably high, which leads to a moment on the planet about the
y-axis. Therefore, it results in a larger elastic deformation of the pin and planet on the GS side, which explains the more significant oval shape on GS in
Figure 7a. On the contrary, the absolute values of
and
for load
100% of the third-stage bearings are quite small, generating a very low additional moment about the
y-axis, so the pressure distribution in
Figure 9c is more homogeneous in the axial direction.
Experiments were performed to investigate the load-carrying capacity of bearings for loads between
20% and 100% with increments of 10% load. The letters A to G are used to identify different pins of the same bearing stage. If pressure sensors fail during the experiments, their values are omitted. A comparison of the experimental and simulation results over the entire load range is summarized in
Figure 10, where the pressures are expressed in dimensionless form relative to the maximum pressure of bearing with
and
for the respective stage. In addition to the measured and predicted sensor pressures, the maximum predicted pressure on the sliding surface is presented. In combination with
Figure 9a,b, the characteristics in
Figure 10 show that maximum predicted pressure is located close to a pressure sensor in the case of the second stage bearing while there is a bigger distance between the highest local pressure load and the sensor location in case of the first stage. The simulation only considers mechanically induced deformation and neglects thermal deformation based on the assumption of sufficiently low-temperature levels in this low-speed application. The predicted pressures at the sensor position for the first- and second-stage bearings in
Figure 10a,b show very good agreement with the measurement data of all pins at the three axial positions over the entire load range. The deviations are in the range of 0.1 to 5.5% for first- and second-stage, except slightly higher values for the generator side at
20% to 60% load of the first-stage bearing. Combined with the calculation of the load carrying capacity of each bearing in less than ten minutes, this confirms the reliability as well as the efficiency of the planet gear bearing code.
3.3. Extended Thermal Deformation Analysis for the Third Stage
More significant differences between measurement and prediction exist for the third-stage bearing.
Figure 11a includes measured and predicted pressures for the third-stage bearing. Deviations become higher with increasing load, particularly for the mid-sensor position.
Table 4 shows that the dimensionless experimental and predicted sliding surface temperatures relative to the oil supply temperature in °C at
100% load increase as the bearing rotational speed rises from a low speed in the first stage to a high speed in the third stage. The oil supply temperature of all bearings in the three stages is the same. The experimental and predicted temperatures of sliding surfaces in the first two stages bearing increase within a range of 18% and 42% of the oil supply temperature, respectively. The previous validation results show that the thermal expansion caused by the increasing temperature has no significant effect on the load-carrying capacity of the first- and second-stage bearings. However, the temperature of the third-stage bearing is much higher and reaches approximately 76% of the oil supply temperature. This temperature increase might incorporate non-negligible thermal deformation of the bearing components. To verify this conjecture, the temperature fields on sliding surfaces of the pin and planet simulated using the THD analysis are applied as external loads to calculate the thermal deformation of the third-stage bearing for loads
20–110%. Since the thermal deformation changes only slightly with modification of the pressure distribution, a one-time calculation of the thermal deformation at the beginning of the analysis is assumed to be sufficient. The shape of the radial thermal deformations is similar for all load cases.
Figure 12 presents exemplary results for
100% in dimensionless form for two views. This three-dimensional thermal deformation is approximately parabolic in the axial direction, i.e., the bearing clearance at the bearing edge becomes larger through the positive value at
0 and 1, while it decreases in the bearing center through the negative value. The total thermal deformation of the pin and planet is regarded as an additional offset crowning, which is used together with the original crowning of the pin to recalculate the pressure for the third-stage bearing in the planet gear bearing code. As shown in
Figure 11b, these pressures provide a significantly improved correspondence with the experimental pressure at the generator side (GS) and in the bearing center (Mid). Moreover, the results indicate that the position of maximum pressure at maximum torque load is slightly shifted away from the sensor location due to the consideration of thermal deformation as the deviation between the calculated sensor and calculated maximum pressure increases from
Figure 11a to
Figure 11b.
Results indicate that an enhancement of the modeling depth by thermal deformation is required for the third-stage bearings. Although the validation results in
Figure 11b still show some deviation on the rotor side (RS), there is a significant improvement in the agreement of measured and predicted results.