# Operating Behavior of Sliding Planet Gear Bearings for Wind Turbine Gearbox Applications—Part II: Impact of Structure Deformation

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Procedure for Consideration of Planet and Pin Deformation

#### 2.2. Investigated Gear Set: Planet Gear Bearing

#### 2.3. Investigated Gear Set: CAD Model and Material Properties

#### 2.4. FEM Approximation for Structure Analysis: Meshing

#### 2.5. FEM Approximation for Structure Analysis: Boundary Conditions and Mesh Forces

## 3. Results

#### 3.1. Mesh and Bearing Loads

_{sc}= 900 kN and a moment load of M

_{sc}= 27.6 kNm. This load case exists for a relative input torque of T

_{r}= 100%. According to the explanations in part I, bearing force and moment loads vary linearly with the relative input torque.

#### 3.2. Verification of the Calculation Procedure

#### 3.3. Deformation Behavior of Components

#### 3.4. Impact of Axial Profiling Model on Bearing Considering Elastic Deformation of Components

_{c}< 0.04 MPa, significant deviations between the results for flexible and rigid geometries exist for constant clearance in axial directions. As shown in Figure 18b, minimum film thickness increases under consideration of deformation. Consequently, the intensity of mixed friction is reduced, and maximum solid contact pressure decreases, according to Figure 19a. Assuming a boundary coefficient of friction of µ = 0.03, to derive solid contact shear stress from solid contact pressure, a notable impact of asperity contacts on maximum temperature and frictional power loss can be observed, starting with a relative load of T

_{r}= 50%, in Figure 19b.

#### 3.5. Modification of the Lubricant Gap by Wear Considering Elastic Deformation of Components

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Jang, J.Y.; Khonsari, M.M. On the characteristics of misaligned journal bearings. Lubricants
**2015**, 3, 27–53. [Google Scholar] [CrossRef] - Jang, J.Y.; Khonsari, M.M. Performance and characterization of dynamically-loaded engine bearings with provision for misalignment. Tribol. Int.
**2019**, 130, 387–399. [Google Scholar] [CrossRef] - Sun, J.; Gui, C.; Li, Z.; Li, Z. Influence of journal misalignment caused by shaft deformation under rotational load on performance of journal bearing. Proc. Inst. Mech. Eng. Part J J. Eng. Tribol.
**2005**, 219, 275–283. [Google Scholar] [CrossRef] - Sun, J.; Gui, C.; Li, Z. An Experimental Study of Journal Bearing Lubrication Effected by Journal Misalignment as a Result of Shaft Deformation Under Load. ASME J. Tribol.
**2005**, 127, 813–819. [Google Scholar] [CrossRef] - Sander, D.E.; Allmaier, H.; Priebsch, H.H.; Reich, F.M.; Witt, M.; Skiadas, A.; Knaus, O. Edge loading and running-in wear in dynamically loaded journal bearings. Tribol. Int.
**2015**, 92, 395–403. [Google Scholar] [CrossRef] - Lahmar, M.; Frihi, D.; Nicolas, D. The Effect of Misalignment on Performance Characteristics of Engine Main Crankshaft Bearings. Eur. J. Mech. A
**2002**, 21, 703–714. [Google Scholar] [CrossRef] - Zhang, X.; Yin, Z.; Dong, Q. An experimental study of axial misalignment effect on seizure load of journal bearings. Tribol. Int.
**2019**, 131, 476–487. [Google Scholar] [CrossRef] - Bouyer, J.; Fillon, M. Improvement of the THD performance of a misaligned plain journal bearing. J. Trib.
**2003**, 125, 334–342. [Google Scholar] [CrossRef] - Hagemann, T.; Ding, H.; Radtke, E.; Schwarze, H. Operating behavior of sliding planet gear bearings in turbine gearbox applications—Part I: Basic relations. Lubricants
**2021**, in press. [Google Scholar] - Desbordes, H.; Fillon, M.; Frene, J.; Chan Hew Wai, C. The Effects of Three-Dimensional Pad Deformations on Tilting-Pad Journal Bearings under Dynamic Loading. J. Tribol.
**1995**, 117, 379–384. [Google Scholar] [CrossRef] - Hopf, G. Experimentelle Untersuchungen an Großen Radialgleitlagern für Turbomaschinen. Ph.D. Thesis, Ruhr University Bochum, Bochum, Germany, 1989. [Google Scholar]
- Hagemann, T.; Kukla, S.; Schwarze, H. Measurement and prediction of the static operating conditions of a large turbine tilting-pad bearing under high circumferential speeds and heavy loads. In Proceedings of the ASME Turbo Expo 2013, San Antonio, TX, USA, 3–7 June 2013. [Google Scholar] [CrossRef]
- Lahmar, M.; Ellagoune, S.; Bou-Saïd, B. Elastohydrodynamic lubrication analysis of a compliant journal bearing considering static and dynamic deformations of the bearing liner. Tribol. Trans.
**2010**, 53, 349–368. [Google Scholar] [CrossRef] - Prölß, M. Berechnung Langsam Laufender und Hoch Belasteter Gleitlager in Planetengetrieben unter Mischreibung, Verschleiß und Deformationen. Ph.D. Thesis, Clausthal University of Technology, Clausthal-Zellerfeld, Germany, 2020. [Google Scholar]
- Profito, F.J.; Zachariadis, D.C.; Dini, D. Partitioned Fluid-Structure Interaction Techniques Applied to the Mixed-Elastohydrodynamic Solution of Dynamically Loaded Connecting-Rod Big-End Bearings. Tribol. Int.
**2019**, 140, 105767. [Google Scholar] [CrossRef] - Habchi, W.; Eyheramendy, D.; Vergne, P.; Morales-Espejel, G. A full-system approach of the elastohydrodynamic line/point contact problem. J. Tribol.
**2008**, 130, 021501. [Google Scholar] [CrossRef] - Oh, K.P. The numerical solution of dynamically loaded elastohydrodynamic contact as a nonlinear complementarity problem. J. Tribol.
**1984**, 106, 88–95. [Google Scholar] [CrossRef] - Craig, R.; Roy, R.; Bampton, M.C.C. Coupling of substructures for dynamic analyses. AIAA J.
**1968**, 6, 1313–1319. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**(

**a**) Complete FE-Model and (

**c**) complete stiffness matrices, (

**b**) reduced FE-Model and (

**d**) reduced stiffness matrices for planet.

**Figure 8.**Radial deformation at 0° degree across the bearing width for different mesh densities of the structure model.

**Figure 11.**Comparison of gap contour in the center of bearing z = −4.7 mm for (

**a**) pin and (

**c**) planet, and at the bearing edge z = −150 mm for (

**b**) pin and (

**d**) planet @ nominal operating conditions (n

_{pl}= 30 rpm, F

_{sc}= 900 kN, T

_{sup}= 60 °C, p

_{sup}= 0.2 MPa, M

_{bear}= 27.6 kNm, β = 7°).

**Figure 12.**Load conditions (

**a**) and gap contour in the center of bearing z = −4.7 mm (

**b**) and at the bearing edge, z = 150 mm (

**c**) @ nominal operating conditions (n

_{pl}= 30 rpm, F

_{sc}= 900 kN, T

_{sup}= 60 °C, p

_{sup}= 0.2 MPa, M

_{bear}= 27.6 kNm, β = 7°).

**Figure 13.**Radial clearance of elastic calculation for variable relative loads (n

_{pl}= 30 rpm, F

_{sc}= 0–900 kN, T

_{sup}= 60 °C, p

_{sup}= 0.2 MPa, M

_{bear}= 0–27.6 kNm, β = 7°).

**Figure 14.**Change of ovalization of the total radial deformation field over the bearing width @ nominal operating conditions (n

_{pl}= 30 rpm, F

_{sc}= 900 kN, T

_{sup}= 60 °C, p

_{sup}= 0.2 MPa, M

_{bear}= 27.6 kNm, β = 7°).

**Figure 15.**Radial deformations of components of planet (

**a**) and pin and carrier (

**b**) in ANSYS @ nominal operating conditions (n

_{pl}= 30 rpm, F

_{sc}= 900 kN, T

_{sup}= 60 °C, p

_{sup}= 0.2 MPa, M

_{bear}= 27.6 kNm, β = 7°).

**Figure 16.**Film thickness of rigid (

**a**,

**b**) and flexible calculation (

**c**,

**d**) @ nominal operating conditions with crowning #2 (n

_{pl}= 30 rpm, F

_{sc}= 900 kN, T

_{sup}= 60 °C, p

_{sup}= 0.2 MPa, M

_{bear}= 27.6 kNm, β = 7°).

**Figure 17.**Pressure distributions of elastic calculation at nominal operating conditions without crowning (

**a**) and with crowning #2 (

**b**) (n

_{pl}= 30 rpm, F

_{sc}= 900 kN, T

_{sup}= 60 °C, p

_{sup}= 0.2 MPa, M

_{bear}= 27.6 kNm, β = 7°).

**Figure 18.**Comparison of maximum pressure (

**a**) and minimum film thickness (

**b**) between rigid and elastic calculation for variable relative loads (n

_{pl}= 30 rpm, F

_{sc}= 0–900 kN, T

_{sup}= 60 °C, p

_{sup}= 0.2 MPa, M

_{bear}= 0–27.6 kNm, β = 7°).

**Figure 19.**Comparison of maximum solid contact pressure (

**a**) and temperature and frictional power (

**b**) between rigid and elastic calculation for variable relative loads without axial crowning (n

_{pl}= 30 rpm, F

_{sc}= 0–900 kN, T

_{sup}= 60 °C, p

_{sup}= 0.2 MPa, M

_{bear}= 0–27.6 kNm, β = 7°).

**Figure 20.**Wear after 560 h run with rigid geometries (

**a**) and under consideration of elastic deformations (

**b**) at nominal operating conditions (n

_{pl}= 30 rpm, F

_{sc}= 900 kN, T

_{sup}= 60 °C, p

_{sup}= 0.2 MPa, M

_{bear}= 27.6 kNm, β = 7°, t = 560 h, no axial crowning).

**Figure 21.**Pressure distribution after 560 h with rigid geometries (

**a**) and under consideration of elastic deformations (

**b**) @ nominal operating conditions (n

_{p}= 30 rpm, F

_{sc}= 900 kN, T

_{sup}= 60 °C, p

_{sup}= 0.2 MPa, M

_{bear}= 27.6 kNm, β = 7°, t = 560 h, no axial crowning).

**Figure 22.**Film thickness of rigid calculation (

**a**) and under consideration of elastic deformations (

**b**) at nominal operating conditions without crowning after 560 h (n

_{pl}= 30 rpm, F

_{sc}= 900 kN, T

_{sup}= 60 °C, p

_{sup}= 0.2 MPa, M

_{bear}= 27.6 kNm, β = 7°, t = 560 h, no axial crowning).

**Figure 23.**Comparison of maximum pressure (

**a**) and minimum film thickness (

**b**) between rigid and elastic calculation for variable relative loads, without crowning, after 560 h (n

_{pl}= 30 rpm, F

_{sc}= 900 kN, T

_{sup}= 60 °C, p

_{sup}= 0.2 MPa, M

_{bear}= 27.6 kNm, β = 7°, t = 560 h, no axial crowning).

Parameter | Value |
---|---|

Geometrical Properties | |

Number of pads, - | 1 |

Nominal diameter, mm | 250 |

Pitch circle diameter, mm | 499 |

Helix angle, degrees | 7 |

Bearing width, mm | 300 |

Angular span of lube oil pocket, degrees | 20.5 |

Width of lube oil pocket, mm | 260 |

Radial clearance, µm | 138 |

Pad sliding surface preload, - | 0.0 |

Static Analysis Parameters | |

Nominal rotational speed, rpm | 30 |

Nominal bearing load, kN | 900 |

Nominal bearing moment, kNm | 27.6 |

Lubricant supply temperature, °C | 60 |

Lube oil supply pressure, MPa | 0.2 |

Lubricant Properties | |

Lubricant | ISO VG 320 |

Lubricant density kg/m³ | 865 @ 40 °C |

Lubricant specific heat capacity kJ/(kg·K) | 2.0 @ 20 °C |

Lubricant thermal conductivity, W/(m·K) | 0.13 |

Planet | Pin | Carrier | |
---|---|---|---|

Parameter | Value | ||

Young’s Modulus, MPa | 210,000 | 210,000 | 176,000 |

Poisson’s Ratio, - | 0.3 | 0.3 | 0.275 |

Coefficient of Thermal Expansion, 10^{−6}/K | 12 | 11.1 | 12.5 |

Isotropic Thermal Conductivity, W/(m·K) | 39.8 | 42.6 | 31.1 |

Tensile Yield Strength, MPa | 850 | 1000 | 420 |

Parameter | Ring Gear Side | Sun Gear Side |
---|---|---|

Radial force ${F}_{r}$, kN | −164 | 164 |

Tangential force ${F}_{t}$, kN | −450 | −450 |

Axial force ${F}_{ax}$, kN | 55.3 | −55.3 |

**Table 4.**Minimum film thickness of rigid and elastic calculation at nominal operating conditions with crowning #2 or without crowning after 560 h.

Parameter | Rigid Calculation | Elastic Calculation |
---|---|---|

Minimum film thickness with crowning #2, µm | 2.6 | 2.748 |

Minimum film thickness with wear after 560 h, µm | 1.55 | 1.75 |

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**MDPI and ACS Style**

Hagemann, T.; Ding, H.; Radtke, E.; Schwarze, H.
Operating Behavior of Sliding Planet Gear Bearings for Wind Turbine Gearbox Applications—Part II: Impact of Structure Deformation. *Lubricants* **2021**, *9*, 98.
https://doi.org/10.3390/lubricants9100098

**AMA Style**

Hagemann T, Ding H, Radtke E, Schwarze H.
Operating Behavior of Sliding Planet Gear Bearings for Wind Turbine Gearbox Applications—Part II: Impact of Structure Deformation. *Lubricants*. 2021; 9(10):98.
https://doi.org/10.3390/lubricants9100098

**Chicago/Turabian Style**

Hagemann, Thomas, Huanhuan Ding, Esther Radtke, and Hubert Schwarze.
2021. "Operating Behavior of Sliding Planet Gear Bearings for Wind Turbine Gearbox Applications—Part II: Impact of Structure Deformation" *Lubricants* 9, no. 10: 98.
https://doi.org/10.3390/lubricants9100098