# Operating Behavior of Sliding Planet Gear Bearings for Wind Turbine Gearbox Applications—Part I: Basic Relations

^{*}

## Abstract

**:**

## 1. Introduction

_{t}and radial F

_{r}component. As the sum of the radial force components is zero, the bearing force F

_{sc}is equal to the sum of the tangential forces. Both components of the mesh force contribute to an oval deformation of the planet. The magnitude of the oval shape of deformation strongly depends on the ratio between mesh forces and rim thickness and influences the lifetime of rolling element bearings [5,6]. In the case of the helical gear in Figure 2b, additional axial force components F

_{ax}exist that involve a moment load about y-axis. This moment load M

_{sc}has to be restored by the contact pressure in the bearing and is a function of the helix angle of the helical gear. An additional moment about the x-axis occurs if the load distribution on the tooth flanks becomes inhomogeneous in the lateral direction and causes eccentric points of application of resulting mesh forces. Figure 2c depicts an assembled helical planet gear with its contact partners in the front view of a planet stage.

## 2. Materials and Methods

#### 2.1. Governing Equations and Bearing Model

_{x}, e

_{y}and angular alignment φ

_{x}, φ

_{y}between the centerlines of planet and pin are determined using stiffness matrix

**C**defined by Equation (5) to minimize the error between mesh forces and bearing contact forces in order to find the static state of equilibrium. The coefficients of

**C**represent the sum of hydrodynamic and solid contact stiffness coefficients in the mixed friction regime while solid contact stiffness is equal to zero in the hydrodynamic regime.

_{X},φ

_{Y}) rigid body displacements relative to the current position described by e

_{x}, e

_{y,}φ

_{x}, and φ

_{y}is assumed according to Equation (6).

#### 2.2. Verification of the Numerical Procedure

#### 2.3. Flow Factors and Solid Contact Pressure

_{q}= 2.83 µm and matches the one earlier presented by the authors in [25]. A turning process generates the surface topography that features small grooves in the circumferential direction. Solid body contact pressure is determined based on the formula proposed by Hu et al. [26]. Consequently, it slightly deviates from the ones presented in [25], which are evaluated differently. The concrete course of the solid body load-carrying capacity has exemplary character in this investigation. Therefore, the contradiction that the theory of Greenwood and Williamson presupposes a normal distribution of the surface roughness that does not exist for the investigated contact surface is accepted here.

#### 2.4. Investigated Sliding Planet Gear Bearing

#### 2.5. Bearing Loads Due to Mesh Forces

_{sc}is equal to the sum of the tangential mesh forces that result from the driving torque T

_{d}and the contact between ring gear and planet gear and between planet gear and sun gear, respectively.

_{n}and the helix angle of the helical gear β, Equation (8) defines the axial and radial components of the mesh forces.

_{sc}that has to be restored by the bearing. This moment load follows according to Equation (9).

_{sc}depends linearly on the driving torque and, additionally, it is a nearly linear function of the helix angle of the helical gear. Restoring this moment by the bearing contact requires an alignment of pin and planet relative to each other. The impact of this effect on planet gear bearing operating behavior is considered as a key or contributing effect throughout the entire subsequent investigations.

_{r}. Here, 100% T

_{r}correspond to the nominal load situation with a bearing force load of 900 kN according to Table 2. While the force load in Figure 8a is independent of the helix angle, the moment load in Figure 8b strongly depends on it. The distribution of tooth forces to more than one contact due to a profile overlap is neglected in this study.

## 3. Results

#### 3.1. Impact of Helix Angle of the Helical Gear on Operating Conditions in the Lubricant Gap

_{sc}results exclusively from M

_{y}while M

_{x}= 0 Nm. Figure 9 shows the local pressure distribution in the lubricant gap for a straight gear (a) and a helical gear (b) with a helix angle β = 7° in two different views. Here, the term “aggregate pressure” refers to the sum of hydrodynamic and asperity contact pressure. While the aggregate pressure in case (a) purely results from hydrodynamic fluid film, the edge loading in case (b) that is caused by the moment load due to the axial component of the mesh forces involves mixed friction, and therefore, a combination of hydrodynamic and solid contact pressure in the bearing. Moreover, the aggregate pressure significantly increases in case (b) and reaches a level that exceeds the strength level of common sliding bearing materials. In addition to these results, Figure 10 includes the characteristic of minimum film thickness, solid contact and hydrodynamic pressure as well as the alignment between planet and pin as a function of the helix angle. A significant rise of the pressure level exists for helix angles β > 4°. As expected, the load moment M

_{y}causes an alignment of the bearing about x- and y-axis due to the cross-coupling effects in the fluid film.

#### 3.2. Impact of Radial Clearance on Load Carrying Capacity

#### 3.3. Impact of Axial Profiling

#### 3.4. Operation at Part-Load and Over-Load Conditions

_{r}. Figure 8 includes the corresponding force and moment loads.

#### 3.4.1. Part-Load Conditions for Homogenous Load Distribution on Tooth Flank

_{r}= 10% to 100%. Results clearly show that the crowning exhibits advantages in the entire investigated load range as maximum pressure decreases and minimum film thickness rises. Moreover, a strictly monotonous characteristic of these parameters can be observed, and operating conditions in the bearing become unfavorably with a rising relative load.

#### 3.4.2. Part-Load and Over-Load Conditions for Constant Tooth Flank Geometry

_{y}is assumed between relative loads T

_{r}= 0 and 100%, and a linear one is present in the overload range. The moment M

_{x}in Figure 15b remains unchanged as no shifts are considered between the meshing points of sun and planet gear and planet and ring gear, respectively. In opposite to the ideal case studied so far, the direction of the resulting moment load varies with driving torque. Figure 15c shows a rise of the resulting moment load due to the eccentricity of the tangential mesh forces depicted in Figure 15d. In particular, the moment loads in the part-load region are much higher, as the yellow curve in Figure 15c illustrates.

#### 3.5. Modification of the Lubricant Gap by Wear

_{sol}are used to evaluate wear according to Archard’s law [27].

_{w}is a function of the contact force F

_{sol}, the sliding distance L and the wear coefficient K. It is assumed that the softer material is located on the stator side provoking a two-dimensional variable distribution of wear on the sliding surface of the bearing without axial crowning. In the concrete case, a wear coefficient of $K=8.5\xb7{10}^{-9}\text{}\mathrm{m}{\mathrm{m}}^{3}/\mathrm{J}$ and an operation time of t = 280 h at nominal load is assumed. These operating conditions cause significant wear according to Figure 17a. Figure 17b shows the characteristic of the maximum aggregate pressure and minimum film thickness during the run. Here, wear increases minimum film thicknesses and decreases maximum pressure monotonously in the first 160 h. After 160 h a nearly constant level of minimum film thickness close to pure hydrodynamic operation is reached that slightly reduces in the further run. The pressure level decreases monotonously throughout the entire investigated time range. Local maximum pressures tend to shift to the middle plane similar to the behavior observed in the results for crowning #1 in Figure 12b. Nevertheless, maximum pressures significantly reduce as wear accompanies regions of only slightly changed film thickness. To show this phenomenon more clearly, the run time is extended to 560 h. Figure 18b depicts an extension of the load-carrying region in the circumferential direction of the bearing due to wear. Therefore, maximum pressure reaches a level comparable to the one predicted for the unworn bearing with crowning #2 in Figure 12d. Though the pressure is more homogenous in the axial direction with the crowning, Figure 18a indicates that the load-carrying region in the circumferential direction, which is enclosed by the two red lines in both partial figures, is smaller than the one of the worn bearing.

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

c | lubricant specific heat |

C_{R} | radial clearance |

d_{1} | pitch diameter |

e | Eccentricity between pin and planet |

F_{0}, F_{1}, F_{2} | viscosity factors |

F | force |

h | film thickness |

M | moment |

n | rotor speed |

p | pressure |

R_{q} | root mean square surface roughness |

T | temperature, torque |

U | surface speed |

u, v, w | flow velocities |

x, y, z | Cartesian coordinates |

X,Y | translational displacement relative to equilibrium position |

Θ | lubricant density ratio |

γ | attitude angle |

η | lubricant dynamic viscosity |

λ | lubricant conductivity |

ρ | lubricant density |

φ_{x,} φ_{y} | alignment angle about x-,y-axis |

φ_{X,} φ_{Y} | alignment angle about x-,y-axis relative to equilibrium position |

ϕ | angular coordinate |

ϕ_{x},ϕ_{z}^{p} | pressure flow factors |

Φ_{x}^{S} | shear flow factor |

## References

- Ragheb, A.; Ragheb, M. Wind turbine gearbox technologies. In Proceedings of the 2010 1st International Nuclear & Renewable Energy Conference (INREC), Amman, Jordan, 21–24 March 2010; pp. 1–8. [Google Scholar]
- Tavner, P.J.; Xiang, J.; Spinato, F. Reliability analysis for wind turbines. Wind Energy
**2007**, 10, 1–18. [Google Scholar] [CrossRef] - Ribrant, J.; Bertling, L. Survey of failures in wind power systems with focus on Swedish wind power plants during 1997–2005. In Proceedings of the 2007 IEEE Power Engineering Society General Meeting, Tampa, FL, USA, 24–28 June 2007; pp. 1–8. [Google Scholar]
- Qiao, W.; Lu, D. A survey on wind turbine condition monitoring and fault diagnosis—Part I: Components and subsystems. IEEE Trans. Ind. Electron.
**2015**, 62, 6536–6545. [Google Scholar] [CrossRef] - Jones, A.; Harris, T.A. Analysis of a Rolling-Element Idler Gear Bearing Having a Deformable Outer-Race Structure. J. Basic Eng.
**1963**, 85, 273–278. [Google Scholar] [CrossRef] - Fingerle, A.; Hochrein, J.; Otto, M.; Stahl, K. Theoretical Study on the Influence of Planet Gear Rim Thickness and Bearing Clearance on Calculated Bearing Life. J. Mech. Des.
**2019**, 142, 031102. [Google Scholar] [CrossRef] - Bouyer, J.; Fillon, M. An Experimental Analysis of Misalignment Effects on Hydrodynamic Plain Journal Bearing Performances. J. Tribol.
**2001**, 124, 313–319. [Google Scholar] [CrossRef] - Sun, J.; Changlin, G. Hydrodynamic lubrication analysis of journal bearing considering misalignment caused by shaft deformation. Tribol. Int.
**2004**, 37, 841–848. [Google Scholar] [CrossRef] - Hili, M.A.; Bouaziz, S.; Maatar, M.; Fakhfakh, T.; Haddar, M. Hydrodynamic and Elastohydrodynamic Studies of a Cylindrical Journal Bearing. J. Hydrodyn.
**2010**, 22, 155–163. [Google Scholar] [CrossRef] - 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]
- 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]
- Hagemann, T.; Schwarze, H. A Model for Oil Flow and Fluid Temperature Inlet Mixing in Hydrodynamic Journal Bearings. J. Tribol.
**2018**, 141, 021701. [Google Scholar] [CrossRef] - Muzakkir, S.M.; Hirani, H.; Thakre, G.D. Lubricant for Heavily Loaded Slow-Speed Journal Bearing. Tribol. Trans.
**2013**, 56, 1060–1068. [Google Scholar] [CrossRef] - Linjamaa, A.; Lehtovaara, A.; Larsson, R.; Kallio, M.; Söchting, S. Modelling and analysis of elastic and thermal deformations of a hybrid journal bearing. Tribol. Int.
**2018**, 118, 451–457. [Google Scholar] [CrossRef] - Xiang, G.; Han, Y.; Wang, J.; Wang, J.; Ni, X. Coupling transient mixed lubrication and wear for journal bearing modeling. Tribol. Int.
**2019**, 138, 1–15. [Google Scholar] [CrossRef] - Garabedian, N.T.; Gould, B.J.; Doll, G.L.; Burris, D.L. The Cause of Premature Wind Turbine Bearing Failures: Overloading or Underloading? Tribol. Trans.
**2018**, 61, 850–860. [Google Scholar] [CrossRef] - Hagemann, T.; Ding, H.; Radtke, E.; Schwarze, H. Operating behavior of sliding planet gear bearings in wind turbine gearbox applications—Part II: Impact of structure deformation. Lubricants
**2021**, in press. [Google Scholar] - Patir, N.; Cheng, H.S. An Average Flow Model for Determining Effects of Three-Dimensional Roughness on Partial Hydrodynamic Lubrication. J. Lubr. Technol.
**1978**, 100, 12–17. [Google Scholar] [CrossRef] - Falz, E. Grundzüge der Schmierungstechnik; Springer: Berlin/Heidelberg, Germany, 1931. [Google Scholar]
- Elrod, H.G. A Cavitation Algorithm. J. Lubr. Technol.
**1981**, 103, 350–354. [Google Scholar] [CrossRef] - Patankar, S.V. Numerical Heat Transfer and Fluid Flow; CRC Press: Boca Raton, FL, USA, 1980; ISBN 9781315275130. [Google Scholar]
- Waltermann, H. Optimierte Thermo-Elasto-Hydrodynamische Berechnungsverfahren für Gleitlager. Ph.D. Thesis, Rheinisch-Westfälisch Technische Hochschule Aachen, Aachen, Germany, 1992. [Google Scholar]
- Hagemann, T.; Zemella, P.; Pfau, B.; Schwarze, H. Experimental and theoretical investigations on transition of lubrication conditions for a five-pad tilting-pad journal bearing with eccentric pivot up to highest surface speeds. Tribol. Int.
**2020**, 142, 106008. [Google Scholar] [CrossRef] - Greenwood, J.A.; Tripp, J.H. The Contact of Two Nominally Flat Rough Surfaces. Proc. Inst. Mech. Eng.
**1970**, 185, 625–633. [Google Scholar] [CrossRef] - Prölß, M.; Schwarze, H.; Hagemann, T.; Zemella, P.; Winking, P. Theoretical and Experimental Investigations on Transient Run-Up Procedures of Journal Bearings Including Mixed Friction Conditions. Lubricants
**2018**, 6, 105. [Google Scholar] [CrossRef] [Green Version] - Hu, Y.; Cheng, H.S.; Arai, T.; Kobayashi, Y.; Aoyama, S. Numerical Simulation of Piston Ring in Mixed Lubrication—A Nonaxisymmetrical Analysis. J. Tribol.
**1994**, 116, 470–478. [Google Scholar] [CrossRef] - Archard, J.F. Contact and Rubbing of Flat Surfaces. J. Appl. Phys.
**1953**, 24, 981–988. [Google Scholar] [CrossRef]

**Figure 2.**Mesh forces and bearing load for (

**a**) straight gear, (

**b**) helical gear, and (

**c**) the helical gear assembled in the planet stage.

**Figure 3.**Numerical grid for temperature evaluation for the (

**a**) journal bearing and (

**b**) planet gear bearing.

**Figure 4.**Cross section of the three-dimensional solution domain of the temperature analysis (

**a**) journal bearing, and (

**b**) planet gear bearing at a certain axial position.

**Figure 5.**Pressure distributions determined by Sun and Changlin [8] (

**a**,

**c**) and the planet gear bearing code (

**b**,

**d**) for misalignment angles of γ = 0.0° (

**a**,

**b**) and γ = 0.03° (

**c**,

**d**).

**Figure 6.**Resulting moment for different levels of misalignment angles determined by Sun and Changlin [8] (

**a**) and the planet gear bearing code (

**b**).

**Figure 9.**Pressure distributions @ nominal operating conditions (

**a**) straight gear (β = 0°) and (

**b**) helical gear (β = 7°) (n

_{pl}= 30 rpm, F

_{sc}= 900 kN, T

_{sup}= 60 °C, p

_{sup}= 0.2 MPa, M

_{sc}= 27.6 kNm (

**b**)).

**Figure 10.**Impact of helix angle of helical gear on (

**a**) maximum pressures, minimum film thickness, and (

**b**) alignment angles between planet and pin (n

_{pl}= 30 rpm, F

_{sc}= 900 kN, T

_{sup}= 60 °C, p

_{sup}= 0.2 MPa).

**Figure 11.**Maximum pressure (

**a**) and maximum temperature on the pin sliding surface (

**b**) for variable relative loads (n

_{pl}= 30 rpm, F

_{sc}= 900 kN, T

_{sup}= 60 °C, p

_{sup}= 0.2 MPa).

**Figure 12.**Impact of axial crowning (

**a**) on the pressure distribution: (

**b**) crowning #1, (

**c**,

**d**) crowning #2 (n

_{pl}= 30 rpm, F

_{sc}= 900 kN, T

_{sup}= 60 °C, p

_{sup}= 0.2 MPa, β = 7°, M

_{sc}= 27.6 kNm).

**Figure 13.**Film thickness @ nominal operating conditions with crowning #2 (n

_{pl}= 30 rpm, F

_{sc}= 900 kN, T

_{sup}= 60 °C, p

_{sup}= 0.2 MPa, β = 7°, M

_{sc}= 27.6 kNm).

**Figure 14.**Maximum pressure (

**a**) and minimum film thickness (

**b**) for variable relative loads (n

_{pl}= 30 rpm, T

_{sup}= 60 °C, p

_{sup}= 0.2 MPa, β = 7°).

**Figure 15.**Load moments M

_{y}(

**a**), M

_{x}(

**b**), and M

_{res}(

**c**) for variable relative loads (n

_{pl}= 30 rpm, T

_{sup}= 60 °C, p

_{sup}= 0.2 MPa, β = 7°); eccentricity of tangential mesh force from the middle plane (

**d**).

**Figure 16.**Maximum aggregate pressure (

**a**) and minimum film thickness (

**b**) for variable relative loads (n

_{pl}= 30 rpm, T

_{sup}= 60 °C, p

_{sup}= 0.2 MPa, β = 7°).

**Figure 17.**Wear after 280 h run (

**a**), maximum aggregate pressure and minimum film thickness (

**b**), and pressure distribution after 280 h run (

**c**,

**d**) (n

_{pl}= 30 rpm, T

_{sup}= 60 °C, p

_{sup}= 0.2 MPa, β = 7°, t = 280 h).

**Figure 18.**Comparison of pressure distribution with crowning #2 (

**a**) and without crowning after 560 h (

**b**) (n

_{pl}= 30 rpm, T

_{sup}= 60 °C, p

_{sup}= 0.2 MPa, β = 7°).

**Table 1.**Bearing parameters [8].

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

Geometrical properties | |

Number of pads, - | 1 |

Nominal diameter, mm | 60 |

Bearing width, mm | 66 |

Radial clearance, µm | 30 |

Pad sliding surface preload, - | 0.0 |

Static analysis parameters | |

Rotational speed, rpm | 3000 |

Lubricant properties | |

Lubricant dynamic viscosity, mPas | 9.0 |

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

Geometrical properties | |

Number of pads, - | 1 |

Nominal diameter, mm | 250 |

Pitch circle diameter, mm | 499 |

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 |

Lubricant supply temperature, °C | 60 |

Lube oil supply pressure, MPa | 0.2 |

Lubricant properties | |

Lubricant | ISO VG 320 |

Lubricant density kg/m^{3} | 865 @ 40 °C |

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

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

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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 I: Basic Relations. *Lubricants* **2021**, *9*, 97.
https://doi.org/10.3390/lubricants9100097

**AMA Style**

Hagemann T, Ding H, Radtke E, Schwarze H.
Operating Behavior of Sliding Planet Gear Bearings for Wind Turbine Gearbox Applications—Part I: Basic Relations. *Lubricants*. 2021; 9(10):97.
https://doi.org/10.3390/lubricants9100097

**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 I: Basic Relations" *Lubricants* 9, no. 10: 97.
https://doi.org/10.3390/lubricants9100097