# Analysis of the Influence of Calculation Parameters on the Design of the Gearbox of a High-Power Wind Turbine

^{*}

## Abstract

**:**

## 1. Introduction

## 2. History of Gearbox Problems

## 3. Types of Epicyclic Geartrains for Use in High-Power Wind Turbines

## 4. Kinematic Analysis of the Epicyclic Geartrain Model 1

${\mathit{Z}}_{\mathit{P}}<{\mathit{Z}}_{\mathit{S}}$ | ${\mathit{Z}}_{\mathit{P}}>{\mathit{Z}}_{\mathit{S}}$ | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

${Z}_{P}/{Z}_{S}$ | 1/6 | 1/5 | 1/4 | 1/3 | 1/2 | 1 | 2 | 3 | 4 | 5 | 6 |

${i}_{i/o}$ | 2.33 | 2.4 | 2.5 | 2.67 | 3 | 4 | 6 | 8 | 10 | 12 | 14 |

${i}_{ap}$ | −1.33 | −1.4 | −1.5 | −1.67 | −2 | −3 | −5 | −7 | −9 | −11 | −13 |

## 5. Analysis of the Multiplier Gearbox Multiplier

_{engra}depend on the diameter of the corresponding shaft d.

**,**the tooth height, and the value of ${h}_{1}$ (the depth of the keyway in the hub). In this work, the influence of this parameter on the value of the pitch diameter will be analyzed.

- (a)
- Coaxiality condition, derived from Equation (15):$${z}_{ring}={2\xb7z}_{pla}+{z}_{sun}$$
- (b)
- Mounting condition—the number of teeth of the sun plus the ring divided by the number of satellites must be a whole number:$$Integernumber=\frac{{Z}_{ring}+{Z}_{sun}}{{n}_{planets}}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}N\in {\mathbb{N}}^{+}$$
- (c)
- Contiguity condition—this translates into $\pi \xb7{D}_{m}>{n}_{planets}\xb7{D}_{planet}$, that is to say,$$\frac{\pi}{2}\xb7\frac{{Z}_{ring}+{Z}_{Sun}}{{Z}_{Planet}+2\mathrm{cos}\beta}>{n}_{planets}$$
- (d)
- Maximum number of planets: ${n}_{planets}=\frac{360\xb0}{2\xb7(90\xb0-arcos\frac{{z}_{planet}+2}{{z}_{ring}+{z}_{sun}})}$
- (e)
- To avoid interference:$${z}_{min}=\frac{2\xb7cos\beta}{\mathrm{s}\mathrm{i}\mathrm{n}{\left({\alpha}_{t}\right)}^{2}}$$

#### Calculation of the Weight of the Epicyclic Gear Model 1

- (a)
- Tension at the base of the tooth: ${\sigma}_{F}\le \frac{{S}_{FP}}{{X}_{F}}$

- (b)
- Surface pressure on the tooth: ${\sigma}_{H}\le \frac{{S}_{HP}}{{X}_{H}^{2}}$

${m}_{n}\left(\mathrm{mm}\right)$ | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 | 60 | 70 | 80 | 90 | 100 |

## 6. Results and Discussion

_{n}, the values of the diameter of the input shaft to the epicyclic gear train of Stage 1, which correspond to the planet carrier (d

_{carr1}), diameter of the sun axis in Stage 1 (d

_{s1}), the pitch diameter of the sun in Stage 1 (D

_{s1}), the number of teeth of the sun (z

_{s1}), of the planets (z

_{p1}) and ring at Stage 1 (z

_{cor1}), the apparent gear ratio in Stage 1 (i

_{ap1}), the tooth width for all Stage 1 gears (b) and a proportional estimate of the weight of the epicyclic gear train (W). It can be noticed that, for any value of m

_{n}, the value of the tooth width is excessive. Moreover, the pitch diameter D

_{s1}is small due to the low value of ${K}_{engr}$—see Equation (9)— which means that the tooth width is too large to comply with the safety coefficients (Equations (24) and (27)) and the weight of the planetary train of the first stage is excessive. In addition, from the normal modulus 25, interference in the sun of Stage 1 is reached—see Equation (19)—so that it is not possible to use larger moduli.

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Carriveau, R. Advances in Wind Power; Rupp: Rijeka, Croatia, 2012; 374p, ISBN 978-953-51-0863-4. [Google Scholar] [CrossRef]
- Liang, J.; Kato, B.; Wang, Y. Constructing simplified models for dynamic analysis of monopile-supported offshore wind turbines. Ocean. Eng.
**2023**, 271, 113785. [Google Scholar] [CrossRef] - Nejad, A.R.; Keller, J.; Guo, Y.; Sheng, S.; Polinder, H.; Watson, S.; Dong, J.; Qin, Z.; Ebrahimi, A.; Schelenz, R.; et al. Wind turbine drivetrains: State-of-the-art technologies and future development trends. Wind Energy Sci.
**2022**, 7, 387–411. [Google Scholar] - Chauhan, A.; Singla, A.; Panwar, N.; Jindal, P. CFD Based Thermo-Hydrodynamic Analysis of Circular Journal Bearing. Int. J. Adv. Mech. Eng.
**2014**, 4, 475–482. [Google Scholar] - Florescu, A.; Barabas, S.; Dobrescu, T. Research on Increasing the Performance of Wind Power Plants for Sustainable Development. Sustainability
**2019**, 11, 1266. [Google Scholar] [CrossRef] - Oyague, F. Gearbox Modeling and Load Simulation of a Baseline 750-kW Wind Turbine Using State-of-the-Art Simulation Codes; Technical Report NREL/TP-500-41160; National Renewable Energy Laboratory: Golden, CO, USA, 2009; 94p. [Google Scholar]
- Zhao, M.; Ji, J. Dynamic Analysis of Wind Turbine Gearbox Components. Energies
**2016**, 9, 110. [Google Scholar] [CrossRef] - ISO 6336:2019; Calculation of Load Capacity of Spur and Helical Gears. International Organization for Standardization: Geneva, Switzerland, 2019.
- Hari Babu, A.V.; Naresh, P.; Madhava, V.; Sudhakar Reddy, M. Minimum Weight Optimization of a Gear Train by Using GA. Int. J. Eng. Trends Adv. Sci.
**2016**, 1, 43–50. [Google Scholar] - Hart, E.; Clarke, B.; Nicholas, G.; Kazemi Amiri, A.; Stirling, J.; Carroll, J.; Dwyer-Joyce, R.; McDonald, A.; Long, H. A review of wind turbine main-bearings: Design, operation, modelling, damage mechanisms. Wind. Energy Sci.
**2019**, 5, 105–124. [Google Scholar] [CrossRef] - Musial, W.D.; Beiter, P.C.; Nunemaker, J.; Heimiller, D.M.; Ahmann, J.; Busch, J. Oregon Offshore Wind Site Feasibility and Cost Study; National Renewable Energy Lab.: Golden, CO, USA, 2019.
- Hau, E. Wind Turbines, Fundamentals, Technologies, Application, Economics; Springer: Berlin/Heidelberg, Germany, 2011. [Google Scholar] [CrossRef]
- Ragheb, A.M.; Ragheb, M. Wind Turbine Gearbox Technologies. In Proceedings of the 2010 1st International Nuclear & Renewable Energy Conference (INREC), Amman, Jordan, 21–24 March 2011. [Google Scholar] [CrossRef]
- Oswald, F.B.; Jett, T.R.; Predmore, R.E.; Zaretsky, E.V. Probabilistic Analysis of Space Shuttle Body Flap Actuator Ball Bearings. Tribol. Trans.
**2008**, 51, 193–203. [Google Scholar] [CrossRef] - Kaiser, S.; Fröhlingsdorf, M. The Dangers of Wind Power. Der Spiegel. 2007. Available online: http://www.spiegel.de/international/germany/0,1518,500902,00.html (accessed on 1 August 2023).
- Tiwari, P.; Kumar, V. Analysis of Hydrodynamic Journal Bearing Using CFD and FSI Technique. Int. J. Eng. Res. Technol.
**2014**, 3, 1210–1215. [Google Scholar] - Nie, M.; Wang, L. Review of condition monitoring and fault diagnosis technologies for wind turbine gearbox. Procedia CIRP
**2013**, 11, 287–290. [Google Scholar] [CrossRef] - Department of Energy. Wind Turbine Testing in the NREL Dynamometer Test Bed. 2010. Available online: http://www.doe.gov/bridge (accessed on 1 August 2023).
- Struggl, S.; Berbyuk, V.; Johansson, H. Review on wind turbines with focus on drive train system dynamics. Wind Energy
**2015**, 18, 567–590. [Google Scholar] [CrossRef] - Tauviqirrahman, M.; Jamari, J.; Wicaksono, A.A.; Muchammad, M.; Susilowati, S.; Ngatilah, Y.; Pujiastuti, C. CFD Analysis of Journal Bearing with a Heterogeneous Rough/Smooth Surface. Lubricants
**2021**, 9, 88. [Google Scholar] [CrossRef] - Rubio, F.; Llopis-Albert, C.; Zeng, S. Best practices and syllabus design and course planning applied to mechanical engineering subjects. Multidisciplinary J. Educ. Soc. Technol. Sci.
**2022**, 9, 123–137. [Google Scholar] [CrossRef] - Llopis-Albert, C.; Rubio, F.; Zeng, S.; Devece, C.; Torner-Feltrer, M.E. Quality assessment program of the teaching activity of the higher education faculty staff. A case study. Multidiscip. J. Educ. Soc. Technol. Sci.
**2023**, 10, 94–113. [Google Scholar] [CrossRef] - Llopis-Albert, C.; Rubio, F.; Zeng, S.; Grima-Olmedo, J.; Grima-Olmedo, C. The Sustainable Development Goals (SDGs) applied to Mechanical Engineering. Multidiscip. J. Educ. Soc. Technol. Sci.
**2022**, 9, 59–70. [Google Scholar] [CrossRef] - Höhn, B.R.; Stahl, K.; Gwinner, P. Light Weight Design for Planetary Gear Transmissions. Gear Technol.
**2013**, 30, 96–103. [Google Scholar] - Vázquez-Hernández, C.; Serrano-González, J.; Centeno, G. A Market-Based Analysis on the Main Characteristics of Gearboxes Used in Onshore Wind Turbines. Energies
**2017**, 10, 1686. [Google Scholar] [CrossRef] - Levai, Z. Structure and Analysis of Planetary Gear Trains. J. Mech.
**1968**, 3, 131–148. [Google Scholar] [CrossRef] - Le, X.C.; Duong, M.Q.; Le, K.H. Review of the Modern Maximum Power Tracking Algorithms for Permanent Magnet Synchronous Generator of Wind Power Conversion Systems. Energies
**2023**, 16, 402. [Google Scholar] [CrossRef] - DIN 3990:1987; Calculation of Load Capacity of Cylindrical Gears. German Institute for Standardisation Registered Association: Berlin, Germany, 1987.

Model 1 | Model 2 | Model 3 | Model 4 |
---|---|---|---|

P(P)N | P(PP)N | P(PP)P | N(PP)N |

Fixed Element | Input | Output | Variant |
---|---|---|---|

Planet carrier | Sun | Ring | 1 |

Ring | Sun | 2 | |

Sun gear | Ring | Planet carrier | 3 |

Planet carrier | Ring | 4 | |

Ring gear | Sun | Planet carrier | 5 |

Planet carrier | Sun | 6 |

Model 1 | ||
---|---|---|

Willis | $\frac{{\omega}_{S}-{\omega}_{C}}{{\omega}_{R}-{\omega}_{C}}=-\frac{{Z}_{R}}{{Z}_{S}}$ | |

Variant | Fixed Ring | |

6 | Input: Planet carrier Output: Sun | ${i}_{ap}=\frac{{\omega}_{S}-{\omega}_{C}}{-{\omega}_{C}}=-\frac{{Z}_{R}}{{Z}_{S}}$ ${i}_{i/o}=\frac{{\omega}_{S}}{{\omega}_{C}}=1+\frac{{Z}_{R}}{{Z}_{S}}=1-{i}_{ap}$ |

Rated power (P): | 7 MW |

$\mathrm{Transmission}\mathrm{ratio}{i}_{i/o}$ | 107 ± 2% |

Optimal rotor speed | 14 rpm |

$\mathrm{Gear}\mathrm{safety}\mathrm{coefficient},{X}_{H}={S}_{HP}/{\sigma}_{H}$ | 1.5 |

Driving machine | Major shocks |

Driven machine | Uniform operation |

Φ (rotor diameter) | 180 m |

Shaft Diameters (mm) | |
---|---|

d_{i1} | 389.15 |

d_{int1} | 216.95 |

d_{i2} | 120.957 |

P | $7\mathrm{M}\mathrm{W}$ |

${\omega}_{carrier1}$ | $14\mathrm{r}\mathrm{p}\mathrm{m}$ |

${\omega}_{sun1}$ | $144.9\mathrm{r}\mathrm{p}\mathrm{m}$ |

${i}_{ap1}$ | $-9.35$ |

${i}_{E1}$ | $10.35$ |

${i}_{T}$ | $107.1429$ |

d_{carr1} | d_{s1} | D_{s1} | z_{s1} | z_{p1} | z_{cor1} | i_{ap1} | b | W | |
---|---|---|---|---|---|---|---|---|---|

m_{n} = 10 | 389.15 | 216.95 | 280.80 | 26 | 109 | 244 | 9.38 | >2500 | No |

m_{n} = 15 | 389.15 | 216.95 | 312.73 | 20 | 84 | 188 | 9.4 | >2500 | No |

m_{n} = 20 | 389.15 | 216.95 | 344.65 | 16 | 67 | 150 | 9.37 | >2500 | No |

m_{n} = 25 | 389.15 | 216.95 | 376.58 | 14 | 59 | 132 | 9.42 | >2500 | No |

m_{n} = 30 | 389.15 | 216.95 | 408.51 | 14 | 55 | 123 | 9.46 | >2500 | No |

d_{carrier1} | d_{s1} | D_{s1} | z_{s1} | z_{p1} | z_{ring1} | i_{ap1} | b_{2} | W_{2} | |
---|---|---|---|---|---|---|---|---|---|

m_{n} = 10 | 389.15 | 216.95 | 376.58 | 35 | 146 | 327 | >2500 | No | |

m_{n} = 15 | 389.15 | 216.95 | 456.39 | 29 | 121 | 271 | >2500 | No | |

m_{n} = 20 | 389.15 | 216.95 | 536.21 | 25 | 105 | 234 | >2500 | No | |

m_{n} = 25 | 389.15 | 216.95 | 616.02 | 23 | 96 | 215 | 742 | 138.44 | |

m_{n} = 30 | 389.15 | 216.95 | 695.83 | 22 | 92 | 206 | 467 | 115.60 | |

m_{n} = 35 | 389.15 | 216.95 | 775.65 | 21 | 88 | 196 | 353 | 109.16 | |

m_{n} = 40 | 389.15 | 216.95 | 855.46 | 20 | 84 | 187 | 305 | 112.61 | |

m_{n} = 45 | 389.15 | 216.95 | 935.27 | 20 | 84 | 187 | 248 | 115.87 | |

m_{n} = 50 | 389.15 | 216.95 | 1015. | 19 | 880 | 178 | 227 | 116.64 | |

m_{n} = 60 | 389.15 | 216.95 | 1174.7 | 18 | 75 | 168 | 178 | 119.39 | |

m_{n} = 70 | 389.15 | 216.95 | 1334.3 | 18 | 75 | 168 | 130 | 118.64 | |

m_{n} = 80 | 389.15 | 216.95 | 1494 | 18 | 75 | 168 | 99 | 117.04 | |

m_{n} = 90 | 389.15 | 216.95 | 1652.6 | 17 | 71 | 159 | 87 | 117.75 | |

m_{n} = 100 | 389.15 | 216.95 | 1813.2 | 17 | 71 | 159 | 69 | 115.84 |

d_{carrier1} | d_{s1} | D_{s1} | z_{s1} | z_{p1} | z_{ring} | i_{ap1} | b_{3} | W_{3} | |
---|---|---|---|---|---|---|---|---|---|

m_{n} = 10 | 389.15 | 216.95 | 429.79 | 40 | 167 | 374 | 9.35 | >2500 | No |

m_{n} = 15 | 389.15 | 216.95 | 536.21 | 34 | 142 | 318 | 9.35 | >2500 | No |

m_{n} = 20 | 389.15 | 216.95 | 642.62 | 30 | 126 | 282 | 9.4 | 611 | 121.41 |

m_{n} = 25 | 389.15 | 216.95 | 749.04 | 28 | 117 | 262 | 9.35 | 391 | 106.66 |

m_{n} = 30 | 389.15 | 216.95 | 855.46 | 27 | 113 | 253 | 9.37 | 274 | 100.65 |

m_{n} = 35 | 389.15 | 216.95 | 961.88 | 26 | 109 | 244 | 9.38 | 230 | 107.01 |

m_{n} = 40 | 389.15 | 216.95 | 1068.3 | 25 | 105 | 235 | 9.4 | 196 | 110.91 |

m_{n} = 45 | 389.15 | 216.95 | 1174.7 | 25 | 105 | 235 | 9.4 | 157 | 112.37 |

m_{n} = 50 | 389.15 | 216.95 | 1281.1 | 24 | 100 | 224 | 9.33 | 139 | 112.15 |

m_{n} = 60 | 389.15 | 216.95 | 1494.0 | 23 | 96 | 215 | 9.34 | 105 | 112.68 |

m_{n} = 70 | 389.15 | 216.95 | 1706.8 | 23 | 96 | 215 | 9.34 | 76 | 110.62 |

m_{n} = 80 | 389.15 | 216.95 | 1919.6 | 23 | 96 | 215 | 9.34 | 57 | 108.85 |

m_{n} = 90 | 389.15 | 216.95 | 2132.5 | 22 | 92 | 206 | 9.36 | 49 | 108.69 |

m_{n} = 100 | 389.15 | 216.95 | 2345.3 | 22 | 92 | 206 | 9.36 | 39 | 107.21 |

d_{carrier1} | d_{s1} | D_{s1} | z_{s1} | z_{p1} | z_{ring} | i_{ap1} | b_{4} | W_{4} | |
---|---|---|---|---|---|---|---|---|---|

m_{n} = 10 | 389.15 | 216.95 | 536.21 | 50 | 209 | 468 | 9.36 | >2500 | No |

m_{n} = 15 | 389.15 | 216.95 | 695.83 | 44 | 184 | 412 | 9.36 | 443 | 105.26 |

m_{n} = 20 | 389.15 | 216.95 | 855.46 | 40 | 167 | 374 | 9.35 | 274 | 95.69 |

m_{n} = 25 | 389.15 | 216.95 | 1015.1 | 38 | 159 | 356 | 9.36 | 194 | 96.22 |

m_{n} = 30 | 389.15 | 216.95 | 1174.7 | 37 | 155 | 347 | 9.37 | 151 | 102.17 |

m_{n} = 35 | 389.15 | 216.95 | 1334.3 | 36 | 151 | 338 | 9.38 | 120 | 104.09 |

m_{n} = 40 | 389.15 | 216.95 | 1494 | 35 | 146 | 327 | 9.34 | 99 | 105.55 |

m_{n} = 45 | 389.15 | 216.95 | 1653.6 | 35 | 146 | 327 | 9.34 | 78 | 105.74 |

m_{n} = 50 | 389.15 | 216.95 | 1813.2 | 34 | 142 | 318 | 9.35 | 64 | 105.99 |

m_{n} = 60 | 389.15 | 216.95 | 2132.5 | 33 | 138 | 309 | 9.36 | 49 | 105.46 |

m_{n} = 70 | 389.15 | 216.95 | 2451.7 | 33 | 138 | 309 | 9.36 | 36 | 104.2 |

m_{n} = 80 | 389.15 | 216.95 | 2771.0 | 33 | 138 | 309 | 9.36 | 27 | 103.05 |

m_{n} = 90 | 389.15 | 216.95 | 3090.2 | 32 | 134 | 300 | 9.37 | 23 | 102.53 |

m_{n} = 100 | 389.15 | 216.95 | 3409.5 | 32 | 134 | 300 | 9.37 | 18 | 101.69 |

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**MDPI and ACS Style**

Rubio, F.; Llopis-Albert, C.; Pedrosa, A.M.
Analysis of the Influence of Calculation Parameters on the Design of the Gearbox of a High-Power Wind Turbine. *Mathematics* **2023**, *11*, 4137.
https://doi.org/10.3390/math11194137

**AMA Style**

Rubio F, Llopis-Albert C, Pedrosa AM.
Analysis of the Influence of Calculation Parameters on the Design of the Gearbox of a High-Power Wind Turbine. *Mathematics*. 2023; 11(19):4137.
https://doi.org/10.3390/math11194137

**Chicago/Turabian Style**

Rubio, Francisco, Carlos Llopis-Albert, and Ana M. Pedrosa.
2023. "Analysis of the Influence of Calculation Parameters on the Design of the Gearbox of a High-Power Wind Turbine" *Mathematics* 11, no. 19: 4137.
https://doi.org/10.3390/math11194137