# Comparison of Selected Parameters of a Planetary Gearbox with Involute and Convex–Concave Teeth Flank Profiles

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Geometry of the Teeth Flanks

_{AC}for the part of the contact path defined using the arc AC and r

_{CE}for the part of the contact path defined by the arc CE. The arc centers S

_{AC}and S

_{CE}are set by the coordinates x

_{AC1}and y

_{AC1}, and x

_{CE2}and y

_{CE2}, respectively. The points A and E define the end points of the path of contact. The geometric parameters affecting the relation between the pressure angle at various points of the path of contact α and the angle of the gear rotation between pressure angles of two arbitrary points ϕ(α) are shown in Figure 2.

_{1}or O

_{2}for each coordinate system, respectively. The division of the contact path into parts divided into circular arcs AC and CE requires the division of all geometric and other gear pair parameters to analogous parts, which is defined according to the corresponding parts of the contact path curve [45].

^{4}module multiples.

## 3. Reduced Planetary Gearbox Parameters

_{23}= 0.45 mm, the module of the output part of the stage had to be m

_{45}= 0.5 mm. The teeth number and the moduli of the gearbox gears are defined in Table 1.

_{035}was defined as:

_{015}was defined as:

_{i}= 0.99 and the efficiency of the external gear contact η

_{e}= 0.98 according to the equations:

_{A}= 1 and the relative input torque T

_{A}= 1. The results of the calculation presented in Table 2 were used for the contact pressures calculation.

_{2}and S

_{1}to the ring gear 5’, and just about 6% of the power flowed directly to the output ring gear 5” and then together to the output B.

## 4. Calculation of Planetary Gearbox Gear Pairs Parameters

#### 4.1. Geometry

_{23}= 0.45 mm and m

_{45}= 0.5 mm. The pressure angle at the contact point C was defined by the value α

_{C}= 20°.

_{a}and dedendum height h

_{f}of the involute external gears R

_{e}was defined considering the gearing module m as:

_{i}was defined to prevent the interference between the planet teeth and ring gear tip surface as:

_{AC}and r

_{CE}and the pressure angle at the contact point C defined by the value α

_{C}. The convex–concave condition of the tooth flank curve describes the inequality [44]:

_{AC}= r

_{CE}= 1.5 mm, which satisfied the Inequality (14).

#### 4.2. Contact Pressures

_{E}depends on the Young‘s moduli E

_{1}, E

_{2}and the Poisson’s ratios μ

_{1}, μ

_{2}of the pinion (indexed by 1) and gear (indexed by 2) material according the equation:

_{1}= μ

_{2}= 0.42 [48] and average Young’s moduli of E

_{1}= E

_{2}= 2930 MPa [49].

_{A}= 1 Nmm, the number of planet gears in each stage as λ = 3, and using the corresponding pressure angle α as:

_{red}was calculated according the radii of curvature of the pinion ρ

_{1}and the gear ρ

_{2}using the equation:

#### 4.3. Specific Sliding Ratios

_{ss1}< 1.0 is recommended, and v

_{ss1}< 0.5 is preferred, for good resistance to macro pitting, micro pitting, and scuffing [50].

_{r1}and the gear v

_{r2}defines the sliding velocity for the pinion v

_{s1}and the gear v

_{s2}, as well as the specific sliding ratios v

_{ss1}and v

_{ss2}according the equations [50]:

_{pinion}= 1 and the gear teeth numbers defined in Table 1.

## 5. Discussion

_{1}, O

_{2}, O

_{4}) is defined in Figure 6a,b,c, as well as in Figure 7a,b,c using the coordinates x = 0 and y = 0. The diameters of the addendum and dedendum circle is the same in both cases. The involute teeth flanks’ path of contact is represented in Figure 6 by the lines declined by the pressure angle α

_{w}= 20° passing through the pitch point C. The convex–concave teeth flanks path of the contact is represented in Figure 7 using circles, whose parts belong to the path of contact consisting of two circular arcs tangentially connected at the pitch point C. The pressure angle, which varies along the path of contact, takes on the value α

_{C}= 20° at the pitch point C lying on the pinion and gear centers’ link. The points A and E represent the start and end point of the path of contact.

_{pinion}between the pressure angles of two arbitrary points (A–E), which is shown in Figure 8 for the involute and convex–concave teeth flanks.

_{C-C}/p

_{INV}between the convex–concave (p

_{C-C}) and the involute (p

_{INV}) teeth flanks values.

_{ss}of the partial pinion-gear pairs that depend on the angle of the pinion rotation ϕ

_{pinion}between the pressure angles of two arbitrary points (A–E) are shown in Figure 9 for the involute and the convex–concave teeth flanks.

_{ssC-C}/v

_{ssINV}between the convex–concave (v

_{ssC-C}) and the involute (v

_{ssINV}) teeth flank values.

_{α}using the involute and convex–concave teeth flanks in the observed gearbox are presented in Table 5.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Kim, G.H.; Lee, J.W.; Seo, T.I. Durability Characteristics Analysis of Plastic Worm Wheel with Glass Fiber Reinforced Polyamide. Materials
**2013**, 6, 1873–1890. [Google Scholar] [CrossRef][Green Version] - Hlebanja, G.; Kulovec, S.; Hlebanja, J.; Duhovnik, J. On endurance of the S-shaped plastic gears. Int. Conf. Power Transm.
**2016**, 79–86. [Google Scholar] - Gola, A. Reliability analysis of reconfigurable manufacturing system structures using computer simulation methods. Eksploat. I Niezawodn. Maint. Reliab.
**2019**, 21, 90–102. [Google Scholar] [CrossRef] - Chen, Z.; Shao, Y. Dynamic simulation of planetary gear with tooth root crack in ring gear. Eng. Fail. Anal.
**2013**, 31, 8–18. [Google Scholar] [CrossRef] - Li, X.; Elasha, F.; Shanbr, S.; Mba, D. Remaining Useful Life Prediction of Rolling Element Bearings Using Supervised Machine Learning. Energies
**2019**, 12, 2705. [Google Scholar] [CrossRef][Green Version] - Kia, S.H.; Henao, H.; Capolino, G.A. Gear tooth surface damage fault detection using induction machine stator current space vector analysis. IEEE Trans. Ind. Electron.
**2015**, 62, 1866–1878. [Google Scholar] [CrossRef] - Sun, W.; Yao, B.; Zeng, N.; Chen, B.; He, Y.; Cao, X.; He, W. An Intelligent Gear Fault Diagnosis Methodology Using a Complex Wavelet Enhanced Convolutional Neural Network. Materials
**2017**, 10, 790. [Google Scholar] [CrossRef][Green Version] - Ruqiang, Y.; Yuning, Q.; Zhoudi, H.; Gao, R.X. Rolling bearing defect severity evaluation using recurrence plot entropy. In Proceedings of the 2011 IEEE International Instrumentation and Measurement Technology Conference, Binjiang, China, 10–12 May 2011. [Google Scholar] [CrossRef]
- Jedliński, Ł.; Jonak, J. A disassembly-free method for evaluation of spiral bevel gear assembly. Mech. Syst. Signal Process.
**2017**, 88, 399–412. [Google Scholar] [CrossRef] - Sawalhi, N.; Randall, R.B. Gear parameter identification in a wind turbine gearbox using vibration signals. Mech. Syst. Signal Process.
**2014**, 42, 368–376. [Google Scholar] [CrossRef] - Karioja, K.; Lahdelma, S.; Litak, G.; Ambrożkiewicz, B. Extracting periodically repeating shocks in a gearbox from simultaneously occurring random vibration. In Proceedings of the 15th International Conference on Condition Monitoring and Machinery Failure Prevention Technologies, CM/MFPT 2018, Nottingham, UK, 10–12 September 2018; pp. 456–464. [Google Scholar]
- Soualhi, A.; Medjaher, K.; Zerhouni, N. Bearing health monitoring based on Hilbert-Huang transform, support vector machine and regression. IEEE Trans. Instrum. Meas.
**2015**, 64, 52–62. [Google Scholar] [CrossRef][Green Version] - Marichal, G.N.; Del Castillo, M.L.; López, J.; Padrón, I.; Artés, M. An Artificial Intelligence Approach for Gears Diagnostics in AUVs. Sensors
**2016**, 16, 529. [Google Scholar] [CrossRef][Green Version] - Ambrożkiewicz, B.; Meier, N.; Guo, Y.; Litak, G.; Georgiadis, A. Recurrence-based diagnostics of rotary systems. IOP Conf. Ser.
**2019**, 710, 012014. [Google Scholar] [CrossRef] - Syta, A.; Jonak, J.; Jedliński, Ł.; Litak, G. Failure diagnosis of a gear box by recurrences. J. Vib. Acoust.
**2012**, 134, 41006. [Google Scholar] [CrossRef] - Hadryś, D.; Bąkowski, H.; Stanik, Z.; Kubik, A. Analysis of shaft wear in turbocharges of automotive vehicles. Transp. Prob.
**2019**, 14, 85–95. [Google Scholar] [CrossRef][Green Version] - Bąkowski, H. Wear mechanism of spheroidal cast iron piston ring-aluminum matrix composite cylinder liner contact. Arch. Metall. Mater.
**2018**, 63, 481–490. [Google Scholar] - Longwic, A.; Nieoczym, A.; Kordos, P. Evaluation of the combustion process in a spark-ignition engine based on the unrepeatability of the maximum pressure. IOP Conf. Ser. Mater. Sci. Eng.
**2018**, 421, 1–9. [Google Scholar] [CrossRef] - Makareviciene, V.; Matijosius, J.; Pukalskas, S.; Vegneris, R.; Kazanceva, I.; Kazancev, K. The exploitation and environmental characteristics of diesel fuel containing rapeseed butyl esters. Transport
**2013**, 28, 158–165. [Google Scholar] [CrossRef] - Balytskyi, A.; Abramek, K.F.; Stoeck, T.; Osipowicz, T. Diagnostic of degradation of the lock seal ring by the loss of combustion engine working gases. Mater. Sci.
**2014**, 50, 156–169. [Google Scholar] [CrossRef] - Ignaciuk, P.; Gil, L. Damages to injectors in diesel engines. Adv. Sci. Technol. Res. J.
**2014**, 8, 58–61. [Google Scholar] - Osipowicz, T.; Abramek, K.F.; Matuszak, Z.; Jaśkiewicz, M.; Ludwinek, K.; Poliak, M. The analysis of technical condition common rail fuel system components. In Proceedings of the 11th International Scientific and Technical Conference on Automotive Safety, Casta Papiernicka, Slovakia, 18–20 April 2018. [Google Scholar] [CrossRef]
- Szpica, D. Investigating fuel dosage non-repeatability of low-pressure gas-phase injectors. Flow Meas. Instrum.
**2018**, 59, 147–156. [Google Scholar] [CrossRef] - Glos, J.; Sejkorova, M. Tribo-diagnostics as an indicator and input for the optimization of vehicles preventive maintenance. In Proceedings of the 11th International Conference on Intelligent Technologies in Logistics and Mechatronics Systems (ITELMS’2016), Panevezys, Lithuania, 28–29 April 2016; pp. 83–89. [Google Scholar]
- Sejkorova, M.; Hurtova, I. Engine oil analysis-effective instrument to evaluate reliability of tractor engines. In Proceedings of the 18th International Scientific Conference Engineering for Rural Development, Jelgava, Latvia, 22–24 May 2019; pp. 971–976. [Google Scholar]
- Stoeck, T.; Abramek, K.F. Application of the polynomial interpolation method for determining performance characteristics of a diesel engine. Metrol. Meas. Syst.
**2014**, 21, 157–168. [Google Scholar] [CrossRef] - Tucki, K.; Mruk, R.; Orynycz, O.; Gola, A. The effects of pressure and temperature on the process of auto-ignition and combustion of rape oil and its mixtures. Sustainability
**2019**, 11, 3451. [Google Scholar] [CrossRef][Green Version] - Figlus, T. A method for diagnosing gearboxes of means of transport using multi-stage filtering and entropy. Entropy
**2019**, 21, 441. [Google Scholar] [CrossRef][Green Version] - Harman, B.N.; Opalic, M.; Veres, M. Deterministic and probabilistic methods in determination of correct mating cylindrical teeth profiles. Strojarstvo
**2011**, 53, 191–197. [Google Scholar] - Veres, M.; Nemcekova, M.; Marinkovic, A. Tooth flanks scoring resistance of non-involute teeth profiles in plane toothed cylindrical gears. FME Trans.
**2009**, 37, 103–106. [Google Scholar] - Hlebanja, G.; Hlebanja, J. S-gears: From Metal to Polymer Solution. In Advanced Gear Engineering; Goldfarb, V., Trubachev, E., Barmina, N., Eds.; Springer International Publishing Switzerland: Cham, Switzerland, 2018; pp. 255–269. [Google Scholar] [CrossRef]
- Zorko, D.; Kulovec, S.; Tavcar, J.; Duhovnik, J. Different teeth profile shapes of polymer gears and comparison of their performance. J. Adv. Mech. Des. Syst. Manuf.
**2017**, 11, 1–5. [Google Scholar] [CrossRef][Green Version] - Hlebanja, G.; Kulovec, S. Development of a Planocentric Gear Box Based on S-Gear Geometry. Kolloqu. Getr. Garch.
**2015**, 11, 205–216. [Google Scholar] - Arnaudov, K.; Karaivanov, D.P. Planetary Gear Trains; CRC Press: Boca Raton, FL, USA; New York, NY, USA; Taylor & Francis Group: London, UK, 2019. [Google Scholar]
- Fiorio, R.; Diez, S.V.; Sánchez, R.; D’hooge, D.R.; Cardon, L. Influence of Different Stabilization Systems and Multiple Ultraviolet A (UVA) Aging/Recycling Steps on Physicochemical, Mechanical, Colorimetric, and Thermal-Oxidative Properties of ABS. Materials
**2020**, 13, 212. [Google Scholar] [CrossRef][Green Version] - García-Domínguez, D.; Claver, J.; Camacho, A.M.; Sebastián, M.A. Considerations on the Applicability of Test Methods for Mechanical Characterization of Materials Manufactured by FDM. Materials
**2020**, 13, 28. [Google Scholar] [CrossRef][Green Version] - Gardyński, L.; Lonkwic, P. Testing polymer rollers memory in the context of passenger lift car comfort. J. Vibroeng.
**2014**, 16, 225–230. [Google Scholar] - Borazjani, S.; Belingardi, G. Development of an innovative design of a composite-sandwich based vehicle roof structure. Compos. Struct.
**2017**, 168, 522–534. [Google Scholar] [CrossRef] - Krzyżak, A.; Kucharczyk, W.; Gaska, J.; Szczepaniak, R. Ablative test of composites with epoxy resin and expanded perlite. Compos. Struct.
**2018**, 202, 978–987. [Google Scholar] [CrossRef] - Krzyżak, A.; Valis, D. Selected reliability measures of composites with natural fibres tested in climatic environment. In Proceedings of the International Conference on Military Technologies (ICMT 2015), Brno, Czech, 19–21 May 2015; pp. 81–87. [Google Scholar]
- Pieniak, D.; Wit-Rusieck, A.M.; Krzyżak, A.; Gil, L.; Krzysiak, Z. Adhesion tests of varnish coatings used on the surface of carbon fiber reinforced polimer compositions. Przemysł Chem.
**2019**, 98, 1619–1622. [Google Scholar] - Jiang, R.; Liu, T.; Xu, Z.; Park, C.B.; Zhao, L. Improving the Continuous Microcellular Extrusion Foaming Ability with Supercritical CO
_{2}of Thermoplastic Polyether Ester Elastomer through In-Situ Fibrillation of Polytetrafluoroethylene. Polymers**2019**, 11, 1983. [Google Scholar] [CrossRef] [PubMed][Green Version] - Gupta, K.; Jain, N.; Laubscher, R. Advanced Gear Manufacturing and Finishing; Academic Press: Cambridge, MA, USA, 2017. [Google Scholar]
- Veres, M.; Bosansky, M.; Gadus, J. Theory of Convex-Concave and Plane Cylindrical Gearing; Slovak University of Technology: Bratislava, Slovak, 2006. [Google Scholar]
- Brumercik, F.; Lukac, M.; Majchrak, M.; Krzysiak, Z.; Krzywonos, L. Teeth geometry and contact pressure calculation of external cycloidal gears. Sci. J. Sil. Univ. Technol. Ser. Transp.
**2018**, 101, 27–35. [Google Scholar] [CrossRef] - Wikimedia Commons. Available online: https://commons.wikimedia.org/wiki/File:Rearview_Mirror_Epicyclic_Gears.jpg (accessed on 1 June 2019).
- Looman, J. Zahnradgetriebe. Grundlagen, Konstruktionen, Anwendungen in Fahrzeugen; Springer: Berlin/Heidelberg, Germany, 1996. [Google Scholar]
- Polymer Science. Typical Poisson’s Ratios of Polymers at Room Temperature. Available online: https://polymerdatabase.com/polymer%20physics/Poisson%20Table.html (accessed on 1 June 2019).
- MatWeb. Overview of Materials for Nylon 6, Cast. Available online: http://matweb.com/search/DataSheet.aspx?MatGUID=8d78f3cfcb6f49d595896ce6ce6a2ef1 (accessed on 1 June 2019).
- Erichello, R. Gear Sliding. In Encyclopedia of Tribology; Wang, Q.J., Chung, Y.-W., Eds.; Springer: Boston, MA, USA, 2013. [Google Scholar] [CrossRef]
- Orokocky, R.; Bosansky, M.; Veres, M. The influence of geometrical parameter to sliding speed in K-K gears. In Proceedings of the 44th Conference of the Departments of Machine Elements and Mechanisms, Prague, Czech Republic, 9–10 September 2003; pp. 240–243. [Google Scholar]

**Figure 2.**Relation between the angles α and ϕr: 1—path of contact, 2—tooth flank profile, and 3—tooth flank profile evolute.

**Figure 3.**Rear-view mirror drive gearbox, reproduced from [46].

**Figure 6.**Geometric models of the gear pairs with involute teeth flanks: (

**a**) gear pair 1-2, (

**b**) gear pair 2-3, and (

**c**) gear pair 4-5.

**Figure 7.**Geometric models of the gear pairs with convex-concave teeth flanks: (

**a**) gear pair 1-2, (

**b**) gear pair 2-3, and (

**c**) gear pair 4-5.

**Figure 8.**Hertzian pressure p, which depends on angle ϕ

_{pinion}in the gear pairs with involute and convex–concave teeth flanks: (

**a**) gear pair 1-2, (

**b**) gear pair 2-3, and (

**c**) gear pair 4-5.

**Figure 9.**Specific sliding ratio v

_{ss}that depends on the angle ϕ

_{pinion}between the gear pairs with involute and convex–concave teeth flanks: (

**a**) gear pair 1-2, (

**b**) gear pair 2-3, and (

**c**) gear pair 4-5.

Gear | Teeth Number z (-) | Module m (mm) |
---|---|---|

Sun gear 1 | 15 | 0.45 |

Planet gear 2 | 18 | 0.45 |

Ring gear 3 | 51 | 0.45 |

Planet gear 4 | 18 | 0.50 |

Ring gear 5 | 48 | 0.50 |

Gearbox Element | Relative Speed ω (-) | Relative Torque T (-) | Relative Power P (-) |
---|---|---|---|

A | 1 | 1 | 1 |

B | −0.0142 | 51.9269 | −0.7376 |

C | 0.0000 | −52.9269 | 0.0000 |

1 | 1.0000 | −1.0000 | −1.0000 |

3 | 0.0000 | 52.9269 | 0.0000 |

5’ | −0.0142 | −48.8223 | 0.6935 |

5’’ | −0.0142 | −3.1046 | 0.0441 |

5 | −0.0142 | −51.9269 | 0.7376 |

S_{1} | 0.2273 | −4.1046 | −0.9329 |

S_{2} | 0.2273 | 4.1046 | 0.9329 |

Gear Pair | Point of the Path of Contact | Contact Pressure p [MPa] | Flank Type Ratio p _{C-C}/p_{INV} (-) | |
---|---|---|---|---|

Involute Teeth Flanks | Convex–Concave Teeth Flanks | |||

1-2 | A | 12.3086 | 3.6178 | 0.2939 |

B_{2}/B_{1} | 5.9996/8.4645 | 3.8418/5.4372 | 0.6407/0.6423 | |

C | 7.9379 | 7.9379 | 1.0000 | |

D_{1}/D_{2} | 7.9999/5.658 | 5.2443/3.6961 | 0.6555/0.6533 | |

E | 7.6604 | 3.4491 | 0.4502 | |

2-3 | A | 42.9729 | 8.7852 | 0.2044 |

B_{2}/B_{1} | 14.2245/20.0888 | 9.1338/12.9432 | 0.6421/0.6443 | |

C | 16.9851 | 16.9851 | 1.0000 | |

D_{1}/D_{2} | 16.2260/11.4516 | 9.5872/6.7698 | 0.5909/0.5912 | |

E | 8.1294 | 6.3054 | 0.7756 | |

4-5 | A | 41.7424 | 7.7110 | 0.1847 |

B_{2}/B_{1} | 12.8809/18.0976 | 7.9674/11.2820 | 0.6185/0.6234 | |

C | 15.3391 | 15.3391 | 1.000 | |

D_{1}/D_{2} | 14.7914/10.4142 | 8.4263/5.9422 | 0.5697/0.5706 | |

E | 7.2949 | 5.6199 | 0.7704 |

Gear Pair | Point of the Path of Contact | Slide Ratio v_{ss} (-) | Flank Type Ratio v _{ssC-C}/v_{ssINV} (-) | ||||
---|---|---|---|---|---|---|---|

Involute Teeth Flanks | Convex–Concave Teeth Flanks | ||||||

Pinion | Gear | Pinion | Gear | Pinion | Gear | ||

1-2 | A | −13.4628 | 0.9308 | −1.1526 | 0.5355 | 0.0856 | 0.5752 |

E | 0.8436 | −5.3937 | 0.5079 | −1.0323 | 0.6021 | 0.1914 | |

2-3 | A | −5.0521 | −0.3770 | −0.2891 | −0.1869 | 0.0572 | 0.4958 |

E | 0.2738 | 0.8348 | 0.1575 | 0.2242 | 0.5752 | 0.2686 | |

4-5 | A | −5.5229 | −0.3595 | −0.2574 | −0.1709 | 0.0466 | 0.4752 |

E | 0.2645 | 0.8467 | 0.1459 | 0.2047 | 0.5518 | 0.2418 |

Gear Pair | Contact Ratio ε_{α} (-) | |
---|---|---|

Involute Teeth Flanks | Convex–Concave Teeth Flanks | |

1-2 | 1.5055 | 1.1567 |

2-3 | 1.6893 | 1.1251 |

4-5 | 1.7017 | 1.1025 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Brumercik, F.; Lukac, M.; Caban, J.; Krzysiak, Z.; Glowacz, A.
Comparison of Selected Parameters of a Planetary Gearbox with Involute and Convex–Concave Teeth Flank Profiles. *Appl. Sci.* **2020**, *10*, 1417.
https://doi.org/10.3390/app10041417

**AMA Style**

Brumercik F, Lukac M, Caban J, Krzysiak Z, Glowacz A.
Comparison of Selected Parameters of a Planetary Gearbox with Involute and Convex–Concave Teeth Flank Profiles. *Applied Sciences*. 2020; 10(4):1417.
https://doi.org/10.3390/app10041417

**Chicago/Turabian Style**

Brumercik, Frantisek, Michal Lukac, Jacek Caban, Zbigniew Krzysiak, and Adam Glowacz.
2020. "Comparison of Selected Parameters of a Planetary Gearbox with Involute and Convex–Concave Teeth Flank Profiles" *Applied Sciences* 10, no. 4: 1417.
https://doi.org/10.3390/app10041417