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A generalised mathematical model of involute gears generated by rack type cutters

Year 2024, EARLY VIEW, 1 - 1

Berat Gürcan Şentürk , Mahmut Fetvacı

https://doi.org/10.2339/politeknik.1364954

Abstract

In this study, a mathematical model of beveloid gears is developed based on the vector expression of the cutting tool. Coordinate transformations, differential geometry and gear theory were applied to obtain the fabricated surfaces. In addition, the ranges of the helical design parameter are also examined. Depending on the angular displacement of the contact point on the line of action of the rack gear mechanism, a generalized expression is proposed to determine the boundary point between the involute profile and the end circle. Numerical examples were prepared to investigate the effect of design parameters on the fabricated gear geometry. For the accuracy and validity of the presented model, a comparison was made with a beveloid gear model in the literature. In addition, undercut and zero topland were investigated in various sections of spur and helical beveloid gears.

Keywords

Beveloid Gears Rack Cutter Undercutting Analysis Addendum Modification

References

  • [1] Liu, C.C. ve Tsay, C.B., “Tooth undercutting of beveloid gears”, J. Mech. Des., 123: 569-576, (2001).
  • [2] Brauer, J., “Analytical geometry of straight conical involute gears”, Mechanism and Machine Theory, 37: 127-141, (2002).
  • [3] Brauer, J., “A general finite element model of involute gears”, Finite Elements in Analysis and Design, 40: 1857-1872, (2004).
  • [4] Brecher, C., Brumm, M. and Henser, J., “Calculation of the tooth root load carrying capacity of beveloid gears”, Gear Technology, 31(4): 52-61, (2014).
  • [5] Mitome, K. I.. “Conical involute gear: part 1. design and production system”, Bulletin of JSME, 26: 299-305, (1983).
  • [6] Sun, R., Song, C., Zhu, C., Liu, S. and Wei, C., “Tooth surface modelling and mesh behaviors for paralleled beveloid gears”, Journal of Mechanical Design, 142: (054501(1-13), (2020).
  • [7] Sun, R., Song, C., Zhu, C., Wang, Y. and Liu, K., “Numerical study on contact force of paralleled beveloid gears using minimum potential energy theory”, The Journal of Strain Analysis for Engineering Design, 56: 249-264, (2021).
  • [8] Şentürk, B.G. and Fetvacı, M.C., “Modelling and undercutting analysis of beveloid gears”, Journal of the Faculty of Engineering and Architecture of Gazi University, 35: 901-916, (2020).
  • [9] Yazar, M. and Özdemir, A. "Eliptik düz dişlilerin bilgisayar destekli tasarımı ve CNC tel erozyon ile imalatı", Politeknik Dergisi, 13: 245-253, (2010).
  • [10] Şentürk B.G., "Beveloid dişlilerin matematik modellenmesi ve dişli temas analizi", Doktora tezi, İstanbul Üniversitesi-Cerrahpaşa Lisansüstü Eğitim Enstitüsü, (2020).
  • [11] Litvin, F.L. and Fuentes, A., “Gear Geometry and Applied Theory”, Cambridge University Press, UK, (2004).
  • [12] Chang, S. L., Tsay, C. B. and Wu, L. I., “Mathematical model and undercutting analysis of elliptical gears generated by rack cutters”, Mechanism and Machine Theory, 31: 879-890, (1996).
  • [13] Yang, S. C., “Mathematical model of a helical gear with asymmetric involute teeth and its analysis”, The International Journal of Advanced Manufacturing Technology, 26: 448-456, (2005).
  • [14] Rajesh, S., Marimuthu, P., Babu, P. D. and Venkatraman, R., “Balanced bending fatigue life for helical gear drives to enhance the power transmission capacity through novel rack cutters”, Engineering Failure Analysis, 144: 106989, (2023).
  • [15] Yang, H. C. and Pai, P. F., “Kinematic performance of a parabolic gear tooth with two parabolic coefficients”, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 231: 4431-4440, (2017).
  • [16] Hsueh-Cheng, Y. andHuang, Z. W., “Using variable modulus to modify a rack cutter and generate a gear pair”, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 236: 5342-5359, (2022).
  • [17] Ni, G., Zhu, C., Song, C., Shi, J. and Liu, S., “Effects of rack-cutter parabolic modification on loaded contact characteristics for crossed beveloid gears with misalignments”, International Journal of Mechanical Sciences, 141: 359-371, (2018).
  • [18] Gaoxiang, N., Chaosheng, S., Zifan, F. and Zhang, Z., “Effects of geometric design parameters and misalignments on contact ellipse of crossed beveloid gears”, Mech. Mach. Theory, 165: 104441, (2021).
  • [19] Cao, B., Li, G., Tao, Y. and Ran, Q., “Robust geometric parameter optimization of a crossed beveloid gear pair with approximate line contact”, Mech. Mach. Theory, 168: 104596, (2022).
  • [20] Zhu, C., Song, C., Lim, T. C. and Peng, T., “Pitch cone design and influence of misalignments on tooth contact behaviors of crossed beveloid gears”, Mech. Mach. Theory, 59: 48-64, (2013).
  • [21] Song, C., Zhou, Y., Zhu, C., Ni, G. and Liu, S., “Loaded Tooth Contact Analysis Of Intersected Beveloid And Cylindrical Involute Gear Pair With Small Shaft Angle”, Journal of Advanced Mechanical Design, Systems and Manufacturing, 12: 1–15, (2018).
  • [22] Brecher, C., Brumm, M. and Henser, J. ‘Validation Of The Tooth Root Load Carrying Capacity Calculation Of Beveloid Gears With Parallel Axes’, International Gear Conference 2014. Woodhead Publishing Limited, (2014).
  • [23] Song, C., Zhou, S., Zhu, C., Yang, X., Li, Z.and Sun, R., “Modeling and analysis of mesh stiffness for straight beveloid gear with parallel axes based on potential energy method”, Journal of Advanced Mechanical Design, Systems, and Manufacturing, 12 : JAMDSM0122, (2018).
  • [24] Sun, R., Song, C., Zhu, C., Liu, S. and Wei, C., “Tooth surface modelling and mesh behaviors for paralleled beveloid gears”, Journal of Mechanical Design, 142: 054501, (2020).
  • [25] Song, C., Zhu, C., Liu, H. and Ni, G, “Dynamic analysis and experimental study of a marine gearbox with crossed beveloid gears”, Mechanism and Machine Theory, 92: 17-28, (2015).
  • [26] Zhu, C., Liu, L., Song, C., Xiang, Y. and Liu, H., “Pitch cone design and tooth contact analysis of intersected beveloid gears for marine transmission”, Mechanism and Machine Theory, 82: 141-153, (2014).
  • [27] Liu, C.-C. and Tsay, C.-B., “Mathematical models and contact simulations of concave beveloid gears”, Journal of Mechanical Design, 124: 753–760, (2002).
  • [28] Komatsubara, H., Mitome, K. I. and Ohmachi, T., “Development of concave conical gear used for marine transmissions (1st report, principle of generating helical concave conical gear)”, JSME Mechanical Systems, Machine Elements and Manufacturing, 45: 371-377. (2002).
  • [29] Liu, C. C., Chen, Y. C. and Peng, Y. L., “Contact pattern simulation and stress analysis of intersected concave conical involute gear pairs generated by shaper cutters”, 14th IFToMM World Congress, Taiwan, 259-264, (2015).
  • [30] Liu, C. C., Chen, Y. C. and Lin, S. H., “Contact stress analysis of straight concave conical involute gear pairs with small intersected angles”, International MultiConference of Engineers and Computer Scientists, Hong Kong, 3: 17-19, (2010).
  • [31] Liu, S., Song, C., Zhu, C., Ni, G. and Ullah, N., “Concave and convex modifications analysis for skewed beveloid gears considering misalignments”, Mechanism and Machine Theory, 133: 127-149, (2019).
  • [32] Batista, M., “Analytical treatment of the geometry of involute gears”, 10.13140/RG.2.2.31057.66404, (2021).
  • [33] Roth, K., “Zahnradtechnik Evolventen-Sonderverzahnungen zur Getriebeverbesserung: Evoloid-, Komplement-, Keilschräg-, Konische-, Konus-, Kronenrad-, Torus-, Wälzkolbenverzahnungen, Zahnrad-Erzeugungsverfahren”, 3: Springer-Verlag, Berlin, (2013).
  • [34] Fetvacı, M., “Profil kaydırmalı silindirik evolvent dişli çarkların matematik modellenmesi”, Politeknik Dergisi, 23: 1339-1346, (2020).
  • [35] Karakoç, B., Uzun G. "Ergiyik Yığma Modelleme Yöntemi ile Üretilen Numunelerde Örme Yönteminin ve Baskı Yönünün Mukavemete Olan Etkisi", Politeknik Dergisi, DOI: 10.2339/politeknik.1262855
  • [36] Günay, M., Gündüz, S., Yılmaz, H., Yaşar, N. ve Kaçar, R., “PLA esaslı numunelerde çekme dayanımı için 3D baskı işlem parametrelerinin optimizasyonu”, Politeknik Dergisi, 23: 73-79. (2020).

Kremayer tipi takımla imal edilen evolvent dişli çarkların genelleştirilmiş matematik modeli

Year 2024, EARLY VIEW, 1 - 1

Berat Gürcan Şentürk , Mahmut Fetvacı

https://doi.org/10.2339/politeknik.1364954

Abstract

Bu çalışmada, kesici takımın vektör ifadesinden hareketle beveloid dişlilerin matematiksel modeli geliştirilmiştir. İmal edilen yüzeyleri elde etmek için koordinat dönüşümleri, diferansiyel geometri ve dişli teorisi uygulanmıştır. Ayrıca, evolvent tasarım parametresinin sınırları da incelenmiştir. Kremayer dişli mekanizmasının kavrama doğrusu üzerindeki temas noktasının açısal yer değiştirmesine bağlı olarak, evolvent profil ile uç daire arasındaki sınır noktasını belirlemek için genelleştirilmiş bir ifade önerilmiştir. Tasarım parametrelerinin imal edilen dişli geometrisi üzerindeki etkisini araştırmak için sayısal örnekler hazırlanmıştır. Sunulan modelin doğruluğu ve geçerliliği için literatürdeki bir beveloid dişli modeli ile karşılaştırma yapılmıştır. Ayrıca, düz ve helisel beveloid dişlilerin çeşitli kesitlerinde alttan kesme ve sivri tepe incelenmiştir.

Keywords

beveloid dişli kremayer takım alttan kesme analizi profil kaydırma

References

  • [1] Liu, C.C. ve Tsay, C.B., “Tooth undercutting of beveloid gears”, J. Mech. Des., 123: 569-576, (2001).
  • [2] Brauer, J., “Analytical geometry of straight conical involute gears”, Mechanism and Machine Theory, 37: 127-141, (2002).
  • [3] Brauer, J., “A general finite element model of involute gears”, Finite Elements in Analysis and Design, 40: 1857-1872, (2004).
  • [4] Brecher, C., Brumm, M. and Henser, J., “Calculation of the tooth root load carrying capacity of beveloid gears”, Gear Technology, 31(4): 52-61, (2014).
  • [5] Mitome, K. I.. “Conical involute gear: part 1. design and production system”, Bulletin of JSME, 26: 299-305, (1983).
  • [6] Sun, R., Song, C., Zhu, C., Liu, S. and Wei, C., “Tooth surface modelling and mesh behaviors for paralleled beveloid gears”, Journal of Mechanical Design, 142: (054501(1-13), (2020).
  • [7] Sun, R., Song, C., Zhu, C., Wang, Y. and Liu, K., “Numerical study on contact force of paralleled beveloid gears using minimum potential energy theory”, The Journal of Strain Analysis for Engineering Design, 56: 249-264, (2021).
  • [8] Şentürk, B.G. and Fetvacı, M.C., “Modelling and undercutting analysis of beveloid gears”, Journal of the Faculty of Engineering and Architecture of Gazi University, 35: 901-916, (2020).
  • [9] Yazar, M. and Özdemir, A. "Eliptik düz dişlilerin bilgisayar destekli tasarımı ve CNC tel erozyon ile imalatı", Politeknik Dergisi, 13: 245-253, (2010).
  • [10] Şentürk B.G., "Beveloid dişlilerin matematik modellenmesi ve dişli temas analizi", Doktora tezi, İstanbul Üniversitesi-Cerrahpaşa Lisansüstü Eğitim Enstitüsü, (2020).
  • [11] Litvin, F.L. and Fuentes, A., “Gear Geometry and Applied Theory”, Cambridge University Press, UK, (2004).
  • [12] Chang, S. L., Tsay, C. B. and Wu, L. I., “Mathematical model and undercutting analysis of elliptical gears generated by rack cutters”, Mechanism and Machine Theory, 31: 879-890, (1996).
  • [13] Yang, S. C., “Mathematical model of a helical gear with asymmetric involute teeth and its analysis”, The International Journal of Advanced Manufacturing Technology, 26: 448-456, (2005).
  • [14] Rajesh, S., Marimuthu, P., Babu, P. D. and Venkatraman, R., “Balanced bending fatigue life for helical gear drives to enhance the power transmission capacity through novel rack cutters”, Engineering Failure Analysis, 144: 106989, (2023).
  • [15] Yang, H. C. and Pai, P. F., “Kinematic performance of a parabolic gear tooth with two parabolic coefficients”, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 231: 4431-4440, (2017).
  • [16] Hsueh-Cheng, Y. andHuang, Z. W., “Using variable modulus to modify a rack cutter and generate a gear pair”, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 236: 5342-5359, (2022).
  • [17] Ni, G., Zhu, C., Song, C., Shi, J. and Liu, S., “Effects of rack-cutter parabolic modification on loaded contact characteristics for crossed beveloid gears with misalignments”, International Journal of Mechanical Sciences, 141: 359-371, (2018).
  • [18] Gaoxiang, N., Chaosheng, S., Zifan, F. and Zhang, Z., “Effects of geometric design parameters and misalignments on contact ellipse of crossed beveloid gears”, Mech. Mach. Theory, 165: 104441, (2021).
  • [19] Cao, B., Li, G., Tao, Y. and Ran, Q., “Robust geometric parameter optimization of a crossed beveloid gear pair with approximate line contact”, Mech. Mach. Theory, 168: 104596, (2022).
  • [20] Zhu, C., Song, C., Lim, T. C. and Peng, T., “Pitch cone design and influence of misalignments on tooth contact behaviors of crossed beveloid gears”, Mech. Mach. Theory, 59: 48-64, (2013).
  • [21] Song, C., Zhou, Y., Zhu, C., Ni, G. and Liu, S., “Loaded Tooth Contact Analysis Of Intersected Beveloid And Cylindrical Involute Gear Pair With Small Shaft Angle”, Journal of Advanced Mechanical Design, Systems and Manufacturing, 12: 1–15, (2018).
  • [22] Brecher, C., Brumm, M. and Henser, J. ‘Validation Of The Tooth Root Load Carrying Capacity Calculation Of Beveloid Gears With Parallel Axes’, International Gear Conference 2014. Woodhead Publishing Limited, (2014).
  • [23] Song, C., Zhou, S., Zhu, C., Yang, X., Li, Z.and Sun, R., “Modeling and analysis of mesh stiffness for straight beveloid gear with parallel axes based on potential energy method”, Journal of Advanced Mechanical Design, Systems, and Manufacturing, 12 : JAMDSM0122, (2018).
  • [24] Sun, R., Song, C., Zhu, C., Liu, S. and Wei, C., “Tooth surface modelling and mesh behaviors for paralleled beveloid gears”, Journal of Mechanical Design, 142: 054501, (2020).
  • [25] Song, C., Zhu, C., Liu, H. and Ni, G, “Dynamic analysis and experimental study of a marine gearbox with crossed beveloid gears”, Mechanism and Machine Theory, 92: 17-28, (2015).
  • [26] Zhu, C., Liu, L., Song, C., Xiang, Y. and Liu, H., “Pitch cone design and tooth contact analysis of intersected beveloid gears for marine transmission”, Mechanism and Machine Theory, 82: 141-153, (2014).
  • [27] Liu, C.-C. and Tsay, C.-B., “Mathematical models and contact simulations of concave beveloid gears”, Journal of Mechanical Design, 124: 753–760, (2002).
  • [28] Komatsubara, H., Mitome, K. I. and Ohmachi, T., “Development of concave conical gear used for marine transmissions (1st report, principle of generating helical concave conical gear)”, JSME Mechanical Systems, Machine Elements and Manufacturing, 45: 371-377. (2002).
  • [29] Liu, C. C., Chen, Y. C. and Peng, Y. L., “Contact pattern simulation and stress analysis of intersected concave conical involute gear pairs generated by shaper cutters”, 14th IFToMM World Congress, Taiwan, 259-264, (2015).
  • [30] Liu, C. C., Chen, Y. C. and Lin, S. H., “Contact stress analysis of straight concave conical involute gear pairs with small intersected angles”, International MultiConference of Engineers and Computer Scientists, Hong Kong, 3: 17-19, (2010).
  • [31] Liu, S., Song, C., Zhu, C., Ni, G. and Ullah, N., “Concave and convex modifications analysis for skewed beveloid gears considering misalignments”, Mechanism and Machine Theory, 133: 127-149, (2019).
  • [32] Batista, M., “Analytical treatment of the geometry of involute gears”, 10.13140/RG.2.2.31057.66404, (2021).
  • [33] Roth, K., “Zahnradtechnik Evolventen-Sonderverzahnungen zur Getriebeverbesserung: Evoloid-, Komplement-, Keilschräg-, Konische-, Konus-, Kronenrad-, Torus-, Wälzkolbenverzahnungen, Zahnrad-Erzeugungsverfahren”, 3: Springer-Verlag, Berlin, (2013).
  • [34] Fetvacı, M., “Profil kaydırmalı silindirik evolvent dişli çarkların matematik modellenmesi”, Politeknik Dergisi, 23: 1339-1346, (2020).
  • [35] Karakoç, B., Uzun G. "Ergiyik Yığma Modelleme Yöntemi ile Üretilen Numunelerde Örme Yönteminin ve Baskı Yönünün Mukavemete Olan Etkisi", Politeknik Dergisi, DOI: 10.2339/politeknik.1262855
  • [36] Günay, M., Gündüz, S., Yılmaz, H., Yaşar, N. ve Kaçar, R., “PLA esaslı numunelerde çekme dayanımı için 3D baskı işlem parametrelerinin optimizasyonu”, Politeknik Dergisi, 23: 73-79. (2020).
There are 36 citations in total.

Details

Primary Language Turkish
Subjects Machine Design and Machine Equipment
Journal Section Research Article
Authors

Berat Gürcan Şentürk 0000-0002-7325-0559

Mahmut Fetvacı 0000-0002-1622-1583

Early Pub Date February 27, 2024
Publication Date
Submission Date September 22, 2023
Published in Issue Year 2024 EARLY VIEW

Cite

APA Şentürk, B. G., & Fetvacı, M. (2024). Kremayer tipi takımla imal edilen evolvent dişli çarkların genelleştirilmiş matematik modeli. Politeknik Dergisi1-1. https://doi.org/10.2339/politeknik.1364954
AMA Şentürk BG, Fetvacı M. Kremayer tipi takımla imal edilen evolvent dişli çarkların genelleştirilmiş matematik modeli. Politeknik Dergisi. Published online February 1, 2024:1-1. doi:10.2339/politeknik.1364954
Chicago Şentürk, Berat Gürcan, and Mahmut Fetvacı. “Kremayer Tipi takımla Imal Edilen Evolvent dişli çarkların genelleştirilmiş Matematik Modeli”. Politeknik Dergisi, February (February 2024), 1-1. https://doi.org/10.2339/politeknik.1364954.
EndNote Şentürk BG, Fetvacı M (February 1, 2024) Kremayer tipi takımla imal edilen evolvent dişli çarkların genelleştirilmiş matematik modeli. Politeknik Dergisi 1–1.
IEEE B. G. Şentürk and M. Fetvacı, “Kremayer tipi takımla imal edilen evolvent dişli çarkların genelleştirilmiş matematik modeli”, Politeknik Dergisi, pp. 1–1, February 2024, doi: 10.2339/politeknik.1364954.
ISNAD Şentürk, Berat Gürcan - Fetvacı, Mahmut. “Kremayer Tipi takımla Imal Edilen Evolvent dişli çarkların genelleştirilmiş Matematik Modeli”. Politeknik Dergisi. February 2024. 1-1. https://doi.org/10.2339/politeknik.1364954.
JAMA Şentürk BG, Fetvacı M. Kremayer tipi takımla imal edilen evolvent dişli çarkların genelleştirilmiş matematik modeli. Politeknik Dergisi. 2024;:1–1.
MLA Şentürk, Berat Gürcan and Mahmut Fetvacı. “Kremayer Tipi takımla Imal Edilen Evolvent dişli çarkların genelleştirilmiş Matematik Modeli”. Politeknik Dergisi, 2024, pp. 1-1, doi:10.2339/politeknik.1364954.
Vancouver Şentürk BG, Fetvacı M. Kremayer tipi takımla imal edilen evolvent dişli çarkların genelleştirilmiş matematik modeli. Politeknik Dergisi. 2024:1-.

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