Asimetrik Dişlilerde Diş Dibi Gerilmelerinin Hesaplanması için Yeni Bir Yöntem

Öz

Günümüzde asimetrik diş profillerine sahip dişli tasarımları, dişlilerin diş kökü gerilmelerinin azaltılmasında önemli çözümler sunmaktadır. Asimetrik diş profiline sahip dişlilerde diş dibi gerilmelerini hesaplamak için sayısal, analitik ve deneysel çalışmalar yapılmasına rağmen literatürde standart veya basitleştirilmiş bir denklem veya ampirik ifadeye rastlanmamıştır. Bu çalışmada hem DIN3990 standardı hem de sonlu elemanlar yöntemi kullanılarak asimetrik diş profillerine sahip dişliler için yeni bir diş dibi gerilmesi hesaplama prosedürü geliştirilmiştir. Simetrik profilli dişlilerin diş dibi gerilmeleri geliştirilen sonlu elemanlar modeli ile belirlenmiş ve sonuçlar DIN 3990 standardı ile karşılaştırılarak doğrulanmıştır. Doğrulanmış sonlu elemanlar modeli kullanılarak, süren taraf basınç açısı 20° ile 30° arasında ve diş sayısı 18 ile 100 arasında değiştirilerek 66 farklı durum incelenmiş ve asimetrik profile sahip dişlilerdeki diş dibi gerilmeleri belirlenmiştir. Analizler sonucunda yeni bir asimetriklik faktör elde edilmiştir. Elde edilen faktörün DIN 3390 formülüne eklenmesiyle asimetrik profilli dişlilerin diş dibi gerilmelerinde kullanılmak üzere yeni bir denklem türetilmiştir. Bu denklem sayesinde tasarımcılar, asimetrik diş profiline sahip dişlilerde sonlu elemanlar analizine ihtiyaç duymadan diş dibi gerilmelerini yüksek hassasiyetle hesaplayabilecektir.

Anahtar Kelimeler

• Diş dibi gerilmesi
• Asimetrik düz dişli çarklar
• DIN3990
• Sonlu elemanlar yöntemi

A Novel Method for Tooth Bending Stress Calculation of Gears with Asymmetric Teeth

Abstract

Today, gear designs with asymmetric tooth profiles offer essential solutions in reducing tooth root stresses of gears. Although numerical, analytical, and experimental studies are carried out to calculate the bending stresses in gears with asymmetric tooth profiles a standard or a simplified equation or empirical statement has not been encountered in the literature. In this study, a novel bending stress calculation procedure for gears with asymmetric tooth profiles is developed using both the DIN3990 standard and the finite element method. The bending stresses of gears with symmetrical profile were determined by the developed finite element model and was verified by comparing the results with the DIN 3990 standard. Using the verified finite element model, by changing the drive side pressure angle between 20° and 30° and the number of teeth between 18 and 100, 66 different cases were examined and the bending stresses in gears with asymmetric profile were determined. As a result of the analysis, a new asymmetric factor was derived. By adding the obtained asymmetric factor to the DIN 3390 formula, a new equation has been derived to be used in tooth bending stresses of gears with asymmetric profile. Thanks to this equation, designers will be able to calculate tooth bending stresses with high precision in gears with asymmetric tooth profile without the need for finite element analysis.

Keywords

• Gear bending stress
• Asymmetric spur gears
• DIN3990
• Finite element method

Kaynakça

1. Cavdar K., Karpat F., Babalik FC. Computer aided analysis of bending strength of involute spur gears with asymmetric profile. Journal of Mechanical Design 2005; 127: 477-484.
2. Costopoulos TN., Spitas CA. Optimum gear tooth geometry for minimum fillet stress using BEM and experimental verification with photoelasticity. Journal of Mechanical Design 2016; 128(5): 1159-1164.
3. Costopoulos TN., Spitas V. Reduction of gear fillet stresses by using one-sided involute asymmetric teeth, Mechanism and Machine Theory 2009; 44: 1524-1534.
4. Demet SM., Ersoyoǧlu AS. Fatigue fracture behaviour of asymmetric spur gear tooth under cyclic loading. Procedia Structural Integrity 2018; 13: 2030-2035.
5. Dharashivkar NS., Sondur VB., Joshi KD. 3D photoelastic and finite element analysis of asymmetric involute spur gear. International Conference on Electrical, Electronics, and Optimization Techniques ICEEOT 2016; 1838-1842.
6. DIN3990. Calculation of load capacity of cylindrical gears: Calculation of tooth strength. 1987.
7. Dogan O., Yılmaz TG. Karpat F. Stress analysis of involute spur gears with different parameters by finite element and graphical method. Journal of the Faculty of Engineering and Architecture of Gazi University 2018; 33:1493-1504.
8. Drago RJ. An Improvement in the conventional analysis of gear tooth bending fatigue strength. American Gear Manufacturers Association 1982; 229-224.

9. Francesco GD., Marini S. Asymmetric teeth: Bending stress calculation, Gear Technology 2007; 24: 52-55.
10. He R., Tenberge P., Xu X., Li H., Uelpenich R., Dong P., Wang S. Study on the optimum standard parameters of hob optimization for reducing gear tooth root stress. Mechanism and Machine Theory 2021; 104128.
11. ISO 6336 – Calculation of load capacity of spur and helical gears - Application for industrial gears, 2002.
12. Kalay OC., Dogan O., Yılmaz TG. Yuce C., Karpat F. A comparative experimental study on the impact strength of standard and asymmetric involute spur gears. Measurement 2020; 172: 108950.
13. Kapelevich A. Geometry and design of involute spur gears with asymmetric teeth. Mechanism and Machine Theory 2000; 35: 117-130.
14. Karpat F., Cavdar K., Babalik FC. Computer aided analysis of involute spur gears with asymmetric teeth. VDI Berichte 2005; 145-163.
15. Karpat F., Dogan O., Ekwaro-Osire S., Yuce C. A novel method for calculation gear tooth stiffness for dynamic analysis of spur gears with asymmetric teeth. ASME International Mechanical Engineering Congress & Exposition 2014; 1-8.
16. Karpat F., Ekwaro-Osire S., Cavdar K., Babalik FC. Dynamic analysis of involute spur gears with asymmetric teeth. International Journal of Mechanical Sciences 2008; 50: 1598-1610.
17. Karpat F., Yuce C., Dogan O. Experimental measurement and numerical validation of single tooth stiffness for involute spur gears. Measurement 2020; 150: 107043.
18. Keçici A., Ünüvar A. Investigation of the effect of pressure angle on gear performance in asymmetric gears. Meccanica 2021; 56: 2919-2933.
19. Kumar VS., Muni V., Muthuveerappan G. Optimization of asymmetric spur gear drives to improve the bending load capacity. Mechanism and Machine Theory 2008; 43: 829-858.
20. Lisle TJ., Shaw BA. Frazer, RC. Frazer. External spur gear root bending stress: A comparison of ISO 6336: 2006,
21. AGMA 2101-D04, ANSYS finite element analysis and strain gauge techniques. Mechanism and Machine Theory 2017; 111: 1-9.
22. Litvin FL., Lian Q., Kapelevich A. Asymmetric modified spur gear drives: Reduction of noise, localization of contact, simulation of meshing and stress analysis. Computer Methods in Applied Mechanics and Engineering 2000; 188: 363-390.
23. Marimuthu P., Muthuveerappan G. Investigation of load carrying capacity of asymmetric high contact ratio spur gear based on load sharing using direct gear design approach. Mechanism and Machine Theory 2016; 96: 52-74.
24. Pedersen NL. Improving bending stress in spur gears using asymmetric gears and shape optimization. Mechanism and Machine Theory 2010; 45: 1707-1720.
25. Pramono AS., Rizal MZ. Influence of asymmetric factor on spur gears to dynamic bending stress. IOP Conf. Series: Materials Science and Engineering 2021; 1034: 012010.
26. Sekar RP., Muthuveerappan G. Estimation of tooth form factor for normal contact ratio asymmetric spur gear tooth. Mechanism and Machine Theory 2015; 90: 187-218.
27. Spitas V., Spitas C. Optimizing involute gear design for maximum bending strength and equivalent pitting resistance. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 2007; 221: 479-488.
28. Thomas B., Sankaranarayanasamy K., Ramachandra S., Kumar SPS. Search method applied for gear tooth bending stress prediction in normal contact ratio asymmetric spur gears. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 2018; 232: 4647-4663.
29. Yılmaz TG., Karadere G., Karpat F. A numerical analysis of hybrid spur gears with asymmetric teeth: stress and dynamic behavior. Machines 2022; 10(11): 1056.
30. Wen Q., Du Q., Zhai X. A new analytical model to calculate the maximum tooth root stress and critical section location of spur gear. Mechanism and Machine Theory 2018; 128: 275-286.