Alın dişlilerin yığılı parametre ve sonlu elemenlar yöntemiyle modal davranışının incelenmesi

Öz

Dişli dinamiğinde, dişli dinamik kuvvetlerinin kesin olarak hesaplanması ve doğru modal davranış tahmini yüksek doğruluklu modelleme yöntemlerini gerektirmektedir. Bu çalışmada, bir alın dişli çiftinin doğrusal modal davranışı, belirlenmiş şartlar altında yığılı parametre ve sonlu elemanlar yöntemiyle incelenmektedir. Bu çalışmanın ana amacı yığılı parametre ve sonlu elemanlar yöntemiyle dişli çiftinin modal analiz sonuçlarını karşılaştırmaktır. Bunun için alın dişli çiftinin altı serbestlik dereceli dinamik modeli yığılı parametre yöntemiyle oluşturulmuştur. Bu yöntemde, dişli gövdeleri rijit kabul edilirken dişli temasları ve yatakları esnek kabul edilmiştir. Sonrasında, alın dişli çiftinin üç boyutlu katı modeli yüksek doğruluklu sayısal hesaplamalar için sonlu elemanlar yöntemiyle oluşturulmuştur. Sonlu elemanlar yönteminde, dişliler esnek üç boyutlu katı elemanlarla modellenmiştir. Bu modelde, dişli çifti temasında doğrusal olmama durumunun kaldırılması için temas rijit olarak basitleştirilmiştir. Doğal frekanslar ve mod şekilleri her iki yöntemde doğrusal modal analiz ile hesaplanmıştır. Her iki yöntemle elde edilen sonuçlar, yüksek frekanslarda görülen esnek dişli gövde modlarının sadece sonlu elemanlar yöntemiyle hesaplanabileceğini göstermektedir. Gövdenin hareket etmediği diş modları ise her iki yöntemle başarılı bir şekilde tespit edilebilmektedir.

Anahtar Kelimeler

• Dişli dinamiği
• alın dişliler
• sonlu elemanlar yöntemi
• yığılı parametre yöntemi
• modal analiz

Modal behaviour investigation of spur gears with lumped parameter and finite element methods

Öz

Computation of gear dynamic forces accurately and correct modal behaviour estimation require highly validated modelling techniques in gear dynamics. In this paper, linear modal behaviour of a spur gear pair under prescribed conditions is investigated with lumped parameter and finite element methods. The main aim of this study is to compare the modal analysis results of the spur gear pair with the lumped parameter and finite element methods. For this purpose, a six degrees of freedom dynamic model of a spur gear pair is created using the lumped parameter method. In this method, the gears are assumed to be rigid disks whereas the gear teeth contacts and bearings are considered as flexible, which are modelled with spring elements. Then, a 3D solid model of the spur gear pair is created using the finite element method for high fidelity numerical analyses. In the finite element method, the gears are modelled with flexible three-dimensional solid elements, which is one of the main differences between the two methods. To remove the nonlinearity in the gear pair system, the contact is simplified with a rigid bonding of nodes in the finite element model. The natural frequencies and mode shapes are calculated by linear modal analysis for both methods. The obtained results from the individual methods show that flexible gear body modes, which are seen at higher frequencies, can only be detected with the finite element method. The tooth modes in which the gear bodies acting as a rigid body can be detected successfully with the two methods.

Anahtar Kelimeler

• Gear dynamics
• spur gears
• finite element method
• lumped parameter method
• modal analysis

Kaynakça

1. Lynwander, P., Gear drive systems: Design and application, 20, CRC Press, (1983).
2. Gregory, R. W., Harris, S. L. & Munro, R. G., Dynamic behaviour of spur gears, Proceedings of the institution of mechanical engineers, 178, 207–218 (1963).
3. Tang, X., Zou, L., Yang, W., Huang, Y. & Wang, H., Novel mathematical modelling methods of comprehensive mesh stiffness for spur and helical gears, Applied Mathematical Modelling, 64, 524–540 (2018).
4. Özgüven, H. N. & Houser, D. R., Mathematical models used in gear dynamics—a review, Journal of Sound and Vibration, 121, 383–411 (1988).
5. Wang, J., Li, R. & Peng, X., Survey of nonlinear vibration of gear transmission systems, Applied Mechanics Reviews, 56, 309–329 (2003).
6. Velex, P., On the Modelling of Spur and Helical Gear Dynamic Behaviour. (2012).
7. Yang, J. & Dai, L., Survey of dynamics of planetary gear trains, International Journal of Materials and Structural Integrity, 1, 302–322 (2008).
8. Chen, S., Tang, J., Li, Y. & Hu, Z., Rotordynamics analysis of a double-helical gear transmission system, Meccanica, 51, 251–268 (2016).

9. Chen, S., Tang, J., Zhou, C. & Hu, Z., Modal and whirling analysis of coupled lateral and torsional vibration of herringbone gear, International Journal of Dynamics and Control, (2014).
10. Wang, C., Wang, S. R., Yang, B. & Wang, G. Q., Dynamic modeling of double helical gears, Journal of Vibration and Control, 24, 3989–3999 (2018).
11. Kang, M. R. & Kahraman, A., An experimental and theoretical study of the dynamic behavior of double-helical gear sets, Journal of Sound and Vibration, 350, 11–29 (2015).
12. Kahraman, A. & Singh, R., Non-linear dynamics of a spur gear pair, Journal of Sound and Vibration, 142, 49–75 (1990).
13. Kahraman, A. & Singh, R., Interactions between time-varying mesh stiffness and clearance non-linearities in a geared system, Journal of Sound and Vibration, 146, 135–156 (1991).
14. Theodossiades, S. & Natsiavas, S., Non-linear dynamics of gear-pair systems with periodic stiffness and backlash, Journal of Sound and Vibration, 229, 287–310 (2000).
15. Litak, G. & Friswell, M. I., Vibration in gear systems, Chaos, Solitons & Fractals, 16, 795–800 (2003).
16. Li, S. & Kahraman, A., A spur gear mesh interface damping model based on elastohydrodynamic contact behaviour, International Journal of Powertrains, 1, 4 (2011).
17. Fang, Y., Liang, X. & Zuo, M. J., Effects of friction and stochastic load on transient characteristics of a spur gear pair, Nonlinear Dynamics, 93, 599–609 (2018).
18. Wei, S., Han, Q. K., Dong, X. J., Peng, Z. K. & Chu, F. L., Dynamic response of a single-mesh gear system with periodic mesh stiffness and backlash nonlinearity under uncertainty, Nonlinear Dynamics, 89, 49–60 (2017).
19. Yang, Y., Cao, L., Li, H. & Dai, Y., Nonlinear dynamic response of a spur gear pair based on the modeling of periodic mesh stiffness and static transmission error, Applied Mathematical Modelling, 72, 444–469 (2019).
20. Yang, Y., Xu, M., Du, Y., Zhao, P. & Dai, Y., Dynamic analysis of nonlinear time-varying spur gear system subjected to multi-frequency excitation, Journal of Vibration and Control, 25, 1210–1226 (2019).
21. Shi, J. fei, Gou, X. feng & Zhu, L. yun., Modeling and analysis of a spur gear pair considering multi-state mesh with time-varying parameters and backlash, Mechanism and Machine Theory, 134, 582–603 (2019).
22. Cirelli, M., Valentini, P. P. & Pennestrì, E., A study of the non-linear dynamic response of spur gear using a multibody contact based model with flexible teeth, Journal of Sound and Vibration, 445, 148–167 (2019).
23. Xiao, Z., Zhou, C., Chen, S. & Li, Z., Effects of oil film stiffness and damping on spur gear dynamics, Nonlinear Dynamics, 96, 145–159 (2019).
24. Park, C. I., Dynamic behavior of the spur gear system with time varying stiffness by gear positions in the backlash, Journal of Mechanical Science and Technology, 34, 565–572 (2020).
25. Liu, F., Jiang, H., Liu, S. & Yu, X., Dynamic behavior analysis of spur gears with constant & variable excitations considering sliding friction influence, Journal of Mechanical Science and Technology, 30, 5363–5370 (2016).
26. Kahraman, A. & Blankenship, G. W., Experiments on nonlinear dynamic behavior of an oscillator with clearance and periodically time-varying parameters, Journal of Applied Mechanics, 64, 217–226 (1997).
27. Velex, P. & Cahouet, V., Experimental and Numerical Investigations on the Influence of Tooth Friction in Spur and Helical Gear Dynamics, Journal of Mechanical Design, 122, 515 (2000).
28. Parker, R. G., Vijayakar, S. M. & Imajo, T., Non-linear dynamic response of a spur gear pair: modelling and experimental comparisons, Journal of Sound and Vibration, 237, 435–455 (2000).
29. Parker, R. G., Agashe, V. & Vijayakar, S. M., Dynamic Response of a Planetary Gear System Using a Finite Element/Contact Mechanics Model, Journal of Mechanical Design, 122, 304 (2002).
30. Abousleiman, V. & Velex, P., A hybrid 3D finite element/lumped parameter model for quasi-static and dynamic analyses of planetary/epicyclic gear sets, Mechanism and Machine Theory, 41, 725–748 (2006).
31. Cooley, C. G., Parker, R. G. & Vijayakar, S. M., A Frequency Domain Finite Element Approach for Three-Dimensional Gear Dynamics, Journal of Vibration and Acoustics, 133, 041004 (2011).
32. Vinayak, H. & Singh, R., Multi-body dynamics and modal analysis of compliant gear bodies, Journal of Sound and Vibration, 210, 171–214 (1998).
33. Ambarisha, V. K. & Parker, R. G., Nonlinear dynamics of planetary gears using analytical and finite element models, Journal of Sound and Vibration, 302, 577–595 (2007).
34. Cooley, C. G. & Parker, R. G., A Review of Planetary and Epicyclic Gear Dynamics and Vibrations Research, Applied Mechanics Reviews, 66, 040804 (2014).