Analysis and optimization of time-varying meshing parameters of spiral bevel gear with consideration of load sensitivity

Abstract

The dynamic response of automotive transmission systems is significantly influenced by the meshing behavior of spiral bevel gears, which are subject to load variations. This paper establishes a meshing model considering load sensitivity to explore the relevant meshing parameters of spiral bevel gears that change with load during the meshing process. Load-bearing Tooth Contact Analysis (LTCA) is then performed using this model to analyze the impact of loading torque on meshing parameters. The actual contact ratio calculation method is discussed, and the time-varying meshing stiffness curve of the gear pair can be obtained. The investigation demonstrates that the meshing stiffness is higher in meshing intervals with a larger contact ratio, which is vital for the robustness of gears under different loads. Based on these findings, an optimization strategy has been developed for the actual contact ratio of the gear pair. The manipulability of tooth surface geometry is leveraged to optimize the angle η between the contact trajectory of the gear tooth surface and the tooth root by adjusting the direction of the tooth surface contact path to achieve the goal of the contact ratio optimization, and the sensitivity of installation errors and friction before and after optimization are analyzed. Simulation results indicate a certain difference between the theoretical and actual contact ratio, and the introduction of the actual contact ratio into dynamic analysis will ensure the more accurate calculation. As the load increases, the actual contact ratio and time-varying meshing stiffness will be increased accordingly, leading to an improved load distribution coefficient and reduced fluctuation values of time-varying meshing stiffness, which promotes transmission stability. This study provides an in-depth analysis of the meshing behavior of spiral bevel gears under varying loading conditions. The findings offer valuable guidance and application for subsequent research on dynamic response of automotive transmission system.

Keywords

Spiral bevel gears load sensitivity contact ratio time-varying meshing stiffness transmission error LTCA

Get full access to this article

View all access options for this article.

References

Liou

C-H

Lin

Oswald

, et al. Effect of contact ratio on spur gear dynamic load. DTIC Document. In: Proceedings of the ASME 1992 design technical conferences. 6th international power transmission and gearing conference: advancing power transmission into the 21st century, Scottsdale, AZ, 13–16 September, 1992.

Gosselin

Cloutier

Nguyen

. A general formulation for the calculation of the load sharing and transmission error under load of spiral bevel and hypoid gears. Mech Mach Theory 1995; 30(3): 433–450.

Peng

. Coupled multi-body dynamic and vibration analysis of hypoid and bevel geared rotor system. PhD Dissertation, University of Cincinnati, Cincinnati, OH, 2010.

Wang

. Gear mesh characteristics and dynamics of hypoid gear rotor system. PhD Dissertation, University of Alabama, Tuscaloosa, AL, 2002.

Wang

. Nonlinear time-varying gear mesh and dynamic analysis of hypoid and bevel geared rotor system. PhD Dissertation, University of Cincinnati, Cincinnati, OH, 2007.

Yang

Lim

. Dynamics of coupled nonlinear hypoid gear mesh and time-varying bearing stiffness systems. SAE Int J Passeng Cars Electron Mech Syst 2011; 4(2): 1039–1049.

Feng

, et al. An improved analytical method for calculating time-varying mesh stiffness of helical gears. Meccanica 2018; 53(4–5): 1131–1145.

Chaari

Baccar

Abbes

, et al. Effect of spalling or tooth breakage on gear mesh stiffness and dynamic response of a one-stage spur gear transmission. Eur J Mech A Solids 2008; 27(4): 691–705.

Wei

Lai

Qin

, et al. Analytical calculation model of tooth profile modification helical gear pair mesh stiffness. Vib Shock 2018; 37(10): 94–101+116.

10.

Ajmi

Velex

. A model for simulating the quasi-static and dynamic behaviour of solid wide-faced spur and helical gears. Mech Mach Theory 2005; 40(2): 173–190.

11.

Liu

. Research on the variation law of meshing stiffness of hypoid gears. J Aerosp Power 2010; 4: 957–962.

12.

Zhu

Chen

Zhu

, et al. Modeling and dynamic analysis of spiral bevel gear coupled system of intermediate and tail gearboxes in a helicopter. Proc IMechE, Part D: J Automobile Engineering 2021; 235(22): 5975–5993.

13.

Tan

Chen

Liang

, et al. Theoretical and experimental analyses of spiral bevel gears with two contact paths. Proc IMechE, Part C: J Mechanical Engineering Science 2018; 232(4): 665–676

14.

Cai

Hayashi

. The linear approximated equation of vibration of a pair of spur gears. J Mech Des 1994; 116(2): 558–564.

15.

Zhang

Qiu

. A method for solving the meshing stiffness of involute spur gears. J Xi’an Univ Archit Technol 1996; 28(2): 134–137.

16.

Wang

Chen

, et al. Optimization of high-efficiency tooth surface accuracy of spiral bevel gears considering machine-tool motion errors. Proc IMechE, Part E: J Process Mechanical Engineering 2023; 237(4): 1394–1406.

17.

Gosselin

Gagnon

Cloutier

. Accurate tooth stiffness of spiral bevel gear teeth by the finite strip method. ASME J Mech Des 1998; 120: 599–605.

18.

Mennem

. Parametrically excited vibrations in spiral bevel geared systems. PhD Dissertation, Purdue University, West Lafayette, IN, 2004, pp.82–90.

19.

Cheng

Lim

. Dynamics of hypoid gear transmission with non.linear time-varying mesh characteristics. J Mech Des 2003; 125(2): 373–382.

20.

Cao

Zhang

Fang

Analysis and design of aerospace spiral bevel gear transmission error. J Aerosp Power 2009; 24(11): 2618–2624.

21.

Cai

Wang

Yuan

. Research on twisting mesh stiffness characteristics of spiral bevel gear based on planning method. Mech Sci Technol 2010; 29(7): 866–870.

22.

Wang

Zhou

, et al. Research on meshing stiffness and parameter vibration stability of spiral bevel gear. J Aerosp Power 2010; 25(7): 1664–1669.

23.

Wang

, et al. Research on transmission error of spiral bevel gear based on finite element method. Vib Shock 2014; 33(14): 165–170.

24.

Vijayakar

. A combined surface integral and finite element solution for a three-dimensional contact problem. Int J Numer Methods Eng 1991; 31: 525–545.

25.

Jiang

Fang

Liu

. Analysis of multi-body gear contact characteristics in planetary transmission. J Mech Eng 2019; 55(15): 174–182.

26.

Fang

. Dynamic load calculation of quasi-hypoid gear transmission. Automot Eng 1994; 16(2): 92–97.

27.

Zhou

Tian

Ding

, et al. Study on time-varying meshing characteristics of quasi-hypoid gears based on finite element method. J Mech Eng 2016; 52(15): 36–43.

28.

Liu

Shi

Chen

, et al. Calculation of time-varying mesh stiffness of quasi-hypoid gear in automobile driving bridge. Vib Shock 2017; 36(20): 240–247.

29.

Zhang

Chen

Huang

. Influence of coincidence on nonlinear dynamic characteristics of gear pair. Comb Mach Tool Autom Manuf Technol 2014; 2: 40–43.

30.

Tang

. A new method for theoretical coincidence algorithm of involute cylindrical gears. Mach Des Manuf 2012; 2(2): 54–55.

31.

Tang

Xiao

. Actual coincidence based on rolling-grinding process. J Changsha Railw Coll 1996; 14(1): 63–67.

32.

Luo

Lin

, et al. Effect of root cutting on coincidence of involute oil pump gears. Mach Transm 2002; 27(3): 48–49.

33.

Yuan

. Determination of actual coincidence of spur gears. Mech Sci Technol 1995; 2: 65–67.

34.

Xiong

. Research on nonlinear dynamic characteristics of high-overlap spur gears considering rough tooth surface effect. PhD Dissertation, Hefei University of Technology, China, 2017.

35.

Lee

Lin

Oswald

, et al. Influence of linear profile modification and loading conditions on the dynamic tooth load and stress of high-contact-ratio spur gears. J Mech Des 1990; 113(4): 473–480.

36.

Kahraman

Blankenship

. Effect of involute contact ratio on spur gear dynamics. J Mech Des 1999; 121(1): 112–118.

37.

Wang

Howard

. Finite element analysis of high contact ratio spur gears in mesh. J Tribol 2005; 127(3): 469–483.

38.

Fang

Jiang

Song

. Performance study of high-overlap ratio gears. Gearing 1987; 1: 27–32.

39.

Niu

Liang

Deng

. A new method for designing low-noise spiral bevel gears with high overlap ratio. Mach Des 1996; 7: 36–37+41–42.

40.

. Design and dynamic analysis of high-overlap ratio planetary gear transmission system. PhD Dissertation, Nanjing University of Aeronautics and Astronautics, China, 2011.

41.

Bao

Zhu

, et al. Dynamic characteristics analysis of high-overlap ratio planetary gear transmission system. Mech Sci Technol 2012; 31(7): 1174–1179.

42.

Bao

Zhu

. Optimization design of parameters for high-overlap ratio planetary gear system. Mach Des Manuf 2011; 12: 41–43.

43.

. Strength and dynamic research on high-overlap ratio planetary gear transmission system. PhD Dissertation, Nanjing University of Aeronautics and Astronautics, China, 2015.

44.

Zhang

Wang

, et al. Study on the generation of spiral bevel gears with spherical involute tooth profile by the tracing line. Proc IMechE, Part C: J Mechanical Engineering Science 2012; 226(C4): 1097–1106.

45.

Liu

G-z

Shi

W-k

Chen

Z-y

. Finite element analysis of transmission error for hypoid gears considering installation error. J Jilin Univ Eng Technol Ed 2018; 48(4): 984–989.