Double circular-arc spiral bevel gears for nutation drive are characterized by their compact structure, small size, large transmission ratio, and exceptional load-bearing capacity. The gear is suitable for industrial robotic joint reducers. Grease lubrication is widely adopted to avoid the need for bulky oil reservoirs due to the spatial constraints in robotic joint reducers. This paper establishes for the first time a grease-mixed elastohydrodynamic lubrication model of double circular-arc spiral bevel gears for nutation drive based on the Ostwald grease lubrication rheological model. The model's accuracy is experimentally validated. The effects of shaft misalignment, rheological index, and rotational speed on the lubrication characteristics of the gear are analyzed. The results show that an appropriate positive shaft misalignment can increase film thickness, suppress the peaks of pressure and temperature rise, and mitigate edge contact effects. Moreover, a decrease in the rheological index weakens the shear-thinning effects of the grease, thereby reducing the film thickness and increasing the risk of asperity squeezing. Boundary lubrication may occur at m = 0.75. A decline in rotational speed n shifts the lubrication regime from full-film to mixed lubrication. At n = 10 rpm, the necking phenomena of the film thickness disappear, resulting in oil film rupture and a transition of the gear into mixed lubrication.
CaiYYaoLZhangJ, et al.Feasibility analysis of using a two-stage nutation drive as joint reducer for industrial robots. J Mech Sci Technol2019; 33: 1799–1807.
2.
LingZZhaoLXiaoD, et al.A novel strain wave gear reducer with double flexsplines. Actuators2023; 12: 13.
3.
TangWHuangYWuY, et al.Geometry design, meshing analysis and error influences of face-hobbed cycloidal bevel gears with double circular-arc profile for a nutation drive. Mech Mach Theory2022; 176: 105001.
4.
ZhongATangWYaoL, et al.A running-in strategy based on an efficient wear simulation model for double circular-arc spiral bevel gears. Proc Inst Mech Eng Part J: J Eng Tribol2024; 238: 224–240.
DowsonDHigginsonGR. Elasto-hydrodynamic lubrication: the fundamentals of roller and gear lubrication. Oxford, New York: Pergamon Press, 1966, p.235.
7.
LarssonR. Transient non-Newtonian elastohydrodynamic lubrication analysis of an involute spur gear. Wear1997; 207: 67–73.
8.
OuyangTHuangGChenJ, et al.Investigation of lubricating and dynamic performances for high-speed spur gear based on tribo-dynamic theory. Tribol Int2019; 136: 421–431.
9.
KolivandMLiSKahramanA. Prediction of mechanical gear mesh efficiency of hypoid gear pairs. Mech Mach Theory2010; 45: 1568–1582.
10.
WangZZhengP. Numerical study on mixed thermal-elastohydrodynamic lubrication of grease lubricating lubricated spiral bevel gear for heavy-duty unmanned aerial vehicle. Proc Inst Mech Eng Part J: J Eng Tribol2023; 237: 85–102.
11.
SunXLiuYZhaoY, et al.EHL Analysis of spiral bevel gear pairs considering the contact point migration due to deformation under load. Math Probl Eng2020; 2020: 2047876.
12.
CaoWHeTPuW, et al.Dynamics of lubricated spiral bevel gears under different contact paths. Friction2022; 10: 247–267.
13.
MohammadpourMTheodossiadesSRahnejatH, et al.Non-Newtonian mixed thermo-elastohydrodynamics of hypoid gear pairs. Proc Inst Mech Eng, Part J: J Engineering Tribol2018; 232: 1105–1125.
14.
PanLYinZLinB, et al.Non-Newtonian thermal elastohydrodynamic mixed lubrication in elliptical contact for nutation-driven double circular-arc spiral bevel gear pair. Proc Inst Mech Eng Part J: J Eng Tribol2024; 238: 1121–1137.
15.
LinBPanLTangJ, et al.Lubrication performance and wear mechanism of double-circular-arc spiral bevel gears for nutation drive in mixed lubrication. Lubr Sci2025; 37: 222–236.
16.
WuMHanXTaoY, et al.A mixed EHL analysis method for grease and formulas for film thickness and asperity load. Tribol Lett2022; 70: 28.
17.
CannPM. Starvation and reflow in a grease-lubricated elastohydrodynamic contact. Tribol Trans1996; 39: 698–704.
18.
LiXGuoFPollG, et al.Grease film evolution in rolling elastohydrodynamic lubrication contacts. Friction2021; 9: 179–190.
19.
PoonSY. An experimental study of grease in elastohydrodynamic lubrication. J Lubr Technol1972; 94: 27–34.
20.
YooJKimK. Numerical analysis of grease thermal elastohydrodynamic lubrication problems using the Herschel-Bulkley model. Tribol Int1997; 30: 401–408.
21.
GaoHQingLMaG, et al.Numerical investigation into effects of rheological properties on grout flow in rock fracture using Herschel-Bulkley model. Eng Geol2024; 329: 107402.
22.
TichyJA. Hydrodynamic lubrication theory for the Bingham plastic flow model. J Rheol1991; 35: 477–496.
23.
SamantaBDasGRayS, et al.Laminar planar hydraulic jump during free surface flow of Bingham plastic liquid. Chem Eng Sci2024; 284: 119505.
24.
WangDYangJWeiP, et al.A mixed EHL model of grease lubrication considering surface roughness and the study of friction behavior. Tribol Int2021; 154: 106710.
25.
WuXTangWZhongA, et al.Kinematic analysis, structural design and prototype verification for the single-stage nutation drive. Proc Inst Mech Eng, Part C: J Mech Eng Sci2025; 239: 4335–4349.
26.
YaoLGuBHaungS, et al.Mathematical modeling and simulation of the external and internal double circular-arc spiral bevel gears for the nutation drive. J Mech Des2010; 132: 021008.
27.
LinZYaoLZhangJ, et al.Tooth contact analysis with latent error of double circular-arc spiral bevel gears for industrial robot joint nutation drive. J Braz Soc Mech Sci Eng2019; 42: 10.
28.
LitvinFLFuentesA. Gear Geometry and Applied Theory. Cambridge: Cambridge University Press, 2004.
29.
VulloV. Spiral bevel gears and hypoid gears. In: VulloV (eds) Gears: volume 1: geometric and kinematic design. Cham: Springer International Publishing, 2020, pp.569–694.
30.
LiuSWangQLiuG. A versatile method of discrete convolution and FFT (DC-FFT) for contact analyses. Wear2000; 243: 101–111.
31.
KogutLEtsionI. Elastic-Plastic contact analysis of a sphere and a rigid flat. J Appl Mech2002; 69: 657–662.
32.
RoelandsCJAVlugterJCWatermanHI. The viscosity-temperature-pressure relationship of lubricating oils and its correlation with chemical constitution. J Basic Eng1963; 85: 601–607.
33.
BairSWinerWO. A rheological model for elastohydrodynamic contacts based on primary laboratory data. J Lubr Technol1979; 101: 258–264.
34.
HoupertLFlamandLBertheD. Rheological and thermal effects in lubricated E.H.D. Contacts. J Lubr Technol1981; 103: 526–532.
35.
ZhangKPengXZhangY, et al.Numerical thermal analysis of grease-lubrication in limited line contacts considering asperity contact. Tribol Int2019; 134: 372–384.
36.
PuWWangJZhuD. Progressive mesh densification method for numerical solution of mixed elastohydrodynamic lubrication. J Tribol2015; 138: 021502.
37.
LiuSWangQLiuG. A versatile method of discrete convolution and FFT (DC-FFT) for contact analyses. Wear2000; 243: 101–111.
38.
WuZXuYLiuK, et al.The study on grease mixed-lubrication model of point contact pair. J Mech Eng2022; 58: 145–153.
39.
BordenetLDalmazGChaomleffelJ, et al.A study of grease film thicknesses in elastorheodynamic rolling point contacts. Lubr Sci1990; 2: 273–284.
40.
WuJ. Simulation of rough surfaces with FFT. Tribol Int2000; 33: 47–58.
