^{1}School of Construction Machinery, Chang'an University, Xi’an710064, China ^{2}Shantui Construction Machinery Co., Ltd., Jining272073, China ^{3}Department of Mechanical Engineering, Northwestern University, Evanston, IL60208, USA ^{4}School of Aeronautics and Astronautics, Sichuan University, Chengdu610065, China ^{5}College of Mechanical Engineering, Chongqing University, Chongqing400044, China

To assess the meshing quality of spiral bevel gears, the static meshing characteristics are usually checked under different contact paths to simulate the deviation in the footprint from the design point to the heel or toe of the gear flank caused by the assembly error of two gear axes. However, the effect of the contact path on gear dynamics under lubricated conditions has not been reported. In addition, most studies regarding spiral bevel gears disregard the lubricated condition because of the complicated solutions of mixed elastohydrodynamic lubrication (EHL). Hence, an analytical friction model with a highly efficient solution, whose friction coefficient and film thickness predictions agree well with the results from a well-validated mixed EHL model for spiral bevel gears, is established in the present study to facilitate the study of the dynamics of lubricated spiral bevel gears. The obtained results reveal the significant effect of the contact path on the dynamic response and meshing efficiency of gear systems. Finally, a comparison of the numerical transmission efficiency under different contact paths with experimental measurements indicates good agreement.

Fund: National Natural Science Foundation of China(52005047);Natural Science Basic Research Plan in Shaanxi Province of China(2020JQ-367);China Postdoctoral Science Foundation(2020M672129);Fundamental Research Funds for the Central Universities, CHD(300102250301)

Corresponding Authors:
Wei PU
E-mail: Pwei@scu.edu.cn

About author: Wei CAO. He received his Ph.D. degree in mechanical engineering from Sichuan University, China, in 2019. Now, he is a lecturer at School of Construction Machinery, Chang’an University. His research interests are tribology, dynamics, and fatigue in transmission systems.|Tao HE. He received his Ph.D. degree in mechanical engineering from Sichuan University, China, in 2017. Now, he is a postdoctor researcher at Northwestern University, IL, USA. His research interests include multi- phsical interfacial science, tribology, and superlubricity of the mechanical transmission and manufacturing systems.|Wei PU. He received his Ph.D. degree in mechanical engineering from Sichuan University, China, in 2017. He currently is a professor at School of Aeronautics and Astronautics, Sichuan University and a visiting scholar in Massachusetts Institute of Technology, USA. His interests include the lubrication and friction in transmission components.|Ke XIAO. He received his Ph.D. degree in mechanical engineering from Chongqing University, China, in 2012. He is an associate research fellow at College of Mechanical Engineering, Chongqing University, China. His research interests are the nonlinear dynamic of flexible drive mechanism and system.

Fig. 1Schematic illustration of contact paths and initial contact point.

Fig. 2Contact and assembling relationship between pinion and gear.

Fig. 3Schematic illustration of spiral bevel gear train.

Fig. 4Dynamic mesh model.

Fig. 5Schematic illustration of tapered roller bearing.

Gear parameter

Pinion (mm)

Gear (mm)

Number of teeth

15

44

Module (mm)

5.8

Tooth width (mm)

43

Average pressure angle (°)

20

Mean spiral angle (°)

30

Shaft angle (°)

90

Face angle (°)

22.17

72.83

Pitch angle (°)

18.82

71.18

Root angle (°)

17.17

67.84

Outside diameter (mm)

100.08

257.08

Hand of spiral

Left

Right

Mass (kg)

1.40

6.20

Inertia (kg·m^{2})

1.23 × 10^{-3}

6.23 × 10^{-2}

Backlash (μm)

75

Tapered roller bearing

13

Number of tapered roller elements, Z

Bearing contact angle, ${\alpha}_{1}$ (°)

15

Effective stiffness of inner ring-rolling-outer ring, ${k}_{\text{n}}$ (N·m^{-1})

4 × 10^{8}

Table 1Gear pair and bearing parameters.

Fig. 6Flowchart of methodology of dynamics and efficiency of spiral bevel gear.

Contact path

Toe contact

Middle contact

Heel contact

$\Delta V$

1.084

0.0248

-1.943

$\Delta H$

-0.113

0.155

1.194

Table 2$\Delta V$ and $\Delta H$ values for different contact paths (mm).

Fig. 7Three contact paths and contact ellipses.

Fig. 8Mesh stiffness and kinematic error in mesh cycle for different contact paths.

Fig. 9Contact radii of pinion and gear for mesh cycle.

Fig. 10Curvature radii along minor and major axis of contact ellipse in mesh cycle.

Fig. 11Frictional moment arm of pinion and gear during engaging cycle.

Fig. 12Maximum and minimum DTE amplitude during pinion speed sweep.

Fig. 13Time histories of DTE at resonant speed.

Fig. 14Maximum and minimum mesh force amplitudes during pinion speed sweep.

Fig. 15Time histories of dynamic mesh force at resonances.

Fig. 16Time histories of maximum Hertzian pressure at resonances.

Fig. 17Contact stress distributions under maximum Hertzian pressure for different contact paths.

Fig. 18Response of radial displacement of pinion and gear under different contact paths.

Fig. 19Response of axial displacement of pinion and gear under different contact paths.

Fig. 20Maximum and minimum radial and axial bearing forces (bearing A) during pinion speed sweep.

Fig. 21Maximum and minimum lateral and axial bearing forces (bearing C) during pinion speed sweep.

Fig. 22Variations in friction coefficient obtained from different models.

Fig. 23Variation in film thickness in mesh cycle under different contact paths.

Fig. 24Predictions of meshing efficiency during mesh cycle under static condition.

Fig. 25Dynamic meshing efficiency during pinion speed sweep.

Fig. 26History of (a) meshing efficiency and (b) dynamic friction coefficient in mesh cycle.

Gear parameter

Pinion (mm)

Gear (mm)

Number of teeth

25

34

Module (mm)

5.0

Tooth width (mm)

30

Average pressure angle (°)

20

Mean spiral angle (°)

35

Shaft angle (°)

90

Face angle (°)

39.63

56.00

Pitch angle (°)

36.33

53.67

Root angle (°)

34.00

20.37

Outside diameter (mm)

105.50

105.50

Hand of spiral

Left

Right

Mass (kg)

1.64

3.81

Inertia (Kg·m^{2})

3.45 × 10^{-3}

1.36 × 10^{-2}

Backlash (μm)

75

Table 3Gear pair parameters.

Contact path

Toe contact

Middle contact

Heel contact

$\Delta V$

0.860

-1.491

-1.644

$\Delta H$

-0.292

0.106

1.370

Table 4$\Delta V$ and $\Delta H$ value for different contact paths (mm).

Effective elastic modulus (GPa)

Density of lubricant (kg/L)

Lubricant viscosity (mm^{2}/s)

Viscosity-pressure coefficient (1/Pa)

RMS roughness (μm)

219.78

0.89

150 (40 °C) 14.7 (100 °C)

2.57 × 10^{-8}

0.5

Table 5Parameters of gear materials, lubricant, and roughness.

Fig. 27Gear transmission system test rig and mounted gears.

Fig. 28Transmission efficiencies of (a) tested results and numerical results: (b) toe contact, (c) middle contact, and (d) heel contact.

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