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Friction  2022, Vol. 10 Issue (2): 247-267    doi: 10.1007/s40544-020-0477-x
Research Article     
Dynamics of lubricated spiral bevel gears under different contact paths
Wei CAO1,2,Tao HE3,Wei PU4,*(),Ke XIAO5
1 School of Construction Machinery, Chang'an University, Xi’an 710064, China
2 Shantui Construction Machinery Co., Ltd., Jining 272073, China
3 Department of Mechanical Engineering, Northwestern University, Evanston, IL60208, USA
4 School of Aeronautics and Astronautics, Sichuan University, Chengdu 610065, China
5 College of Mechanical Engineering, Chongqing University, Chongqing 400044, China
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Abstract  

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.



Key wordsspiral bevel gears      contact path      dynamic response      friction      meshing efficiency     
Received: 25 July 2020      Published: 17 January 2022
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, Changan 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.
Cite this article:

Wei CAO,Tao HE,Wei PU,Ke XIAO. Dynamics of lubricated spiral bevel gears under different contact paths. Friction, 2022, 10(2): 247-267.

URL:

http://friction.tsinghuajournals.com/10.1007/s40544-020-0477-x     OR     http://friction.tsinghuajournals.com/Y2022/V10/I2/247

ΔV, ΔJ, ΔHAssembling parameters
Lgr, RgrAxial and radial projections of initial point of gear
pp,pgUnit vectors along pinion and gear axes, respectively
jp,jgUnit vectors normal to pp and pg, respectively
tp,tgUnit tangential vectors of pinion and gear, respectively
Rbp,RbgPosition vectors of pinion and gear, respectively
np,ngUnit normal vectors of pinion and gear, respectively
aminor,bmajorUnit vectors along minor and major axes of contact ellipse, respectively
Rzx,RzyCurvature radii along aminor and bmajor, respectively
ΔγShaft angle (angle between pp and pg)
ϕp, ϕgRotational angles of pinion and gear, respectively
M(pp(g),ϕp)Rotational matrix of pinion with angle ϕp about pp(g)
M(pg,ϕg)Rotational matrix of pinion with angle ϕg about pg
ΔRdDistance vector
M(Δγ)jTransformation matrix
θtp,θtgAngular increments of cutter for pinion and gear machining, respectively
qp,qgCradle rotations for pinion and gear machining, respectively
Ue,VsEntraining and sliding velocity vectors, respectively
km(t)Mesh stiffness
cm(t)Mesh damping
bGear backlash
em(t)Kinematic transmission error
phMaximum Hertzian pressure
tTime
xi (i=p,g)Displacement component
θp,θgPinion and gear rotational angles during meshing, respectively
δd(t) Dynamic transmission error (DTE)
Rp,Rg Contact radii of pinion and gear, respectively
Fm(t) Dynamic mesh force
Fba, Fbr Axial and radial bearing loads, respectively
Z Number of tapered rollers
φl Half-loaded area angle of bearing
α1Bearing contact angle
knStiffness due to assembly of inner ring- outer ring roller elements
δmaxMaximum bearing deflection in direction of resultant force vector
M,K,C,FMass, stiffness, damping, and force matrices, respectively
Ip,Ig Rotational inertia of pinion and gear about its axis, respectively
mp,mg Masses of pinion and gear, respectively
Tp,Tg Torques acting on pinion and gear, respectively
Tpf,Tgf Friction torques of pinion and gear, respectively
f Friction force
fv,fb Viscous shear friction and boundary friction, respectively
τL Limiting shear stress of lubricant
ξ Friction coefficient of dry contact
Wa Load shared by asperities
Aa Asperity contact area
(ηGβGσG)Roughness parameter
(σG/βG) Average asperity slope
hc Film thickness
σ Composite root mean square roughness
Λ=hcσ Film thickness ratio
E Equivalent elastic modulus, 1E=12(1ν12E1+1ν22E2)
ν1,ν2 Poisson’s ratio of bodies 1 and 2
α Viscosity-pressure coefficient
η Equivalent viscosity of lubricating oil
G Limiting elastic shear modulus
τ Shear stress
p Pressure
Tc Temperature
θe Lubricant flow entrainment angle
Rpf,Rgf Moment arms of pinion and gear, respectively
Tpf,Tgf Total frictional torques of pinion and gear, respectively
μ Friction coefficient
k k-th meshing gear pair
ηe Meshing efficiency
Fro Rolling friction force
CT Thermal reduction factor
SRR Slide-to-roll ratio, SRR=| Ue |/| Vs |
β Temperature-viscosity coefficient
Kf Heat conduction coefficient
τ¯ Average viscous shear stress
γ˙ Shear rate of lubricant
 Nomenclature
Fig. 1 Schematic illustration of contact paths and initial contact point.
Fig. 2 Contact and assembling relationship between pinion and gear.
Fig. 3 Schematic illustration of spiral bevel gear train.
Fig. 4 Dynamic mesh model.
Fig. 5 Schematic illustration of tapered roller bearing.
Gear parameterPinion (mm)Gear (mm)
Number of teeth1544
Module (mm)5.8
Tooth width (mm)43
Average pressure angle (°)20
Mean spiral angle (°)30
Shaft angle (°)90
Face angle (°)22.1772.83
Pitch angle (°)18.8271.18
Root angle (°)17.1767.84
Outside diameter (mm)100.08257.08
Hand of spiralLeftRight
Mass (kg)1.406.20
Inertia (kg·m2)1.23 × 10-36.23 × 10-2
Backlash (μm)75
Tapered roller bearing13
Number of tapered roller elements, Z
Bearing contact angle, α1 (°)15
Effective stiffness of inner ring-rolling-outer ring, kn (N·m-1)4 × 108
Table 1 Gear pair and bearing parameters.
Fig. 6 Flowchart of methodology of dynamics and efficiency of spiral bevel gear.
Contact pathToe contactMiddle contactHeel contact
ΔV1.0840.0248-1.943
ΔH-0.1130.1551.194
Table 2 ΔV and ΔH values for different contact paths (mm).
Fig. 7 Three contact paths and contact ellipses.
Fig. 8 Mesh stiffness and kinematic error in mesh cycle for different contact paths.
Fig. 9 Contact radii of pinion and gear for mesh cycle.
Fig. 10 Curvature radii along minor and major axis of contact ellipse in mesh cycle.
Fig. 11 Frictional moment arm of pinion and gear during engaging cycle.
Fig. 12 Maximum and minimum DTE amplitude during pinion speed sweep.
Fig. 13 Time histories of DTE at resonant speed.
Fig. 14 Maximum and minimum mesh force amplitudes during pinion speed sweep.
Fig. 15 Time histories of dynamic mesh force at resonances.
Fig. 16 Time histories of maximum Hertzian pressure at resonances.
Fig. 17 Contact stress distributions under maximum Hertzian pressure for different contact paths.
Fig. 18 Response of radial displacement of pinion and gear under different contact paths.
Fig. 19 Response of axial displacement of pinion and gear under different contact paths.
Fig. 20 Maximum and minimum radial and axial bearing forces (bearing A) during pinion speed sweep.
Fig. 21 Maximum and minimum lateral and axial bearing forces (bearing C) during pinion speed sweep.
Fig. 22 Variations in friction coefficient obtained from different models.
Fig. 23 Variation in film thickness in mesh cycle under different contact paths.
Fig. 24 Predictions of meshing efficiency during mesh cycle under static condition.
Fig. 25 Dynamic meshing efficiency during pinion speed sweep.
Fig. 26 History of (a) meshing efficiency and (b) dynamic friction coefficient in mesh cycle.
Gear parameterPinion (mm)Gear (mm)
Number of teeth2534
Module (mm)5.0
Tooth width (mm)30
Average pressure angle (°)20
Mean spiral angle (°)35
Shaft angle (°)90
Face angle (°)39.6356.00
Pitch angle (°)36.3353.67
Root angle (°)34.0020.37
Outside diameter (mm)105.50105.50
Hand of spiralLeftRight
Mass (kg)1.643.81
Inertia (Kg·m2)3.45 × 10-31.36 × 10-2
Backlash (μm)75
Table 3 Gear pair parameters.
Contact pathToe contactMiddle contactHeel contact
ΔV0.860-1.491-1.644
ΔH-0.2920.1061.370
Table 4 ΔV and ΔH value for different contact paths (mm).
Effective elastic modulus (GPa)Density of lubricant (kg/L)Lubricant viscosity (mm2/s)Viscosity-pressure coefficient (1/Pa)RMS roughness (μm)
219.780.89150 (40 °C)
14.7 (100 °C)
2.57 × 10-80.5
Table 5 Parameters of gear materials, lubricant, and roughness.
Fig. 27 Gear transmission system test rig and mounted gears.
Fig. 28 Transmission efficiencies of (a) tested results and numerical results: (b) toe contact, (c) middle contact, and (d) heel contact.
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