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.
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Abbreviations
- ΔV, ΔJ, ΔH :
-
Assembling parameters
- L gr, R gr :
-
Axial and radial projections of initial point of gear
- p p, p g :
-
Unit vectors along pinion and gear axes, respectively
- j p, j g :
-
Unit vectors normal to pp and pg, respectively
- t p, t g :
-
Unit tangential vectors of pinion and gear, respectively
- R bp, R bg :
-
Position vectors of pinion and gear, respectively
- n p, n g :
-
Unit normal vectors of pinion and gear, respectively
- a minor, b major :
-
Unit vectors along minor and major axes of contact ellipse, respectively
- R zx, R zy :
-
Curvature radii along aminor and bmajor, respectively
- Δγ :
-
Shaft angle (angle between pp and pg)
- ϕ p, ϕ g :
-
Rotational angles of pinion and gear, respectively
- \({\bf{M}}\left( {{\bf{p}}_{\rm{p}}^{({\rm{g}})},{\phi _{\rm{p}}}} \right)\) :
-
Rotational matrix of pinion with angle ϕp about \({\bf{p}}_{\rm{p}}^{({\rm{g}})}\)
- M(p g, ϕ g):
-
Rotational matrix of pinion with angle ϕg about pg
- ΔR d :
-
Distance vector
- M(Δγ)j :
-
Transformation matrix
- θ tp, θ tg :
-
Angular increments of cutter for pinion and gear machining, respectively
- q p, q g :
-
Cradle rotations for pinion and gear machining, respectively
- U e, V s :
-
Entraining and sliding velocity vectors, respectively
- k m (t):
-
Mesh stiffness
- c m (t):
-
Mesh damping
- b :
-
Gear backlash
- e m (t):
-
Kinematic transmission error
- p h :
-
Maximum Hertzian pressure
- t :
-
Time
- x i (i = p, g):
-
Displacement component
- θ p, θ g :
-
Pinion and gear rotational angles during meshing, respectively
- δ d (t):
-
Dynamic transmission error (DTE)
- R p, R g :
-
Contact radii of pinion and gear, respectively
- F m (t):
-
Dynamic mesh force
- F ba F br :
-
Axial and radial bearing loads, respectively
- Z :
-
Number of tapered rollers
- φ 1 :
-
Half-loaded area angle of bearing
- α1 :
-
Bearing contact angle
- k n :
-
Stiffness due to assembly of inner ring-outer ring roller elements
- δ max :
-
Maximum bearing deflection in direction of resultant force vector
- M, K, C, F :
-
Mass, stiffness, damping, and force matrices, respectively
- I p, I g :
-
Rotational inertia of pinion and gear about its axis, respectively
- m p, m g :
-
Masses of pinion and gear, respectively
- T p, T g :
-
Torques acting on pinion and gear, respectively
- T pf, T gf :
-
Friction torques of pinion and gear, respectively
- f :
-
Friction force
- f v, f b :
-
Viscous shear friction and boundary friction, respectively
- τ L :
-
Limiting shear stress of lubricant
- ξ :
-
Friction coefficient of dry contact
- W a :
-
Load shared by asperities
- A a :
-
Asperity contact area
- (η G β G σ G):
-
Roughness parameter
- (σ G/β G):
-
Average asperity slope
- h c :
-
Film thickness
- σ :
-
Composite root mean square roughness
- \(\Lambda = {{{h_{\rm{c}}}} \over \sigma }\) :
-
Film thickness ratio
- E′ :
-
Equivalent elastic modulus, \({1 \over {{E^\prime }}} = {1 \over 2}\left( {{{1 - v_1^2} \over {{E_1}}} + {{1 - v_2^2} \over {{E_2}}}} \right)\)
- ν 1, ν 2 :
-
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
- T c :
-
Temperature
- θ e :
-
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
- η e :
-
Meshing efficiency
- F ro :
-
Rolling friction force
- C T :
-
Thermal reduction factor
- SRR :
-
Slide-to-roll ratio, \(SRR = {{\left| {{{\bf{U}}_{\rm{e}}}} \right|} \mathord{\left/{\vphantom {{\left| {{{\bf{U}}_{\rm{e}}}} \right|} {\left| {{{\bf{V}}_{\rm{s}}}} \right|}}} \right. \kern-\nulldelimiterspace} {\left| {{{\bf{V}}_{\rm{s}}}} \right|}}\)
- β :
-
Temperature-viscosity coefficient
- K f :
-
Heat conduction coefficient
- \(\overline \tau \) :
-
Average viscous shear stress
- \(\dot \gamma \) :
-
Shear rate of lubricant
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Acknowledgements
The present study was founded by the National Natural Science Foundation of China (Grant Nos. 52005047 and 51875369), Natural Science Basic Research Plan in Shaanxi Province of China (Grant Nos. 2020JQ-367 and 2020JQ-345), China Postdoctoral Science Foundation (Grant No. 2020M672129), and the Fundamental Research Funds for the Central Universities, CHD (Grant No. 300102250301).
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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 multiphsical 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.
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Cao, W., He, T., Pu, W. et al. Dynamics of lubricated spiral bevel gears under different contact paths. Friction 10, 247–267 (2022). https://doi.org/10.1007/s40544-020-0477-x
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DOI: https://doi.org/10.1007/s40544-020-0477-x