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Design and evaluation of spur gear pairs with improved bearing capacity and lubrication performance

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Abstract

The tooth form is the most influential factor on the bearing capacity and operation performance of a gear drive. Tooth profiles of spur gears can be designed based on control of relative curvature of conjugate profiles, but how to conceive appropriate control strategies and appraise gear pairs accurately needs to be investigated in depth. This paper presents the control strategies through analysis of the effect mechanisms of relative curvature on the bearing capacity and lubrication performance. Evaluation approaches of the gear pairs are then proposed by establishing the analytical models (AMs). In case studies, gear pairs with optimal comprehensive performance are designed through five control strategies. The AMs for contact and bending capacity are validated with the finite element models. The effects of the control strategies are discussed through comparative analysis. Results reveal that the control strategies lead to remarkable enhancement of bearing capacity and lubrication performance.

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Data availability statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Abbreviations

r 1, r 2 :

Radii of pitch circles of pinion and gear

Z 1, Z 2 :

Tooth numbers of pinion and gear

m :

Module of gear pair

p :

Pitch on pitch circle

i :

Gear ratio

φ 1, φ 2 :

Rolling angles of pinion and gear

α :

Instantaneous pressure angle at pitch point

r :

Distance from origin O to point N′

r′ :

The first derivate of r with respect to α

(x 1*,y 1*):

Coordinate of start point of top fillet for pinion

(x 2*,y 2*):

Coordinate of start point of top fillet for gear

α 1*, α 2*:

Included angle between normal of top fillet at start point and X3-axis

α f1,α f2 :

Included angle between normal of top fillet at any point and x3-axis

k 1 , k 2 :

Curvature of Σ1 and Σ2

K r :

Relative curvature of Σ1 and Σ2

ε :

Contact ratio

\(K_{{\rm{min }}}^{\rm{i}}\) :

Minimum relative curvature of involute gear pair

\(K_{{{\rm{B}}_1}}^{\rm{i}}\) :

Relative curvature at end point of meshing for involute gear pair

\(r_{\rm{M}}^{\rm{i}}\) :

Distance from point M to pitch point

\(r_{{{\rm{B}}_1}}^i\) :

Distance from point B1 to pitch point

S a :

Thickness of tooth top

\({\varphi _{{{\rm{B}}_2}}},{\varphi _{{{\rm{B}}_1}}}\) :

Rolling angles from pitch point to start point and end point

\(K_{\rm{L}}^{\rm{i}}\) :

Relative curvature for LPSTC of involute gear pair

\(K_{\rm{H}}^{\rm{i}}\) :

Relative curvature for HPSTC of involute gear pair

p b :

Pitch on base circle of involute gear

σ Hmax :

Maximum contact stress of tooth surfaces

ν 1,ν 2 :

Poisson’s ratio of material

E1, E 2 :

Young’s modulus of material

F n :

Normal contact force between engaged teeth

B :

Face width of tooth

R :

Relative curvature radius of tooth profiles, r = 1/Kr

T d :

Driving torque on pinion

F t :

Tangential component of Fn

F r :

Radial component of Fn

β :

Intersection angle between Fn and Fr

θ s :

Central angle for half tooth thickness on pitch circle

φc :

Rolling angle of pinion while meshing from pitch point to HPSTC

αc :

Pressure angle at pitch point when meshing at HPSTC

l :

Arm of normal contact force

sb :

Length of dangerous cross section

δ :

Angle of action for normal contact force

γ :

Angle between tangent of root fillet and x-axis

γ s :

Factor of stress concentration

ρ f :

Curvature radius at tangency point D

σ Fmax :

Maximum bending stress

W b :

Bending resistance coefficient for dangerous cross section

p H :

Contact pressure between engaged teeth

b H :

Semi-Hertzian contact width \({b_{\rm{H}}} = \sqrt {4{F_{\rm{n}}}R/\left( {\pi B{E^\prime }} \right)} \)

E′:

Reduced Young’s modulus of materials, 1 / E′ = 0.5 \(\left( {\left( {1 - v_1^2} \right)/{E_1} + \left( {1 - v_2^2} \right)/{E_2}} \right)\)

ξ 1, ξ 2 :

Sliding coefficient for pinion and gear

n :

Rotate speed of pinion

t :

Running time of gear drive

S :

Sliding distance of teeth over one meshing cycle

L :

Total sliding distance during running of gear drive

h w :

Wear depth of tooth surface

I h :

Wear rate of material

T f :

Flash temperature rise between engaged tooth surfaces

f :

Average friction coefficient

w :

Normal load per length at contact point

v s :

Relative sliding speed of engaged tooth surfaces

ρ 1, ρ 2 :

Density of pinion and gear

c 1, c 2 :

Specific heat capacity of pinion and gear

λ 1, λ 2 :

Coefficients of thermal conductivity of pinion and gear

v 1, v 2 :

Tangential speed of pinion and gear

ω 1 :

Angular speed of pinion

R a :

Roughness of tooth surface

η 0 :

Ambient viscosity of lubricant oil

v Σ :

Sum of tangential speed of pinion and gear

H fmin :

Dimensionless minimum oil-film thickness Hfmin = hfmin/r

h fmin :

Minimum oil-film thickness

G :

Dimensionless parameter related to material, G = ζE′

ζ :

Pressure-viscosity coefficient

U :

Dimensionless parameter related to velocity, U = η0ve/(E′R)

v e :

Entraining velocity

W :

Dimensionless parameter related to load, w = w/(E′R)

h fa :

Average minimum oil-film thickness

∣ξ∣1max :

Maximum magnitude of sliding coefficient for pinion

∣ξ∣2max :

Maximum magnitude of sliding coefficient for gear

h fmin :

Minimal oil-film thickness

T fmax :

Maximum flash temperature rise

T fa :

Average flash temperature rise

h w 1max, h w 2max :

Maximum wear depths of pinion and gear

h w 1a, h w 2a :

Average wear depths of pinion and gear

w 1,w 2,…,w 12 :

Weight coefficient

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Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant Number 52075238) and the National Key Laboratory of Science and Technology on Helicopter Transmission (Nanjing University of Aeronautics and Astronautics, Grant Number HTL-A-21G04).

Author information

Authors and Affiliations

  1. College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Lei Liu, Zhiwei Chen & Rong Kong

  2. National Key Laboratory of Science and Technology on Helicopter Transmission, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Lei Liu

Authors
  1. Lei Liu
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  2. Zhiwei Chen
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  3. Rong Kong
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Corresponding author

Correspondence to Lei Liu.

Additional information

Lei Liu is an Associate Professor of the College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics. He received his Ph.D. in Mechanical Engineering from the Nanjing University of Science and Technology. His research interests include gear transmission and dynamics of mechanical system.

Zhiwei Chen is a Master’s student in the College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics. He received his Bachelor’s degree in Mechanical Engineering from Yangzhou University. His research interest is gear transmission.

Rong Kong is a Master’s student in the College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics. He received his Bachelor’s degree in Mechanical Engineering from the North China Institute of Aerospace Engineering. His research interest is gear transmission.

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Liu, L., Chen, Z. & Kong, R. Design and evaluation of spur gear pairs with improved bearing capacity and lubrication performance. J Mech Sci Technol 37, 1349–1364 (2023). https://doi.org/10.1007/s12206-023-0221-3

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  • DOI: https://doi.org/10.1007/s12206-023-0221-3

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