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Time-varying mesh characteristics of a spur gear pair considering the tip-fillet and friction

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Abstract

Considering the effects of the tip-fillet and friction between teeth surfaces, an improved analytical model for time-varying mesh stiffness (TVMS) calculation is further developed based on our previous works (Ma et al. in Mech Mach Theory, 98:64–80, 2016) in which the effects of extended tooth contact, revised fillet-foundation stiffness and nonlinear contact stiffness are taken into account. The proposed model is also verified by comparing the TVMS with that obtained from finite element method. Combined effects of the tip-fillet and friction on the TVMS, loaded transmission error and loaded sharing factor are discussed. The results show that TVMS between double- and single-tooth engagements becomes smooth under the small radii of the tip-fillet, however, an abrupt change of the TVMS from double-tooth contact to single-tooth contact can be observed under the large radii of the tip-fillet. The range of the double-tooth engagement decreases and the single-tooth engagement increases with the increase of the radius of the tip-fillet. In addition, the tooth-surface friction mainly affects the mesh stiffness in single-tooth engagement region due to the change of friction force direction near pitch point, and the extent of variation increases with the increase of friction coefficient. The friction has a larger effect on the TVMS than the tip-fillet for the selected parameters of tip-fillet and friction. This study can provide a basis for the structural design of the spur gears.

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Abbreviations

ETC:

Extended tooth contact

FE:

Finite element

IAM:

Improved analytical method

LSF:

Loaded sharing factor

LTE:

Loaded transmission error

STE:

Static transmission error

TVMS:

Time-varying mesh stiffness

2D:

Two dimensional

a :

The center distance of the gear pair

A y1, A y2 :

The cross-section area

D x , D y :

The coordinates of point D in x and y directions

E :

Young’s modulus

\(\left( {E_{\text{d}}^{i} } \right)_{j} \left( {i = a,b} \right)\) :

The tooth deflection of the ith tooth pair at meshing point j

E f :

The amount of modification of the tip-fillet along the line of action

\(\left( {E_{\text{f}}^{i} } \right)_{j} \left( {i = a,b} \right)\) :

The tooth profile error of the ith tooth pair at meshing point j with tip-fillet

\(E_{\text{r}}^{\text{gear}}\) :

The loaded transmission error of the gear

\(\left( {E_{\text{r}}^{i} } \right)_{j} \left( {i = a,b} \right)\) :

The static transmission error of the ith tooth pair at meshing point j

\(\left( {E_{\text{s}}^{a} } \right)_{j}\) :

The spacing error of the tooth pair a at meshing point j

f :

The friction force on tooth surface

f h , f v :

The vertical and the horizontal components of the friction force

F :

The total meshing force

F h , F v :

The vertical and the horizontal components of the meshing force

F i (i = 1, 2):

The meshing force of the ith tooth pair

\(F_{j}^{i} \left( {i = a,b} \right)\) :

The meshing force of the ith tooth pair at meshing point j

F r, F t :

The gear meshing force in radial and tangential directions

G :

Shear modulus

I y1, I y2 :

The cross section moment of inertia

j :

The meshing point

k :

The total mesh stiffness of the gear pair

k an,i (n = 1, 2) (i = 1, 2):

The axial compressive stiffness of the ith tooth pair, subscript n denotes the driving and driven gear

k a_f :

The axial compressive stiffness considering friction

k bn,i (n = 1, 2) (i = 1, 2):

The bending stiffness of the ith tooth pair, subscript n denotes the driving and driven gear

k b_f :

The bending stiffness considering friction

k fn (n = 1, 2):

The stiffness of fillet-foundation, subscript n denotes the driving gear and driven gear

k f11, k f21 :

The fillet-foundation stiffness of the driving gear and driven gear during single-tooth engagement

k hi (i = 1, 2):

The Hertzian contact stiffness of the ith meshing tooth pair

k sn,i (n = 1, 2) (i = 1, 2):

The shear stiffness of the ith tooth pair, subscript n denotes the driving gear and driven gear

k s_f :

The shear stiffness considering friction

k tn,i (n = 1, 2) (i = 1, 2):

The stiffness of tooth of the ith tooth pair, subscript n denotes the driving gear and driven gear

k tooth :

The total mesh stiffness of meshing tooth pairs

\(k_{\text{tooth}}^{i} \left( {i = \, 1,2} \right)\) :

The mesh stiffness of the ith tooth pair

L :

The tooth face width

m :

Module of the gear pair

N :

The number of meshing tooth pairs

O 1 :

The center of the driving gear

O c :

The center of the gear tip-fillet

\(Q_{j}^{i} \left( {i = \, a,b} \right)\) :

The teeth deformation of the ith tooth pair under unit force at contact point j

r ai (i = 1, 2):

The radius of the addendum circle of the driving gear and driven gear

r bi (i = 1, 2):

The radius of the base circle of the driving gear and driven gear

r c :

The radius of the gear tip-fillet

r int :

Hub radius of the gear pair

S a, S r :

The separation distances of tooth pairs c and a along the line of action

t P :

The time between the beginning point of the single-tooth engagement and the pitch point

T :

The torque applied to the driving gear

x DE, y DE :

The coordinates of the point on the involute curve DE in x and y directions

x Oc , y Oc :

The coordinates of the point O c in x and y directions

x β :

The distance between the contact point and the central line of the tooth

y β :

The distance between the contact point and original point in the horizontal direction

y 1, y 2 :

The horizontal coordinates of arbitrary point at transition curve and involutes curve

z :

The number of teeth

Ω 1 :

The rotating angular velocity of the driving gear

α :

The pressure angle of the pitch circle

α 1 :

The pressure angle corresponding to point A (see Fig. 3)

α 2 :

The pressure angle corresponding to point D (see Fig. 3)

α c :

The angle corresponding to the point O c (see Fig. 4)

α p :

The pressure angle corresponding to point E (see Fig. 4)

β :

The operating pressure angle

β 2 :

The angle corresponding to the point D (see Fig. 4)

β c :

The angle corresponding to the point O c (see Fig. 4)

γ :

The angular displacement of arbitrary point at the transition curve

θ 1 :

The rotational angle of driving gear

θ b :

The half tooth angle on the base circle of the gear

θ B :

The rotational angle at the beginning point of the single-tooth engagement

θ P :

The rotational angle of the pitch point

λ 1N , λ 2N (N = 1, 2):

The correction coefficient of the fillet-foundation stiffness of the driving gear and driven gear, N denotes the number of meshing tooth pairs

μ :

The friction coefficient

τ :

The angular displacement of arbitrary point at the involute curve

τ c :

The meshing angle at the position of the involute starting point

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Acknowledgments

This project is supported by the Joint Funds of the National Natural Science Foundation and the Civil Aviation Administration of China (Grant No. U1433109), the Fundamental Research Funds for the Central Universities (Grant Nos. N150305001 and N140301001) and State Key Laboratory for Strength and Vibration of Mechanical Structures (Grant No. SV2015-KF-08) for providing financial support for this work.

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Authors and Affiliations

  1. School of Mechanical Engineering and Automation, Northeastern University, Shenyang, 110819, Liaoning, People’s Republic of China

    Hui Ma, Mengjiao Feng, Zhanwei Li, Ranjiao Feng & Bangchun Wen

  2. State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, Xi’an, 710049, Shanxi, People’s Republic of China

    Hui Ma

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  1. Hui Ma
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Correspondence to Hui Ma.

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Ma, H., Feng, M., Li, Z. et al. Time-varying mesh characteristics of a spur gear pair considering the tip-fillet and friction. Meccanica 52, 1695–1709 (2017). https://doi.org/10.1007/s11012-016-0502-3

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