Abstract
Gear wear is one of the most common gear failures, which changes the mesh relationship of normal gear. A new mesh relationship caused by gear wear affects meshing excitations, such as mesh stiffness and transmission error, and further increases vibration and noise level. This paper aims to establish the model of mesh relationship and reveal the vibration characteristics of external spur gears with gear wear. A geometric model for a new mesh relationship with gear wear is proposed, which is utilized to evaluate the influence of gear wear on mesh stiffness and unloaded static transmission error (USTE). Based on the mesh stiffness and USTE considering gear wear, a gear dynamic model is established, and the vibration characteristics of gear wear are numerically studied. Comparison with the experimental results verifies the proposed dynamic model based on the new mesh relationship. The numerical and experimental results indicate that gear wear does not change the structure of the spectrum, but it alters the amplitude of the meshing frequencies and their sidebands. Several condition indicators, such as root-mean-square, kurtosis, and first-order meshing frequency amplitude, can be regarded as important bases for judging gear wear state.
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Abbreviations
- a 1, a 2 :
-
Tip angles of gear and pinion, respectively
- a ω :
-
Center distance of two gears
- A 1, A 2 :
-
Amplitude of long period and short period, respectively
- A 1perfect, A 1 :
-
Meshing points on the perfect profile and the worn profile of the driving gear, respectively
- A 2perfect, A 2 :
-
Meshing points on the perfect profile and the worn profile of the driven gear, respectively
- c :
-
Time-varying mesh damping
- c 1, c 2 :
-
Root angles of gear and pinion or supporting damping of gear and pinion
- e(t):
-
Unloaded static transmission error
- e A1(t), e A2(t):
-
Long and short period errors, respectively
- e error(t):
-
Random error
- e wear(t):
-
Gear failure error caused by gear wear
- f 1, f m :
-
Rotational frequency and meshing frequency, respectively
- F :
-
Force
- g 1(x 1):
-
Perfect profile function
- h, h x :
-
Heights of the tooth profile at positions d and x, respectively
- h A1_x1, h A2_x2 :
-
Wear depth in the perfect meshing point A1perfect and A2perfect, respectively
- H :
-
Surface hardness
- I 1 I 2 :
-
Moments of inertia of gear and pinion, respectively
- k :
-
Mesh stiffness
- k 1, k 2 :
-
Supporting stiffnesses of gear and pinion, respectively
- k a, k b, k h, k s :
-
Axial compressive stiffness, bending stiffness, Hertzian contact stiffness, shear deformation stiffness, respectively
- k(t):
-
Time-varying mesh stiffness
- K :
-
Dimensionless wear coefficient or mesh stiffness
- m 1, m 2 :
-
Masses of two gears
- n th :
-
The nth tooth pair
- N 1 N 2 :
-
Ideal meshing line
- NUM:
-
Number of discrete meshing points and number of
- Data_NUM:
-
experimental set, respectively
- O 1, O 2 :
-
Geometric center of two gears
- rand:
-
Random number in the range [1, 0]
- R 1, R 2 :
-
Radii of the pitch circle
- R A1perfect, R A1 :
-
Radii of the meshing points A1perfect and A1, respectively
- R A2perfect, R A2 :
-
Radii of the meshing points A2perfect and A2, respectively
- R b1, R b2 :
-
Radii of the base circle
- R f1 :
-
Radius of the root circle
- s :
-
Sliding distance
- t 1, t 2 :
-
Tangents of two meshing points
- T :
-
Moment
- T 1 :
-
Input torque of the gear
- T 2 :
-
Load torque of the pinion
- Threshold :
-
Threshold of the tolerable error of rotational angle
- V :
-
Worn volume
- W :
-
Normal force
- x :
-
Signal
- x 1, x 2 :
-
Abscissa values of the coordinate systems X1OY1 and X2O2Y2 or translational DOFs of gear and pinion, respectively
- x m :
-
Relative displacement of two gears
- XOY :
-
Fixed global coordinate system
- X 1 OY 1 :
-
Local coordinate systems rotating with the driving and
- X 2 O 2 Y 2 :
-
driven gears, respectively
- y 1, y 2 :
-
Ordinate values of the coordinate systems X1OY1 and X2O2Y2 or translational DOFs of gear and pinion, respectively
- z 1, z 2 :
-
Tooth number of two gears
- Z 1, Z 2, Z 3, Z 4 :
-
Tooth number in experiments
- α 2, α 1 :
-
Angles of the base circle and the perfect meshing point A1perfect on the single-tooth profile, respectively
- α̲2, α̲1 :
-
Angles of the base circle and the perfect meshing point A2perfect on the single-tooth profile, respectively
- γ 1perfect, γ 1 :
-
Angles of the perfect meshing point A1perfect and the worn meshing point A1, respectively
- γ 2perfect, γ 2 :
-
Angles of the perfect meshing point A2perfect and the worn meshing point A2, respectively
- θ :
-
Angular displacement
- θ 1, θ 2 :
-
Acute angle between tangent t1 and OO2 and the acute angle between tangent t2 and OO2, or rotational DOFs of gear and pinion, respectively
- ϕ 1, ϕ 2 :
-
Rotational angles of gear and pinion
- Δϕ 2 :
-
Angle difference between two pairs of gears for the same ϕ1
- ψ 1, ψ 2 :
-
Angle between OO2 and OA1 and the angle between OO2 and O2A2 respectively
- δ 1, δ 2 :
-
Acute angle between tangent t1 and OA1 and the acute angle between tangent t2 and O2A2, respectively
- λ 1, λ 2 :
-
Acute angle between tangent t1 and axis X1 and the acute angle between tangent t2 and axis X2, respectively
- ε a :
-
Tolerable error of the center distance
- μ, σ :
-
Mean and standard deviation of the signal, respectively
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Acknowledgements
This paper was supported by the National Key R&D Program of China (Grant No. 2018YFB1702400) and the National Natural Science Foundation of China (Grant No. 52075414).
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Shen, Z., Yang, L., Qiao, B. et al. Mesh relationship modeling and dynamic characteristic analysis of external spur gears with gear wear. Front. Mech. Eng. 17, 9 (2022). https://doi.org/10.1007/s11465-021-0665-z
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DOI: https://doi.org/10.1007/s11465-021-0665-z