RESEARCH ARTICLE

Mesh relationship modeling and dynamic characteristic analysis of external spur gears with gear wear

  • Zhixian SHEN 1,2 ,
  • Laihao YANG 1,2 ,
  • Baijie QIAO , 1,2 ,
  • Wei LUO 1,2 ,
  • Xuefeng CHEN 1,2 ,
  • Ruqiang YAN 1,2
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  • 1. School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an 710049, China
  • 2. State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an 710061, China

Received date: 05 Jul 2021

Accepted date: 11 Nov 2021

Published date: 15 Mar 2022

Copyright

2022 Higher Education Press 2022

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.

Cite this article

Zhixian SHEN , Laihao YANG , Baijie QIAO , Wei LUO , Xuefeng CHEN , Ruqiang YAN . Mesh relationship modeling and dynamic characteristic analysis of external spur gears with gear wear[J]. Frontiers of Mechanical Engineering, 2022 , 17(1) : 9 . DOI: 10.1007/s11465-021-0665-z

Nomenclature

a1, a2 Tip angles of gear and pinion, respectively
aω Center distance of two gears
A1, A2 Amplitude of long period and short period, respectively
A1perfect, A1 Meshing points on the perfect profile and the worn profile of the driving gear, respectively
A2perfect, A2 Meshing points on the perfect profile and the worn profile of the driven gear, respectively
c Time-varying mesh damping
c1, c2 Root angles of gear and pinion or supporting damping of gear and pinion
e(t) Unloaded static transmission error
eA1(t), eA2(t) Long and short period errors, respectively
eerror(t) Random error
ewear(t) Gear failure error caused by gear wear
f1, fm Rotational frequency and meshing frequency, respectively
F Force
g1(x1) Perfect profile function
h, hx Heights of the tooth profile at positions d and x, respectively
hA1_x1, hA2_x2 Wear depth in the perfect meshing point A1perfect and A2perfect, respectively
H Surface hardness
I1, I2 Moments of inertia of gear and pinion, respectively
k Mesh stiffness
k1, k2 Supporting stiffnesses of gear and pinion, respectively
ka, kb, kh, ks 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
m1, m2 Masses of two gears
nth The nth tooth pair
N1N 2 Ideal meshing line
NUM, Data_NUM Number of discrete meshing points and number of experimental set, respectively
O1, O2 Geometric center of two gears
rand Random number in the range [1, 0]
R1, R2 Radii of the pitch circle
RA 1perfect, RA1 Radii of the meshing points A1perfect and A1, respectively
RA 2perfect, RA2 Radii of the meshing points A2perfect and A2, respectively
Rb1, Rb2 Radii of the base circle
Rf1 Radius of the root circle
s Sliding distance
t1, t2 Tangents of two meshing points
T Moment
T1 Input torque of the gear
T2 Load torque of the pinion
Threshold Threshold of the tolerable error of rotational angle
V Worn volume
W Normal force
x Signal
x1, x2 Abscissa values of the coordinate systems X1OY1 and X2O 2 Y2 or translational DOFs of gear and pinion, respectively
xm Relative displacement of two gears
XOY Fixed global coordinate system
X1OY1, X2O2Y2 Local coordinate systems rotating with the driving and driven gears, respectively
y1, y2 Ordinate values of the coordinate systems X1OY1 and X2O 2 Y2 or translational DOFs of gear and pinion, respectively
z1, z2 Tooth number of two gears
Z1, Z2, Z3, Z4 Tooth number in experiments
α2, α1 Angles of the base circle and the perfect meshing point A1p er fe ct on the single-tooth profile, respectively
α 2, α1 Angles of the base circle and the perfect meshing point A2p er fe ct on the single-tooth profile, respectively
γ 1p er fe ct, γ 1 Angles of the perfect meshing point A1 pe rf ec t and the worn meshing point A1, respectively
γ 2p er fe ct, γ 2 Angles of the perfect meshing point A2p er fe ct and the worn meshing point A2, respectively
θ Angular displacement
θ1, θ2 Acute angle between tangent t1 and OO 2 and the acute angle between tangent t2 and OO 2, or rotational DOFs of gear and pinion, respectively
φ1, φ2 Rotational angles of gear and pinion
Δφ2 Angle difference between two pairs of gears for the same φ 1
ψ 1, ψ 2 Angle between OO2 and OA 1 and the angle between OO2 and O2A 2, respectively
δ 1, δ 2 Acute angle between tangent t1 and OA 1 and the acute angle between tangent t2 and O2A 2, respectively
λ1, λ2 Acute angle between tangent t1 and axis X1 and the acute angle between tangent t2 and axis X2, respectively
εa Tolerable error of the center distance
µ, σ Mean and standard deviation of the signal, respectively

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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