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

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.

Graphical abstract

Keywords

gear wear / mesh relationship / mesh stiffness / transmission error / vibration characteristics

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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, h_x	Heights of the tooth profile at positions d and x, respectively
h_{A1\mathrm{\_}x1}, h_{A2\mathrm{\_}x2}	Wear depth in the perfect meshing point A_{\text{1perfect}} and A_{\text{2perfect}}, respectively
H	Surface hardness
I_1, I_2	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
m_1, m_2	Masses of two gears
n^{\text{th}}	The nth tooth pair
N_1{N}_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
R_{A\text{1perfect}}, R_{A1}	Radii of the meshing points A_{\text{1perfect}} and A_1, respectively
R_{A\text{2perfect}}, R_{A2}	Radii of the meshing points A_{\text{2perfect}} and A_2, respectively
Rb1, Rb2	Radii of the base circle
R_{\text{f}1}	Radius of the root circle
s	Sliding distance
t1, t2	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 X_{1}O{Y}_1 and X_2{O}_2{Y}_2 or translational DOFs of gear and pinion, respectively
x_{\text{m}}	Relative displacement of two gears
XOY	Fixed global coordinate system
X1OY1, X2O2Y2	Local coordinate systems rotating with the driving and driven gears, respectively
y_1, y_2	Ordinate values of the coordinate systems X_{1}O{Y}_1 and X_2{O}_2{Y}_2 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 A_{1\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{e}\mathrm{c}\mathrm{t}} on the single-tooth profile, respectively
{\alpha }_2^{\prime }, {\alpha }_1^{\prime }	Angles of the base circle and the perfect meshing point A_{2\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{e}\mathrm{c}\mathrm{t}} on the single-tooth profile, respectively
{\gamma }_{1\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{e}\mathrm{c}\mathrm{t}}, {\gamma }_1	Angles of the perfect meshing point A_{1\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{e}\mathrm{c}\mathrm{t}} and the worn meshing point A_1, respectively
{\gamma }_{2\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{e}\mathrm{c}\mathrm{t}}, {\gamma }_2	Angles of the perfect meshing point A_{2\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{e}\mathrm{c}\mathrm{t}} and the worn meshing point A_2, respectively
θ	Angular displacement
θ1, θ2	Acute angle between tangent t_1 and O{O}_2 and the acute angle between tangent t_2 and O{O}_2, or rotational DOFs of gear and pinion, respectively
φ1, φ2	Rotational angles of gear and pinion
\mathrm{\Delta }{\phi }_2	Angle difference between two pairs of gears for the same {\phi }_1
{\psi }_1, {\psi }_2	Angle between O{O}_2 and O{A}_1 and the angle between O{O}_2 and O_2{A}_2, respectively
{\delta }_1, {\delta }_2	Acute angle between tangent t_1 and O{A}_1 and the acute angle between tangent t_2 and O_2{A}_2, respectively
λ1, λ2	Acute angle between tangent t_1 and axis X_1 and the acute angle between tangent t_2 and axis X_2, 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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