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A new dynamic model for vibration analysis of a ball bearing due to a localized surface defect considering edge topographies

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Abstract

When a small initial defect occurs on the races of a ball bearing such as spalls and pits caused by fatigue, the defect will progress around the race in the direction of the ball motion until catastrophic failure happens due to impacts between the ball and the defect edges. Undesirable impulsive excitations can be caused when the orbiting ball strikes the defect edges. The impulse is depended on the shape and sizes of the defect, which can be used to detect and diagnose the defects in bearing systems. To understand characteristics of an impulse caused by a localized surface defect in a ball bearing, a new dynamic model is proposed to investigate the vibration response of a ball bearing due to a localized surface defect on its races, which can consider effects of defect edge topographies. Based on the defect edge topographies and the sizes of the defect, a new contact model for modeling contact relationships between the ball and the defect edges is also developed according to Hertzian elastic contact theory, which can be used to determine changes in the excitations including the time-varying deflection excitation and the time-varying contact stiffness excitation caused by the defect. The proposed model is applied to investigate effects of the defect edge topographies on the contact stiffnesses between the ball and the defect edges, and the vibration response of a ball bearing with a localized surface defect on its races. The results from the proposed model are compared with the available results from the previous models in the literature, which reveals the superiority of the proposed model. It is also shown that numerical results can provide some guidance for the ball bearing defect diagnosis and detection.

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Abbreviations

\(B\) :

Defect width (mm)

\(C_\mathrm{{r}}\) :

Internal radial clearance of the ball bearing (mm)

\(c\) :

Damping coefficient (Ns/m)

\(D\) :

Pitch diameter (mm)

\(D_\mathrm{{i}}\) :

Inner race diameter (mm)

\(D_\mathrm{{o}}\) :

Outer race diameter (mm)

\(d\) :

Ball diameter (mm)

\(E_{1},\,E_{2}\) :

Elastic modulus associated with each contact body (MPa)

\(\nu _{1},\,\nu _{2}\) :

Poisson’s ratio associated with each contact body (MPa)

\(E_\mathrm{{eq}}\) :

Equivalent modulus of elasticity (MPa)

\(e\) :

Elliptical eccentricity parameter

\(F_\mathrm{{r}}\) :

Radial force applied on the ball (N)

\(H\) :

Defect depth (mm)

\(H_{1},\,H_{2},\,H_{3},\,H_{4}\) :

Time-varying deflection excitations (mm)

\(j\) :

Ball number

\(K\) :

Total contact stiffness (N/mm)

\(K_\mathrm{{e}}\) :

Time-varying contact stiffness between the ball and races (N/mm)

\(K_\mathrm{{i}}\) :

Contact stiffness between the ball and inner race without defect (N/mm)

\(K_\mathrm{{o}}\) :

Contact stiffness between the ball and outer race without defect (N/mm)

\(K_\mathrm{{p}}\) :

Contact stiffness between the ball and smooth surface (N/mm)

\(K_\mathrm{{ts}}\) :

Total contact stiffness between the ball and races with defect (N/mm)

\(K_\mathrm{{t0}}\) :

Contact stiffness between the ball and races with first type defect (N/mm)

\(k\) :

First and second kind of elliptical integral

\(L\) :

Defect length (mm)

\(l\) :

Length of the small surfaces at the defect edges (mm)

\(N_\mathrm{{f}}\) :

Number of samples

\(n_\mathrm{{d}}\) :

Load-deflection exponent for the first defect type

\(n_\mathrm{{s}}\) :

Numbers of the contact surfaces between the ball and defect edges

\(Q_{x},\,Q_{y}\) :

Components of external force applying on the shaft (N)

\(R\) :

Radius of the ball (mm)

\(r_\mathrm{{o}}\) :

Radius of curvature of the outer race (mm)

\(r_\mathrm{{i}}\) :

Radius of curvature of the inner race (mm)

\(s\) :

Number of the contact surfaces

\(t\) :

Time (s)

\(x,\,y\) :

Displacement responses in \(X\)- and \(Y\)-direction (mm)

\(Z\) :

Number of the ball

\(\alpha \) :

Contact angle (\(^{\circ }\))

\(\gamma \) :

Elevation angle of the small surfaces at defect edges (rad)

\(\delta _\mathrm{{dr}}\) :

Contact deformation between the ball and defect (mm)

\(\delta _\mathrm{{hr}}\) :

Contact deformation between the ball and healthy race (mm)

\(\theta _{0}\) :

Initial angular offset of the defect to the \(j\)th ball (rad)

\(\theta _\mathrm{{d}}\) :

Arc length of the defect in the tangential direction (rad

\(\theta _\mathrm{{h}}\) :

Half of the arc length of the defect (rad)

\(\theta _{dj}\) :

Race contact angle (rad)

\(\Delta \,\theta \) :

Arc length caused by the defect with different edge topographies (rad)

\(\lambda _{j}\) :

Loading zone parameter of the \(j\)th ball

\(\varepsilon \) :

Second kind of elliptical integral

\(\xi _\mathrm{{d}}\) :

Ratio of the defect length to its width

\(\xi _\mathrm{{bd}}\) :

Ratio of the ball size to the defect minimum size

\(\Sigma \,\rho \) :

Curvature sum

\(\omega _\mathrm{{c}}\) :

Speed of the cage (rad/s)

\(\omega _\mathrm{{r}}\) :

Speed of the shaft (rad/s)

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Acknowledgments

The authors are grateful for the financial support provided by the National Natural Science Key Foundation of China under Contract No. 51035008 and the Fundamental Research Funds for the Central Universities under Grant Number 0903005203236.

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Authors and Affiliations

  1. State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing, 400030, People’s Republic of China

    Jing Liu & Yimin Shao

  2. College of Mechanical Engineering, Chongqing University, Chongqing, 400030, People’s Republic of China

    Jing Liu

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  1. Jing Liu
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Correspondence to Yimin Shao.

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Liu, J., Shao, Y. A new dynamic model for vibration analysis of a ball bearing due to a localized surface defect considering edge topographies. Nonlinear Dyn 79, 1329–1351 (2015). https://doi.org/10.1007/s11071-014-1745-y

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