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A new model for spall-rolling-element interaction

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Abstract

Evaluation of the spall size of a radially loaded rolling-element bearing is required for the assessment of the bearing damage severity and estimation of its remaining useful life. A new multi-body, nonlinear dynamic model of the interaction between the rolling-element and a spalled outer race is presented. The study focuses on the physics of rolling-element contact in a broader range of spall sizes than has been investigated to date, with intermittent contact between rolling-element and the outer race, which is a relevant defect size for diagnostics and prognosticsquery. The analysis is performed in several time intervals according to periods of rolling-element/race contact and periods when the rolling-element is not connected with the outer race. An explicit expression of the spall size as function of the time-to-impact has been developed by considering radial load, shaft speed, and gravity. The expression of the spall size was used in a sensitivity study of the effect of parameters such as geometry and radial load. The results obtained from the new model are in good agreement with a well-established general bearing model. The acceleration of the outer ring during the rolling-element/spall interaction with intermittent rolling-element race connection is a novel contribution, which allows verification of the model from direct observations using vibration sensors mounted on the structure

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Abbreviations

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

Radial acceleration

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

RE bearing clearance diameter

c :

Damping coefficient of each of the RE-raceways contact

\(c_{\mathrm {B}}\) :

Damping coefficient of the bearing

\(c_{\mathrm {RE}}\) :

Damping coefficient of each RE

\(c_{0}\) :

Speed of sound in a solid material

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

Bearing pitch diameter

d :

Spall depth

E :

Elasticity modulus

\(\mathbf {E}\) :

Trailing edge relative to the outer ring centre

F :

Force applied to the RE by the raceway

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

Force applied to the RE that enters the spall

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

Normal force applied to an RE

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

Force applied to the RE that is within the loading zone but does not interact with the spall

g :

Gravitational acceleration

e :

Coefficient of restitution

h :

Vertical distance between the centre of the RE and the outer ring surface

\(\mathbf {I}\) :

Moment of inertia tensor

j :

Index of the RE that is within the loading zone

K :

Linearized bearing stiffness

k :

Hertzian contact stiffness coefficient

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

Equivalent Hertzian contact stiffness coefficient

\(k_{\mathrm {RE-i}}\) :

Hertzian contact stiffness between the RE and the inner raceway

\(k_{\mathrm {RE-o}}\) :

Hertzian contact stiffness between the RE and the outer raceway

m :

Mass

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

Mass of the inner ring

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

Mass of the outer ring

\(m_{\mathrm {RE}}\) :

Mass of the RE

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

Number of the REs in the bearing

\(\hat{n}\) :

Unit vector in the impact direction

n :

Hertzian contact exponent coefficient

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

Location of the outer ring in the \(\hat{n}\) direction relative to the impact location

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

Location of the RE in the \(\hat{n}\) direction relative to the impact location

\(\mathbf {R}\) :

RE centre location relative to the leading edge; Location of a general body

\(\mathbf {R}_{\mathrm {dis}}\) :

Location of the RE centre at the moment of its disconnection from the raceways

\(\mathbf {R}_{\mathrm {imp}}\) :

Location of the RE centre at the moment of its impact on the trailing edge

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

The distance between the centre of the RE and the outer ring

\(R_{\mathrm {RE}}\) :

Radius of the RE

\(\mathbf {r}\) :

General location

t :

Timescale from the RE’s disconnection from the raceways

\(t_{\mathrm {b}}\) :

Timescale from the beginning of the RE-spall interaction

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

Time interval between the disconnection of the RE from the raceways and its impact on the spall floor

\(t_{\mathrm {dis}}\) :

Time interval between the beginning of the RE-spall interaction and the RE’s disconnection

\(t_{\mathrm {exit}}\) :

Time interval between the disconnection of the RE from the raceways and it exit from the spall

\(t_{\mathrm {imp}}\) :

Time interval between the disconnection of the RE from the raceways and its impact on the trailing edge

\(\hat{X}\) :

Unit vector in the vertical direction

\(\hat{x}\) :

Unit vector in the direction vertical to the leading edge

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

Vertical location of the inner ring

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

Vertical location of the outer ring

\(\hat{Y}\) :

Unit vector in the horizontal direction

\(\hat{y}\) :

Unit vector in the direction parallel to the leading edge

\(\delta \) :

Hertzian contact force deflection

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

Hertzian contact force RE deflection into the inner ring

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

Distance between the raceways’ centres

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

Deflection of the RE into the inner ring during its rotation around the leading edge

\(\delta _{\psi _{j}}\) :

Deflection of an RE into both of the rings

\(\dot{\delta }^{\left( -\right) }\) :

Impact speed

\({\varOmega }\) :

Angular velocity of the body system

\(\omega \) :

Angular speed

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

Angular speed of the cage

\(\psi \) :

Azimuth of an RE

\(\psi _{\mathrm {dis}}\) :

Azimuth of an RE at the disconnection from the raceways

\(\psi _{\mathrm {imp}}\) :

Azimuth of an RE at the impact with the trailing edge of the spall

\(\rho \) :

Material density

\({\varDelta }_{\mathrm {s}}\) :

Spall size

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Acknowledgments

We gratefully acknowledge Prof. Michael Lipsett for his constructive comments.

Author information

Authors and Affiliations

  1. Pearlstone Center for Aeronautical Engineering Studies and Laboratory for Mechanical Health Monitoring, Department of Mechanical Engineering, Ben-Gurion University of the Negev, P.O. Box 653, 8410501, Beer Sheva, Israel

    G. Kogan & J. Bortman

  2. R.K. Diagnostics, P.O. Box 101, 20103, Gilon, D.N. Misgav, Israel

    R. Klein

Authors
  1. G. Kogan
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  2. J. Bortman
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  3. R. Klein
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Corresponding author

Correspondence to J. Bortman.

Additional information

We gratefully acknowledge the invaluable support of the Pearlstone Foundation.

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Kogan, G., Bortman, J. & Klein, R. A new model for spall-rolling-element interaction. Nonlinear Dyn 87, 219–236 (2017). https://doi.org/10.1007/s11071-016-3037-1

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