Abstract

This paper proposes a model of concentrated parameters for a rolling bearing operating in dynamic conditions with and without localized defect. The rolling bearing is modeled as a Z + 2 df degree of freedom (DOF) system, where Z is the number of rolling elements. The radial displacement of these rolling elements is considered in this model. In the analytical formulation, the contact force between the balls and races is considered as non-linear spring-dampers, whose stiffnesses are obtained applying Hertzian elastic contact deformation theory. The equations of motion are formulated using Lagrange's equation, considering the characteristics of the individual components of a rolling bearing, such as rotor, rolling elements, and inner and outer race. The Runge-Kutta method is used to solve the non-linear differential equations of motion. The simulation is accomplished by MATLAB and SIMULINK. To validate the simulated model, we have designed a testbed to carry out. The frequency components of the signal generated by the model in simulation and the experimentally obtained signal are compared. The results achieved experimentally demonstrate the validity of the mathematical model presented here. The model provides a powerful tool to predict the satisfactory behavior of this system.

References

1.
Harris
,
T. A.
,
Rolling Bearing Analysis
, 4th ed.,
Wiley
,
Hoboken, NJ
,
2001
.
2.
Changsen
,
W.
,
Analysis of Rolling Element Bearings
,
Mechanical Engineering
,
London
,
1991
.
3.
Wang
,
H.
and
Chen
,
P.
, “
Fault Diagnosis for a Rolling Bearing Used in a Reciprocating Machine by Adaptive Filtering Technique and Fuzzy Neural Networks
,”
WSEAS Trans. Syst.
, Vol.
7
, No.
1
,
2008
, pp.
1
6
. Available at http://www.wseas.us/e-library/transactions/systems/2008/2007-526N.pdf
4.
Cao
,
Y.
and
Altintas
,
Y.
, “
A General Method for the Modeling of Spindle-Bearing Systems
,”
J. Mech. Design
, Vol.
126
,
2004
, pp.
1089
1104
. https://doi.org/10.1115/1.1802311
5.
Walters
,
C. T.
, “
The Dynamics of Ball Bearings
,”
J. Lubr. Eng.
, Vol.
93
,
1971
, pp.
1
10
. https://doi.org/10.1115/1.3451516
6.
Wensing
,
J.
,
1998
, “
On the Dynamics of Ball Bearings
,” Ph.D. thesis,
University of Twente
, The Netherlands.
7.
Dietl
,
P.
,
Wensing
,
J.
, and
van Nijen
,
G.
, “
Rolling Bearing Damping for Dynamic Analysis of Multi-Body Systems—Experimental and Theoretical Results
,”
J. Multi-Body Dyn.
, Vol.
214
, No.
1
,
2000
, pp.
33
43
. https://doi.org/10.1155/S1023621X02000234
8.
Sarangi
,
M.
,
Majumdar
,
B. C.
, and
Sekhar
,
A. S.
, “
On the Dynamics of Elastohydrodynamic Mixed Lubricated Ball Bearings. Parts I and II
,”
J. Eng. Tribol.
, Vol.
219
,
2005
, pp.
411
433
. https://doi.org/10.1243/135065005X34071
9.
Gupta
,
P. K.
,
Advanced Dynamics of Rolling Element Bearings
,
Springer Verlag
,
Berlin
,
1984
.
10.
Gupta
,
T. C.
,
Gupta
,
K.
, and
Sehgal
,
D. K.
, “
Nonlinear Vibration Analysis of an Unbalanced Flexible Rotor Supported by Ball Bearings With Radial Internal Clearance
,”
Proceedings of the 53rd ASME Turbo Expo 2008
, Vol.
5
,
Part A, Berlin
,
Germany
, June 9–13,
2008
, pp.
1289
1298
.
11.
Tiwari
,
M.
,
Gupta
,
K.
, and
Prakash
,
O.
, “
Experimental Study of a Rotor Supported by Deep Groove Ball Bearing
,”
Int. J. Rotat. Mach.
, Vol.
8
, No.
4
,
2002
, pp.
243
258
. https://doi.org/10.1080/1023-620291910879
12.
Kennel
,
J. W.
and
Bupara
,
S. S.
, “
A Simplified Model of Cage Motion in Angular Contact Bearings Operating in the EHD Lubricating Regime
,”
J. Lubr. Technol.
,
1978
, Vol.
101
, pp.
395
401
. https://doi.org/10.1115/1.3453196
13.
Meeks
,
C.
, “
Ball Bearing Dynamic Analysis Using Computer Methods and Correlation With Empirical Data
,”
International Tribology Conference
, Melbourne, Australia, Dec 2–4,
1987
.
14.
Fukata
,
S.
,
Gad
,
E.
,
Kondou
,
T.
,
Ayabe
,
T.
, and
Tamura
,
H.
, “
On the Radial Vibrations of Ball Bearings (Computer Simulation)
,”
Bull. JSME
, Vol.
44
,
1985
, pp.
83
111
.
15.
Meeks
,
C.
and
Tran
,
L.
, “
Ball Bearing Dynamic Analysis Using Computer Methods. Part I Analysis
,”
J. Tribol.
, Vol.
118
,
1996
, pp.
52
58
. https://doi.org/10.1115/1.2837092
16.
Harsha
,
S. P.
, “
Non-Linear Dynamic Response of a Balanced Rotor Supported on Rolling Element Bearings
,”
Mech. Syst. Signal Proc.
, Vol.
19
,
2005
, pp.
551
578
. https://doi.org/10.1016/j.ymssp.2004.04.002
17.
Harsha
,
S. P.
, “
Nonlinear Dynamic Analysis of a High-Speed Rotor Supported by Rolling Element Bearings
,”
J. Sound Vib.
, Vol.
290
,
2006
, pp.
65
100
. https://doi.org/10.1016/j.jsv.2005.03.008
18.
Harsha
,
S. P.
,
Sandeep
,
K.
, and
Prakash
,
R.
, “
Effects of Preload and Number of Balls on Nonlinear Dynamic Behaviors of Ball Bearing System
,”
Int. J. Nonlin. Sci. Numer. Sim.
, Vol.
4
, No.
3
,
2003
, pp.
265
278
.
19.
Nataraj
,
C.
and
Harsha
,
S. P.
, “
The Effect of Bearing Cage Run-Out on the Nonlinear Dynamics of a Rotating Shaft
,”
Commun. Nonlin. Sci. Numer. Sim.
, Vol.
13
,
2008
, pp.
822
838
. https://doi.org/10.1016/j.cnsns.2006.07.010
20.
Upadhyay
,
S. H.
,
Harsha
,
S. P.
, and
Jain
,
S. C.
, “
Analysis of Nonlinear Phenomena in High Speed Ball Bearings due to Radial Clearance and Unbalanced Rotor Effects
,”
J. Vib. Control
, Vol.
16
, No.
1
,
2010
, pp.
65
88
. https://doi.org/10.1177/1077546309104188
21.
Purohit
,
R. K.
, “
Dynamic Analysis of Ball Bearings With Effect of Preload and Number of Balls
,”
Int. J. Appl. Mech. Eng.
, Vol.
11
, No.
1
,
2006
, pp.
77
91
. Available at http://www.ijame.uz.zgora.pl/ijame_files/archives/v11PDF/n1/77-91_Article_05.pdf
22.
Feng
,
N.
,
Hahn
,
E.
, and
Randall
,
R.
, “
Simulation of Vibration from a Rolling Element Bearing Defect
,”
International Conference on Health and Usage Monitoring
, Melbourne, Australia, Feb 19–20,
2001
, pp.
37
52
.
23.
Bhattacharyya
,
K.
and
Mukherjee
,
A.
, “
Modeling and Simulation of Centerless Grinding of Ball Bearings
,”
Sim. Modell. Pract. Theory
, Vol.
14
,
2006
, pp.
971
988
. https://doi.org/10.1016/j.simpat.2006.05.003
24.
Klang
,
Y.
,
Shen
P.
,
Huang
,
Ch.
,
Shyr
,
S.
, and
Chang
,
Y.
, “
A Modification of the Jones-Harris Method for Deep-Groove Ball Bearings
,”
Tribol. Int.
, Vol.
39
,
2006
, pp.
1413
1420
. https://doi.org/10.1016/j.triboint.2005.12.005
25.
Rubio
,
H.
,
Garcia-Prada
,
J. C.
, and
Laniado
,
E.
, “
Dynamic Analysis of Rolling Bearing System Using Lagrangian Model vs. Fem Code
,”
Proceedings of the 12th IFToMM World Congress
, Besançon, France, June 18–21,
2007
.
26.
Chen
,
M.
,
Meng
,
G.
, and
Wu
,
B.
, “
Nonlinear Dynamics of a Rotor-Ball Bearing System With Alford Force
,”
J. Vib. Control
, Vol.
18
, No.
1
,
2012
, pp.
17
27
. https://doi.org/10.1177/1077546311405701
27.
Dougdag
,
M.
,
Ouali
,
M.
,
Boucherit
,
H.
,
Titouche
,
N. E.
, and
Djaoui
,
M.
, “
An Experimental Testing of a Simplified Model of a Ball Bearing: Stiffness Calculation and Defect Simulation
,”
Meccanica
, Vol.
47
,
2012
, pp.
335
354
. https://doi.org/10.1007/s11012-011-9434-0
28.
Gao
,
X. H.
,
Huang
,
X. D.
,
Wang
,
H.
, and
Chen
,
J.
, “
Modelling of Ball-Raceway Contacts in a Slewing Bearing With Non-Linear Springs. Proceedings of the Institution of Mechanical Engineers, Part C
,”
J. Mech. Eng. Sci.
, Vol.
225
, No.
4
,
2011
, pp.
827
831
. https://doi.org/10.1177/09544062JMES2454
29.
Kappaganthu
,
K.
and
Nataraj
,
C.
, “
Nonlinear Modeling and Analysis of a Rolling Element Bearing With a Clearance
,”
Commun. Nonlin. Sci. Numer. Simul.
, Vol.
16
,
2011
, pp.
4134
4145
. https://doi.org/10.1016/j.cnsns.2011.02.001
30.
Slavič
,
J.
,
Brković
,
A.
, and
Boltežar
,
M.
, “
Typical Bearing-Fault Rating Using Force Measurements: Application to Real Data
,”
J. Vib. Control
, Vol.
17
, No.
14
,
2011
, pp.
2164
2174
. https://doi.org/10.1177/1077546311399949
31.
Tadina
,
M.
and
Boltezar
,
M.
, “
Improved Model of a Ball Bearing for the Simulation of Vibration Signals because of Faults during Run-Up
,”
J. Sound Vib.
, Vol.
330
,
2011
, pp.
4287
4301
https://doi.org/10.1016/j.jsv.2011.03.031
32.
Ahmad
,
R.
,
Saeed
,
A.
,
Anoushiravan
,
F.
, and
Hamid
,
M.
, “
Nonlinear Dynamic Modeling of Surface Defects in Rolling Element Bearing Systems
,”
J. Sound Vib.
, Vol.
319
,
2009
, pp.
1150
1174
. https://doi.org/10.1016/j.jsv.2008.06.043
33.
SIMULINK; Dynamic Simulation for MATLAB
. (
1999
).
The Math Works
,
Natick, MA
.
34.
Braut
,
S.
,
Žigulić
,
R.
, and
Butković
,
M.
, “
Numerical and Experimental Analysis of a Shaft Bow Influence on a Rotor to Stator Contact Dynamics
,”
J. Mech. Eng.
, Vol.
54
, No.
10
,
2008
, pp.
693
706
. Available at http://cat.inist.fr/?aModele=afficheN&cpsidt=20807799
35.
Bai
,
C. Q.
,
Zhang
,
H. Y.
, and
Xu
,
Q. Y.
, “
Effects of Axial Preload of Ball Bearing on the Nonlinear Dynamic Characteristics of a Rotor-Bearing System
,”
Nonlinear Dynam.
, Vol.
53
, No.
3
,
2008
, pp.
173
190
. https://doi.org/10.1007/s11071-007-9306-2
36.
Li
,
B.
,
Chow
,
M.
,
Tipsuwan
,
Y.
, and
Hung
,
J. C.
, “
Neural-Network-Based Motor Rolling Bearing Fault Diagnosis
,”
IEEE Trans. Ind. Electron.
, Vol.
47
, No.
5
,
2000
, pp.
1060
1069
. https://doi.org/10.1109/41.873214
37.
Liu
,
T.
,
Ordukhani
,
F.
, and
Jani
,
D.
, “
Monitoring and Diagnosis of Roller Bearing Conditions Using Neural Networks and Soft Computing
,”
Int. J. Knowl.-Based Intell. Eng. Syst.
, Vol.
9
, No.
2
,
2005
, pp.
149
157
. https://doi.org/10.1007/11553939
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