This research presents an analytical model to investigate the stability due to the ball bearing waviness in a rotating system supported by two ball bearings. The stiffness of a ball bearing changes periodically due to the waviness in the rolling elements as the rotor rotates, and it can be calculated by differentiating the nonlinear contact forces. The linearized equations of motion can be represented as a parametrically excited system in the form of Mathieu’s equation, because the stiffness coefficients have time-varying components due to the waviness. Their solution can be assumed as a Fourier series expansion so that the equations of motion can be rewritten as the simultaneous algebraic equations with respect to the Fourier coefficients. Then, stability can be determined by solving Hill’s infinite determinant for these algebraic equations. The validity of this research is proven by comparing the stability chart with the time responses of the vibration model suggested by prior research. This research shows that the waviness in the ball bearing generates the time-varying component of the stiffness coefficient, whose frequency is called the frequency of the parametric excitation. It also shows that the instability takes place from the positions in which the ratio of the natural frequency to the frequency of the parametric excitation corresponds to i/2 $i=1,2,3,….$

1.
Jones
,
A. B.
,
1960
, “
A General Theory of Elastically Constrained Ball and Radial Roller Bearings Under Arbitrary Load and Speed Conditions
,”
ASME J. Basic Eng.
,
82
, pp.
309
320
.
2.
Harris, T. A., 1991, Rolling Bearing Analysis, 3rd ed., John Wiley & Sons, Inc.
3.
Hamrock, B. J., and Dowson, D., Ball Bearing Lubrication—The Elastohydrodynamics of Elliptical Contacts, John Wiley & Sons, Inc.
4.
Yhland
,
E.
,
1992
, “
A Linear Theory of Vibrations Caused by Ball Bearings With Form Errors Operating at Moderate Speed
,”
ASME J. Tribol.
,
114
, pp.
348
359
.
5.
Aktu¨rk
,
N.
,
Uneeb
,
M.
, and
Gohar
,
R.
,
1997
, “
The Effects of Number of Balls and Preload on Vibrations Associated with Ball Bearings
,”
ASME J. Tribol.
,
119
, pp.
747
753
.
6.
Aktu¨rk
,
N.
,
1999
, “
The Effect of Waviness on Vibrations Associated with Ball Bearings
,”
ASME J. Tribol.
,
121
, pp.
667
677
.
7.
Jang
,
G. H.
, and
Jeong
,
S. W.
,
2002
, “
Nonlinear Excitation Model of Ball Bearing Waviness in a Rigid Rotor Supported by Two or More Ball Bearings Considering Five Degrees of Freedom
,”
ASME J. Tribol.
,
124
, pp.
82
90
.
8.
Newland, D. E., 1989, Mechanical Vibration Analysis and Computation, Longman Scientific and Technical.
9.
Hayashi, C., 1985, Nonlinear Oscillations in Physical Systems, Princeton University Press, Princeton, NJ.
10.
Nayfeh, A. H., and Mook, D. T., 1979, Nonlinear Oscillations, John Wiley & Sons, Inc.
11.
Wardle
,
F. P.
,
1988
, “
Vibration Forces Produced by Waviness of the Rolling Surfaces of Thrust Loaded Ball Bearing: Part 1—Theory
,”
Proc. Inst. Mech. Eng., Part C: Mech. Eng. Sci.
,
202
(
C5
), pp.
305
312
.