Instrumentation & Measurement Magazine 24-3 - 43

Fig. 1. (a) Bearing structure, (b) Kinematics of a bearing; (c) Kinematics analysis of the bottom roller.

bearing component is damaged, the interaction of defects in
the bearing will produce impulses whenever the defect strikes
other bearing components. These impulses excite the natural
frequencies of the bearing system and bearing housing structures, resulting in an increase in the vibratory energy at this
element's specific frequencies.
These defect characteristic frequencies can be calculated
from kinematics analysis. Consider a bearing, shown in Fig.
1, which has a stationary outer ring and rotating inner ring (a
general case for most bearings). Assume that bearing has pure
rolling contact, sound installation, and stable operating conditions, with the following parameters:
d = rolling element diameter (in m),
D = pitch diameter (in m),
Z = number of rolling elements
α = angle of contact (in deg).
If the speed of the rotating inner ring is n rpm, or ωr =
2πn/60 rad/sec, or fr = n/60 Hz, these characteristic frequencies can be derived by the following procedures:
v1
Dd
(m/sec)	
 r
2
4

v
2

	

(1)

 
f 
v
d
d


f
Cage Frequency:
 1   (rad/sec), or
 1   (Hz).
D/2
D
2 
2
D
Consider the effect of the angle of contact, the modified cage
frequency will be:
c


fc

	

2

r

c

fr 

d
 1  cos   (Hz)	
2
D


r

(2)

 D
v
d
Rolling Element Spinning Frequency:


 1   (rad/sec),
2d 
d/2
D
f D
d

f
or
 1   (Hz). Consider the effect of the angle of contact,
2d 
D
the modified rolling element spinning frequency will be:
r

2

e

r

e


fe

	


fr D 
d2
2
 1  2 cos   (Hz)	
2d 
D


(3)

Z 
d


Z
Outer Race Defect Frequency: 
 1   rad/sec, or
2 
D
Zf 
d
f

 1   (Hz).
2 
D
Consider the effect of the contact angle, the modified outer
race defect frequency will be:
od

od

c

r

r

	

May 2021	


fod

Zfr 

d
 1  cos   (Hz)	
2 
D


(4)

Zr 

Inner Race Defect Frequency:   Z(   )  2  1  Dd  rad/sec, or


Zf 
d
f

 1  .
2 
D
Consider the effect of the angle of contact, the modified inner race defect frequency will be:
id

r

c



r

id


fid

	

Zfr 

d
 1  cos   (Hz)	
2 
D


(5)

 D
d
Rolling Element Defect Frequency:  2

 1   rad/sec, or
d 
D
f D
d
f

 1  .
d 
D
Consider the effect of the angle of contact, the modified rolling element defect characteristic frequency will be:
ed

ed

	

e

r

r


fed 2
fb

Dfr
d



d2
2
 1  2 cos   (Hz)	
D



(6)

Considerations in Frequency Analysis
While the computation of bearing characteristic frequencies is
straightforward, several factors can modify the vibration spectra related to bearing defects.
◗◗ The characteristic frequencies in (1)-(6) depend on the
defect location, the bearing geometry, and the speed of
rotation. The bearing parameters are available from the
bearing manufacturer. But care should be taken that, even
for a same bearing model, these parameters may vary
among manufacturers. In the case that bearing parameters are not available, inner race and outer race defect
characteristic frequencies can be approximated as 60%
and 40% of Zfr, respectively. This approximation is possible because the ratio of d/D is relatively constant for most
bearings.
◗◗ The vibration of rolling element bearings can be transmitted well to the bearing casing, which can be best
measured by using accelerometers. For bearings that
provide axial support, axial measurement often provides
a good indicator to defect because machines are usually
more flexible in this direction.
◗◗ For a bearing with a stationary inner ring and a rotating
outer ring, change the signs in the parentheses of (4) and
(5), respectively.
◗◗ Some of these characteristic frequencies will appear
on the vibration spectrum of a healthy bearing. This is

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