Fault Diagnosis
Gear Vibration
When gear teeth go into and out of mesh, they create cyclic
forces and vibrations. These vibrations occur at the gear mesh
frequency (GMF), which is given by:
GMF  N
where N is the number of teeth and  is the angular velocity. For
example, a 40 tooth gear mounted on a shaft rotating at 3600
rpm would have a GMF of (40×3600/60) = 2400 Hz. These
periodic forces produce harmonics in the spectrum according to
the Fourier series. Therefore, vibrations are expected to occur
not only at the GMF, but also at its harmonics.
Amplitude modulation. This occurs in gears with excessive backlash or eccentricity. The result
is a high-frequency vibration signal (carrier wave) whose amplitude is varying at a lower
frequency (modulating wave), as shown below:
1.5
1
0.5
0
-0.5
-1
-1.5
0
0.5
1
1.5
2
2.5
3
Variation in the amplitude of vibration, or amplitude modulation, can be caused by varying
loads, misalignment, improper backlash and eccentricity in the shaft, teeth or gear. If one event
occurs during each revolution, then the modulating frequency is the speed of the problem
gear. This particular example represents gear vibrations at the GMF of 150 rad/s, modulated by
the speed of shaft rotation of 5 rad/s. The time signal can be expressed as:
x  1  0.5cos  5t  cos 150t 
This time signal leads to distinctive sidebands in the spectrum of gear vibration, as explained
here. Using the trigonometric identity:
cos  cos   12 cos      12 cos    
we can write the amplitude-modulated time signal as:
x  cos 150t   14 cos 155t   14 cos 145t 
Therefore, the Fourier analysis reveals a spectrum that has the GMF plus and minus sidebands,
as indicated below.
These sidebands also occur at the harmonics of the GMF, and the entire spectrum may appear as
indicated below.
Furthermore, when two or more events occur during each revolution (such as tooth-to-tooth
spacing errors or eccentricity errors), the modulating frequency will be two or more times the
speed of rotation of the faulty gear, resulting in a family of sidebands around the GMF and its
harmonics.
Bearing Faults
Damage may occur in the inner ring, rolling element or outer ring. Discrete faults give rise to a
series of bursts at a rate corresponding to the contacts with the rolling elements. The pulse
patterns are shown below. Note that inner race faults rotate in and out of the loaded zone giving
amplitude modulation.
Damaged inner ring
Damaged rolling element
Damaged outer ring
These faults are distinguished by their frequency of occurrence. Consider a rolling bearing
having a fixed outer ring, as shown below.
Rb
s
Rs
Ro
From kinematics, Rs s  2Rb b , since we are assuming rolling without slipping. The angular
velocity of the cage is related by Rs  Rb  c  Rb b . If there is a defect on the outer race, it
happens at a frequency of
Rb Rs
Rs
d  Nc  N
s  N
s
2  Rs  Rb 
 Rs  Rb  2Rb
where N is the number of balls in the bearing. If there is a defect on the inner race, then it will
appear at a frequency of
2  Rs  Rb   Rs
Rs  2 Rb
s
2  Rs  Rb 
2  Rs  Rb 
If there is a defect on the ball, the calculation is a little more difficult. When the defect makes a
complete rotation with respect to the outer race, it will have rotated an angle of

 R  Rb 
2Rb
R 

2    2 
 2 1  b   2  o
Ro
Ro 

 R0 
and it will generate 2 pulses as it contacts the inner and outer races. Accordingly, we have 2
 R  Rb 
 Ro  Rb 
pulses every  o
 revolution of the ball, or one pulse every 
 revolution,
 R0 
 2 R0 
therefore the frequency of a defective ball is
2Ro
Ro Rs
R  2Rb Rs
d 
b 
s  s
s
Ro  Rb
Ro  Rb Rb
Rs  Rb Rb
d  N s  c   N
s  N
Misalignment
When the signals from two proximity probes are combined together in a two-channel
oscilloscope or vibration analyzer, the orbital motion of the shaft can be observed. Misalignmetn
appears not only at the 1X frequency but also at 2X and 3X.