**Deep
Understanding of Roller/Ball Bearing Defect Frequencies**

As a
condition monitoring technician/analyst, you may have already known the roller/ball
bearing frequencies such FTF, BSF, BPFO and BPFI, and some of you may even
remember the formulae or have an Excel spread sheet to calculate these
frequencies. You may have also noted that the theoretical defect frequencies
may not exactly match the defect frequencies in real vibration spectra. Why? Are these formulae are always correct?
Under what conditions will these formulae give reasonable results? To answer
these questions, we need to know how these defect frequencies are derived.

Let’s look at
a roller or ball bearing with its outer raceway fixed and the inner raceway
free to rotate. The pitch diameter is denoted with D, and roller diameter with
d. The contact angle in this example is zero. The
inner race is rotating at an angular speed of ω. The number of
balls/rollers is **n**.

**Assumption1 :** All rolling elements are rolling without any slips between the
roller/ball and raceways.

This
assumption implies:

(1): No
clearances between each ball/roller and either the inner or outer raceway.

(2): The
friction on the contact points between a ball and any a raceway should be big
enough to prevent any slipping.

**Assumption 2:** The rotational axis of each
ball/roller keeps the same angle with the shaft, i.e. parallels with the shaft
if the contact angle is zero, or keeps the same angle with the shaft that
equals to the contact angle.

This assumption holds with cylindrical rollers, but may not be true
for balls, particularly when the ball is at the non-loading zone.

**3.
Cage rate or Fundamental Frequency**

To find the
cage rotating speed, we only need to know the speed of a point on the cage. We
can select a special point C as this point. We say this point is special
because both cage and the ball/roller have the same speed. To find out the
speed at Point C, let’s look at the ball/roller in the light green colour. Due
to our assumptions, point B on this ball/roller is stationary, and point A
should have the same speed as the contact point on the inner raceway, i.e.

_{}.

The roller is
actually doing the so called planar movement, which can be treated as rigid
body rotating about an instant axis at point B. Therefore, the speed at point C
can be easily obtained as

_{}

Now we’ve got the speed at point C on the cage, the angular speed of
the cage can be derived as

**4.
Ball Pass Frequency Outer race (BPFO)**

Imagine that
there is a defect on the outer raceway, when the cage has one revolution, all ** n ** balls/rollers will
pass the defect once. This means that the ball pass the outer race defect
frequency is ** n ** times the cage rate,
which gives

This is the
frequency that all balls pass one defect on the outer raceway, so it is also
the outer race defect frequency.

**5.
Ball Spin Frequency (BSF)**

To find out
the ball spin speed, let us check a single ball marked as light green in the
figure shown on the right hand side. When
a ball spins for one revolution, the ball moves from poison P1 to P2, and the
path on the outer raceway this ball travelled is marked as read colour. From
position P1 to position P2, the cage swept an angle

Now we know
the angle the ball rotated, that is, (a full revolution).
To find the angular speed of the ball, we need to find out the time it spent.
To this end, we check the cage. It rotated an angle
at a angular speed of _{}. The time used to rotate such an angle is

The angular speed of the ball can be obtained as

Is the ball/roller
spin frequency is the ball defect frequency? The answer is no. Why? In one
revolution of the ball, the defect hits the raceways TWICE, one on the outer
raceway (the impacting sensed by the accelerometer is stronger), and the other
on the inner raceway (this impacting sensed by the accelerometer is weaker).
Therefore, the ball defect frequency should be TWO TIMES the ball spin
frequency. The ball pass frequency is

**6.
Ball Pass Frequency Inner race (BPFI)**

Now we’ve got
two components that rotate around the centre point O of the bearing, one is the
inner race, which has an angular speed of ω, and another one is the cage with a
speed of _{}. Both components rotate at the same direction but with
different speeds. There are several ways to calculate the BPFI. Here, we
actually don’t need to derive the BPFI in a lengthy way, but “discover” it by
changing your observing point.

To better
understand the speed that the balls pass a point on the inner raceway,
imagine that you close your eyes and jumped onto the cage and rotate together
with the cage. Now, open your eyes, what you can see when standing on the cage?
The cage stands still, all balls rotate around a fixed point on the cage, and
the outer race way is now rotating (at the cage rotational speed)! Of course,
the inner race is still rotating, but at a lower angular speed, which is the
speed of the inner race relative to the cage

The above
formula is actually the frequency the inner raceway pass one ball. Multiplying
by the number of roller elements, we get the BPFI, i.
e.

BPFI is also the inner race defect frequency.