Bearing Vibration by Forced Rotation

While a bearing performs by rotation, this rotation generates vibration as well.
Vibration is affected by RPM, and its frequency, is called "Vibration by forced rotation"

Calculation of vibration by forced rotation in a ball bearing

In a ball bearing vibration is generated in axial, radial, and rotating directions. In certain bearing applications, vibration can have a significant effect.
This vibration sometimes causes other parts of the assemblies to resonate as vibration energy is emitted.
Understanding the bearing application characteristics well is an important part of selecting the right bearing specification for any application.

Formula for Bearing Inner Ring Rotation

Vibration caused by ball revolution (fa)
1 1 - Dw cos α0 fr
2 Dpw
Vibration caused by retainer rotation (fb) same as fa
Vibration caused by ball rotation (fc)
1 Dpw - Dw cos2 α0 fr
2 Dw Dpw
Vibration caused by ball pass (fd) Zfa
Z (fr - fa)
Vibration caused by inner ring raceway dents or bumps (fe) 
Vibration in Axial direction (fet) nZ (fr - fa)
Vibration in Radial direction (fer) fet ± fr
Vibration caused by outer ring raceway dents or bumps (ff)nZfa
Vibration caused by ball surface dents or bumps (fg) 
Vibration in Axial direction (fgt) 2nfc
Vibration in Radial direction (fgr) fgt ± fa

Formula for Bearing Outer Ring Rotation

Vibration caused by ball revolution (Fa)
1 1 + Dw cos α0 Fr
2 Dpw
Vibration caused by retainer rotation (Fb) same as Fa
Vibration caused by ball rotation (Fc)
1 Dpw - Dw cos2 α0 Fr
2 Dw Dpw
Vibration caused by ball pass (Fd) ZFa
Z (Fr - Fa)
Vibration caused by inner ring raceway dents or bumps (Fe)nZFa
Vibration caused by outer ring raceway dents or bumps (Ff) 
Vibration in axial direction (Fft) nZ (Fr - Fa)
Vibration in radial direction (Ffr) Fft ± Fr
Vibration caused by ball surface dents or bumps (Fg) 
Vibration in axial direction (Fgt) 2nFc
Vibration in radial direction (Fgr) Fgt ± Fa
Dw: Ball diameter (mm) Z: number of balls
DPW: Pitch circle diameter (mm) n: Integer number
α0: Nominal contact angle (°) fr: Inner ring rotation speed (Hz)
   Fr: Outer Ring rotation speed (Hz)

To simplify, cos α0 = 1 could be used.

The calculations below are examples.

example.1 : When the inner ring of an R-1560X2ZZ bearing is rotated at 1800RPM, vibration caused by ball or retainer revolution is calculated as follows:

fa = 1 1 - 2.778 × 1 × 30 = 11 Hz
2 10.5

As the difference in each ball gets bigger, vibration in the rotating direction also increases.
(Figure 1, 2)

example.2 : The amplitude of vibration at the vibration position calculated above for this R-1560X2ZZ bearing increases when the inner and outer ring raceways deform to hexagonal, heptagonal, and octagonal shapes. (Figure 3, 4, 5, and 6)

These calculations are very helpful to analyze bearing vibration, speed fluctuation, noise, and so on.

Normal vibration in rotating direction Normal bearing vibration in rotating direction : Figure 1 Figure 1

Vibration in rotating direction if the difference in each ball is huge Bearing Vibration in rotating direction if the difference in each ball is huge : Figure 2 Figure 2

Outer ring raceway deformation (Triangle) Bearing Outer ring raceway deformation (Triangle) : Figure 3 Figure 3

Inner ring raceway deformation (hexagonal shape) Bearing Inner ring raceway deformation (hexagonal shape) : Figure 4 Figure 4

Inner ring raceway deformation (heptagonal shape) Bearing Inner ring raceway deformation (heptagonal shape) : Figure 5 Figure 5

Inner ring raceway deformation (octagonal shape) Bearing Inner ring raceway deformation (octagonal shape) : Figure 6 Figure 6