# Some Pump Related Calulations You May Want to Know

Motor bearing frequencies for pumps are calculated based on the running speed and the bearing geometry.  For rolling element bearings, the following formulas are used:
• Ball Pass Frequency Outer Race (BPFO) = Nb/2 x S x (1 - (Bd/Pd x cos(th))
• Ball Pass Frequency Inner Race (BPFI) = Nb/2 x S x (1 - (Bd/Pd x cos(th))
• Fundamental Train Frequency (FTF) = S/2 x (1 - (Bd/Pd x cos(th))
• Ball Spin Frequency (BSF) = Pd/2Bd x S x (1 - (Bd/Pd x cos(th))
Where:
• Nb = number of rolling elements
• S = speed (revolutions per second, in Hz)
• Bd = ball diameter
• Pd = pitch diameter
• th = contact angle (degrees)
The following guidelines can be used as a quick reference:
• Ball Pass Frequency Outer Race (BPFO) = Nb x S x 0.4
• Ball Pass Frequency Inner Race (BPFI) = Nb x S x 0.6
• Fundamental Train Frequency (FTF) = S x 0.4
• Ball Spin Frequency (BSF) = S x 1.6
For example, a centrifugal pump with eight blades operating at 1,800 RPM with 19 rolling elements in each bearing will have the following frequencies:
• Fundamental Train Frequency: 12Hz
• 1x Motor Speed: 30Hz
• Ball Spin Frequency: 48Hz
• 2x Motor Speed: 60Hz
• Ball Pass Frequency Outer Race (BPFO): 228Hz
• Blade Pass Frequency (FPF): 240Hz
• Ball Pass Frequency Inner Race (BPFI): 342Hz
The vibration spectrum will display a peak at each frequency noted above.  The actual frequency will be slightly lower as the speed slows under load with motor slip of a few percent.  The analysis should be performed over a frequency band accommodating this range of speed.

For variable speed pumps, the motor speed will vary over a range of speeds based on the required pressure, flow rate, or peak efficiency.

Each peak is analyzed for a trend in amplitude against pre-set warning and alarm levels.  Specific information may be available from individual manufacturers, or through operational specifications.  The following guidelines provide a useful start for analyzing the FFT max amplitudes in each band for a pump:

General (rotation and bearing locations including motor speed, FTF, BSF, BPFO, and BPFI)
 Level Amplitude (max) Warning 0.10 in/s Alarm 0.25 in/s Shutdown 0.62 in/s