Bearing Vibration Frequency Calculator
- BPFI (inner race)
- Hz
- Result
- BSF (rolling element)
- Hz
- Result
- FTF (cage)
- Hz
- Result
This calculator works out the characteristic vibration frequencies of a rolling-element bearing from its geometry and rotational speed. You enter the bearing’s pitch diameter, rolling element diameter, number of rolling elements, and contact angle, along with the shaft speed, and you get back the four fundamental frequencies: BPFO, BPFI, BSF, and FTF. These values are used to diagnose bearing faults in rotating machinery.
Formula
The four bearing vibration frequencies are calculated using the standard rolling-element bearing equations. Each formula assumes the outer race is stationary and the inner race rotates with the shaft.
- FTF (Fundamental Train Frequency) – the rotational frequency of the cage (and thus the rolling elements’ center) around the bearing:
FTF = (N / 60) × 0.5 × (1 − (d/D) × cos α) - BPFO (Ball Pass Frequency, Outer race) – the rate at which rolling elements pass a single point on the outer race:
BPFO = (N / 60) × (n/2) × (1 − (d/D) × cos α) - BPFI (Ball Pass Frequency, Inner race) – the rate at which rolling elements pass a single point on the inner race:
BPFI = (N / 60) × (n/2) × (1 + (d/D) × cos α) - BSF (Ball Spin Frequency) – the rotational speed of a single rolling element about its own axis:
BSF = (N / 60) × (D/(2d)) × (1 − ((d/D) × cos α)²)
Where:
- N = shaft speed in revolutions per minute (rpm)
- n = number of rolling elements (balls or rollers)
- D = pitch diameter (mm) – the diameter of the circle passing through the centers of all rolling elements
- d = rolling element diameter (mm)
- α = contact angle (degrees) – the angle between the line joining the two raceway contact points and a plane perpendicular to the bearing axis
All frequencies are in Hertz (Hz), which is cycles per second.
Worked Example
Using the default values from the calculator above: D = 50 mm, d = 10 mm, n = 12, α = 0°, N = 1800 rpm.
First, convert shaft speed to revolutions per second: 1800 / 60 = 30 Hz.
Compute the geometric factor: (d/D) × cos α = (10/50) × cos 0° = 0.2 × 1 = 0.2.
Now calculate each frequency:
- FTF = 30 × 0.5 × (1 − 0.2) = 30 × 0.5 × 0.8 = 12.00 Hz
- BPFO = 30 × (12/2) × (1 − 0.2) = 30 × 6 × 0.8 = 144.00 Hz
- BPFI = 30 × (12/2) × (1 + 0.2) = 30 × 6 × 1.2 = 216.00 Hz
- BSF = 30 × (50/(2×10)) × (1 − 0.2²) = 30 × 2.5 × 0.96 = 72.00 Hz
These match the calculator’s outputs exactly.
How to Measure Bearing Vibration
In practice, you don’t calculate these frequencies from scratch every time – you measure the vibration signal and compare its spectrum against these theoretical values. A typical setup uses:
- An accelerometer mounted on the bearing housing (or as close as possible)
- A data acquisition system that samples the signal at a rate at least twice the highest frequency of interest (Nyquist criterion)
- Software that performs a Fast Fourier Transform (FFT) to convert the time-domain signal into a frequency spectrum
Once you have the spectrum, you look for peaks at the BPFO, BPFI, BSF, or FTF values (and their harmonics). If a peak appears at, say, 144 Hz in the example above, that suggests an outer-race defect. The amplitude of the peak often correlates with the severity of the fault.
How to Choose the Right Bearing Geometry
To use this calculator, you need the bearing’s geometry. This information is usually available from the bearing manufacturer’s datasheet or from the bearing’s part number. For common bearings, you can also measure the dimensions directly using precision tools:
- Pitch diameter (D): Measure the distance from the center of one rolling element to the center of the opposite rolling element, passing through the bearing center. Alternatively, average the inner and outer raceway diameters.
- Rolling element diameter (d): Measure a single ball or roller with a micrometer. For rollers, use the average diameter.
- Number of elements (n): Count them directly.
- Contact angle (α): This is often specified in the datasheet. For deep-groove ball bearings under radial load, it is close to 0°. For angular-contact bearings, it can be 15°, 25°, or 40°.
If you don’t have the datasheet, you can often find the geometry online by searching the bearing part number (e.g., 6205). Many suppliers list pitch diameter and ball diameter.
Common Mistakes and Misconceptions
A frequent error is mixing up units – the calculator uses mm for diameters and rpm for speed, but if you measure in inches, you must convert to mm (1 inch = 25.4 mm) or adjust the formulas accordingly. Another common mistake is using the shaft speed in Hz instead of rpm in the formulas – always convert rpm to Hz by dividing by 60.
Another misconception is that the vibration frequency equals the rotational speed of the shaft. In fact, bearing defect frequencies are almost always higher than shaft speed. For example, a 12-element bearing at 1800 rpm has BPFO at 144 Hz, which is 4.8 times the shaft speed (30 Hz).
Also note that these formulas assume zero slip. In reality, rolling elements can slip slightly under load, so measured frequencies may differ by 1–2% from theoretical values. This is normal and should be accounted for when setting alarm thresholds.
Limitations of the Formulas
The standard formulas above are based on a simplified kinematic model. They assume:
- Pure rolling (no sliding) between elements and raceways
- No deformation of the races or elements under load
- A constant contact angle
- Outer race stationary, inner race rotating
In real bearings, especially under heavy load or with clearance, these assumptions may not hold perfectly. The formulas are still excellent for identifying the frequency bands to look for, but for precise diagnostics, you may need to calibrate against a known-good bearing or use more advanced models that account for slip and load distribution.
Frequently Asked Questions
What is BPFO and BPFI?
BPFO stands for Ball Pass Frequency Outer race – the rate at which rolling elements pass a specific point on the outer raceway. BPFI is the same for the inner raceway. A defect on the outer race produces vibration at BPFO, while a defect on the inner race produces vibration at BPFI. Because the inner race rotates, BPFI is always higher than BPFO for the same bearing.
Why is the inner race frequency higher than the outer race?
The inner race rotates with the shaft, so a defect on it moves relative to the sensor. Each time a rolling element passes the defect, it produces a pulse, but the defect also moves, so the pulse rate is the sum of the cage speed and the relative motion. The outer race is stationary, so the pulse rate is just the cage speed times the number of elements. Hence BPFI = BPFO + FTF × n.
Can I use this calculator for roller bearings?
Yes, the same formulas apply to cylindrical and tapered roller bearings, provided you use the mean roller diameter and the correct contact angle. For tapered roller bearings, the contact angle is typically 15–30°, and the pitch diameter is the mean cone diameter. The formulas are slightly less accurate for rollers because of the line contact, but they are still a good approximation.
What if I don’t know the contact angle?
For most deep-groove ball bearings under radial load, the contact angle is very small (near 0°). You can set it to 0 and get a good estimate. For angular-contact bearings, check the bearing number or datasheet – common values are 15°, 25°, and 40°. If you’re unsure, use 0° as a starting point and compare the predicted frequencies with the measured spectrum to infer the actual angle.
How do I convert the frequency to orders?
Orders are multiples of the shaft rotational frequency. To convert a frequency in Hz to orders, divide by the shaft speed in Hz. For example, in the worked example, BPFO = 144 Hz and shaft speed = 30 Hz, so BPFO = 4.8 orders. Many vibration analyzers display spectra in orders, which is convenient because it doesn’t change with speed.