Understanding FTF — Fundamental Train Frequency

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FTF (Fundamental Train Frequency — also called cage frequency or retainer frequency) is one of the four fundamental bearing fault frequencies. It represents the rotational speed of the bearing cage (the separator or retainer that holds the rolling elements in place and keeps them evenly spaced). The cage orbits around the bearing carrying the rolling elements with it, completing one revolution in the time it takes the whole set of rolling elements to travel once around the raceways. FTF is the lowest of the four bearing frequencies — typically 0.35× to 0.48× shaft speed, and therefore always sub-synchronous. Although cage defects themselves are rare, FTF is diagnostically important as the modulation frequency that creates sidebands around the other bearing frequencies, particularly BSF.

1. Definition: What FTF Represents

Every rolling-element bearing has a cage that traps the balls or rollers in pockets and shepherds them around the annulus between the inner and outer races. As the inner race turns with the shaft, it drags the rolling elements around, and the cage moves at their collective orbital speed. Because that orbital speed is roughly the average of the stationary outer race (zero) and the rotating inner race (shaft speed), the cage circulates at only about 40% of shaft speed. This orbital rate is the Fundamental Train Frequency — the slowest, gentlest rhythm in the bearing, but one that underlies the diagnosis of rolling-element faults.

2. Mathematical Calculation

Formula

FTF is derived from the bearing geometry and the shaft speed. Strictly, it is the cage speed seen from the rotating inner race; with a stationary outer race and rotating inner race it is:

FTF = (n / 2) × [1 − (Bd / Pd) × cos β]

Variables

• n = shaft rotational frequency in Hz (i.e. RPM ÷ 60).
• Bd = ball or roller diameter.
• Pd = pitch diameter (the diameter of the circle through the centres of the rolling elements).
• β = contact angle.

Simplified Form

For bearings with a zero contact angle (β = 0°, cos β = 1):

• FTF ≈ (n / 2) × [1 − Bd / Pd]
• For a typical bearing with Bd/Pd ≈ 0.2, this yields FTF ≈ 0.4 × n.
• Rule of thumb: FTF is about 0.4× shaft speed — 40% of the shaft frequency.

Typical Range

• FTF typically falls between 0.35× and 0.48× shaft speed, depending on geometry.
• Example: at 1800 RPM (30 Hz), FTF ≈ 12 Hz (0.4× shaft speed).
• It is always sub-synchronous (below 1× running speed).
• It is the lowest of the four bearing fault frequencies.

These calculations are part of any bearing-defect study; a bearing defect frequency calculator computes FTF alongside BPFO, BPFI and BSF directly from the geometry, which is far faster and less error-prone than working the formula by hand for every bearing on a machine.

3. Physical Significance

Cage Motion

The cage’s rotation is dictated by the rolling elements it carries:

• The rolling elements roll without slipping between the inner and outer races.
• The cage moves at the average velocity of the rolling-element centres.
• That velocity is approximately the midpoint between the stationary outer race (0) and the rotating inner race (shaft speed).
• Hence the cage circulates at roughly 40% of shaft speed.

The small departure from a clean 0.5× ratio — and the fact that real cages can experience minor slip — is exactly why FTF is irrational relative to running speed and never lands on a tidy harmonic.

Function of the Cage

• Spacing: maintains even spacing between the rolling elements.
• Guidance: keeps each rolling element on its proper orbital path.
• Lubrication: can help distribute lubricant through the bearing.
• Separation: prevents adjacent rolling elements from rubbing against one another.

4. When FTF Appears in Vibration Spectra

Direct Cage Defects

A primary FTF peak appears when the cage itself is damaged:

• Broken cage: a fractured or cracked cage structure.
• Worn pockets: excessive clearance between the cage and the rolling elements.
• Cage rubbing: the cage contacting the races or seals.
• Frequency: a direct FTF peak, often with harmonics.
• Rarity: cage-only defects are uncommon, accounting for under about 5% of bearing failures.

As Sideband Modulation (the More Common Role)

Far more often, FTF reveals itself as sideband spacing around BSF rather than as a peak of its own:

• When a rolling-element defect is present, BSF is active.
• The defective ball’s impact severity rises and falls as it orbits in and out of the load zone.
• That variation occurs at the cage orbital frequency — FTF.
• The result is sidebands at BSF ± FTF, BSF ± 2×FTF, BSF ± 3×FTF, and so on.
• This pattern is a reliable diagnostic fingerprint for rolling-element defects, and is sharpened by envelope analysis.

In Bearing Instability

• Sub-synchronous vibration from bearing-induced instability can appear near FTF.
• It may point to inadequate preload or excessive bearing clearance.
• It is distinguished from a true cage defect by its character — continuous and broadband rather than the discrete, repetitive impacting of a damaged cage.

5. Cage Defect Diagnosis

Symptoms of Cage Problems

• A peak at the FTF frequency in the vibration spectrum.
• Harmonics at 2×FTF, 3×FTF and beyond.
• Amplitude that is often erratic or variable rather than steady.
• Audible clicking or rattling in many cases.
• Periodic impacts sometimes visible in the time waveform.

Causes of Cage Defects

• Improper lubrication: inadequate lubrication causing cage wear.
• High-speed operation: excessive centrifugal force on the cage.
• Contamination: particles damaging the cage material or its pockets.
• Overheating: thermal distortion or softening of the cage material.
• Fatigue: high-cycle fatigue in thin cage sections.
• Installation damage: a cage bent or knocked during mounting.

6. Practical Importance and Relationship to the Other Bearing Frequencies

FTF as a Diagnostic Marker

FTF’s chief practical value lies in the spacing it imposes on sidebands:

• 1× sidebands: point to inner-race defects (modulation by shaft rotation as the defect passes through the load zone).
• FTF sidebands: point to rolling-element defects (modulation by the cage’s orbital motion).
• Pattern recognition: the sideband spacing alone often identifies the defect type at a glance.
• Advanced diagnosis: understanding FTF is what lets an analyst interpret an otherwise confusing bearing spectrum correctly.

In Automated Diagnostics

• Modern analysers compute all four bearing frequencies automatically from the bearing model.
• Software flags peaks at BPFO, BPFI, BSF and FTF.
• Automatic sideband detection uses FTF and 1× as the search spacings.
• Severity is graded from peak amplitude and harmonic content.

Frequency Hierarchy

The four bearing frequencies, in ascending order of magnitude:

• Lowest: FTF (≈ 0.4× shaft speed).
• Low–medium: BSF (≈ 2–3× shaft speed).
• Medium: BPFO (≈ 3–5× shaft speed).
• Highest: BPFI (≈ 5–7× shaft speed).

Mathematical Relationships

• All four frequencies stem from the same bearing geometry.
• Knowing one frequency and the bearing type lets you back-calculate the others.
• The ratios between them are fixed for a given bearing model, providing built-in cross-verification.
• Notably, for a bearing with Z rolling elements, BPFO + BPFI = Z × shaft speed and BPFO = Z × FTF — handy identities for sanity-checking a diagnosis.

In the field, these frequencies are only useful if your instrument can resolve them cleanly at the machine’s actual running speed. A portable two-channel analyser such as the Balanset-1A captures the spectrum and time waveform directly in the machine’s own bearings, so the slow FTF rhythm and the BSF ± FTF sideband family it generates can be picked out on site — and, where the underlying issue turns out to be excessive unbalance loading the bearing rather than a true cage fault, corrected then and there. To map every bearing tone onto the spectrum before you start, feed the bearing geometry into a bearing defect frequency calculator and overlay the predicted FTF, BSF, BPFO and BPFI lines.

FTF, then, may be the lowest and least frequently observed of the bearing fault frequencies, but it is far from unimportant. Its role as the modulation frequency for rolling-element defects, and its occasional signalling of genuine cage problems, make a working grasp of FTF essential to complete and accurate bearing condition assessment.

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