How pinion runout propagates through a geartrain
A pinion is 15 µm eccentric relative to its bearing. What does the output shaft actually see, and which stage's runout matters for position and which for noise?
Runout is eccentricity of the tooth field with respect to the axis the gear actually turns about. It is not a tooth defect. Every tooth can be perfect and the gear can still be 20 µm off center, in which case the effective pitch radius rises and falls once per revolution and the mesh alternately tightens and loosens.
Two consequences follow, and they behave differently as they travel down a reduction. One is an angular position error. The other is a speed ripple. Confusing them is why runout arguments in a gearmotor go in circles.
δφ ≈ e / r, Δω/ω ≈ e / r
- δφ
- peak angular position error at that gear's own shaft, rad
- Δω/ω
- fractional speed ripple produced at that mesh
- e
- radial eccentricity of the tooth field, m
- r
- pitch radius of that gear, m
Position error is divided by everything downstream
An angular error introduced at the input shaft is reduced by every stage between it and the output, because the output turns that many times slower. An error introduced at the last gear is not reduced at all.
Take an illustrative train with a 25:1 total reduction. The input pinion has a 1.5 mm pitch radius and 15 µm of runout, which is 0.010 rad, or 0.57 degrees, at its own shaft. Referred to the output that becomes 0.023 degrees. The final gear has a 6 mm pitch radius and the same 15 µm of runout, which is 0.0025 rad, or 0.143 degrees, and it arrives at the output undivided. The final gear contributes six times the position error of the pinion, despite being the more accurate part in relative terms.
Speed ripple is not divided at all
A fractional speed modulation survives a reduction unchanged. If the input shaft speed varies by one percent, the output shaft speed varies by one percent, because the ratio is a constant multiplier. What the reduction changes is the frequency: the input's once-per-revolution ripple arrives at the output as 25 cycles per output revolution.
This is why the input stage is the one you hear and the output stage is the one you measure. Human hearing responds to the frequency band the input stage lands in, and a slow once-per-output-revolution wobble is usually below anything audible while being exactly what a position measurement records.
| Stage | Position error at the output | Frequency at the output | What it shows up as |
|---|---|---|---|
| Motor shaft and input pinion | Divided by the whole ratio. Usually negligible | High: once per input revolution | Audible modulation of the mesh tone, velocity ripple, current ripple |
| Intermediate stages | Divided by the ratio downstream of that stage | Intermediate | Sidebands around the mesh tones, sometimes a beat between stages |
| Final gear and output shaft | Full amplitude, undivided | Once per output revolution | Position error, dosing or indexing inaccuracy, a slow wobble |
Where runout is introduced
Runout is rarely a gear cutting error. It is usually introduced after the teeth are formed, which is why component inspection of the gear alone can pass a part that is eccentric in the assembly.
- Bore to tooth field. Any concentricity error between the mounting bore and the tooth field carries straight into the assembly.
- Press fit onto a shaft. A press that is not square, or a shaft with a burr, tips or displaces the gear. This is a common source and it is invisible in incoming inspection of the loose gear.
- Bearing clearance. A shaft that can move radially in its bearing adds an eccentricity that varies with load direction rather than staying fixed to the gear.
- Molded gear geometry. Shrinkage around a gate, ejector pin marks, and a non-round bore all produce eccentricity. A molded gear can also be round and still run out if it is molded onto or pressed onto an insert.
- Re-chucking during manufacture. Any operation that removes a part from its cutting reference and returns it introduces the concentricity error of that fixture.
How to measure it
-
Measure runout on the assembly, not only on the component
A ball or pin seated in successive tooth spaces, indicated while the shaft turns in its own bearings, measures what the mesh actually sees. Runout measured on an inspection arbor measures the gear, which is a different question.
-
Take a double-flank composite check where it is available
Total composite error contains the once-per-revolution runout component and the tooth-to-tooth component in one trace, and the two are separable by eye.
-
Use an order analysis on the running unit
With a speed reference, once-per-revolution content at each shaft is directly identifiable, and it tells you which stage carries the eccentricity without disassembly.
-
Sort and tear down the extremes
Runout is a distribution, not a constant. Comparing the best and worst units of a lot is how you learn which operation controls it.
What to do about it
- For position accuracy, control the last stage. It is one gear, one bore and one press operation, and it carries the error undivided.
- For noise, control the first stage. Its errors land in the audible band and are modulated onto the loudest mesh tone in the train.
- Clock the gears if two sit on one shaft, so their eccentricities oppose rather than add.
- Control the press operation, not just the parts. Squareness, seating force and shaft finish decide how much of a good gear's accuracy survives assembly.
- State runout on the drawing, referenced to the bearing journals, rather than assuming a quality grade covers it. The grade covers the gear. The assembly is what runs.
Runout changes the load, not only the motion
Eccentricity varies the effective center distance once per revolution, which means it varies backlash once per revolution by the same relation that governs any center-distance change. A gear with 20 µm of runout tightens and loosens its mesh by about 15 µm of backlash every turn.
That produces a cyclic load as well as a cyclic position error. On the tight part of the revolution the mesh runs with reduced clearance and higher friction, so drag torque and motor current rise; on the loose part they fall. The result is a current ripple at the same frequency as the runout, which is why a current trace with a speed reference can identify an eccentric gear without any mechanical measurement at all.
It also concentrates wear. The tight portion of each revolution carries the higher load, so the flanks in that angular sector see more contact stress than the rest. A gear that shows polished or worn flanks over part of its circumference and not the rest is an eccentric gear, whatever the inspection report says.