Gears · Backlash equivalence
Gear Backlash Calculator
Calculate spur gear circular, normal, and angular backlash from center-distance increase, or find the equivalent center-distance change.
Reference calculator #007
Choose the known backlash quantity
Inputs stay in your browser. Values are normalized to canonical units before calculation.
Calculated output
Results
- Normal backlash, jn
- Equivalent center-distance increase, Δa
- Gear 1 angular backlash, θ₁
- Gear 2 angular backlash, θ₂
Valid spur-gear backlash equivalence result
- The model covers one external involute spur-gear pair with parallel axes and zero helix angle.
- Circumferential backlash is measured along the operating pitch-circle tangent; angular backlash is the free rotation of one gear while its mate is fixed.
- The center-distance relationship is a small-increment equivalence at the entered operating transverse pressure angle.
- Any entered circular backlash is treated as total mesh backlash, not as the tooth-thickness reduction assigned to either individual gear.
- Tooth-thickness deviations, runout, mounting error, deflection, temperature, lubrication, wear, and manufacturing tolerances are not included.
Calculation engine: gear-backlash/1.0.0
Spur gear backlash formulas
This calculator uses explicit backlash definitions for one external involute spur-gear pair. It can solve from center-distance increase or reverse the same relationship to find an equivalent center-distance increase from known circular backlash.
| Symbol | Meaning | Unit |
|---|---|---|
jt |
Total circumferential backlash at the operating pitch circles | mm or in |
jn |
Normal backlash | mm or in |
Δa |
Increase from the zero-backlash reference center distance | mm or in |
α |
Operating transverse pressure angle | degrees |
d₁, d₂ |
Operating pitch diameters | mm or in |
θ₁, θ₂ |
Free rotation of one gear while its mate is fixed | degrees or arcmin |
Worked center-distance example
For Δa = 0.1 mm, α = 20°, d₁ = 40 mm, and d₂ = 80 mm:
- Circular backlash:
jt = 2 × 0.1 × tan 20° = 0.072794 mm. - Normal backlash:
jn = 0.072794 × cos 20° = 0.068404 mm. - Gear 1 angular backlash:
θ₁ = 2 × 0.072794 / 40 = 12.512375 arcmin. - Gear 2 angular backlash:
θ₂ = 2 × 0.072794 / 80 = 6.256187 arcmin.
The smaller gear has twice the angular free movement because its pitch diameter is half as large. These are per-gear values measured while the other gear is fixed; they should not be added together as a single shaft result without a defined transmission coordinate system.
Worked circular-backlash example
For jt = 0.005 in, α = 20°, d₁ = 2 in, and d₂ = 4 in, the equivalent center-distance increase is approximately 0.006869 in, normal backlash is 0.004698 in, and angular backlash is approximately 17.188734 arcmin at gear 1 and 8.594367 arcmin at gear 2.
How to use the gear backlash calculator
Choose the known quantity. Use Center-distance increase only when the change is referenced to the zero-backlash geometry and is small relative to the gear diameters. Use Circular backlash for a specified or measured tangential value at the operating pitch circle.
Enter both operating pitch diameters because the same linear backlash corresponds to different angular play at each shaft. Unit selectors convert the physical values; URL parameters preserve the original values and units for reproducible sharing.
Backlash is not a universal allowance
This tool does not calculate a recommended minimum, nominal, or maximum backlash. ISO 21771-2:2025 addresses relationships between backlash, tooth thickness, center distance, and tooth deviations, but its public scope explicitly does not provide guidance for selecting tooth thickness or its tolerance. The published edition is currently marked for revision, with ISO/CD 21771-2 under development.
Actual backlash can include tooth-thickness deviations, pitch and profile errors, runout, center-distance tolerance, bearing clearance, shaft and housing deflection, temperature, lubrication film, wear, and measurement uncertainty. A zero result is mathematically permitted but triggers a warning because it is not evidence that an assembly will run without binding.
Engineering scope and limitations
The V1 calculator excludes:
- helical, internal, bevel, worm, hypoid, rack, planetary, and crossed-axis gears;
- profile shift, detailed involute operating geometry, tooth-thickness limits, cutter data, and accuracy grades;
- manufacturing tolerances, runout, composite error, measurement uncertainty, and acceptance limits;
- temperature, deflection, lubrication, wear, friction, noise, efficiency, dynamics, load, and strength;
- multi-stage accumulated lost motion, motor- or encoder-side backlash, compliance, and signed directions.
Use the result as a transparent geometric equivalence, then validate the complete tolerance stack and measurement method before approving a design.
Frequently asked questions
How does center distance change spur gear backlash?
For the small-increment external spur-gear relationship used here, circular backlash change is jt = 2Δa tan α. Increasing center distance increases backlash; reducing it decreases backlash and can increase binding risk.
What is circular gear backlash?
Circular or circumferential backlash is the free tangential distance at the operating pitch circle between the non-driving tooth flanks. This calculator treats it as total mesh backlash.
How do you convert linear backlash to angular backlash?
For a gear with operating pitch diameter d, angular backlash in radians is θ = 2jt/d while the mating gear is held fixed. The same linear backlash produces more angular movement on the smaller gear.
What is the difference between circular and normal backlash?
Circular backlash is measured along the operating pitch-circle tangent. Normal backlash is measured normal to the tooth flank; for this spur-gear model, jn = jt cos α.
Does this calculator recommend the correct gear backlash?
No. Selecting backlash requires tooth-thickness limits, accuracy grade, runout, center-distance tolerance, temperature, lubrication, deflection, speed, load, material, manufacturing process, and measurement requirements.
References and review status
Reviewed . References support formula and scope checks; they do not imply endorsement, certification, or standards conformity.
- KHK USA — A Primer on Backlash and Its Purpose in Gear Designs — Technical reference used to cross-check center-distance, circumferential, and normal backlash relationships.
- Boston Gear — Engineering Information — Independent manufacturer reference used to cross-check center-distance and backlash change at 14.5° and 20° pressure angles.
- ISO 21771-2:2025 — Tooth Thickness and Backlash — Official scope source for cylindrical involute gear tooth-thickness and backlash relationships; ISO currently lists a replacement committee draft.