Gear Train Speed Torque Mesh Force Calculator

The Gear Train Speed, Torque and Mesh Force Calculator rates one involute spur-gear mesh at common pitch. From the two tooth counts and the driver pitch diameter it returns the tooth-count ratio, the driven pitch diameter, the driven speed, the nominal centre distance and the rotation relationship. From the driver torque or the driver power it returns both shaft torques and both shaft powers at an entered mesh efficiency, and the three nominal pitch-point force components: tangential, radial and normal.

It runs from either of two input bases. Enter driver torque with driver speed, or driver power with driver speed. Whichever of those two fields is not in use is hidden and takes no part in the result. It handles an external mesh, where the gears counter-rotate and the centre distance is half the sum of the pitch diameters, and an internal mesh, where they turn the same way and the centre distance is half the difference.

It works in SI units only and there is no unit selector. Torque in newton metres, power in kilowatts, pitch diameter in millimetres, speed in revolutions per minute. Nothing converts on your behalf and nothing detects a wrong unit.

It is kinematics and mesh statics, not a gear rating. There is no tooth-root or contact stress, no material, hardness or quality grade, no service or dynamic factor, and no check for undercut, interference or contact ratio. The forces it reports are a nominal starting point for a rating calculation, not a substitute for one.

Purpose and model boundary

This model evaluates one steady involute spur-gear mesh. From tooth counts, driver pitch diameter, driver speed and either driver torque or power, it calculates ratio, driven speed, torque, power, pitch geometry and nominal pitch-point force components. It is a kinematic and static-force calculator, not a gear-rating or tooth-design system.

The spreadsheet remains the calculation authority. The page sends the named inputs to the calculation service and displays the workbook's outputs, chart and status without duplicating gear logic in browser code.

Inputs and units

Tooth counts must be positive integers. An internal mesh requires z2 > z1 so the derived pitch-diameter difference and centre distance are positive.

Governing relationships

The tooth-count ratio is R = z2/z1. The workbook uses n2 = n1/R, d2 = d1R, and P2 = ηP1. With ω1 = 2πn1/60, the inactive driver quantity is derived from P1 = T1ω1; driven torque is T2 = T1Rη.

At the driver pitch circle, nominal force magnitudes are

• tangential force Ft = 2T1/d1, with d1 converted to metres;
• radial force Fr = Ft tan φ;
• normal force Fn = Ft/cos φ.

External gears counter-rotate and use C = (d1 + d2)/2. Internal gears co-rotate and use C = (d2 − d1)/2.

Calculation sequence

1. Validate the listed basis and mesh type, integer tooth counts, active load input, speed, pitch diameter, pressure angle and efficiency.
2. Derive the tooth-count ratio, driven pitch diameter and centre distance.
3. Resolve driver power and torque from the selected input basis.
4. Calculate driven speed, efficiency-adjusted power and torque.
5. Calculate tangential, radial and normal nominal mesh forces at the pitch point.
6. Return the rotation relationship, torque comparison chart and workbook status.

Outputs and interpretation

Primary outputs are tooth-count ratio, driven speed, driven torque, driven power and tangential mesh force. Detail outputs show driver power and torque, driven pitch diameter, centre distance, radial and normal force, and rotation relationship. The chart compares positive driver and driven torque magnitudes; it does not encode rotation sign.

Force components are nominal static pitch-point values. They are not peak dynamic tooth loads, bearing reactions or allowable design loads.

Validation and status logic

The workbook has no intermediate CHECK state for this calculator. An invalid state also returns NOT AVAILABLE for the rotation relationship and zeroed protected numeric outputs.

Assumptions and limitations

• The model represents one parallel-axis spur mesh with a common transverse circular pitch and force applied at the nominal pitch point.
• Efficiency is a user-entered lumped power ratio in (0,1]; no loss or vendor efficiency curve is inferred.
• Reported torque and forces are positive magnitudes. Direction appears only through the external/internal relationship label.
• AGMA or other rating, root stress, contact stress, pitting, scuffing, fatigue, wear, backlash, interference, undercut, profile shift, lubrication and dynamic factors are not evaluated.
• Helical, bevel, worm, hypoid, planetary multi-mesh, rack-and-pinion, noncircular and crossed-axis gears are outside scope.
• No material, hardness, quality grade, service factor, bearing reaction, shaft size or manufacturing tolerance is selected.

Restrictions and non-computing states

The active input basis determines whether entered power or torque is used. A pressure angle of zero is permitted and produces zero radial force; 90° is excluded because the normal-force expression is singular. Internal geometry requires a larger driven gear. Any NOT VALID state supersedes formula residues.

Errors and warnings

A rejected entry means the request did not conform to the published input rules. NOT VALID is the workbook's non-computing state; this model does not emit a CHECK warning. A calculation-service failure is an availability error and must not be interpreted as a mechanical result.

No gear catalogue, rating table, allowable stress number or material database is reproduced, and none is embedded. The calculator holds no gear data: the tooth counts, the driver pitch diameter, the pressure angle and the mesh efficiency are all user-entered, and the shipped values, including the 17.3 degree pressure angle and the 0.937 efficiency, are synthetic numbers describing no real gear. The workbook implements the ordinary textbook form of the spur-mesh ratio, speed, torque and pitch-point force relations, plus exact public unit definitions, rather than any one publication's treatment. Its cited sources are below.

• NASA. Gear Design Simplified, for the tooth-count and pitch-diameter ratio, the speed and torque scaling, and the tangential and separating force relations. https://ntrs.nasa.gov/api/citations/19860005142/downloads/19860005142.pdf
• NASA. High-Speed Helical Gear Train Test Facility, whose public 10,000 revolutions per minute, 630 inch pounds-force, 3.5 inch, 20 degree spur-gear force example is recomputed independently as an arithmetic anchor. https://ntrs.nasa.gov/api/citations/20100015488/downloads/20100015488.pdf
• NIST. Guide to the SI, Appendix B.9: Factors for Units Listed by Kind of Quantity or Field of Science, for the exact inch pound-force to newton metre conversion. https://www.nist.gov/pml/special-publication-811/nist-guide-si-appendix-b-conversion-factors/nist-guide-si-appendix-b9

Naming these organisations identifies the source only; no endorsement or approval is implied.

Gear rating methods, allowable stress numbers, overload and dynamic factors, material data, quality grades and tooth-system geometry are not supplied here. Take them from AGMA rating practice and the gear manufacturer.

Additional source notes migrated from Methodology

The governing design standard, manufacturer data and qualified gear-design review control real equipment selection and rating.