How the calculator works, and what it does not do

This is a static copy of the chapter for search engines. The interactive version has animated figures, check questions, and buttons that load the example into the calculator.

8. How the calculator works, and what it does not do

Where every number comes from

What	Source
Standard and shifted spur geometry, inv α_w	KHK Gears, Calculation of Gear Dimensions
σ_H0, Z_H, Z_E, Z_ε, ε_α	ISO 6336-2, as reproduced by MITCalc
Cross-check on Z_E·Z_H	Maitra, Handbook of Gear Design, via drivetrainhub
Willis equation	epicyclic gearing, standard kinematics
Harmonic and cycloidal ratios	strain wave gearing / cycloidal drive

The involute function is inverted by bisection rather than by table lookup, so a shift of any size works and there is no interpolation error. The tooth drawn on screen is a true involute computed from the base circle, not an arc approximation.

How it was checked

node test-gears.cjs runs 109 checks: inv 20° against the tabulated 0.014904, the 20/40 pair's contact ratio against a hand calculation, Z_H = 2.495 and Z_E = 189.8 against the published values, the ISO factors against Maitra's independent formulation of the same stress, the Willis equation balancing for every configuration, the 200/202 harmonic example returning exactly −100, and the scaling laws — σ_H with √F_t and with 1/√b — holding to three decimal places.

What is deliberately not modelled

Tooth root bending strength. The honest reason: the Lewis form factor is published only as a chart image in every accessible source, and copying numbers that cannot be checked is not something this project does. Deriving the critical section from the generated profile was tried and rejected — it reversed the trend with tooth count, which a test caught. Doing it properly needs the trochoidal root that the rack actually cuts and the ISO 6336-3 auxiliary-angle iteration for Y_F. GearLab rates flanks, not roots. For a real design the root check is not optional.

Helical gears. Spur only: β = 0, so Z_β = 1 and there is no overlap ratio and no axial force.
The ISO load factors K_v, K_Hβ, K_Hα. Only K_A is applied. Real gears see more than the nominal load.
The permissible stress side. σ_Hlim is used directly; the life, lubricant, roughness, velocity, size and work-hardening factors are not.
Scuffing, wear and micropitting, which at high speed or poor lubrication can govern before pitting does.
Internal gears, bevels, worms and racks.
Manufacturing tolerances and the quality grade, which set the real backlash and the dynamic factor.

Use it as a design comparator. It will tell you reliably whether wider teeth or a bigger module helps more, whether a shift rescues your pinion, and what a planetary against a harmonic drive costs you in backlash and efficiency. It will not tell you that a gearbox is safe to build.