Nominal calculations and technical references

Revision of 23 October 2026. The formulas were implemented in the project without copying code or tables from the references. engineering.py contains the calculations; geometry.py, assemblies.py and geartrains.py produce the geometry.

Conventions

Dimensions in mm, forces in N, torque in N·m, speed in rpm and power in W. The signed ratio is i = n input / n output: a negative value means opposite directions. Output torque is reported as a magnitude, not as a reaction vector.

Efficiency is an assumption supplied by the user, not a result of the model. Fatigue sizing, Hertzian pressure, ISO 6336/AGMA checks, bearing life and elastic simulation are not implemented. A configured torque therefore does not represent an allowable torque. Harmonic and worm-wheel pairs remain geometric studies.

Cylindrical teeth: normal and transverse geometry

For helical and double-helical gears, the entered module and pressure angle are normal:

mt = mn / cos β
αt = atan(tan αn / cos β)
d = z mt
db = d cos αt
da = d + 2 mn (ha* + x)
df = d − 2 mn (hf* − x)
sn = mn (π/2 + 2 x tan αn)

For spur teeth, β = 0. Addendum and dedendum use mn, including in helical gears: the previous revision incorrectly used mt for the heights. For an unshifted internal ring gear, the internal tip diameter is d − 2mn and the root diameter is d + 2,5mn.

Industrial source: HPC Europe, Precision gears — Technical information, page 132. Profile shift x and coefficients ha*/hf* are available for individual gears. In the new external pairs, x1 = x and x2 = −x: the zero sum preserves the theoretical centre distance while changing thicknesses and heights. For planetary gears, internal ring gears and studies, an independent profile shift that would make the pairs incompatible is not exposed.

The configured tangential backlash is the pair's total nominal backlash, divided between the two gears. The external root is now the envelope of the rounded tip of a normal rack: the transverse tip section is elliptical in helical gears. Profile shift x actually translates the rack; undercut trims the fillet at its intersection with the usable involute branch. The normal tip radius is min(0,3mn, limit preserving a half-land of 0,025mn). This is not a selected commercial tool. The internal ring gear instead uses a circular fillet tangent to the involute and root, explicitly not generated by a pinion-cutter. Methods, parameters and limits are in the per-part metadata; exported metadata reports the profile method, domain and limits.

Contact and centre distance

For balanced external pairs, the working centre distance can be increased:

a0 = (d1 + d2)/2
αw = acos(a0 cos αt / a)
g1 = sqrt(ra1² − rb1²)
g2 = sqrt(ra2² − rb2²)
εα = (g1 + g2 − a sin αw) / (π mt cos αt)
εβ = b active |sin β| / (π mn)
ε total = εα + εβ
Δj = 2a [inv(αw) − inv(αt)], where inv(α) = tan α − α
j working = j nominal × a/a0 + Δj

For an internal pair, a0 = (d2 − d1)/2 and the numerator of εα becomes g1 − g2 + a sin αw; ra2 is the internal tip radius. The historical contact_transverse/contact_total values retain the nominal tip/base calculation. profile_contact_transverse/profile_contact_total instead intersect the two actual involute intervals after trimming/blend; the report exposes both and warns if the second total is less than one. The corrected calculation excludes contact on fillets and does not check 3D collisions or chamfers. These are not loaded-contact analyses. Source: Drivetrain Hub, Geometry / Helical Gears, Gear Mesh and Contact Ratio sections.

The generator rejects total contact ratio < 1 and involute interference in the new pairs. For double-helical gears, b active is (width − gap)/2, even when the gap is zero: the two halves are not treated as one continuous helix. Δj is additional to the backlash already configured.

Undercut and internal interference

The indicative standard-rack condition is x minimum = ha* − zv sin²αn / 2, with zv = z for spur teeth and zv ≈ z/cos³β for helical gears. The virtual-tooth check remains a design approximation. The geometric warning now uses the actual rounded-rack threshold x_min=(hf*mn−ρ(1−sinαn)−rp sin²αt)/mn, distinct from the indicative virtual-tooth limit; the profile records whether the branch was trimmed. Source for the standard limit and the role of profile shift: KHK, Technical Data, sections 3.5–3.6.

The new internal pairs require a tooth-count difference > 9, in addition to the base-circle check. This is a conservative generator filter; the KHK note on the limit of 9 concerns standard teeth at 20° and is not a guarantee for other angles. Cutting interference and tool profile remain to be checked. Source: KHK, Internal ring gears — design and considerations.

The old 18/18/54 spur planetary preset can enter the area below the base circle in the planet-ring pair. It is retained for compatibility, with an explicit warning and theoretical contact ratio in the report. It is not declared interference-free. A 24/24/72 combination avoids this specific warning and retains the 4:1 ratio; this does not make it a strength-rated design.

Kinematics and forces

External pair: i = −z2/z1
Fixed-axis internal pair: i = +z2/z1
Compound with two external stages: i = (z2/z1)(z4/z3)
Fixed-ring planetary: i = 1 + zr/zs
P input = T input × n input × 2π/60
n output = n input / i
|T output| = T input × |i| × η overall
Ft = 2000 |T| / d working
Fr = Ft tan αw
Fa = Ft tan β

In the compound train, η stage = √η overall is an explicit assumption for estimating intermediate torque. The coaxial variant requires equal centre distances; input and output shafts are axially separated. The two intermediate gears must be rigidly connected, but the mechanical connection has not yet been designed. Teaching sources: JNTUA/JNTUK, Gear Trains, lectures by Prof. N. Mohan Rao, University of Arkansas, Introduction to Mechanical Design and Manufacturing — Gearing Systems. The latter sometimes uses the reciprocal ratio convention; here, i = n input/n output is always explicit.

The forces are nominal references on the driving gear, not complete support reactions. Fr and Ft are magnitudes; the sign of Fa follows the pinion's reference hand and does not by itself determine the direction of the reaction in the assembly. For double-helical gears, Fa is the conservative full-torque reference for a single helix; overall cancellation requires symmetric loads. Independent cross-check example: T = 100 N·m, d = 40 mm, αw = 21,515° → Ft = 5000 N, Fr ≈ 1971,1 N. Source: Drivetrain Hub, Strength / Force Analysis.

For planetary gears, the coaxiality, equal-spacing and inter-planet distance conditions are retained. The sun-planet and planet-ring contact ratios of each stage are reported; load sharing is not invented by simply dividing torque by the number of planets. Reference: Drivetrain Hub, Geometry / Planetary Gears.

Cycloidal

The profile remains the inward normal offset of the roller-centre trajectory. Profile clearance, output-hole clearance, assembly axial clearance and distance between the two discs are distinguished. The output-hole radius is r pin + eccentricity + output clearance. If an old configuration does not specify the new values, the old profile clearance and old axial clearance are used, respectively.

Adaptive sampling compares the midpoint and quarter points of each interval with the chord. This is a numerical threshold, not a manufacturing tolerance or a proven global bound. The curvature check uses at least 4096 samples and a 2% margin against offset cusps; it is not an analytical proof of cusp absence for every parameter. The ratio is −(N rollers − 1).

The separation of clearances is motivated by Chen et al., ASME Journal of Mechanical Design, 2017, Analysis of a Cycloid Speed Reducer Considering Tooth Profile Modification and Clearance-Fit Output Mechanism. Normal profile modifications are discussed in A New Tooth Profile Modification Method of Cycloidal Gears in Precision Reducers for Robots, Applied Sciences 2020. Their optimisation and elastic load-sharing model are not implemented: clearance alone does not allow stress or accuracy under load to be estimated.

Worm and strain wave

For the worm pair, m is the axial worm module: lead L = π m starts, γ = atan(m starts / worm diameter), mn = m cos γ and αn = atan(tan αaxial cos γ). The wheel uses the conjugate screw/hob envelope, with an extended generating tip for root clearance and explicit generating thickness for backlash. ZA and ZI, both hands and independent wheel bores are supported within the checked geometric domain. Signed ratio follows world +X worm / +Z wheel, with the wheel centre on negative Y: i = hand × wheel teeth / starts. Geometry alone does not establish efficiency or self locking. See worm equations and primary references.

For a two-lobe strain wave, zc = zf + 2 and 2 ωwave + zf ωflex − zc ωring = 0. Fixed ring gives i = −zf/2; fixed flex gives i = +(zf+2)/2. The rigid ring is generated from the complete matching tooth sweep under the prescribed pitch-arc/normal ellipse map. The reference tooth-thickness allowance and numerical cell-enclosure relief are separate reported quantities. The same immutable material map drives preview and discrete interference poses. This is not an elastic, fatigue or loaded contact solution. See strain-wave equations and references.

Reading results and exporting

Configurations and files → Calculation report shows results and warnings. Each bill-of-materials export also produces .calculations.txt and .calculations.json. The bill-of-materials JSON and CAD assembly metadata retain the analysis section. Operating data is collected in the expandable Operating conditions and loads panel and does not change the geometry.

Kinematic arrangements

The ratio is input/output. Planetary, cycloidal and strain wave solve all six member permutations with zero speed for the fixed member. The two-stage planetary propagates speed and sign between the two stages. Parallel-axis, internal-ring and worm pairs allow input and output to be reversed; for the worm, this does not guarantee backdrivability. See IO-MODELS.md.