Model assumptions and verification
What changes when you move a parameter?
Edit a number or move a slider. Start by doubling power: tangential force and bending stresses double, while contact stress increases by √2. Changing the illustrative material preset or reference stresses changes only the comparison ratios; the estimated stresses retain the fixed steel-pair elasticity used by this model.
Equations and units
m, b and d are in mm; P in kW; n₁ in rpm; T in N·m; forces in N; stresses in MPa. The torque conversion uses the rounded coefficient 9550. Bending is evaluated separately for each gear. Distinguish circular-pitch y from module-based Y = πy.
dᵢ = m zᵢ
T = 9550 P / n₁
Fₜ = 2000 T / d₁
yᵢ = 0.154 − 0.912 / zᵢ
Yᵢ = π yᵢ
σFᵢ = Fₜ K_F / (b m Yᵢ)
u = z₂ / z₁
Z_H = √[2 / (sin20° cos20°)]
σH = Z_E Z_H √[Fₜ K_H (u+1) / (b d₁ u)]
Contact uses a Hertz line-contact approximation with the two curvature radii at the 20° pitch point. Fixed values are Z_E = 191 √MPa for a steel pair and K_F = K_H = 1.25. The tooth ratio u = z₂/z₁ may be below one. Fₙ = Fₜ/cos20° and Fᵣ = Fₜtan20°. Arrows show forces before the load factor; contact stress includes K_H.
Check the default example
m=3 mm, z₁=20, z₂=40, b=30 mm, P=5 kW, n₁=1500 rpm. T=31.8333 N·m, d₁=60 mm, d₂=120 mm, Fₜ=1061.1111 N, y₁=0.1084 and y₂=0.1312.
| Quantity | Calculated value |
|---|
| Driver bending σF₁ | 43.2762 MPa |
| Driven bending σF₂ | 35.7556 MPa |
| Pitch-point contact σH | 500.9270 MPa |
Using the illustrative 200 / 550 MPa references gives comparison ratios of 4.62 for driver bending, 5.59 for driven bending and 1.10 for contact. These are comparisons, not design acceptance.
Example: double the module
Holding every other input fixed, m=3→6 mm doubles both pitch diameters and halves the tangential force. Bending becomes one quarter (10.8191 / 8.9389 MPa), and contact stress one half (250.4635 MPa). Doubling face width instead halves bending and multiplies contact stress by 1/√2: these changes are not equivalent.
What this model cannot establish
This is an educational, early-comparison model. It omits undercut, profile shift and modification, load sharing from contact ratio, sliding, uneven face loading, detailed dynamics, lubrication, wear, plasticity and fatigue life. Below 18 teeth, values are flagged as extrapolations that do not model undercut. Real design requires the actual tooth profile and load history, qualified material and heat-treatment data, and a separate assessment using applicable standards.
Frequently asked questions
Why does the Lewis form factor contain π?
y = 0.154 − 0.912/z is a circular-pitch approximation for 20° full-depth teeth. Since circular pitch p = πm, σF = FₜK_F/(bpy) = FₜK_F/(bmY), where Y = πy. Using y directly as the module-based Y makes the stress π times too large.
Does a comparison ratio above one mean safe?
No. Reference stresses are illustrative or user-assumed values. Ratios only compare them with the simplified estimates; they do not establish service life, assembly safety or standards compliance.
Do animation and results use the same conditions?
Normally yes. The optional load demo varies only the diagram forces between 0.45 and 1.55 times the input load. Result cards and the chart keep the input conditions. The demo does not predict actual tooth-mesh loading.
Sources and calculation code
Page 8 of the lecture notes below defines the circular-pitch Lewis equation and 20° full-depth factor. Equations (4) and (7) of the author-hosted paper describe the bending and pitch-point cylindrical-contact models. The diagram is our schematic. The illustrative material references are not allowables certified by these sources.
