Spur gear design: Lewis and Buckingham equations

Why it matters

Spur gears are the simplest positive drive: they set the speed ratio in gearboxes, servo reducers, printers, robots and machine tools. A gear pair must survive two different failure modes - tooth breakage by bending and surface pitting by contact stress - while running at speed with manufacturing errors that add dynamic load. The Lewis and Buckingham equations are the classic way to size the module and face width against these.

Key ideas

Geometry. Module m = d/z (mm), the standard size parameter (preferred values 1, 1.25, 1.5, 2, 2.5, 3, 4, 5, 6, 8, 10 mm). Circular pitch p = π·m. For 20° full-depth involute teeth: addendum = m, dedendum = 1.25·m. Velocity ratio i = n₁/n₂ = z₂/z₁ = d₂/d₁. Centre distance a = m·(z₁ + z₂)/2. Minimum pinion teeth to avoid interference: about 18 for 20° full depth (theoretically 17).

Forces. The tooth force acts along the line of action (pressure line) at pressure angle φ. Its components at the pitch point: tangential Ft = 2T/d = P/v (does the work) and radial Fr = Ft·tan φ (pushes the gears apart and loads the shafts and bearings).

Bending - Lewis equation. Lewis modelled a tooth as a cantilever of uniform strength (a parabola inscribed in the tooth profile) with the full load at the tip, carried by one tooth. The resulting beam strength is Sb = σb·b·m·Y, where Y is the Lewis form factor (depends on tooth count and tooth system; for 20° full depth Y ≈ 0.484 − 2.87/z). More teeth → larger Y → stronger tooth for the same module.

• Weaker member: if both are the same material, the pinion (fewer teeth, smaller Y) is weaker; in general compare σb·Y for pinion and gear and design the smaller one.
• Face width: usually b = 8m to 12m (often 10m). Wider faces suffer from misalignment and uneven load across the face.
• Lewis neglects stress concentration at the root fillet, radial compression, load sharing and dynamic effects - corrections come next.

Dynamic effects. Tooth spacing and profile errors and tooth deflection cause impacts that raise the load above Ft.

• Velocity factor (Barth) for preliminary design: Cv = 3/(3 + v) for ordinary cut gears, v < 10 m/s; Cv = 6/(6 + v) for 10-20 m/s (v in m/s). Effective load P_eff = Cs·Ft/Cv, with Cs the service factor (shock).
• Buckingham's dynamic load for precise design: the incremental dynamic load is Pd = 21v·(C·e·b + Ft)/(21v + √(C·e·b + Ft)) (SI form used in Indian texts: v in m/s, C in N/mm², e in mm, b in mm, forces in N). C is the deformation factor (depends on E of both materials and tooth form - from the data book) and e is the sum of tooth errors (from the grade of manufacture). Then P_eff = Cs·Ft + Pd. The design requires Sb ≥ n·P_eff.

Surface durability - Buckingham wear strength. Repeated Hertzian contact stress causes pitting. Wear strength Sw = b·Q·dp·K, with ratio factor Q = 2z₂/(z₂ + z₁) (external gears) and load-stress factor K = σc²·sin φ·cos φ·(1/E₁ + 1/E₂)/1.4. Increase surface hardness (case hardening) to raise K. The design must satisfy both Sb and Sw ≥ n·P_eff.

Failure modes: tooth breakage (bending fatigue), pitting and spalling (contact fatigue), scoring (lubricant film breakdown at high speed and load), abrasive wear, and plastic flow.

Formulas

m = d / z · p = π·m · a = m·(z₁ + z₂)/2 · i = z₂ / z₁

• m: module (mm); d: pitch diameter (mm); z: number of teeth; p: circular pitch (mm); a: centre distance (mm); i: gear ratio.

v = π·d·n / 60 · Ft = P / v = 2T / d · Fr = Ft·tan φ

• v: pitch-line velocity (m/s, d in m, n in rpm); P: power (W); T: torque (N·m); φ: pressure angle.

Sb = σb·b·m·Y · Y ≈ 0.484 − 2.87 / z (20° full depth)

• Sb: beam strength (N); σb: permissible bending stress (N/mm²); b: face width (mm); Y: Lewis form factor.

Cv = 3 / (3 + v) (v < 10 m/s) · Cv = 6 / (6 + v) (10 ≤ v < 20 m/s) · P_eff = Cs·Ft / Cv

• Cv: Barth velocity factor; Cs: service factor; P_eff: effective load (N).

Pd = 21v·(C·e·b + Ft) / (21v + √(C·e·b + Ft)) · P_eff = Cs·Ft + Pd

• Pd: incremental dynamic load (N); C: deformation factor (N/mm²); e: total tooth error (mm).

Sw = b·Q·dp·K · Q = 2z₂/(z₂ + z₁)

• Sw: wear strength (N); dp: pinion pitch diameter (mm); K: load-stress factor (N/mm²).

Worked examples

Example 1 (standard). A 20° full-depth steel pinion with 20 teeth, m = 3 mm, b = 30 mm, transmits 5 kW at 1440 rpm. Permissible bending stress 140 MPa; Cs = 1. Using the Barth factor, find the factor of safety in bending.

1. dp = 3 × 20 = 60 mm; v = π·d·n/60 = π × 0.060 × 1440/60 = 4.524 m/s.
2. Ft = P/v = 5000/4.524 = 1105 N.
3. Y = 0.484 − 2.87/20 = 0.3405.
4. Sb = σb·b·m·Y = 140 × 30 × 3 × 0.3405 = 4290 N.
5. Cv = 3/(3 + 4.524) = 0.3987; P_eff = Ft/Cv = 1105/0.3987 = 2772 N.
6. FoS = 4290/2772 = 1.55.

Example 2 (GATE level). A 20-tooth 20° full-depth pinion transmits 10 kW at 1000 rpm. Face width b = 10m, permissible bending stress 100 MPa, service factor 1.5, Barth factor Cv = 3/(3 + v). Find the smallest standard module (try 4 and 5 mm).

1. Y = 0.3405 (as above). Sb = σb·(10m)·m·Y = 100 × 10 × m² × 0.3405 = 340.5·m² N.
2. m = 4 mm: d = 80 mm, v = π × 0.080 × 1000/60 = 4.189 m/s, Ft = 10,000/4.189 = 2387 N, Cv = 3/7.189 = 0.4173, P_eff = 1.5 × 2387/0.4173 = 8581 N; Sb = 340.5 × 16 = 5448 N < 8581 N - fails.
3. m = 5 mm: d = 100 mm, v = 5.236 m/s, Ft = 1910 N, Cv = 3/8.236 = 0.3643, P_eff = 1.5 × 1910/0.3643 = 7865 N; Sb = 340.5 × 25 = 8513 N > 7865 N - OK.
4. m = 5 mm, b = 50 mm, dp = 100 mm (then check wear strength and the Buckingham dynamic load).

Common mistakes

• Using the gear instead of the pinion (or vice versa) without comparing σb·Y.
• Mixing the Lewis factor based on module (Y) with the one based on circular pitch (y = Y/π).
• Computing v with d in mm and forgetting to convert to m.
• Forgetting Cv or Cs and getting an unconservative design.
• Checking only bending strength - case-hardened gears usually fail by pitting first.
• Picking a non-standard module from the calculation instead of rounding up to the next standard value.

For GATE ME

Typical questions: tangential and radial forces from power and speed, the Lewis beam strength or the required module, which of pinion or gear is weaker, centre distance and gear ratio, and interference/minimum teeth concepts. Dynamic-load and wear-strength formulas appear mainly as concepts. Practise the force analysis because it also feeds shaft and bearing problems.

Quick check

1. m = 4 mm, z = 25. Pitch diameter?
2. Ft = 2 kN, φ = 20°. Radial force?
3. Same material for pinion and gear - which is weaker in bending and why?
4. v = 6 m/s. Barth factor for ordinary cut gears? Answers: 1. 100 mm. 2. 728 N. 3. The pinion - fewer teeth give a smaller form factor Y. 4. 3/9 = 0.333.