Taper, Thread and Gear Calculations
Taper, thread and gear questions are formula-recognition questions with a small calculation. Write the formula, substitute in consistent units, then check whether the answer is an angle, a ratio or a length.
Taper
- Half-angle: tan α = (D − d) ÷ (2L).
- Included angle = 2α.
- Taper ratio = (D − d) ÷ L, often written as 1:x where x = L ÷ (D − d).
- Sine bar: sin θ = h ÷ L.
Threads
- Metric coarse tap drill ≈ D − P; external metric thread depth ≈ 0.6134P.
- Pitch = 1 ÷ TPI for inch threads; lead = pitch × number of starts.
- Pitch diameter and minor diameter are not the nominal diameter.
Gears
- Module m = PCD ÷ N; circular pitch = πm; addendum = m; dedendum = 1.25m.
- Centre distance = m(N₁ + N₂) ÷ 2; gear ratio = N₂ ÷ N₁.
Worked example
A taper has D = 40 mm, d = 30 mm and L = 100 mm. tan α = (40 − 30) ÷ (2 × 100) = 0.05. α = tan⁻¹ 0.05 = 2.862°. Included angle = 5.724°. Taper ratio = 10 ÷ 100 = 1:10, conicity 0.1. On a 200 mm sine bar, the slip gauge height for the half-angle would be 200 × sin 2.862° = 9.99 mm.
Traps
- Forgetting the 2 in (D − d)/(2L).
- Reporting the half-angle as the included angle.
- Using TPI where metric pitch is required.
- Confusing module with diametral pitch.
- Using nominal diameter instead of pitch diameter in thread strength checks.