Tools › Mechanical Design › Gears › Spur and Helical Gear Geometry › Example

20/40-tooth helical pair, 2.5 mm module, 20°, 15° helix: dimensions, span and pin measurements, contact ratio and sliding

Tool	Spur and Helical Gear Geometry
Source	Buckingham, E. (1928). Spur Gears: Design, Manufacture, and Inspection. McGraw-Hill, New York (Chapter III eqs. 40–44, 53–58; Chapter IX eqs. 115–126); Coy, J. J., Townsend, D. P., and Zaretsky, E. V. (1985). Gearing. NASA Reference Publication 1152 (eqs. 7–17, 21); Naval Education and Training Program Management Support Activity (1990). Machinery Repairman. NAVEDTRA 12204, ch. 15.
Level	Undergraduate

Problem

An external helical pair of 20 and 40 teeth, normal module 2.5 mm, normal pressure angle 20°, helix angle 15°, standard rack (h_a* = 1, h_f* = 1.25), no profile shift, face width 30 mm. Find the transverse pressure angle and module, the reference, base, tip and root diameters, the zero-backlash centre distance and working pressure angle, the chordal caliper settings, the span measurement over k teeth, the measurement over 4.32 mm balls, the transverse, overlap and total contact ratios and the specific sliding at the ends of the path of contact.

Inputs to Enter

z_1 20, z_2 40, m_n 2.5 mm, α_n 20°, β 15°, h_a* 1.0, h_f* 1.25, x_1 = x_2 = 0, b 30 mm; k and the pins automatic (k 3 / 5, d_p 1.728 m_n = 4.32 mm)

Open the Spur and Helical Gear Geometry tool (loaded with this example)

Answers

tan α_t = tan 20°/cos 15° → α_t = 20.647°, m_t = 2.588 mm, β_b = 14.076°; d_1 = 51.764 mm, d_b1 = 48.439 mm, d_a1 = 56.764 mm, d_f1 = 45.514 mm; d_2 = 103.528 mm, d_b2 = 96.878 mm, d_a2 = 108.528 mm, d_f2 = 97.278 mm; p_bt = 7.609 mm, p_x = 30.35 mm
a = m_t (z_1 + z_2)/2 = 77.646 mm; with x_1 + x_2 = 0, inv α_wt = inv α_t = 0.016453 and α_wt = α_t, a_w = 77.646 mm (clearance 0.625 mm)
Chordal thickness (virtual gear, z_v1 = 22.19): 3.924 mm at a chordal height of 2.569 mm (pinion); 3.926 / 2.535 mm (gear)
Span over 3 pinion teeth W_3 = m_n cos α_n [π (3 − 0.5) + 20 inv α_t] = 19.224 mm (over 5 gear teeth 34.758 mm); over two 4.32 mm balls: inv α_M = 0.029859, α_M = 24.969°, M_1 = 57.753 mm (gear 109.591 mm)
Path of contact 11.877 mm (A at −6.206 mm, E at +5.671 mm from P), ε_α = 1.561, ε_β = b sin β/(π m_n) = 0.989, ε_γ = 2.550; specific sliding ζ_1 = −3.19 at the pinion root (A), ζ_2 = −1.35 at the gear root (E); z_min = 2 cos β/sin²α_t = 15.54, no undercut

Notes

The tool's default inputs are this pair; press Submit. The case is illustrative (stated on the page); every value is reproduced by test_gear_geometry.php against an independent numpy construction from the involute curve (References/Gear-Geometry/checks/), and the method is checked on Buckingham's 1928 worked problems, the NAVEDTRA 12204 caliper example and the NASA TM 83725 contact ratios.