Gear fundamentals: interpreting geometry and drawing parameters

What a gear pair actually does

A gear pair transfers rotary motion and torque through contacting tooth flanks. For two external gears, the shafts turn in opposite directions. If a 20-tooth driver turns a 40-tooth driven gear, the driven speed is halved; its torque ideally doubles. Real bearings, seals and sliding contact introduce losses, so the output torque is lower than this ideal value.

Do not confuse tooth profile with a complete gear specification. The profile is the shape of the tooth in a stated section. A usable drawing also identifies the module system, tooth count, pressure angle, helix angle, profile shift, face width, material, heat treatment, accuracy requirements and inspection references.

Module, tooth count and reference circles

Module expresses tooth size in millimetres; tooth count tells you how many teeth fit around the gear. For a spur gear, the reference diameter is d = mz. Two gears with the same module do not necessarily have the same diameter. A larger module generally gives larger teeth, but does not by itself establish permissible torque.

For a helical gear, distinguish the normal section, perpendicular to the tooth direction, from the transverse section, perpendicular to the shaft. With normal module mₙ and helix angle β, mₜ = mₙ/cos β and d = zmₜ. A drawing stating only “module 2” is incomplete unless the intended system is clear.

The reference circle defines nominal geometry. The base circle generates the involute; the tip circle passes through the tooth tips; the root circle passes through the tooth roots. Under the unshifted normal full-depth assumptions used here, dₐ = d + 2mₙ, d_f = d − 2.5mₙ, and d_b = d cos αₜ. The reference circle is not always the operating pitch circle.

Pressure angle, helix angle and backlash

Pressure angle describes the direction of the tooth normal relative to the pitch-circle tangent. It affects the balance between tangential and separating forces. Increasing it is not a universal strength improvement: contact ratio, bearing loads, noise and manufacturing constraints must also be considered.

Helix angle changes transverse geometry and produces axial thrust. Progressive engagement can help smoothness, but does not guarantee a quiet assembly. Face width influences load sharing; shaft and housing deflection can prevent the whole width from carrying load evenly.

Backlash is clearance between mating tooth flanks, not an accuracy grade and not the same as tooth-tip clearance. It accommodates lubrication, thermal growth and manufacturing variation. Profile shift changes tooth thickness and root geometry; it must be specified for the pair rather than treated as a cosmetic diameter adjustment. The interactive tooth-root transition is schematic and is not a cutting profile.

Spiral bevel gears: define the cone before calculating the load

For intersecting shafts at 90°, the pitch cones share an apex. Using outer transverse module mₑ, d₁ = mₑz₁, d₂ = mₑz₂, δ₁ = atan(z₁/z₂), δ₂ = 90° − δ₁, and outer cone distance Rₑ = √(d₁² + d₂²)/2. Face width is measured along the cone generator, not along the shaft.

The mean section lies at Rₘ = Rₑ − b/2; its diameters scale by Rₘ/Rₑ. Use the mean diameter when making the tangential-force estimate shown here. This example limits b ≤ Rₑ/3 and does not calculate tooth-height allocation, mounting distance, cutter settings or contact patterns. The entered efficiency is an assumption, not a prediction.

Planetary gearing: identify the fixed member

A simple planetary set comprises a sun, planets, a ring and a carrier. With the same module and standard centre geometry, z_r = z_s + 2z_p. Their speeds obey Willis' equation: z_s(n_s − n_c) + z_r(n_r − n_c) = 0. The ratio therefore depends on which member is fixed and which is driven.

For equally spaced planets, check assembly phasing and planet-to-planet clearance as well as tooth counts. Passing these checks does not prove adequate strength or equal load sharing. Carrier stiffness, tolerances, support conditions and floating arrangements influence how the load is actually divided.

References