Spur gears

For two standard unshifted external spur gears, module 2 mm, 20° pressure angle and 20/40 teeth give 40/80 mm reference diameters and 60 mm center distance. At an assumed 5 N·m on the pinion, ideal tooth force is 250 N tangential, about 91 N radial, and zero axial. Check material, tooth geometry and duty separately for capacity. KHK, Calculation of gear dimensions, §§4.1–4.2.

What the teeth connect

External spur gears have straight axial teeth and reverse rotation between parallel shafts. Internal gears require separate center-distance and interference checks. SDP/SI, Gears, spur-gear overview, KHK, Calculation of gear dimensions, §§4.1–4.2.

Match module or diametral pitch, pressure angle and tooth system. Verify bores, keys, hubs and shaft support separately. KHK, Spur gears, selection and application guidance.

Reference geometry and pitch conventions

For a standard, unshifted, full-depth metric spur pair at its standard center distance, the following relationships apply. All lengths use millimeters; z is tooth count and m is module.

These are geometric relationships, not manufacturing tolerances. Profile shift changes tooth thickness and other dimensions; working pitch circles and operating pressure angle can differ from reference values. Standard outside-diameter arithmetic should therefore not be applied blindly to modified gears. KHK, Calculation of gear dimensions, Tables 4.1–4.3.

Diametral pitch, Pd, counts teeth per inch of reference diameter. Its conversion is m = 25.4/Pd: larger module means larger teeth, whereas larger diametral pitch means smaller teeth. For example, module 2 corresponds to 12.7 diametral pitch, not 2 diametral pitch. Pressure angle is a separate parameter; matching tooth spacing alone is insufficient. SDP/SI, Elements of Metric Gear Technology, Table 1-5, pp. T-16–T-17.

Pressure angle, contact ratio and undercut

The pressure angle relates the line of tooth force to the tangent of the pitch circle. For the same tangential load, a larger operating pressure angle increases the radial separating force. It is therefore relevant to bearings and housing stiffness as well as tooth shape. KHK, Gear forces, Table 12.1 and §12.1.

Transverse contact ratio is the active length of the line of action divided by base pitch. A value between one and two means that engagement alternates between one and two tooth pairs. It describes overlap, rather than the number of teeth on either gear. SDP/SI recommends at least 1.2, with a 1.1 minimum at worst tolerances; these are source-specific recommendations. SDP/SI, §3.2.1, p. T-32.

Generating a small-tooth-count pinion can undercut the root, removing useful involute and weakening the tooth. Positive profile shift can address undercut, but changes the geometry that must be checked with its mate. A minimum tooth count quoted without cutter, pressure angle and shift assumptions is incomplete. SDP/SI, §§4.3–4.6, pp. T-36–T-40.

Backlash and assembly accuracy

Backlash is the clearance between mating flanks when the opposite flanks are in contact. Circumferential backlash is measured along a pitch circle; normal backlash is measured between flanks, and angular backlash is rotational lost motion with the other gear held stationary. A value is ambiguous without its direction and measurement condition. KHK, Gear backlash, §6.1.

Tooth thinning or an increased center distance can create backlash. Changing center distance also changes mesh conditions, so using it to remove play requires more than checking free rotation at one angular position. Runout can vary backlash around a revolution. KHK specifically cautions that zero clearance can prevent smooth rotation. Spring-loaded split gears address reversal play through preload; they have their own load and friction implications. Backlash is consequently different from tooth accuracy or elastic deflection under load. KHK, §§6.3–6.5.

Materials, manufacture and load capacity

Material and processing route belong in the part definition. KHK's catalog distinguishes cut S45C steel SS gears, hardened and ground SSG gears, cut nylon PS gears and injection-molded acetal DS gears. Those families do not share one strength, accuracy or lubrication assumption. A precision-grade claim should retain its standard and edition; a cut or ground finish alone does not define a complete tolerance class. KHK, Spur gears, product comparison and specification tables.

Tooth-root bending and tooth-surface durability are separate limits. The same catalog lists them separately, with calculation assumptions. Face width, mating material, heat treatment, cycles, speed and duty affect applicability. Neither module nor steel grade alone supplies an allowable torque, and the shaft attachment must transmit the selected torque too. KHK, Spur gears, strength-selection guidance.

A bounded module-2 example

Assume two standard, unshifted external gears with m = 2 mm, reference pressure angle 20°, and z1 = 20, z2 = 40. At standard center distance:

Reference diameters: d1 = 40 mm; d2 = 80 mm
Tip diameters:       da1 = 44 mm; da2 = 84 mm
Center distance:     a = 60 mm
Speed ratio:         n1/n2 = z2/z1 = 2

These are derived geometry and kinematics, not an identified supplier pair. With an assumed input of 1,000 rpm, output speed is 500 rpm in the opposite direction. KHK, Calculation of gear dimensions, §4.1 and Tables 4.1–4.2.

If the pinion transmits an assumed T1 = 5 N·m, its tangential tooth force is Ft = 2000T1/d1 = 250 N. At this center distance, radial separating force is Fr = Ft tan 20° ≈ 91 N. The ideal spur mesh has no axial tooth-force component. These nominal forces exclude dynamic amplification, misalignment and friction; they neither verify tooth capacity nor determine bearing reactions without the shaft-support geometry. KHK, Gear forces, Table 12.1 and §12.1.

Lubrication and damage in service

Lubrication reduces sliding friction and limits temperature rise. Grease, splash lubrication and forced circulation serve different operating arrangements. Selection depends on speed, load, temperature, lubricant delivery and enclosure. Too much grease can cause drag and heating; too little oil or an unsuitable running oil level can impair lubrication and cooling. KHK's speed bands are guidance with exceptions, rather than universal boundaries. KHK, Lubrication of gears, §§13.1–13.2.

Wear, pitting, plastic deformation and tooth fracture describe different damage. Repeated contact loading can cause surface fatigue; overload, mounting problems and lubrication failure can also contribute. Increasing backlash, vibration or temperature can accompany progressive damage, but none alone identifies the cause. Inspect tooth contact, lubricant condition, debris, alignment and the load history before assigning a mechanism. Geometry calculations above do not establish fatigue life, efficiency, noise, maintenance interval or service suitability. KHK, Damage to gears, §14.1 and damage classification.