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Spur and helical gear terminology

Spur and helical gear drawings, catalogues and standards share one vocabulary - ISO 21771's - and most of it is circles. Every term below is shown on the drawing of one pair, with its value, so the words and the numbers can be read together.

The pair is a spur pair, 20 and 40 teeth, module 3 mm, face width 30 mm, no profile shift, dimensioned by the designer to ISO 21771. Where the two gears differ the value is written pinion / wheel; helical values are for a 19/38 pair at 18.195° in the same centre distance.

Transverse section of a 20/40 spur pair in mesh with the tip, working pitch, base and root diameters, the line of action, the path of contact A-E, the base pitch and the working pressure angle dimensioned — The transverse section - the plane of rotation - with the pinion below and the wheel above. The line of action runs through the pitch point C at α_wt; contact starts at A and ends at E.

Size, module and pitch

The module is the tooth's size: the reference diameter divided by the number of teeth, on a spur gear. A helical gear has two, measured normal to the tooth and in the plane of rotation, and the normal one is the one the tool and the standards use.

Term	Meaning	Example
Number of teeth z	Pinion first: the pinion is the smaller gear.	20 / 40
Ratio u	z₂ / z₁.	2
Normal module m_n	The size of the tooth normal to it - the module of the hob. ISO 54 lists preferred values.	3 mm
Transverse module m_t	The module in the plane of rotation: m_n / cos β. Equal to m_n on a spur gear.	3 mm (helical: 3.158)
Pitch p_t	From one tooth to the next along the reference circle: π m_t.	9.425 mm
Base pitch p_bt	From one flank to the next along the line of action: p_t cos α_t. Mating gears must have the same base pitch.	8.856 mm
Pressure angle α_n	The basic rack's flank angle - 20° for every ISO 53 rack.	20°
Face width b	The length of the teeth along the axis.	30 mm

The diameters

Six circles, of which only the tip and the root can be measured on the gear. The reference circle belongs to the gear and the tool; the working pitch circle belongs to the pair.

Term	Meaning	Example
Reference diameter d	z m_n / cos β - the circle the tool rolls on. The 'pitch diameter' of catalogues.	60 / 120 mm
Tip diameter d_a	Over the tooth tips: the blank's outside diameter.	66 / 126 mm
Root diameter d_f	At the bottom of the tooth spaces. As built, cut thinner for backlash, the root is a little smaller: d_fE.	52.5 / 112.5 mm (as built 52.225 / 112.225)
Base diameter d_b	The circle the involute is unwound from: d cos α_t. No involute exists inside it.	56.382 / 112.763 mm
Working pitch diameter d_w	The circles that roll on each other in mesh. Equal to d unless the shifts change the centre distance.	60 / 120 mm
Tip form diameter d_Fa	Where the involute ends at the tip - below d_a when the tip is chamfered.	66 / 126 mm

The tooth

Axial section of the 20/40 pair with bores of 20 and 40 mm, tip, reference and root diameters, face width and centre distance dimensioned — The axial section through both axes, here with bores of 20 and 40 mm: the material from the bore to the tip, the root as a hidden line, the reference cylinder dash-dot.

The basic rack sets the depths: ISO 53 gives an addendum of 1.0 m and a dedendum of 1.25 m, so the clearance is 0.25 m. A profile shift moves both the tip and the root outward by x m without changing the depth.

Term	Meaning	Example
Addendum h_a	From the reference circle out to the tip.	3 mm
Dedendum h_f	From the reference circle in to the root.	3.75 mm
Tooth depth h	Addendum plus dedendum.	6.75 mm
Working depth h_w	The depth both gears' teeth share.	6 mm
Tip clearance c	The gap between one gear's tip and the other's root.	0.75 mm
Profile shift x	How far the tool was moved away from the gear, in modules. 0 here.	0 / 0
Tooth thickness s_n	Along the reference circle, normal to the tooth, before backlash: π m_n / 2 at x = 0.	4.712 mm
Tip thickness s_an	At the tip circle, normal to the tooth.	2.085 / 2.282 mm

The mesh

Two involute gears touch only on the line of action, the common tangent of their base circles. How much of it is used - the path of contact - against the base pitch is the contact ratio.

Term	Meaning	Example
Centre distance a, a_w	a = (d₁ + d₂)/2, the reference one; a_w the working one, different when the shifts do not cancel.	90 mm
Working pressure angle α_wt	The angle of the line of action to the common tangent at C.	20°
Pitch point C	Where the working pitch circles touch, on the line of centres.	-
Line of action T₁T₂	The common tangent of the two base circles: every contact lies on it.	-
Path of contact g_α	The part of the line of action between the two tip form circles, A to E.	14.482 mm
Transverse contact ratio ε_α	g_α / p_bt: how many pairs are in contact on average.	1.635
Overlap ratio ε_β	b sin β / (π m_n), helical gears only.	0 (helical: 0.994)
Total contact ratio ε_γ	ε_α + ε_β.	1.635 (helical: 2.510)

Helical gear terms

A helical gear is a spur gear twisted along its axis. In the plane of rotation everything above still applies, with the transverse values; normal to the teeth, the normal module and pressure angle are the tool's.

Term	Meaning	Example
Helix angle β	The tooth's angle to the axis on the reference cylinder.	0 (helical: 18.195°)
Base helix angle β_b	The same on the base cylinder, a little smaller.	0 (helical: 17.063°)
Hand	Right or left, as a screw thread. A pair of external gears has opposite hands.	-
Virtual number of teeth z_v	z / cos³ β: the spur gear the helical one behaves like normal to its teeth - for limits and strength.	20 (helical: 22.16)

Backlash and measurement

The nominal gear has no backlash; the finished one gets it from thinner teeth and from the housing's tolerance, and the tooth thickness is checked by one of two measurements.

Term	Meaning	Example
Tooth thickness deviations A_sne, A_sni	How much thinner than nominal the tooth is cut: DIN 3967 deviation series (a–h) and tolerance series (21–30).	−70 / −130 µm (26cd)
Centre distance allowance A_a	The housing's tolerance on a_w, DIN 3964 field js5–js11.	±17.5 µm (js7)
Normal backlash j_n	The play between the flanks, normal to them, from both of the above.	0.120 – 0.256 mm
Span measurement W_k	Over k teeth with a disc micrometer; checks the tooth thickness.	k 3: 22.916 – 22.859 mm
Dimension over balls M_dK	Over two balls in opposite spaces; checks the tooth thickness.	66.974 – 66.836 mm
Flank tolerance class	ISO 1328-1:2013, classes 1 to 11: pitch, profile, helix and runout tolerances.	class 6: f_pT ±9 µm

See every one of these on your own pair.

Open the designer

Questions

What is the difference between the reference diameter and the pitch diameter?

The reference diameter d = z m_n / cos beta belongs to one gear and its tool; catalogues often call it the pitch diameter. The working pitch diameter d_w belongs to the pair in mesh, and differs from d when the profile shifts change the centre distance.

What is the difference between normal and transverse module?

The normal module m_n is measured normal to the tooth and is the module of the hob; the transverse module m_t = m_n / cos beta is in the plane of rotation. On a spur gear they are equal.

What is the contact ratio of a spur gear?

The path of contact divided by the base pitch, ISO 21771 (90): on average how many pairs of teeth are in contact. The 20/40 module 3 example has 1.635.