Gears and Racks – parameters, formulas, and selection: module, number of teeth, and pitch diameter

A gear wheel meshing with a rack converts rotary motion into linear motion. For straight teeth, the gear pitch diameter is d = m × z, where m denotes the module and z the number of teeth. The rack displacement per one revolution of the gear can be calculated using the formula L = π × d = π × m × z.

What are a gear and rack in a linear drive?

The gear, also referred to as a pinion in such a system, meshes with a straight rack fitted with teeth. Rotation of the gear causes linear movement of the rack or — when the rack is stationary — movement of the drive assembly along its length.

A rack-and-pinion system is used in applications where the following are required:

• a large travel range,
• high linear speed,
• the transmission of significant forces,
• the ability to join multiple rack sections,
• high drive stiffness,
• operation without being limited by the length of a lead screw.

Typical applications include machine-tool axes, CNC machines, robotic gantries, pick-and-place devices, conveyors, loading systems, and heavy-duty positioning modules.

Unlike a ball screw, a rack can be built from successive segments, making it well suited to axes several or even a dozen meters long. However, it requires precise installation, proper lubrication, and control of backlash.

What are the basic parameters of a gear?

The most important geometric parameters are module, number of teeth, pitch diameter, pitch, tooth width, and pressure angle. Each affects the dimensions, load capacity, and way the gear meshes with the rack.

What is the module of a gear?

The gear module defines the size of its teeth. It is denoted by the symbol m and specified in millimeters.

The module can be calculated using the formula:

m = d / z

where:

• m — gear module in mm,
• d — pitch diameter in mm,
• z — number of teeth.

A gear and rack can mesh correctly only if they have a matching module, pressure angle, and tooth form. Similar tooth dimensions alone do not guarantee compatibility.

A larger module generally means:

• larger teeth,
• higher load-carrying capacity,
• a larger gear diameter for the same number of teeth,
• fewer teeth per a given rack length,
• larger overall system dimensions.

However, the module should not be selected solely on the basis of the required force. Material, tooth width, surface hardness, lubrication, dynamics, and the expected service life of the drive are also important.

What does the number of gear teeth mean?

The number of teeth, denoted by z, directly affects the gear diameter and the distance traveled by the rack during one revolution.

With a constant module:

• a greater number of teeth increases the pitch diameter,
• one revolution of the gear produces greater linear displacement,
• at the same torque, the force on the rack decreases,
• linear speed increases at the same rotational speed.

A small gear provides greater linear force at a given torque, but it may have less favorable meshing conditions with the rack. For standard involute gearing with a 20° pressure angle and no profile shift, approximately 17 teeth is generally considered sufficient to avoid undercutting at the tooth base. Gears with fewer teeth may require profile correction or an individual geometry check.

What is the pitch diameter of a gear?

The pitch diameter is the theoretical diameter at which the gear meshes with the rack without slip. It is one of the most important calculation dimensions, although it does not directly correspond to the physical edge of the gear.

For a straight-tooth gear:

d = m × z

where:

• d — pitch diameter in mm,
• m — module in mm,
• z — number of teeth.

For a standard, uncorrected external gear, the tip diameter can be estimated as:

da = m × (z + 2)

For a helical gear, it must be established whether the documentation specifies the normal or transverse module. For the normal module mn, the pitch diameter is:

d = mn × z / cos β

where β denotes the helix angle. The basic concepts and geometric relationships for cylindrical involute gears are described in ISO 21771-1:2024.

What is the pitch of a gear and rack?

Circular pitch is the distance between corresponding points of two adjacent teeth, measured along the gear pitch circle or the rack pitch line.

For straight teeth:

p = π × m

For example, for module m = 2:

p = π × 2 ≈ 6,283 mm

This means that successive rack teeth are spaced approximately 6.283 mm apart along the pitch line.

The theoretical tooth thickness at the pitch line in a zero-backlash system is:

s = p / 2 = π × m / 2

The actual thickness may differ due to the required backlash, manufacturing tolerances, profile correction, and installation conditions.

What does gear width mean?

Tooth width, usually denoted by b, is the tooth dimension measured in the axial direction of the gear. For a rack, the corresponding dimension is the width of the active toothed surface.

A greater width can increase load capacity, but only if the load is distributed evenly across the entire tooth surface. Inaccurate installation, structural deflection, or misalignment may cause contact to occur only at one edge.

Tooth width should be selected together with:

• modułem,
• material,
• hardness,
• accuracy class,
• shaft support method,
• expected torque,
• lubrication conditions.

How to calculate the module and pitch diameter of a gear?

The basic relationships can be summarized in a short table.

Parameter	Formula for straight teeth	Unit
Module	m = d / z	mm
Pitch diameter	d = m × z	mm
Tip diameter of a standard gear	da = m × (z + 2)	mm
Circular pitch	p = π × m	mm
Theoretical tooth thickness	s = π × m	mm
Displacement per revolution	L = π × d	mm/rev
Linear speed	v = π × d × n / 60 000	m/s
Linear force	F = 2 × T / d	N, gdy T podano w Nmm

For torque given in Nm and diameter in mm, the following formula is convenient:

F = 2000 × T / d

The result is a theoretical value. Actual selection must account for efficiency, friction, acceleration, shock loads, and the required safety factor.

Step-by-step gear calculation example

Assume that the following have been selected:

• module m = 2,
• number of teeth z = 24,
• gear speed n = 60 rpm,
• shaft torque T = 12 Nm.

Step 1. Calculate the pitch diameter

d = m × z
d = 2 × 24
d = 48 mm

Step 2. Calculate the tip diameter

da = m × (z + 2)
da = 2 × 26
da = 52 mm

Step 3. Calculate the pitch

p = π × m
p = π × 2
p ≈ 6,283 mm

Step 4. Calculate the displacement per revolution

L = π × d
L = π × 48
L ≈ 150,8 mm/rev.

Step 5. Calculate the linear speed

At 60 rpm, the gear makes one revolution per second, therefore:
v ≈ 150,8 mm/s = 0,151 m/s

Step 6. Calculate the theoretical linear force

F = 2000 × T / d
F = 2000 × 12 / 48
F = 500 N

With an assumed mechanical efficiency of 90%, the usable force would be approximately:

Fᵤ = 500 × 0,9 = 450 N

These calculations help with preliminary geometry selection, but they do not replace verification of tooth strength, surface contact stresses, service life, and dynamic conditions. Methods for calculating the load capacity of spur and helical gears are described in the ISO 6336 series.

Available from AMG racks with helical and straight teeth and gears obejmują moduły od m1 do m12, dzięki czemu geometria może zostać dopasowana do lekkich osi pozycjonujących oraz cięższych układów liniowych.

What is the difference between straight and helical teeth?

A straight-tooth gear has teeth arranged parallel to the axis of rotation. In helical gearing, the tooth line is inclined at a specified angle.

Feature	Straight tooth	Helical tooth
Start of contact	Across a large part of the width simultaneously	Progressive, gradual contact
Smoothness of operation	Good at moderate speed	Usually higher
Noise	May increase at high speed	Usually lower
Load-carrying capacity	Dependent on module and width	A greater number of teeth may be in contact
Axial force	Essentially absent due to the gearing	Present and loads the bearings
Installation	Simpler	More demanding
Cost	Usually lower	Usually higher
Typical application	Simple axes, lower speeds, easy service	High-speed and precision axes, noise reduction

When should you choose a straight-tooth rack?

Straight teeth are suitable when the priorities are:

• simple construction,
• easier installation,
• no additional axial force,
• moderate speed,
• favorable solution cost,
• easy component replacement.

They are often used in machines where noise and smoothness at high speed are not the primary criteria.

When should you choose a helical-tooth rack?

Helical teeth are worth considering when the following are required:

• smoother meshing,
• smaller force fluctuations,
• higher motion speed,
• reduced vibration and noise,
• a greater number of teeth remaining in contact simultaneously,
• high motion accuracy.

However, the axial force acting on the gear and transmission shaft must be taken into account. The rack and gear must have matching normal module, helix angle, pressure angle, and tooth-line direction.

The AMG range of racks and gears AMG obejmuje zarówno wykonania z zębem prostym, jak i skośnym, co pozwala dopasować układ do dynamiki oraz dokładności osi.

Which standards apply to gears and racks?

Standards specify, among other things, tooth geometry, tolerance classes, and load-capacity calculation methods. Technical documentation should specify not only the standard number, but also its part, edition, and scope of application.

The most frequently referenced documents are:

• ISO 53:1998 — defines the basic rack reference profile for cylindrical involute gears used in general and heavy engineering;
• ISO 21771-1:2024 — describes concepts and geometric parameters of cylindrical involute gears and gear pairs;
• ISO 1328-1:2013 — specifies the classification of tolerances for tooth flank surfaces;
• ISO 1328-2:2020 — concerns composite radial deviations measured in double-flank contact;
• ISO 6336 — covers load-capacity calculations for cylindrical spur and helical gears.

ISO 1328-1:2013 remains a current standard following its confirmation in 2024, while ISO 1328-2:2020 was confirmed in 2025.

A lower quality-class number generally means smaller permissible deviations and higher manufacturing precision. However, designations from different standards systems should not be compared without checking how they are converted.

Which racks and gears are available from AMG?

AMG offers GearKo components intended for building complete linear drives, including in combination with planetary gearboxes from the same manufacturer.

Parameter	Available range
Module	m1–m12
Quality class	ISO 4–10
Tooth type	Straight or helical
Materials	Carbon steel, alloy steel, stainless steel
Coatings	Nickel plating, black oxide
Rack mounting	Available screw-hole options
Gear mounting/bore configuration	Spline, keyway, shaft version, specified bore diameter
Compatibility	Gears matched to GearKo modules
Special versions	Custom-made versions

The range includes modules m1–m12 and ISO 4–10 quality classes. Straight and helical teeth, various steel grades, protective coatings, and several methods of connecting the gear to the shaft are available. Exact dimensions, tooth width, segment length, and load-capacity parameters must be determined for the specific version.

How to select a gear and rack for a linear drive?

Selection should start with the requirements of the entire axis, not with the module alone. The force, speed, accuracy, travel length, and environmental conditions must be determined.

1. Calculate the required linear force

For a horizontal axis, take into account:

• moving mass,
• acceleration,
• guide resistance,
• friction,
• forces resulting from the technological process.

The simplified relationship is:

F = m × a + Fₒ

where Fₒ denotes the remaining resistances and process forces.

For a vertical axis, the weight must additionally be taken into account:

F = m × (g + a) + Fₒ

An appropriate safety factor must be added to the working value.

2. Select the gear diameter

A smaller gear provides greater force at the same torque, but produces less displacement per revolution. A larger gear increases linear speed, but requires greater torque to achieve the same force.

A compromise must be found between:

• force,
• speed,
• positioning resolution,
• permissible rotational speed,
• overall dimensions,
• minimum number of teeth.

3. Select the module and tooth width

The module should provide adequate tooth-root and tooth-flank strength. In dynamic applications, shock loads and changes in direction of motion must also be taken into account.

The final verification should include:

• tooth-root stresses,
• surface contact stresses,
• fatigue life,
• service factor,
• number of cycles,
• lubrication conditions.

4. Determine the required accuracy class

Higher manufacturing accuracy is required in axes where the following are important:

• low positioning error,
• constant speed,
• low vibration level,
• quiet operation,
• synchronization of multiple drives,
• machining or measurement with a tight tolerance.

In a less demanding conveyor or transport system, using the highest class may not provide a proportional benefit.

5. Take backlash into account

Excessive backlash reduces accuracy when changing direction. Insufficient backlash can cause jamming, overheating, and accelerated wear.

Actual backlash is affected by:

• rack and gear tolerances,
• distance from the gear axis to the rack pitch line,
• straightness of installation,
• thermal expansion,
• structural deflection,
• joining of rack segments.

In precision axes, preload adjustment or solutions with two preloaded gears can be used. However, the backlash-compensation method must take bearing capacity and permissible bearing load into account.

6. Select the material and surface protection

Carbon steel may be sufficient under standard conditions. Alloy steel and suitable heat treatment are used for higher loads and when greater service life is required.

In humid or corrosive environments, consider:

• stainless steel,
• nickel plating,
• black oxide,
• additional protection,
• lubricant compatible with the operating conditions.

7. Plan installation and joining of sections

The surface beneath the rack should be straight, rigid, and carefully prepared. Particular attention is required when joining successive segments, because a pitch error at the joint can cause impacts and periodic tooth overloads.

When inquiring about racks with helical and straight teeth and gears, it is useful to provide the load, speed, acceleration, axis length, expected accuracy, operating conditions, and gearbox parameters. This makes it easier to select the module, tooth quality, material, and gear version.

Selecting a rack and gear requires analysis of the entire axis

The module, number of teeth, and pitch diameter define the basic geometry of the system, but they do not by themselves determine its service life. Drive torque, tooth width, material, installation accuracy, lubrication, and the method of joining rack sections are equally important.

For preliminary selection, prepare:

• required linear force,
• maximum speed,
• acceleration,
• travel length,
• moving mass,
• number of cycles,
• expected accuracy,
• environmental conditions,
• motor and gearbox parameters.

Such data make it possible to select a solution that provides not only the required speed, but also the required stiffness, service life, and positioning repeatability.

FAQ

How do you calculate the pitch diameter of a gear?

For a straight-tooth gear, the pitch diameter is calculated using d = m × z, where m is the module in millimeters and z is the number of teeth. A gear with module 2 and 20 teeth has a pitch diameter of 40 mm.

What is the difference between a straight-tooth and helical gear?

A straight-tooth gear has teeth parallel to the axis and does not generate an axial force resulting from the gearing. A helical tooth meshes progressively, which can make operation smoother and quieter, but it generates an axial force that requires appropriate bearing support.

How do you measure or calculate the gear module?

If the pitch diameter and number of teeth are known, the module is m = d / z. Measuring only the outside diameter allows the module to be estimated. For a standard gear without profile correction, the relationship m ≈ da / (z + 2) can be used.

Which standards apply to gears and racks?

The key standards include ISO 53 concerning the basic rack profile, ISO 21771-1 describing geometry, ISO 1328 specifying tolerance classes, and the ISO 6336 series concerning load-capacity calculations for spur and helical gears.

How do you select a rack and gear for a linear drive?

The required force, speed, acceleration, travel length, accuracy, number of cycles, and environmental conditions must be determined. Based on these, the gear diameter, module, tooth width, material, quality class, and backlash-control method are selected.

How many teeth should a gear have?

The number of teeth depends on the required diameter, linear speed, force, and profile geometry. For standard, uncorrected 20° involute gearing, approximately 17 teeth is commonly taken as a practical limit for avoiding undercutting. A smaller number may require profile correction.