Rack and Pinion Calculator

Feed Force · Velocity · Torque · Pitch Diameter

Rack and Pinion Calculator

Select calculation:

Calculate feed force

Calculate feed velocity

Calculate torque

Calculate pitch diameter

Calculate rotational speed

Torque M [N·m]

Module m [mm]

Number of pinion teeth z

Efficiency η [0…1]

Formulas & Symbols

Key Formulas

Pitch diameter:
d = m · z
m = module [mm], z = number of pinion teeth

Feed force:
F = (2 · M · η) / d
M = torque [N·m], η = efficiency, d = pitch diameter [m]

Feed velocity:
v = π · d · n / 60
d [mm], n [rpm] → v [mm/s]

Torque from force:
M = F · d / 2
F [N], d [m] → M [N·m]

Speed from velocity:
n = (60 · v) / (π · d)
v [mm/s], d [mm] → n [rpm]

Symbol Reference

F	Feed force [N]
M	Torque [N·m]
v	Feed velocity [mm/s]
n	Rotational speed [rpm]
m	Module [mm] – standardized size
z	Number of pinion teeth
d	Pitch diameter [mm]
η	Efficiency [0…1]
t	Circular pitch = π · m

Rack and Pinion – Fundamentals

What is a rack and pinion?

A rack is a linear gear – a straight bar with regularly spaced teeth. Combined with a pinion (a small circular gear), it forms the classic rack-and-pinion mechanism, which converts rotary motion into linear motion – or vice versa.

The system is one of the oldest and most reliable drive principles in engineering. It is found in CNC machines, vehicle steering systems, elevators, gates, rack railways, and countless automation applications.

Advantages

Direct force transmission without slip
High positioning accuracy
Unlimited stroke by extending the rack
Simple, robust construction
Long service life with proper lubrication
Zero-backlash operation possible (preloaded pinion)

Disadvantages

Limited gear ratio in one stage
Generates lateral (radial) forces on the guide
Noise at high speeds
Regular lubrication required

Operating Principle

When the pinion rotates by one full revolution, the rack advances by exactly one pitch circumference of the pinion:

Travel per revolution:
s = π · d = π · m · z
s = travel [mm], d = pitch diameter [mm]

The module m is the key standardized quantity in gear engineering. It defines the ratio of pitch diameter to tooth count and thereby determines tooth height and tooth spacing. Standard modules per ISO 54: 0.5 – 0.8 – 1 – 1.25 – 1.5 – 2 – 2.5 – 3 – 4 – 5 – 6 – 8 – 10 …

Detailed Formula Derivations

1. Pitch diameter d

The pitch circle divides the tooth into addendum and dedendum and is the effective diameter:

d = m · z
Example: m = 2 mm, z = 20 → d = 40 mm

2. Feed velocity v

The linear speed of the rack equals the tangential velocity of the pinion at its pitch circle:

v = π · d · n / 60
n in [rpm], d in [mm] → v in [mm/s]
Rearranged: n = (60 · v) / (π · d)

3. Feed force F

The torque M at the pinion creates a tangential force at the pitch circle. This tangential force is the usable feed force, reduced by efficiency:

F = (2 · M · η) / d
M in [N·m], d in [m], η = efficiency (typically 0.90–0.98 for spur rack)

4. Torque M

Reverse calculation: determine the required drive torque from a known rack force:

M = F · d / 2
F in [N], d in [m] → M in [N·m]

Tooth Geometry – Additional Parameters

Parameter	Formula
Circular pitch t	t = π · m
Addendum h_a	h_a = m
Dedendum h_f	h_f = 1.25 · m
Full tooth height h	h = 2.25 · m
Tip diameter d_a	d_a = m · (z + 2)
Root diameter d_f	d_f = m · (z − 2.5)

Module m	Tooth height h	Pitch t
1 mm	2.25 mm	3.14 mm
2 mm	4.5 mm	6.28 mm
3 mm	6.75 mm	9.42 mm
4 mm	9.0 mm	12.57 mm
5 mm	11.25 mm	15.71 mm
8 mm	18.0 mm	25.13 mm

Worked Example – CNC Gantry Router

Given:
Module m = 2 mm, pinion teeth z = 20, speed n = 1000 rpm,
Torque M = 12 N·m, efficiency η = 0.95

Step 1: Pitch diameter

d = m · z = 2 mm · 20 = 40 mm

Step 2: Travel per revolution

s = π · d = π · 40 mm ≈ 125.66 mm

Step 3: Feed velocity

v = π · 40 mm · 1000 / 60 ≈ 2094 mm/min ≈ 34.9 mm/s

Step 4: Feed force

F = (2 · 12 N·m · 0.95) / 0.04 m = 570 N

Result: The axis moves at ~2094 mm/min and delivers a feed force of 570 N.

Applications

Machine Tools & CNC

CNC milling machines (X/Y/Z axes)
Plotter drives
Laser scanner positioning
Welding robot linear axes

Automotive & Transport

Rack-and-pinion steering (cars)
Electric power steering (EPS)
Rack railways (mountain trains)
Aircraft flap actuators

Automation

Gate and sliding door drives
Elevators & lifts
Stacker cranes
Pick-and-place systems

Frequently Asked Questions

The module m is a normalized quantity: m = d / z (pitch diameter / tooth count). The circular pitch t is the arc distance from one tooth flank to the next along the pitch circle: t = π · m. Two gears can only mesh if they have the same module.

Larger module → taller teeth → higher load capacity, but larger overall size. For light positioning applications, m = 1–2 mm is sufficient. For heavy machine tools with forces above 5 kN, m ≥ 4 mm is recommended. Always prefer standardized modules per ISO 54 for interchangeability.

With a standard pressure angle of 20°, the pinion transmits not only a tangential force but also a radial force (≈ 34% of the tangential force). This radial force pushes the pinion away from the rack and loads the linear guide. Helical racks can reduce this effect but introduce axial forces instead.

Individual rack segments are typically 0.5–2 m long. For longer strokes, segments are butted end-to-end and bolted to a base structure. With careful alignment, total lengths of 10 m and more are achievable (e.g., in large CNC gantries or automated storage/retrieval systems).

Spur rack-and-pinion systems achieve efficiencies of η = 0.90–0.98 – significantly higher than worm gears. Losses occur from tooth mesh friction and bearing losses. With good lubrication and a helical rack, values close to 0.98 are achievable.

Summary

Rack + pinion convert rotary motion into linear motion (and vice versa).
The module m is the standardized quantity – both parts must have the same module.
d = m · z determines the pitch diameter and thus the travel per revolution (π · d).
Feed force follows directly from torque and pitch diameter: F = 2M·η / d.
Velocity is v = π · d · n / 60 (n in rpm, d in mm → v in mm/s).
Efficiencies of 90–98 % make the system highly efficient.
Used in CNC machines, car steering, rail systems, and automation.

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Rack and Pinion Calculator