Transforming rotation into translation and reverse – Rack & Pinion

  1. Definition of a motion transformer
  2. Motion transformer types
  3. Focus: Rack & Pinion
  4. Focus: Crank & Rod system
  5. Quiz
  6. Glossary

Rack & Pinion


The main function of a rack and pinion system is to transform a rotational motion into a translational motion (or vice versa).

This system consists of a toothed wheel called a “pinion” and a toothed rod called a “rack”. When the pinion turns, its teeth mesh with the teeth of the rack and drive the rack in a translational motion.

Rack and pinion in motion



  • This type of mechanism allows to transform the movement without sliding between the parts.
  • It is a reversible system.
  • The strength of this system is relatively great.


  • The gears that are used may require extensive lubrication.
  • This mechanism requires precise adjustment because of the teeth between the wheel and the rack.
  • There is a lot of wear.
  • It is not a cyclic movement; it is a finite movement (you have to stop when you reach the end of the rack).


A rack and pinion system is used in the steering mechanism of cars.

Steering mechanism


How to dimension this system?

Main operation:

Rack and pinion schema

N = Rotation frequency in rpm (revolutions per minute)

T = Torque in m.N (newton-metre)

V = Speed in m/s (meter per second)

F = Force (Newton)


m = modulus

p = step

dpinion = pitch diameter of the pinion

Zpinion = number of pinion teeth

dpinion = m x Zpinion

p = step

m = modulus

R = radius


Source: Les engrenages %5BPart_4%5D.pdf

This table gathers all the standard module sizes that we can find in the shops.

Mechanical characteristic:

Performance diagram of a pinion rack system

Rotation: P = C . ω

With : P (W), C (m.N), ω (rad/s)

Translation : P = F . V

With : P (W), F (N), V (m/s)

Efficiency : Ps / Pe or Pu / Pa

With : Pu = Useful power

Pa = Absorbed power

The efficiency of this system is very good (>95%). It often takes the value 1 in mechanical models.

Rack speed :

Srack = Rpinion x ωpinion

Rack displacement :

Drack = Rpinion x Ꝋpinion

Srack in m/s

ωpinion : pinion rotation speed in rad/s

Rpinion in m

Ꝋpinion in m

System diagram:

System diagram

Source: Les engrenages %5BPart_4%5D.pdf