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Rack and Pinion Joint

In one sentence: Link a rotating pinion to a sliding rack so that rotation of one automatically drives translation of the other — and vice versa.

Overview

Workbench: Assembly
Menu: Assembly → Joints → Rack and Pinion
Shortcut: Q

A Rack and Pinion joint couples a Revolute joint (the pinion) to a Slider joint (the rack) by a pitch-radius ratio. Rotating the pinion by one full revolution (2π radians) translates the rack by 2π × pitch radius. This is a coupled joint — it adds a kinematic ratio between two DOF, not a new geometric constraint.

Intuition

A coupled joint does not introduce new geometric constraints between the parts. The parts must already be independently constrained by their own joints; the coupled joint just says: "when the pinion rotates by X, the rack must move by Y × X."

Think of a car's steering rack: the pinion (attached to the steering wheel shaft) rotates, and the rack (connecting to the front wheels) slides left or right. The ratio between the steering wheel rotation and the lateral rack travel is set by the pinion's pitch radius.

When to use it

  • Rack-and-pinion steering mechanisms.
  • Linear actuators driven by a rotating motor (3-D printer Z-axis, CNC gantry).
  • Any mechanism that converts rotary motion to linear motion via a toothed gear.

When NOT to use it

  • Don't use this if the rotary-to-linear conversion is on the same axis — that is a Screw joint, not a Rack and Pinion.
  • Don't add this before the prerequisite joints exist — the pinion must have a Revolute joint and the rack must have a Slider joint before this coupled joint can be added.

Prerequisites

Before adding this joint:

  1. The pinion must have a Revolute joint connecting it to the frame.
  2. The rack must have a Slider joint connecting it to the frame.

Step-by-step

  1. Press Q or choose Assembly → Joints → Rack and Pinion.
  2. Click a geometry on the pinion (the rotating part) — typically the cylindrical face of the gear wheel.
  3. Click a geometry on the rack (the sliding part) — typically a face defining the slide direction.
  4. In the task panel, set the Pitch Radius (radius of the pinion pitch circle, in mm).
  5. Click OK.

Parameters

Parameter Type Default Description
Reference 1 Link — Geometry on the pinion part
Reference 2 Link — Geometry on the rack part
Pitch Radius Length — Effective radius of the pinion pitch circle (mm)

The rack travel per revolution = 2π × Pitch Radius.

Python API

Minimal example

import FreeCAD as App

doc = App.ActiveDocument
asm = doc.getObject("Assembly")

joint = asm.Joints.newObject("Assembly::JointObject", "RackPinionCoupling")
joint.JointType = "RackPinion"
joint.Reference1 = (doc.getObject("Pinion"), ["Face2"])
joint.Reference2 = (doc.getObject("Rack"), ["Face1"])
joint.Radius = 15.0   # 15 mm pitch radius → 94.25 mm travel per revolution
doc.recompute()

Common mistakes and pitfalls

Coupled joint requires prerequisite kinematic joints first

Adding the Rack and Pinion joint before the pinion has a Revolute joint and the rack has a Slider joint will either fail or produce incorrect results. Always add kinematic joints first.

Pitch radius vs actual gear radius

The pitch radius is the radius of the pitch circle — typically smaller than the outer radius of the gear teeth. Using the outer radius gives the wrong travel ratio. Use the pitch circle radius (module × tooth count / 2).

See also