Pinion driving the rack
0position 0 mm2000 mm
↺ 60pinion rpm
→
0.38rack m/s
→
25.0push kN
Interactive Gear Guide
How a rack and pinion works
A gear rolling along a straight toothed bar — the simplest way to turn rotation into precise linear motion. Change the pinion, the speed and the torque, and watch the rack respond live.
Playground
Design a rack drive
A rack is a gear with an infinite radius: its teeth sit on a straight line, so every turn of the pinion pushes the rack exactly one pitch circumference. Size the pinion, set its speed and torque, and see the travel, linear speed and forces update in real time.
Rack tooth profile normal section
Travel per revolution377.0 mm/revπ · d
Rack speed v0.377 m/s22.6 m/min · π d n / 60 000
Push force Fₜ25.0 kN2 T / d
Separating force Fᵣ9.10 kNFₜ tan αₜ — pushes the rack away
Axial thrust Fₐ0 Nstraight rack — no axial thrust
Pinion pitch / outside dia.120 / 132 mmm Z / cos β · d + 2m
Rack pitch p18.85 mmπ m
Full stroke5.31 rev · 5.3 s106 rack teeth
Contact ratio ε1.77teeth in mesh, on average
Undercut check✓ no undercutmin. teeth ≈ 17 — 2 cos β / sin² αₜ
Estimates for a standard involute rack and pinion (ISO 53 basic rack profile), normal module m, before friction and dynamic factors. Real drives add profile shift, pitch tolerances across rack joints and a strength check — talk to our engineers for a manufacturing spec.
Start with the basicsHow a gear pair works — the interactive gear guide →
Reference
The rack and pinion, explained
How a straight gear turns rotation into travel, the forces it puts into your machine, and the details that decide whether a long rack runs smoothly.
The rack
A gear whose pitch circle has grown to an infinite radius — so it becomes a straight line and the involute flanks become straight, angled sides. That makes racks simple to cut and inspect.
The pinion Z
The round gear that meshes with the rack. Either the pinion turns in place and the rack slides, or the rack is fixed and the pinion travels along it on a carriage — the maths is identical.
Rack pitch p
The distance from one rack tooth to the next — exactly the pinion's circular pitch.
p = π · m
Travel per revolution
One turn of the pinion rolls its whole pitch circle along the rack — Z teeth, one pitch each.
s = π · d = π · m · Z
Linear speed v
Rack speed follows directly from pinion speed — no slip, no creep, no stretch like a belt or chain.
v = π · d · n / 60 000 [m/s]
Push force Fₜ
Torque on the pinion becomes a straight-line force at the pitch line. A smaller pinion pushes harder for the same torque.
Fₜ = 2 T / d
Separating force Fᵣ
The angled flanks try to push the rack and pinion apart. The rack must be bolted to a rigid base, and a travelling pinion needs guide rollers to hold the mesh.
Fᵣ = Fₜ · tan α
Straight vs helical β
Helical racks engage gradually — quieter, smoother and stronger, but with an axial thrust the guides must take. The tooth spacing along the rack stretches to p / cos β.
Fₐ = Fₜ · tan β
The basic rack profile
Every standard involute gear is defined by the rack that would mesh with it (ISO 53, DIN 867). It is also the shape of the hob and rack cutter that generate the teeth.
ha = 1 m · hf = 1.25 m · s = p / 2
Undercut & minimum teeth
If the pinion has too few teeth, the generating rack cuts away the root of each tooth and weakens it. Below this limit, use profile shift or more teeth:
Zₘᵢₙ = 2 / sin² α (≈ 17 at 20°)
Joining rack sections
Long strokes are built from rack sections butted end to end. The ends are machined so the pitch carries across the joint, then set with a gauge piece before the bolts and dowels go in.
- Pitch error across the joint
- Backlash set by mounting height
- Alignment along the full length
Where you meet them
Anywhere a heavy load has to move in a straight line, over any length.
- Sluice & dam gates
- Machine tools & gantries
- Lifts, hoists & platforms
- Stacker & transfer cars
- Sliding gates & steering
Need a rack — or a matching pinion?
We cut straight and helical racks in lengths to suit your machine, with matching pinions, for cement, mining, sugar and steel plants across Egypt.