Rack and Pinion Calculator — Linear Drive Force, Tooth Strength (ISO 6336-based) & Motor Torque

Governing standard: ISO 6336-based· ISO 6336-based tooth strength (ISO 6336-2 pitting, ISO 6336-3 Method B root) of pinion and rack, the rack as the z → ∞ limit of the gear engine · duty forces and drive torque by Newtonian mechanics (estimate)

Page last updated 14 October 2026

Video walkthrough (3:30)

Read the transcript

Here's a gantry axis, 250 kg moving at 2 m/s on a rack and pinion. The tool says it's SAFE, and the rack flank decides.

Now stand the same axis up vertical. Same parts, and it fails.

This is the MechanixCalc rack and pinion calculator. It works out the drive force in every phase of the move, the ISO 6336-based tooth strength of pinion and rack, and the motor torque. Let's go through it, step by step.

A rack and pinion turns motor torque into a straight-line force. The pinion pushes on the rack teeth, and those teeth carry the load in every phase.

A tooth can fail in two ways. The root can crack in bending, or the flank can pit under contact pressure. The tool checks both, on the pinion and on the rack.

Step one is the application. A horizontal axis carries 250 kg at 2 m/s, with 3 m/s² up and down, and a 400 N process force on the forward stroke.

The chart shows the force in every phase of the move. The peak is 1162 N, while accelerating forward.

Step two is the rack and pinion. Module 3, 20 teeth, helical, and 30 mm wide. This helix angle makes the feed a round 200 mm per turn.

Step three is the materials. A case-hardened pinion runs on a quenched and tempered rack. The flank roughness is 6.5 µm, a ground pinion on a milled rack.

Step four is the drive. With an 8 to 1 gearbox, the motor needs 5.36 N·m at its peak. The chart shows its torque in every phase.

Step five shows the checks. The roots are far from their limit. The rack flank governs with a safety factor of 1.61, so the verdict is SAFE.

The tool also gives the max feed force, 2084 N. That's the most this mesh carries with every check met.

Now raise the process force to 1400 N. The peak becomes 2162 N, just past the max feed force.

The verdict changes to CHECK. The rack flank's safety factor drops to 1.18, below its line of 1.2.

The fix is a wider rack. At 35 mm the flank is back to 1.27, and the verdict turns green.

Now go back to the start, and stand the axis vertical. Gravity acts in every phase, and the peak force grows to 3602 N.

The verdict is FAIL. The rack flank's safety factor is only 0.91.

A rack and pinion never locks itself, so the tool asks for a brake too. It needs at least 19.5 N·m at the motor.

The fix is one module up. With module 4, the flank safety factor is 1.23, and the verdict turns green.

A bigger brake puts its full torque on the teeth, so the tool checks that too. With the fitted 40 N·m brake, the static check still holds.

What the tool does not check is listed too: the rack screws, the pinion shaft and bearings, and the gearbox rating. So silence never reads as a pass.

Every input, every formula and every check goes into a PDF report. A report your reviewer can follow.

That's a rack and pinion drive, checked. Try it on your own axis, on MechanixCalc.com.

The MechanixCalc rack and pinion calculator sizes a motor-driven linear axis on a gear rack. Enter what the axis moves — the mass, the axis angle, guide friction, a process force, the speed, acceleration, deceleration and stroke — and the rack and pinion size: module, number of teeth, helix angle, face width, profile shift and quality grade. The tool computes the drive force in every motion phase (accelerate, constant speed and decelerate in both directions, plus the hold force on an inclined axis), rates the teeth at the largest of them, and sizes the gearbox and motor.

Tooth strength is not a separate rule of thumb: the pinion AND the rack are rated for root bending and flank pitting with the same ISO 6336-based engine as the cylindrical-gear calculator, with the rack as its exact z → ∞ limit — the rack's own load-pass count, an open-gearing face-load factor and, for a reversing drive, the fully reversed bending factor. Each member gets its own material, so a soft rack that fails in pitting is caught. The drive step returns the pinion speed and torque, the feed per revolution, the peak and RMS motor torque including the reflected inertia, the inertia ratio and, for a vertical axis, the holding-brake torque — a rack and pinion is never self-locking.

What this calculator does

Drive force in all seven motion phases — accelerate, constant speed, decelerate (both directions) and hold — with gravity, guide friction and a working-stroke process force
ISO 6336-based root bending (Method B) and pitting of BOTH pinion and rack, the rack as the exact z → ∞ limit of the gear engine
Each member rated with its own material and its own life count — rack passes, never pinion cycles divided by an infinite ratio
Reversing drives rated with the fully reversed bending factor Y_M 0.7
Pitch diameter, feed per revolution π·d, pinion speed and torque, contact ratio and undercut check for straight and helical racks
Two or more pinions sharing the load with a load-share factor
Motor torque through the gearbox — peak and RMS over the cycle, with pinion, gearbox and rotor inertia and the inertia ratio
Holding-brake torque for inclined and vertical axes
Max feed force per pinion at the current size — compare it with a catalogue's permissible feed force only after matching K_A, safety factors, speed and life
Emergency-stop and brake-engagement check as a static peak load, kept out of the motor sizing
Dimensioned sketch of the pinion on the rack with the pitch line, module, pitch diameter, mounting height and tooth forces
Branded PDF report and Excel export with the method and what is not checked

Method & formulas

Drive force per motion phase

The rack carries whatever force the carriage needs in each phase of its motion. On a horizontal axis that is mostly the inertia force m·a; on an inclined or vertical axis gravity acts in every phase and the hold force is the weight component itself. Guide friction is applied only where it adds to the force, so an over-estimated friction coefficient can never relieve a peak, and a process force acts on the forward (working) stroke. The largest force over all phases — often acceleration, or braking on the way down — rates the teeth, divided between the pinions with a load-share factor when two or more drive the axis. Each check is then taken at its worst speed, from standstill up to the peak speed. An emergency stop and the engagement of a holding brake are checked separately, as static peak loads that do not enter the motor sizing.

Drive force

F = m·a + m·g·sin θ·SF_v ± μ·m·g·cos θ ± F_proc, F_t = F_max / n · K_γ

where m = moving mass; a = the phase's acceleration (signed); θ = axis angle; SF_v = vertical load factor; μ = guide friction; F_proc = process force (forward stroke); n = pinions; K_γ = load-share factor.

Geometry and kinematics

The pinion's pitch circle rolls on the rack's pitch line, so one revolution advances the carriage by π·d. A helical rack runs smoother and carries more than a straight one at the same module, but adds an axial force on the rack fasteners and the pinion bearings. Below about 17 teeth a straight pinion without profile shift is undercut; on a rack mesh the contact then cannot start before the pinion's base-circle tangent point, which shortens the contact ratio.

Pitch diameter, feed and speed

d = m_n·z / cos β, s = π·d, n = 60 000·v / (π·d), T = F_t·d / 2

where m_n = normal module (mm); z = pinion teeth; β = helix angle; v = traverse speed (m/s); n = pinion speed (rpm); T = pinion torque.

Rack-mesh contact ratio

εα = [√(r_a² − r_b²) − r·sin α_t + min(m_n(1 − x)/sin α_t, r·sin α_t)] / (π·m_t·cos α_t)

where r, r_a, r_b = pinion pitch, tip and base radii; x = pinion profile shift; α_t = transverse pressure angle; m_t = m_n / cos β.

Tooth strength of pinion and rack (ISO 6336-based)

The pinion and the rack are rated by the same ISO 6336-based engine as the cylindrical-gear calculator, with every term that depends on the mating gear's tooth count taken to its exact limit for a rack: the working pressure angle equals the transverse pressure angle, the rack's flank is flat (the relative radius of curvature is the pinion's), and the rack's tooth-root form factor is ISO 6336-3 Method B in its analytic rack limit. Two rack facts are added: each rack tooth is loaded once per pass over it, so its life count is the number of passes, and a reversing drive bends every root in both directions, which takes the fully reversed bending factor Y_M = 0.7 on the permissible root stress. Pitting is often decided by the rack, which is usually the softer member.

Root and contact stress

σ_F = F_t/(b·m_n)·Y_F·Y_S·Y_β·Y_B·K_A·K_V·K_Fβ·K_Fα,   σ_H = Z_H·Z_E·Z_ε·Z_β·√(F_t·K_A·K_V·K_Hβ·K_Hα / (b·d))

where For a rack the (u+1)/u term of the contact stress is 1. SF = σ_FG·Y_M / σ_F (CHECK below 1.4), SH = σ_HG / σ_H (CHECK below 1.2), each member with its own limits.

Gearbox, motor and brake

The pinion torque passes through the gearbox to the motor: divided by the mesh and gearbox efficiency while the motor drives, multiplied by it when the load back-drives the motor. The inertia of the rotor, the gearbox and the pinion, and the carriage mass reflected through the gearbox, add the acceleration torque. The tool returns the peak torque and the RMS torque over the whole cycle for the motor's continuous rating, and the inertia ratio as a tuning guideline. A rack and pinion is never self-locking, so an inclined or vertical axis needs a holding brake sized from the static hold torque with a margin.

Motor torque

T_m = F/n·K_γ·r / (i·η) + (J_m + J_gb + J_pin/i²)·a·i / r

where r = pitch radius (m); i = gearbox ratio; η = mesh × gearbox efficiency (× instead of ÷ when back-driven); J = inertias (kg·m²).

Worked example

A horizontal gantry axis moves 250 kg at 2 m/s with 3 m/s² acceleration and a 400 N process force, on a module 3, 20-tooth helical pinion (β 19°31′41″, face width 30 mm) over a 3000 mm stroke with 0.5 s dwells, through an 8:1 gearbox. The ground pinion runs on a milled rack (flank roughness Rz 6.5 µm, the pair mean).

Given

Moving mass / speed / acceleration250 kg / 2 m/s / 3 m/s²
Guide friction / process forceμ 0.005 / 400 N (forward stroke)
Pinion / rackm 3, z 20, β 19°31′41″, b 30 mm, quality 7, K_A 1.25, 16MnCr5 pinion on a 42CrMo4 rack
Stroke / dwell3000 mm / 0.5 s
Gearbox / inertiasi 8, η 0.95; motor 5·10⁻⁴, gearbox 1·10⁻⁴ kg·m²

Result

VerdictSAFE — the rack flank governs (SH 1.61)
Peak force / max feed force / feed per revolution1162 N / 2084 N / 200 mm
Motor torque (peak / RMS) at 4800 rpm5.36 / 2.73 N·m

Peak force (forward acceleration): F = 250·3 + 0.005·250·9.807 + 400 = 1162 N.
Pitch diameter d = 3·20 / cos 19.53° = 63.66 mm, so the feed per revolution is π·d = 200 mm and the pinion runs at 60 000·2 / 200 = 600 rpm.
Pinion torque T = 1162 · 0.03183 = 37.0 N·m; at the motor, with losses and the inertia torque: 5.36 N·m peak, 2.73 N·m RMS, at 4800 rpm.
The drive reverses (−762 N on the return acceleration), so the roots are rated with Y_M 0.7: root SF 8.23 (pinion) / 4.91 (rack); pitting, governed near standstill and with the milled rack's roughness factor Z_R 0.90, SH 3.24 / 1.61 — SAFE. The mesh would carry up to 2084 N per pinion with every check met. The load-to-motor inertia ratio is 7.9 : 1, within the usual 10 : 1 tuning guideline.

Frequently asked questions

How do I calculate the force on a rack and pinion?

Add up what the carriage needs in each phase of its motion: the inertia force m·a while accelerating or braking, the weight component m·g·sin θ on an inclined axis, the guide friction and any process force. The largest of these — often acceleration on a horizontal axis, or braking on the way down on a vertical one — is the tangential force the teeth must carry, divided between the pinions if more than one drives the axis.

How is the tooth strength of a gear rack checked?

With the same method as a gear pair: root bending and flank pitting by an ISO 6336-based method, the rack being the limit of a gear with infinitely many teeth. This tool rates both members with the ISO 6336-based gear engine in that limit, with the rack's own material and its own load-pass count, and with the fully reversed bending factor when the drive reverses.

What is the feed per revolution of a rack and pinion?

One pinion revolution moves the carriage by the pitch circumference, π·d = π·m_n·z / cos β. A helix angle of 19°31′41″ (19.528°) makes cos β = 3π/10, so the feed becomes a round 10·m·z/3 mm — 200 mm for a module 3, 20-tooth pinion.

Is a rack and pinion self-locking?

No. Unlike a lead screw with a small lead angle, the rack and pinion mesh back-drives freely — do not rely on gearbox friction as a safety brake. A vertical or inclined axis needs a holding brake; the tool gives the static hold torque at the motor, the brake torque with your margin, and checks the teeth for the load when that brake engages.

Straight or helical rack?

A helical rack has more teeth in contact at once, so it runs quieter and carries more at the same module, but it creates an axial force F_a = F_t·tan β that the rack fasteners and the pinion bearings must take. A straight rack has no axial force and is simpler to join and align.

How do I size the motor for a rack and pinion axis?

Take the pinion torque F_t·d/2 for each phase, divide by the gearbox ratio and efficiency, and add the acceleration torque of the rotor, gearbox, pinion and the carriage mass reflected through the gearbox. Compare the peak with the motor's peak torque and the RMS over the whole cycle with its continuous rating. The tool does this for every phase and also reports the inertia ratio.

What is the maximum feed force of my rack and pinion?

The tool reports the largest tangential force per pinion that the current module, face width, materials and duty carry with every fatigue check met — root SF at least 1.4 and flank SH at least 1.2 — and the axis value for several pinions. Compare it with the peak drive force. It includes K_A, the reversed-bending factor and these safety factors, so before comparing it with a rack catalogue's permissible feed force, match the catalogue's own K_A, safety factors, speed and life.

What does this calculator not check?

The rack fasteners and dowels, the rack bar bending between its screws, the pinion shaft, key and bearings, the gearbox rating, backlash and positioning accuracy, and wear of unhardened racks are among the items listed under 'Not checked' on the page, in the PDF and in the API, so silence never reads as a pass.

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