Rack and Pinion Calculator: Travel, Speed and Force

How to use the calculator

1. Choose Motor to rack when motor torque and speed are known, or Rack to motor when the required rack force and speed are known.
2. Select metric or imperial units. Metric geometry uses module in millimetres; imperial geometry uses diametral pitch in teeth per inch.
3. Enter the pinion tooth count and pitch. Use the reference or pitch diameter from the supplier's drawing when checking the geometry.
4. Enter the transmission ratio as motor RPM divided by pinion RPM. A value of 5 means a 5:1 speed reduction.
5. Enter a combined mechanical efficiency. Use 100% only to calculate the ideal loss-free upper bound.
6. Review the substituted equation, warnings, and limitations before using the result for preliminary sizing.

Pinion geometry and rack travel

Rack and pinion equations

For a metric spur pinion, reference diameter is d_p = m z. For an imperial pinion, d_p = z / P_d. Travel per pinion revolution is the reference-circle circumference:

s_rev = pi d_p

With transmission ratio i = motor RPM / pinion RPM and combined efficiency eta:

n_p = n_m / i and T_p = T_m i eta

Rack speed and steady tangential force are then:

v = pi d_p n_p and F_t = 2 T_p / d_p

Use consistent units. In SI, torque is N m and pitch diameter is metres, which produces force in newtons. The calculator converts imperial inputs to SI internally before solving.

Worked examples

Direct-drive example from the original calculator

For a 20-tooth, module 2 pinion driven at 5 N m and 1,000 rpm, with a 1:1 ratio and ideal 100% efficiency:

• Pitch diameter: 2 mm x 20 = 40 mm
• Travel per revolution: pi x 40 mm = 125.664 mm/rev
• Rack speed: 125.664 mm/rev x 1,000 rev/min = 125,664 mm/min
• Steady tangential force: 2 x 5 N m / 0.040 m = 250 N

This reproduces the original page result. It is an ideal calculation because the efficiency is 100%.

Same motor with a 5:1 reduction and 90% efficiency

The pinion speed becomes 200 rpm and pinion torque becomes 22.5 N m. Rack speed falls to 25,132.7 mm/min while calculated tangential force rises to 1,125 N. The rack mechanical power is 90% of the motor mechanical power in this simplified model.

Reverse sizing example

For a target of 500 N at 100 mm/s with the same 40 mm pitch diameter, a 5:1 reduction, and 90% efficiency, the required pinion torque is 10 N m. The corresponding motor requirement is 2.222 N m at approximately 238.7 rpm, or 55.56 W of mechanical shaft power. A real motor must still be checked on its torque-speed curve and for acceleration, thermal duty, and service conditions.

Design tradeoffs the equation exposes

At fixed pinion torque, a larger pitch diameter increases rack travel and speed per revolution but reduces tangential force. A speed-reducing gearbox moves the tradeoff in the other direction: it raises pinion torque and reduces pinion speed. Efficiency reduces transmitted torque and output power; it does not change the geometric travel per pinion revolution.

That relationship is only the start of a drive selection. The force needed to move a machine can include guide friction, gravity, process force, acceleration, cable drag, seals, and external disturbances. Calculate those loads separately, then check pinion and rack capacity, gearbox permissible torque, shaft and bearing loads, stiffness, backlash, lubrication, and duty cycle against manufacturer data.

Common mistakes

• Entering motor stall torque together with normal running RPM even though those values cannot occur at the same operating point.
• Treating a 10:1 reduction as 0.1 when this calculator defines ratio as motor RPM divided by pinion RPM.
• Using 100% efficiency as a prediction instead of an ideal upper bound.
• Using module and diametral pitch as if they were the same quantity. They are reciprocal pitch systems related by m = 25.4 / P_d.
• Calling the calculated tangential force the allowable rack load without checking tooth strength, contact stress, support, and supplier ratings.
• Ignoring acceleration and reflected inertia in a reversing or servo axis.

Assumptions and limitations

The calculation is deterministic for compatible spur-rack geometry at the reference circle. It assumes steady motion, constant ratio, rigid components, and a user-entered efficiency that represents transmission losses at the operating point.

It does not calculate tooth-root bending stress, pitting resistance, pressure-angle separating force, backlash, helical axial force, shaft or bearing capacity, guide friction, rack deflection, acceleration torque, shock, reversing duty, lubrication, wear, thermal limits, braking, holding, or compliance with a code or safety standard.

Standard 20-degree full-depth involute pinions below 18 teeth may undercut. Profile shift, pressure angle, and non-standard tooth forms can change that result, so confirm low-tooth-count geometry with the gear supplier rather than treating 18 as a universal prohibition.

Displayed results are rounded for readability. Input tolerances, motor data, efficiency uncertainty, and real load variation usually matter more than additional decimal places.

How to verify a preliminary result

1. Confirm tooth count, module or diametral pitch, pressure angle, and reference diameter from the rack and pinion drawings.
2. Confirm motor torque at the intended RPM from the motor torque-speed curve.
3. Confirm transmission ratio, efficiency, and permissible output torque from the gearbox manufacturer.
4. Calculate gravity, friction, process, and acceleration forces for the actual moving system.
5. Check rack, pinion, shaft, bearings, guides, fasteners, and supporting structure against manufacturer ratings and the applicable design standard.
6. Measure force, speed, backlash, temperature, and stopping behavior on the assembled machine before release.

Engineering references

1. KHK Gears: Calculation of Gear Dimensions - module, reference diameter, and rack travel per revolution.
2. APEX Dynamics: High Precision Rack and Pinion Catalog - rack force, pinion torque, speed, and selection relationships.
3. OMRON: Servo Selection Technical Guide - independent rack-and-pinion load-torque equation.
4. Oriental Motor: Gearhead Technical Reference - output torque, ratio, and transmission efficiency.
5. NIST SP 811 Appendix B.9 - force and torque conversion factors.
6. ISO 54:1996 and ISO 1122-1:1998 - standardized module context and gear geometry terminology.
7. SDP/SI: Spur Gear Technical Information - undercut behavior for standard 20-degree involute gears.

References accessed July 28, 2026. Supplier data for the actual rack, pinion, gearbox, and motor takes precedence over generic preliminary calculations.

Rack and pinion calculator walkthrough

This short walkthrough shows the original direct-drive calculation. The V2 interface adds ratio, efficiency, units, and reverse sizing, but the 1:1 ideal equations shown in the video remain the same.

Play

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