Calculate a rack and pinion drive
Feed force, pinion torque and drive power for a rack and pinion drive — from module, number of teeth and the actual load, with either the drive speed or the feed rate as the given quantity.
Calculate a rack and pinion drive
Enter geometry and load — results appear immediately.
Gearing
Load
Formulas
d₀ = m · z s_U = π · d₀ v = s_U · n / 1000 F = m_L · a + m_L · g · (sin α + μ · cos α) + F_P M = F · d₀ / (2000 · η) P = M · n / 9550
g = 9.81 m/s² · Efficiency acts on torque and power, not on the kinematics.
What the model covers
Calculated are the gravity component, guide friction, acceleration and process force. Not included are the inertia of rotating parts, breakaway torque and seal friction. The kinematics say nothing about tooth root strength — the calculator gives the required torque, not the permissible load on the gearing.
How a rack and pinion drive is sized
A rack and pinion turns rotation into straight-line motion. Two things define it: the geometry of pinion and rack, and the force that actually has to be overcome.
The geometry follows from module m and number of teeth z: the pitch diameter is d₀ = m · z, and per pinion revolution the carriage travels the circumference s_U = π · d₀. A pinion with module 4 and 20 teeth therefore has an 80 mm pitch diameter and moves the carriage about 251 mm per revolution.
The required feed force is made up of four parts: the acceleration force m·a, the gravity component m·g·sin α on an inclined axis, the guide friction m·g·μ·cos α, and any process force. Only then does the pinion torque follow as M = F · d₀ / (2000 · η) — the factor 2000 converts the diameter in millimetres to the radius in metres.
The friction coefficient is a required field as soon as a mass is entered. That is deliberate: silently assuming μ = 0 would systematically understate the force, and the result would still look plausible.
Frequently asked questions
Which friction coefficient should I use?
That depends on the guide, not on the rack and pinion: recirculating-element carriages sit well below plain-bearing ones, and contamination, lubrication and preload shift the value further. Take the figure from your guide's data sheet — and when in doubt, the less favourable one.
Why is the calculated torque lower than what the motor has to deliver?
Because the calculator does not include the inertia of the rotating parts. Pinion, coupling, gearbox and motor rotor all have to be accelerated as well. That share comes from the moment of inertia calculator and adds to the load torque determined here.
Does the calculator tell me whether my rack can take the force?
No. It gives the required torque, not the permissible load on the gearing. Tooth root strength depends on material, hardening, gear quality and duty cycle, and is verified separately. Send us your values and we will check that too.
Note: These calculation tools are intended for initial orientation only and do not replace a binding design by qualified engineering personnel. All results must be verified by appropriate engineering calculations before use in safety-relevant applications. Technische Antriebselemente GmbH accepts no liability for damage arising from the use of these results.
Have the rack and pinion sized for the calculated feed force?
Feed force and pinion torque are settled — what remains open is gear quality, material, hardening and lubrication, and the kinematics say nothing about tooth root strength. Carry your values into the enquiry and we will size the rack, the pinion and the matching gearbox.
Matching products & guides
Gear racks
Straight- and helical-toothed racks in various modules and lengths — milled, hardened or ground.
ProductSpur gears and pinions
Matching pinions for the rack — as a catalogue item or a custom part to drawing.
ProductPlanetary gearboxes
For when the pinion torque exceeds what the motor delivers directly — low backlash and compact.
CalculatorMoment of inertia calculator
The share this calculator deliberately leaves out: the inertia of the rotating parts.
CalculatorGear geometry calculator
Pitch, tip, root and base circle of the pinion in detail.
GuideCalculating the gear module
Determine the right module from number of teeth, pitch diameter or centre distance — with the standard module series.