Calculation
Tmotor = F·d/(2·η)
At the ideal pinion, Tp = F·r; meshing, bearing and gearbox losses raise the torque actually required at the drive.
Tp = F·d/2; Tmotor = Tp/η
Required drive torque of a rack-and-pinion drive
At the ideal pinion, Tp = F·r; meshing, bearing and gearbox losses raise the torque actually required at the drive.
MINTSI
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Result
Select a target and calculate.
Understand the inputs
- Required drive torque Tmotor — Torque to be supplied at the drive (motor or gearbox output shaft), including losses.
- Required feed force F — Tangential force at the rack needed for feed, acceleration and any process load.
- Pinion pitch diameter d — Pitch diameter of the meshing pinion; halved it gives the effective lever arm.
- Drive efficiency η — Overall efficiency of meshing, bearings and any upstream gearbox between motor and pinion.
Example
F=5,000 N, d=50 mm and η=0.9 give an ideal pinion torque Tp=F·d/2=125 N·m and required drive torque Tmotor=125/0.9≈138.9 N·m.
Assumptions and limits
Steady, quasi-static view with constant η; acceleration torque from moving masses, backlash, preload and dynamic load peaks are excluded and must be added separately.
Understand Required drive torque of a rack-and-pinion drive
This calculator determines the torque a drive must actually supply at the pinion shaft of a rack-and-pinion drive to generate a required feed force. It extends the plain lever-arm relation Tp=F·r with the drive efficiency that many simple rules of thumb omit.
What does this quantity describe?
At the ideal, lossless pinion, Tp=F·r=F·d/2 with pitch radius r=d/2. Meshing friction, bearing friction and any upstream gearbox consume additional power, so the drive itself must supply a torque increased by efficiency η: Tmotor=Tp/η=F·d/(2η).
Formula and variables
Tmotor = F·d/(2·η)
| Symbol / input | Meaning |
|---|---|
| Required drive torque Tmotor | Torque to be supplied at the drive (motor or gearbox output shaft), including losses. |
| Required feed force F | Tangential force at the rack needed for feed, acceleration and any process load. |
| Pinion pitch diameter d | Pitch diameter of the meshing pinion; halved it gives the effective lever arm. |
| Drive efficiency η | Overall efficiency of meshing, bearings and any upstream gearbox between motor and pinion. |
Choose the inputs correctly
F is the force actually needed at the rack for feed, acceleration and process loads. d is the pitch diameter of the meshing pinion. η is the overall efficiency between motor shaft and pinion, typically 0.85 to 0.95 for a single-stage, ball-bearing-supported mesh.
How to use the calculator
Take F from the application's load analysis, including friction, acceleration and any process forces. Take d from the chosen pinion datasheet. Assume η conservatively, or take it from manufacturer data if an upstream gearbox is present.
Worked example
F=5,000 N, d=50 mm and η=0.9 give an ideal pinion torque Tp=F·d/2=125 N·m and an actually required drive torque Tmotor=125/0.9≈138.9 N·m.
Understand the result and units
The smaller η, the larger the margin above the ideal pinion torque; at η=0.7 it is already about 43% more than at η=0.9. Assuming too high an η undersizes the drive.
F is a force, d a length and η dimensionless. Tmotor is output as a torque.
Where the relation comes from
The base relation Tp=F·r is widely accepted drive-engineering knowledge, see e.g. linearmotiontips.com; that sizing guide explicitly flags efficiency as an open point of simplified rules of thumb. This calculator fills exactly that gap without inventing manufacturer-specific mesh or gearbox data.
Typical applications
Preliminary sizing of servo drives for rack-and-pinion axes in machine tools, gantry systems and handling axes, before adding acceleration torque and peak loads from moving inertia.
Assumptions, limits and common mistakes
Steady, quasi-static view with constant efficiency. Acceleration torque from moving masses, backlash and mesh preload, dynamic load peaks, and a separate reduction step for an upstream gearbox are excluded and must be added separately.
Common mistake: Do not confuse pinion radius with pitch diameter; d is the full diameter, and d/2 appears in the numerator. Do not assume η=1 when a gearbox or lubricated mesh with noticeable losses is actually present.
Frequently asked questions
Does the calculator account for accelerating moving masses?
No. F must already include all static and dynamic force components at the rack; an additional acceleration torque from rotating drive-part inertia must be added separately.
What efficiency should I use?
For a single-stage, ball-bearing-supported mesh, 0.85 to 0.95 is typical; with an additional upstream gearbox, overall efficiency drops by that gearbox's own efficiency.
Why does the result differ from simple online calculators?
Many simple calculators only give the ideal pinion torque Tp=F·r without efficiency; this calculator additionally gives the torque actually required at the drive, increased by η.