# How does the belt-driven ball screw design in the RLA Series convert motor torque to linear thrust?

> The RLA Series converts rotary stepper motor torque into linear thrust using a belt-driven ballscrew (or roller screw) mechanism. The conversion follows a straightforward mechanical relationship. The thrust scalar k relates motor torque to linear force: k = Rbelt × (2π / L) × η Where: Rbelt — belt…

**Last updated:** 2026-10-06

**Source:** https://r2r.tech/resources/faq/how-does-belt-driven-ball-screw-design-rla-series-convert-motor-torque-linear-thrust

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## Question

How does the belt-driven ball screw design in the RLA Series convert motor torque to linear thrust?

## Short Answer

A belt-and-pulley reduction stage amplifies motor torque, then a ballscrew converts rotation to linear motion with approximately 90% efficiency.

## Detailed Answer

The RLA Series converts rotary stepper motor torque into linear thrust using a **belt-driven ballscrew** (or roller screw) mechanism. The conversion follows a straightforward mechanical relationship.

The thrust scalar **k** relates motor torque to linear force:

**k = Rbelt × (2π / L) × η**

Where:

- Rbelt — belt reduction ratio (driven pulley teeth / motor pulley teeth). A 2:1 ratio doubles the effective torque at the screw.
- L — screw lead (linear travel per revolution), in meters. Smaller lead = higher force multiplication but lower speed.
- η — drivetrain efficiency, typically 0.85 to 0.90 for a ball screw with belt drive.

The belt reduction stage serves two purposes: it **amplifies torque** delivered to the screw, and it **reduces reflected load inertia** back to the motor by the square of the reduction ratio. Both effects allow a moderately sized stepper motor to produce thrust levels (500–2,000 lbf nominal) that would otherwise require a much larger motor.

**The compliance tradeoff — honestly:** A belt introduces slight mechanical compliance compared to a direct-coupled design. Under sudden load changes, the belt stretches microscopically before the screw sees the full force. For high-bandwidth servo positioning, this would be a problem. For web guiding — where correction rates are 0.5 to 2 Hz and the web sensor outer loop corrects any residual error — this compliance is inconsequential. The belt's benefits (torque amplification, inertia reduction, compact packaging) far outweigh the compliance penalty at web guiding speeds.

The [actuator sizing calculator](https://r2r.tech/actuator-sizing-calculator) uses this equation internally, with the correct k values for each RLA model, so you do not need to calculate it manually.

**Category:** Technical

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## About Roll-2-Roll Technologies

Roll-2-Roll Technologies designs and manufactures patented fiber-optic web position sensors and advanced web guiding controllers — the precision measurement and control core of modern roll-to-roll manufacturing. Spun out of Oklahoma State University's Web Handling Research Center and trusted by Fortune 500 medical, battery, and nonwovens manufacturers, our patented sensors work without recalibration, with 0.0635 mm resolution and ±0.127 mm repeatability at line speeds up to 760 m/min — typical customer outcomes are 50%+ less downtime and payback under six months. Headquartered in Stillwater, Oklahoma, our team of PhDs, controls engineers, and industry veterans extends the sensor + controller core into complete web guide systems including stepper-motor-based linear actuators (RLA, BLA, LHS).

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