Useful force and feed speed in rack transmissions
You know your motor's torque and the pinion module. From these two figures you can derive everything: the force that moves the load, the feed speed, the position resolution. The formulas are few and direct.
From motor to linear motion
The rack transmission converts the rotary motion of the pinion into linear translation of the carriage (or of the fixed rack with a moving carriage). The two fundamental relations are:
Feed speed: v = π × m × z × n (where n is the pinion's rotational speed in rev/s)
Tangential force: Ft = 2 × Mt / d (where Mt is the driving torque on the pinion and d = m × z is the pitch diameter)
The efficiency of the rack transmission is generally high: η = 0.95 - 0.98, thanks to the reduced sliding on the involute profiles.
Table of the main formulas
| Parameter | Formula | Unit |
|---|---|---|
| Feed speed | $v = \pi \times m \times z \times n$ | m/s (with n in rev/s) |
| Advance per revolution | $s/rev = \pi \times m \times z$ | mm/rev |
| Tangential force | $Ft = \frac{2 \times Mt}{(m \times z)}$ | N (Mt in N·mm) |
| Radial force | $Fr = Ft \times tan(20°) = 0.364 \times Ft$ | N |
| Resultant force on the tooth | $Fn = \frac{Ft}{cos(20°)} = 1.064 \times Ft$ | N |
| Transmitted power | $P = Ft \times v$ | W |
| Torque required at the pinion | $Mt = \frac{Ft \times m \times z}{2}$ | N·mm |
Allowable useful force in nylon
The allowable tangential force for a rack depends on the module, the tooth width, the operating speed and the working temperature.
As a reference for a first sizing: for racks in PA6+30%GF (polyamide 6 reinforced with 30% glass fiber), module 2, width 20 mm, speed below 0.5 m/s and temperature below 40°C, the continuous useful force is in the order of 150-300 N.
For accurate calculations, verify the tooth-root strength according to ISO 6336 (the international standard for calculating the load-carrying capacity of gears) adapted to the properties of the polymer material, and the Hertzian pressure on the tooth flank (the contact pressure generated between the surfaces of meshing teeth). Both values depend on the specific material and the operating conditions.
Complete numerical example
Scenario: workpiece carriage on a linear guide. Data: mass to translate M = 15 kg, maximum acceleration a = 2 m/$s^2$, maximum speed v = 0.8 m/s, pinion m = 2, z = 20 in POM-C.
Calculation of the total force:
- Inertia force: F_inertia = 15 × 2 = 30 N
- Friction force: F_friction = 15 × 9.81 × 0.1 (guide friction coefficient) = 14.7 N
- Total force: F_total = 44.7 N ≈ 45 N
Driving torque required: Mt = F × d/2 = 45 × (2 × 20)/2 = 45 × 20 = 900 N·mm = 0.9 N·m
Pinion rotational speed: n = v / (π × m × z) = 0.8 / (π × 2 × 20 × 0.001) = 0.8 / 0.1257 = 6.36 rev/s = 382 rpm
Advance per revolution: the figure to keep handy
The advance per revolution is the value to keep within reach when working with incremental encoders on the motor, that is, the sensors that count the revolutions and let you calculate the linear displacement of the carriage.
- Pinion m = 2, z = 20: advance = π × 2 × 20 = 125.7 mm/rev
- Pinion m = 3, z = 20: advance = π × 3 × 20 = 188.5 mm/rev
These values let you quickly calculate the position resolution of the linear axis as a function of the encoder resolution.
Peripheral speed and thermal limits in nylon
Technopolymer racks, both PA6 and POM-C, have a peripheral-speed limit linked to the heat generated by friction between the teeth. Above 2-3 m/s, the heat produced can lead to accelerated wear or localized melting of the profile.
The limit isn't the same for all nylons. PA6+GF (glass-fiber reinforced) and the graphite versions withstand higher speeds than non-reinforced PA6 or standard POM-C. Before ruling out nylon for a high-speed application, verify the specific grade and the real thermal conditions.
For speeds above 2-3 m/s with significant loads, assess whether the nylon profile can be sized for that condition or whether it's preferable to turn to toothed-belt transmissions, which in that range operate with less heat development.
If your application combines high speed, variable load or high ambient temperature, write to us at the technical office: we verify together which material grade and which geometry withstand the real conditions.