cyclogramFrom carriage speed to motor speed
A timing belt is a drum: one turn of the drive pulley moves the belt, and the carriage clamped to it, by the pitch circumference π·D. The pitch diameter is where the belt's cords run — the tooth count times the pitch, over π.
k = π · D / i travel per motor turn n_pulley = v / (π · D) · 60 rpm n_motor = n_pulley · i D = z · pitch / π z teethOn Ø50 mm pulleys one turn moves 157 mm, so the default move's 0.6 m/s needs only 229 rpm. A belt axis turns its motor slowly, and that is where its trouble starts.
The inertia the motor sees
Everything the belt carries is a mass on a drum of radius D/2: the carriage, the payload and the belt itself, which moves at the same speed. The two pulleys add their own inertia; the page estimates them as solid cylinders of the pitch diameter and width, in aluminium or steel, unless you type the catalogue figure.
J_mass = (m + m_belt) · (D / 2)² / i² J_pulleys = 2 · m_pulley · D² / 8 / i² m_pulley = ρ · π/4 · D² · w J_load = J_mass + J_pulleys + J_coupling + J_gearbox20 kg and a 0.3 kg belt on Ø50 mm pulleys: J_mass = 20.3 · 0.025² = 1.27 × 10⁻² kg·m². Two solid aluminium pulleys 32 mm wide weigh 0.17 kg each and add 1.06 × 10⁻⁴ kg·m². The motor sees 1.28 × 10⁻² kg·m² — about 170 times what the same load looks like on a 10 mm ball screw.
Torque, and the force in the belt
The torques follow the same rule as a screw's, with the pulley radius in place of p/2π. The force in the belt is what the carriage needs — gravity, friction, the external force and the mass times the acceleration — and it is what the belt's catalogue rates. Belt tension does not reach the motor: it loads the bearings.
T_static = (m·g·sin α + μ·m·g·cos α + F) · D / (2 · η · i) T_accel = T_static + (J_mass + J_pulleys) · α / η + J_trans · α / η_t + J_rotor · α F_belt = m·g·sin α + μ·m·g·cos α + F ± (m + m_belt) · aFor the default move, friction costs 0.545 N·m at the motor; accelerating takes 2.60 N·m, cruising 0.545 N·m, braking −1.12 N·m, and the RMS over the cycle is 1.17 N·m. The belt carries 19.6 N of friction and 92.7 N at the peak. Type the belt's allowable force in step 2 and step 3 checks it.
Why belt axes need a gearbox
The same 400 W servo that drove the ball screw comfortably now drives 492 times its own rotor inertia. The torque is there — the peak needs 2.60 of the 3.82 N·m the motor has — but no servo loop settles a load like that cleanly. The load's inertia falls with the square of a gearbox's ratio, so the ratio that brings it down to the limit of 10 is
i = √( (J_load / J_rotor) / 10 ) = √( 492 / 10 ) ≈ 7.0The page suggests about 7.1 : 1. With a 7.1 : 1 gearbox the ratio is 9.76 — just under the limit, so the check reads Close — and an 8 : 1 gearbox brings it to 7.7. The motor then turns at 1627 rpm instead of 229, and the peak torque drops to 0.411 N·m.
Worked example
| Quantity | By hand | Result |
|---|---|---|
| Travel per turn | π · 50 mm | 157 mm |
| Pulley and motor speed | 0.6 / 0.157 · 60 | 229 rpm |
| Moved mass and belt at the motor | 20.3 · 0.025² | 1.27 × 10⁻² kg·m² |
| Both pulleys | 2 · 0.17 · 0.05² / 8 | 1.06 × 10⁻⁴ kg·m² |
| Torque accelerating / cruising / braking | α = 3.6 / 0.025 = 144 rad/s² | 2.60 / 0.545 / −1.12 N·m |
| RMS torque over 1 s | √((2.60² + 0.545² + 1.12²) / 6) | 1.17 N·m |
| Belt force at the peak | 20.3 · 3.6 + 19.62 | 92.7 N |
| Inertia ratio, direct drive | 1.28 × 10⁻² / 2.6 × 10⁻⁵ | 492 |
Questions
Does belt tension add to the motor torque?
No. Pretension loads the bearings and stretches the belt; the motor sees only the forces that move the load, so it does not appear in the torque.
Which pulley diameter do I enter?
The pitch diameter, where the belt's tension cords run — not the tips of the teeth. It is the tooth count times the pitch over π: 30 teeth on a 5 mm pitch is 47.7 mm.
How do I find the belt's mass?
Belt catalogues give a mass per metre for each width; multiply it by the length of the whole loop. It is usually small next to the carriage, and 0 is a fair start.
Why does the page suggest a gearbox for my belt axis?
Because the inertia ratio is far over the limit. A belt reflects the load through the pulley radius, which is large next to a screw's lead, and the reflected inertia falls with the square of a gearbox ratio.
What is the allowable belt force?
The largest peripheral force the belt may carry, as its catalogue states it for the width, the pitch and the teeth in mesh. Step 3 compares the force at the peak of the move with it.
Sources
- Oriental Motor, Technical Reference, selection calculations — the belt-and-pulley load torque and inertia. The test suite works their belt-conveyor and belt-driven-table examples and meets the printed answers within 0.5 %.
- Yaskawa, SigmaSelect help manual, the mechanism equations — the travel per motor turn πD/i.
Related guides
- Axis sizing calculatorThe whole guided flow: motion, mechanism, motor, regeneration, and the move as a bar.
- Inertia ratioThe load's inertia at the motor through a screw, a belt or a gearbox, and the ratio that fixes it.
- Servo motor sizingPeak torque at speed, RMS over the cycle, maximum speed and inertia ratio, each with its margin.
- Move time and motion profilePeak speed, acceleration and jerk from a distance and a move time; trapezoid, triangle and S-curve.
- Brake resistor sizingThe energy an axis returns while braking, what the drive's bus takes, and the resistor for the rest.
Every term on this page, defined: the axis and motor sizing glossary.