How to size a ball screw motor: calculator

From table speed to motor speed

A ball screw turns rotation into travel: one turn of the screw moves the nut by the lead p. A gearbox of ratio i between the motor and the screw divides that again, so one motor turn moves the table k:

k = p / i travel per motor turn n_screw = v / p · 60 rpm n_motor = n_screw · i

The default case moves 20 kg over 200 mm in 0.5 s, in equal thirds of accelerating, cruising and decelerating. That needs a peak speed of 0.6 m/s, and on a 10 mm lead the screw turns at 0.6 / 0.01 · 60 = 3600 rpm.

The inertia the motor sees

Seen from the motor, a mass moved by a screw behaves like a flywheel on the shaft: it is the mass on a drum of radius p/2π. The screw adds its own inertia — a solid steel cylinder unless you type the catalogue figure — and the coupling and a gearbox sit on the motor shaft as they are.

J_mass = m · (p / 2π)² / i² J_screw = m_screw · d² / 8 / i² m_screw = ρ · π/4 · d² · L, steel ρ = 7850 kg/m³ J_load = J_mass + J_screw + J_coupling + J_gearbox

For 20 kg on a 10 mm lead, J_mass = 20 · (0.01 / 2π)² = 5.07 × 10⁻⁵ kg·m². A Ø16 × 500 mm screw weighs 0.79 kg and adds 2.53 × 10⁻⁵ kg·m², so the motor sees 7.59 × 10⁻⁵ kg·m² — 2.92 times the rotor of the 400 W servo it is tried with.

Torque in every phase

The forces on the carriage — gravity along the axis, the guide friction, an external force — reach the motor through the lead and the efficiencies; a preloaded nut adds its own drag. While the axis accelerates, the inertias need torque on top, driven through the efficiency. While it decelerates, the load drives the motor back and the efficiency works the other way.

T_static = (m·g·sin α + μ·m·g·cos α + F) · p / (2π · η · i) + μ₀ · F₀ · p / (2π · i · η_t) α_motor = a / k · 2π T_accel = T_static + (J_mass + J_screw) · α / η + J_trans · α / η_t + J_rotor · α T_decel = T_static − ((J_mass + J_screw) · α · η + J_trans · α · η_t + J_rotor · α) T_dwell = T_gravity, or 0 with the brake holding T_rms = √( ∫ T² dt / t_cycle ) over the whole cycle: the moves, the dwell, the rest

Flat, with μ = 0.1 and η = 0.9, friction alone costs 0.0347 N·m. Accelerating at 3.6 m/s² the motor must give 0.284 N·m, cruising 0.0347 N·m, braking −0.179 N·m, and over the 1 s cycle the RMS is 0.138 N·m. Stand the same axis up vertically and gravity adds 0.347 N·m to every phase and to the dwell: the peak becomes 0.597 N·m and the RMS 0.378 N·m.

The screw's own limits

A long, thin screw whips at its critical speed, and the balls in the nut have a speed limit of their own, given as D·N — the nominal diameter times the screw speed. The default screw's critical speed is 8214 rpm with one fixed and one supported end; the page allows 80 % of it, 6572 rpm, and the screw turns at 3600. Its D·N is 57 600, under the usual 70 000. The critical speed guide works through both.

Choosing the lead

The lead is the screw's gear ratio. Doubling it from 10 to 20 mm halves the screw speed — 1800 rpm instead of 3600, and half the D·N — but every newton on the carriage costs twice the torque, and the moving mass looks four times heavier to the motor, because its inertia goes with p². A short lead suits heavy loads and fine positioning, a long one speed. Step 3 shows what the lead you pick does to the screw, step 4 what it does to the motor.

Worked example

Quantity	By hand	Result
Peak speed	0.2 / (1/12 + 1/6 + 1/12)	0.6 m/s
Peak acceleration	0.6 / (1/6)	3.6 m/s²
Screw and motor speed	0.6 / 0.01 · 60	3600 rpm
Load inertia at the motor	5.07 × 10⁻⁵ + 2.53 × 10⁻⁵	7.59 × 10⁻⁵ kg·m²
Inertia ratio	7.59 × 10⁻⁵ / 2.6 × 10⁻⁵	2.92
Torque accelerating / cruising / braking	see above	0.284 / 0.0347 / −0.179 N·m
RMS torque over 1 s	√((0.284² + 0.0347² + 0.179²) / 6)	0.138 N·m
Peak torque the servo has at 3600 rpm	3.82 · 3000 / 3600	3.18 N·m
Margins: peak, RMS, speed, inertia ratio	(limit − required) / limit	91 %, 89 %, 40 %, 71 %
Braking energy per cycle	0.179 · 377 / 2 · 1/6	5.61 J

The 400 W generic servo: 1.27 N·m rated, 3.82 N·m peak, 3000 rpm rated, 6000 rpm maximum, 2.6 × 10⁻⁵ kg·m² rotor. g = 9.81 m/s².

Questions

What efficiency should I use for a ball screw?

About 0.9 for a ball screw. A trapezoidal lead screw is far lower, 0.3 to 0.5, and some of them hold a vertical load by friction alone. Enter the catalogue figure when you have it.

Does the nut preload add torque?

Yes. A preloaded nut drags with roughly μ₀ · F₀ · p / 2π at the screw, μ₀ about 0.1 to 0.3 by the catalogues, and the page adds it to every phase. Type the preload under “More about the screw”.

How do I size a ball screw for a vertical axis?

Choose Vertical in step 1. Gravity then appears in every phase and in the dwell, where the motor holds the load. With a motor brake switched on in step 4 the brake can hold the dwell instead, and the RMS torque falls.

What inertia ratio is acceptable for a ball screw axis?

Up to 10 for a standard servo is the usual rule. A ball screw on a direct-coupled servo rarely comes near it — the default case is 2.92 — but a heavy load on a long lead can.

What limits the speed of a ball screw?

Two things: the critical speed, from the root diameter, the length between the bearings and the end supports, and the D·N value the nut allows. Step 3 checks both.

Sources

Oriental Motor, Technical Reference, selection calculations — the ball-screw load torque and the inertia of a mass driven by a screw. The page's test suite works its two published ball-screw examples and meets the printed answers within 0.5 %.
Ball-screw catalogues of NSK, THK and Bosch Rexroth — the critical-speed support factors and the D·N limit.

Related guides

Every term on this page, defined: the axis and motor sizing glossary.

Ball screw motor sizing