Ball Screw Axis Sizing: Motor Torque, Screw Limits and Parts

Motion

A trapezoidal move: accelerate to the speed (a triangle when the stroke is too short), run, stop, pause; then the same back. That gives THK's six phases (A15-46), each with its travel and time.

Loads on the nut

(THK A15-46; the incline as Oriental Motor H-3).

Screw limits (THK 515-1E)

• Critical speed $N_1 = \lambda_2\, d_1/\ell^2 \cdot 10^7$ min⁻¹, λ₂ = 15.1 fixed–floating, 3.4 fixed–free (formula 7, factor 0.8 included); ℓ is the distance between the bearings, or from the fixed bearing to the free end.
• Nut speed $N_2 = 70\,000/D$ with D the ball-centre diameter (A15-33; HIWIN M33).
• Buckling $P_1 = 20\,d_1^4/\ell^2 \cdot 10^4$ N, fixed–fixed between the guided nut and the fixed bearing as THK and HIWIN take it, ℓ the distance between the bearings (formula 5, factor 0.5); root section $P_2 = 116\,d_1^2$ (formula 6).
• Static safety $f_S = C_{0a}/F_{a,max}$ (formula 30).
• Life $L_{10} = (C_a/(f_W F_m))^3\cdot 10^6$ revolutions with $F_m = \left(\sum F_a^3 \ell / \sum \ell\right)^{1/3}$ of the larger direction (formulas 32–35, A15-50); hours at the mean speed of the whole cycle, pauses included.

Motor

• Torque in each phase $T = F_u P_h/(2\pi\eta_1) + T_o + J\,\dot\omega$, with $J$ = rotor + screw (a steel cylinder) + coupling + $m(P_h/2\pi)^2$ (THK 45–49; like THK and Oriental Motor, the inertia is outside the efficiency). At rest the load turns the screw with $F P_h \eta_2/2\pi$.
• The largest torque × the safety factor must stay under the lowest point of the motor's curve from standstill up to full speed. The curve ends at its last published point: a faster screw fails. The picture shows the whole path and, while it plays, where the drive is now.
• Servos: the RMS torque of the cycle, and any torque held for long (running at constant speed, holding at rest), must stay under the continuous torque — the rated torque, at most 2/3 of the instantaneous curve as in JMC's charts. Steppers run at constant current, so they have no RMS check.
• Inertia ratio (screw + coupling + load over rotor) at most 30 for closed-loop steppers (Oriental Motor, αSTEP) and 10 for servos (THK). Where the rotor inertia is not published, the torque checks say so instead of passing.
• A vertical or inclined axis needs a motor with a brake (or a counterweight): a ball screw does not hold its load.

Length and parts

Threaded length = stroke + nut + free travel at both ends; overall length adds the machined ends. The calc takes the shortest Dold length that is long enough, the bearing units of the screw's size, and the coupling of the chosen type whose bores fit the motor shaft and the screw journal and that takes the motor's largest torque (its curve or its holding torque). A length Dold does not sell fails the check.

Combinations

The same checks run for every Dold screw with every motor (with your screw length for your screw); the table lists the cheapest that meet all of them.

• Load ratings of the screws are not Dold's. Dold states no Ca, C0a, root or ball-centre diameter. The calc uses those of nuts with the same DIN 69051 dimensions (TBI SFNU) and of HIWIN rolled screws of the same size and lead; the table names the source of each. SFU1204-3 has none, so its life and static safety are not checked.
• Motor curves are read from the plotted points in the JMC catalogue (±0.01 N·m) at the supply it names; a brake variant has the curve of the motor without brake. The rotor inertia of the IHSVC57 servos is not published.
• Only axial loads on the nut: moments and side loads go to the guides. The support bearings are not checked (no ratings published), nor resonance or the stiffness of the axis.
• Buckling uses the distance between the bearings, a little more than the longest span from the nut to the fixed bearing (safe side).
• The coupling's inertia is 0 where it is not stated (Dold RB, XB).
• Prices as listed by Dold incl. 19 % VAT, from snapshots of 2025-11 to 2026-10; check them in the shop.

One axis with a rolled ball screw of 12–20 mm in Dold's standard machining, driven directly by the motor (no belt or gear), from horizontal to vertical. Smooth moves: the load factor and the static safety factor stand for vibration and impact (THK: f_W 1–1.2 up to 0.25 m/s, 1.2–1.5 up to 1 m/s; f_S 2 without, 5 with impacts). THK's factors 0.8 (critical speed) and 0.5 (buckling) are in the limits. Positioning accuracy (P7: 52 µm per 300 mm), axial play and thermal elongation are shown, not checked against a target.