Sizing a servo motor for a ball screw axis comes down to three numbers: maximum speed, peak torque while accelerating, and RMS torque over a cycle. This guide follows the formulas and selection examples in THK's general catalogue «Ball Screw» (515-1E). The numbers were recalculated with PortDock's calculator and agree with the catalogue within 1 %.
1. Axial load Fa
The force along the screw. Horizontally it is guide friction plus the force to accelerate; vertically the weight is added (THK equations 18–29, α = maximum speed ÷ acceleration time).
Horizontal: accel Fa1 = μ·m·g + f + m·α, constant Fa2 = μ·m·g + f, decel Fa3 = μ·m·g + f − m·α Vertical (up): Fa = m·g + f (± m·α) Vertical (down): Fa = m·g − f (∓ m·α) THK horizontal example: m 80 kg, μ 0.003, f 15 N, 1 m/s, 0.15 s → α 6.67 m/s² Fa1 = 550 N, Fa2 = 17 N, Fa3 = −516 N THK vertical example: m 50 kg, f 20 N, 0.3 m/s, 0.2 s → α 1.5 m/s² up 585 · 510 · 435 N, down 395 · 470 · 545 N
A horizontal axis carries very little at constant speed (17 N) and a lot while accelerating or braking; a vertical axis always carries its weight (about 490 N), even at constant speed.
2. Motor speed and lead
NM = V (m/s) × 1000 × 60 ÷ lead Ph (mm) ÷ A (A = screw turns ÷ motor turns, 1 for direct drive) 1 m/s, 20 mm lead → 3,000 min⁻¹ 1 m/s, 40 mm lead → 1,500 min⁻¹ 0.3 m/s, 10 mm lead → 1,800 min⁻¹
THK's horizontal example works out that a 3,000 min⁻¹ rated motor needs a lead of at least 20 mm for 1 m/s, then, after checking critical speed and DN value, chooses a 20 mm shaft with a 40 mm lead. The same formula gives the encoder resolution needed: 0.02 mm minimum feed on a 20 mm lead is 20 ÷ 0.02 = 1,000 p/rev.
3. Constant-speed torque
T1 = Fa × Ph ÷ (2π × η) × A (N·mm, η ball screw efficiency 0.9–0.95) Horizontal example: 17 × 40 ÷ (2π × 0.9) = 120 N·mm Vertical example (up): 510 × 10 ÷ (2π × 0.9) = 900 N·mm Vertical example (down): 470 × 10 ÷ (2π × 0.9) = 830 N·mm
Turned around, it gives the thrust a torque can deliver: Fa = 2π × η × T ÷ Ph. 1 N·m (1000 N·mm) with a 10 mm lead and 0.9 efficiency is about 565 N. For a preloaded screw add the preload torque, and add support bearing and oil seal friction too (THK equation 45).
4. Load inertia and acceleration torque
J = m × (Ph ÷ 2π)² × A² × 10⁻⁶ + JS × A² (kg·m²) ω′ = 2π × NM ÷ (60 × t1) (rad/s²) T3 = (J + Jm) × ω′ (Jm motor inertia) Horizontal example: J = 80 × (40 ÷ 2π)² × 10⁻⁶ + 1.48 × 10⁻⁴ = 3.39 × 10⁻³ kg·m² ω′ = 2π × 1500 ÷ (60 × 0.15) = 1,047 rad/s² T3 = (3.39 + 1) × 10⁻³ × 1,047 ≈ 4.60 N·m
Screw shaft inertia JS is the dimension table's inertia per unit length (1.23 × 10⁻³ kg·cm²/mm in the example) times shaft length (1,200 mm). Doubling the lead quadruples the mass term — a longer lead lowers motor speed but can raise the torque needed to accelerate the same mass.
5. Accel, constant and decel torque, and peak torque
| Phase | Horizontal example (N·mm) | Vertical example up / down (N·mm) |
|---|---|---|
| Accelerating | constant + T3 = 4,720 (catalogue 4,730) | 1,098 (1,100) / 636 (630) |
| Constant speed | 123 (120) | 902 (900) / 832 (830) |
| Decelerating | constant − T3 = −4,474 (−4,490) | 707 (700) / 1,028 (1,030) |
On the horizontal axis the deceleration torque is negative (the motor brakes). On the vertical axis the weight tries to turn the screw on the way down, so T3 is subtracted while accelerating down and added while decelerating — stopping on the way down (1,030 N·mm) takes more than stopping on the way up (700). The largest value (4,730 horizontal, 1,100 N·mm vertical) must be within the motor's instantaneous maximum torque.
6. RMS torque — compare with rated torque
Trms = √(Σ T² × t ÷ Σ t) Horizontal example: accel 0.15 s, constant 0.85 s, decel 0.15 s, dwell 2.6 s √((4730² × 0.15 + 120² × 0.85 + 4490² × 0.15 + 0) ÷ 3.75) = 1,305 N·mm Vertical example: up 0.2·1.8·0.2 s, down 0.2·1.8·0.2 s, dwell 7.6 s (holding 40 kg without the work, 658 N·mm) = 743 N·mm
The RMS torque must be within the motor's rated torque. An axis with large acceleration torque and short dwell, like the horizontal example, has a high RMS torque; on a vertical axis the holding torque at rest counts too (a brake holding the load removes that part).
7. Inertia ratio
THK's examples state the inertia at the motor (3.39 × 10⁻³ horizontal, 1.58 × 10⁻⁴ kg·m² vertical) and add that the motor normally needs an inertia of at least one tenth of that, though the value varies by motor maker. Use the allowable inertia ratio from your servo motor and drive manual.
In PortDock
- «Ball Screw Torque & Motor Sizing Calculator» web tool: mass, friction, speed, acceleration time, lead, gear ratio and inertia → axial load, motor speed, accel/constant/decel torque, holding torque (vertical), peak and RMS torque. The defaults are THK's horizontal example.
- «Encoder Pulses ↔ RPM Calculator» web tool: servo encoder resolution, quadrature and speed versus pulse frequency.
- Modbus Master: poll a Modbus-capable servo drive's monitor registers (load ratio, effective torque, speed and so on — addresses from the drive manual) to put the calculated values next to the running values.
Published 2026-10-24