Ball Screw Torque & Motor Sizing Calculator
Ball screw axial load, motor speed, accel and steady torque, load inertia and RMS torque from mass, speed and lead. Horizontal or vertical, THK formula.
Formulas (THK catalogue)
- Horizontal: Fa = μ·m·g + f ± m·α (+ accelerating forward, − decelerating; signs flip going back)
- Vertical: going up Fa = m·g + f ± m·α, going down Fa = m·g − f ∓ m·α
- Constant-speed torque T1 = Fa × Ph ÷ (2π × η) × A (A = 1 ÷ gear ratio)
- Motor speed NM = V × 1000 × 60 ÷ Ph ÷ A
- J = m × (Ph ÷ 2π)² × A² × 10⁻⁶ + JS × A², ω′ = 2π × NM ÷ (60 × t1)
- Acceleration torque T3 = (J + Jm) × ω′, accelerating = constant + T3, decelerating = constant − T3
- RMS torque Trms = √(Σ T² × t ÷ Σ t)
THK selection examples with this calculator (high-speed transfer / vertical conveyance)
| Item | Horizontal example | Vertical example | Catalogue (horizontal / vertical) |
|---|---|---|---|
| Max axial load Famax | 551 N | 585 N | 550 / 585 N |
| Motor speed | 1,500 min⁻¹ | 1,800 min⁻¹ | 1,500 / 1,800 |
| Load inertia at motor J | 3.39×10⁻³ kg·m² | 1.58×10⁻⁴ kg·m² | 3.39×10⁻³ / 1.58×10⁻⁴ |
| Uniform-speed torque | 123 N·mm | 902 / 832 N·mm | 120 / 900·830 |
| Torque while accelerating (peak) | 4,720 N·mm | 1,098 N·mm | 4,730 / 1,100 |
| RMS torque Trms | 1,302 N·mm | 744 N·mm | 1,305 / 743 |
Horizontal: 80 kg, μ 0.003, f 15 N, 1 m/s, 0.15 s acceleration, 40 mm lead, screw 1.476 kg·cm², motor 1×10⁻³ kg·m², cycle 0.15·0.85·0.15·2.6 s. Vertical: 50 kg (40 kg at rest), f 20 N, 0.3 m/s, 0.2 s acceleration, 10 mm lead, screw 0.312 kg·cm², motor 5×10⁻⁵ kg·m², cycle 0.2·1.8·0.2 s up and down + 7.6 s dwell. The catalogue rounds g and intermediate values (angular acceleration 1,047 → 1,050 and so on), so results differ by less than 1 %.
How to read it
- The peak torque (usually while accelerating) must be within the servo motor's instantaneous maximum torque, and the RMS torque within its rated torque (the criteria in THK's examples).
- On a vertical axis the weight tries to turn the screw on the way down, so stopping on the way down (decelerating) needs more torque, and holding at rest needs torque too. That is why brake motors are used.
- Recommended load-to-motor inertia ratios differ by motor maker. THK's example says the motor inertia normally needs to be at least one tenth of the load inertia, adding that the value varies by manufacturer.
- A longer lead lowers motor speed for the same travel speed, but raises the torque for the same force and the load inertia (lead²).
- Efficiency 0.9 is the low end of THK's 0.9–0.95 for ball screws. Support bearing and oil seal friction (other torque) is not included, so leave margin.
FAQ
What is the ball screw torque formula?
Axial load Fa (N) × lead Ph (mm) ÷ (2π × efficiency η) gives the constant-speed torque in N·mm. THK's vertical example: 510 N × 10 mm ÷ (2π × 0.9) ≈ 900 N·mm (0.9 N·m). While accelerating, add the inertia torque (J + Jm) × angular acceleration.
How do I get ball screw thrust from torque?
Rearranged: thrust Fa = 2π × η × T ÷ Ph. 1 N·m (1000 N·mm) with a 10 mm lead and efficiency 0.9 gives about 565 N.
What motor speed do I need?
NM = speed (m/s) × 60 000 ÷ lead (mm). 1 m/s with a 20 mm lead is 3,000 min⁻¹; with 40 mm it is 1,500 min⁻¹.
Why is a vertical axis different?
Going up the screw carries weight + resistance; going down, weight − resistance. Decelerating on the way down, weight and inertia act together and torque rises (1,030 N·mm in THK's vertical example).
To convert motor speed to drive frequency, use the VFD Hz ↔ RPM Calculator. For encoder pulses and rpm, use the Encoder Pulses ↔ RPM Calculator. To follow the formulas and THK's examples step by step, see the guide Ball screw torque and servo motor sizing. If a timing belt reduces between motor and screw, use the Timing Belt Length & Center Distance Calculator to pick the belt.
Basis
Formulas are from THK's general catalogue «Ball Screw» (515-1E) A15 «Calculating the Axial Load» (equations 18–29), «Considering the Rotational Torque» (45–49) and «Considering the Driving Motor» (52, 53); the two examples are B15 «Example Ball Screw Selections» (PDFs on THK's technical site). Tests check that this calculator reproduces their values within 1 %. g = 9.806 65 m/s². For screw and motor inertia, permissible speed, buckling load and other checks, see each catalogue.