Mechatronics, CNC Linear Motion Axes, Precision Ball Screws & Servo Sizing
CNC Ball-Screw Lead, RPM & Servo Motor Torque Formula
Calculate Peak Rapid-Traverse Ball-Screw Speed (RPM), External Thrust Load Torque T_load (N·m), Double-Nut Preload Drag Torque T_pre (N·m), and Continuous Servo Motor Torque (N·m).
High-lead 20 mm ball screws halve rapid RPM to avoid whirling—yet DOUBLE continuous servo motor torque!
Precision recirculating ball screws convert rotary AC servo torque into zero-backlash linear axis motion at ~90%–95% mechanical efficiency eta (vs ~35%–50% on sliding Acme lead screws)! Selecting Ball-Screw Lead p (mm axial travel per revolution) governs an inescapable mechatronic trade-off: Rapid Traverse Speed V_rapid (m/min) requires Screw Shaft Speed N_max = (V_rapid * 1,000) / p RPM (bounded by DN ball-recirculation ratings and shaft whirling), whereas pushing Axial Cutting/Gravity Thrust F_ax (N) through gear reduction i demands Motor Load Torque T_load = (F_ax * (p / 1,000)) / (2 * PI() * eta * i) N·m PLUS Zero-Backlash Nut Preload Drag Torque T_pre = (0.05 * F_preload * (p / 1,000)) / (2 * PI() * i) N·m!
Verify on a 30 m/min CNC Machining-Center X-Axis—Rapid_Traverse V_rapid = 30.0 m/min (B2), Ball_Screw_Lead p = 10.0 mm (B3 -> Peak Screw Speed = 3,000.0 RPM in E2), Axial_Cutting_Thrust F_ax = 4,500.0 N (B4), Ball_Screw_Efficiency eta = 0.90 (B5), Direct_Coupling_Ratio i = 1.00 (B6), and Nut_Preload_Force F_pre = 900.0 N (20% of dynamic load, B7): External Thrust Torque hits 7.958 N·m (E3), Nut Preload Drag adds 0.072 N·m (E4), and Total Servo Holding/Cutting Torque equals 8.029 N·m (E5, matching an 11 N·m continuous AC servo motor)!
| CNC Ball-Screw Axis KPI (V=30 m/min, Lead=10 mm, Fax=4,500 N, Eff=90%, Fpre=900 N) | Formula | Verified Output |
|---|---|---|
| Peak Rapid Screw Speed (RPM) | (Rapid_m_min * 1000) / Screw_Lead_mm | 3,000.0 RPM (1,500.0 @ 20 mm lead) |
| External Thrust Torque Tload (N·m) | (Fax * (Lead/1000)) / (2*PI()*Eff*Gear) | 7.958 N·m (15.915 N·m @ 20 mm lead) |
| Nut Preload Drag Torque Tpre (N·m) | (0.05 * Fpre * (Lead/1000)) / (2*PI()*i) | 0.072 N·m (0.143 N·m @ 20 mm lead) |
| Total Motor Shaft Torque (N·m) | T_load_Nm + T_preload_Nm | 8.029 N·m (16.059 N·m @ 20 mm lead!) |
ISO 3408 Ball-Screw Thrust + Preload Servo Motor Torque Formula
Validate Rapid_Speed > 0, Screw_Lead p > 0, Axial_Thrust >= 0, Efficiency eta in [0.75, 0.98], Gear_Ratio i > 0, and Preload_Force >= 0!
=LET(vr_m, B2, ld_s, B3, fx_n, B4, ef_m, B5, gr_i, B6, fp_n, B7, IF(OR(vr_m <= 0, ld_s <= 0, fx_n < 0, ef_m < 0.75, ef_m > 0.98, gr_i <= 0, fp_n < 0), "Invalid Ball-Screw Input", LET(tl_d, (fx_n * (ld_s / 1000)) / (2 * PI() * ef_m * gr_i), tp_r, (0.05 * fp_n * (ld_s / 1000)) / (2 * PI() * gr_i), ROUND(tl_d + tp_r, 3))))High-Lead 20 mm Anti-Whirling Screw Trade-Off Check (Set Lead p = 20.0 mm)
Switch to a high-helix 20.0 mm lead screw (B3 = 20) to protect a long bed from critical whirling: Peak Screw Speed E2 drops safely in half from 3,000.0 RPM to 1,500.0 RPM, while Total Direct-Drive Servo Torque E5 doubles from 8.029 N·m to 16.059 N·m (requiring a 2:1 timing-belt pulley or larger frame servo)!
=IF(ISTEXT(E5), E5, ROUND((B4 * (B3 / 1000)) / (2 * PI() * B5 * B6), 3))Try the example
Use the original values or change them to test your understanding.