Ball Screws Calculator — L10h Life, Critical Speed & Buckling (ISO 3408)

Ball screw life is calculated by the rolling-element fatigue model of ISO 3408-5: L10 = (C_a / F_A)^3 × 10^6 revolutions, where C_a is the dynamic axial load rating and F_A is the equivalent axial load on the most heavily loaded ball track. For a preloaded double-nut assembly the external load Fa is shared between two opposed ball tracks according to Hertzian contact theory (δ ∝ F^(2/3)). The loaded track carries F_A = F_pr·(1+u)^1.5 and the opposing (unloaded) track carries F_B = F_pr·(1−u)^1.5, where u is found by bisection from the relation Fa/F_pr = (1+u)^1.5 − (1−u)^1.5. Once the opposing track lifts off (at Fa ≥ 2^1.5·F_pr ≈ 2.83·F_pr), only the loaded track carries load and F_A = Fa.

Using the single-track equivalent load F_A (rather than a fixed 0.33·F_pr heuristic) means the life of a preloaded screw is never overstated at light external loads. Life in hours is L10h = L10 / (60 × n), where n is the rotational speed in RPM.

The critical (whirling) speed of a ball screw shaft is the rotational speed at which the first lateral bending resonance occurs. It depends on the root (minor) diameter and the unsupported span length, and is lowered by the end-fixity condition. MechanixCalc uses the standard industry formula n_c = f × d_root / L² × 10^7 (rpm, mm) with fixity factors f = 21.9 for fixed-fixed, 15.1 for fixed-floating and 3.4 for fixed-free mounting — the values published by THK, NSK and Rexroth. The speed ratio n / n_c is displayed as a percentage; operation above 80% of critical speed triggers a warning.

Column buckling is checked by Euler's formula applied to the root diameter. The effective length factor follows the same end-fixity conditions: fixed-fixed uses a factor of 0.5 (Fcr = 4π²EI/L²), fixed-floating uses 0.707 (Fcr = 2π²EI/L²) and fixed-free uses 2 (Fcr = π²EI/(4L²)). The buckling ratio Fa/Fcr must remain well below 1.0; the calculator warns at 50% of the buckling load.

The torque the drive motor must supply equals the total drive force (external axial load plus preload drag) multiplied by the lead and divided by the screw efficiency. The preload drag is included as an allowance of 0.33 × F_pr added to Fa — this ensures the torque and drive-force outputs are mutually consistent (drive force = torque × 2π / lead). Heat generation is the power lost in the screw: P_loss = ω × T × (1 − η), where ω is the angular velocity in rad/s, T is the drive torque and η is the fractional efficiency.