KEVOS® MAKEOS · Power Transmission
Ball-Screw Axis Stiffness, Preload and Axial Frequency
How stiff is this ball-screw axis, how far will it deflect under load, is the nut preload right, and what is the axial natural frequency?
Axial stiffness of a ball-screw feed axis: screw shaft stiffness for thrust bearings at one end or both ends, combined in series with the nut and support bearing stiffnesses, deflection under the largest axial load, the preload needed so the preloaded nut does not unload (load ÷ 2.83), preload drag torque, preload as a share of the dynamic rating and the axial natural frequency of the moving mass on the axis stiffness.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Nominal screw diameter | mm | Nominal (ball circle) diameter of the screw. |
| Screw root diameter | mm | Root diameter of the screw thread. |
| Lead | mm | Lead of the screw. |
| Thrust bearing layout | choice | Whether thrust is taken at one end only or at both ends. |
| Span (between thrust bearings, or thrust bearing to farthest nut position) | mm | For both ends: distance between the thrust bearings (the screw is weakest with the nut at mid-span). For one end: distance from the thrust bearing to the farthest nut position. |
| Elastic modulus | GPa | Elastic modulus of the screw steel. |
| Nut stiffness (catalogue, at the preload) | N/µm | Axial stiffness of the ball nut at its preload (from the catalogue). |
| Support bearing stiffness | N/µm | Combined axial stiffness of the support bearings and housing. |
| Largest axial load | N | Largest axial load in service (cutting plus acceleration). |
| Allowable elastic deflection | µm | Elastic deflection you accept at the largest load. |
| Nut preload | N | Preload applied to the nut. |
| Dynamic load rating | kN | Dynamic load rating of the ball nut. |
| Largest preload as a share of the dynamic rating | % | Upper limit on preload as a share of the dynamic rating (your design value). |
| Moving mass (table, workpiece, fixture) | kg | Mass moved by the axis. |
| Least axial natural frequency wanted | Hz | Least axial natural frequency you need for the servo tuning. |
Method and formulas
- Screw stiffness k_s = A E ÷ L (one end), or 4 A E ÷ L at mid-span (both ends), with A = π d_r² ÷ 4.
- Axis stiffness 1 ÷ k = 1 ÷ k_s + 1 ÷ k_n + 1 ÷ k_b; deflection = F ÷ k.
- A preloaded double nut unloads at about 2.83 F_p, so F_p ≥ F ÷ 2.83.
- Preload drag torque T_p = K F_p l ÷ 2π with K = 0.05 ÷ √(tan β) and tan β = l ÷ (π d₀); axial frequency f = √(k ÷ m) ÷ 2π.
- Example: the axis has 208.93 N/µm of stiffness and deflects 14.359 µm under 3000 N; the axial frequency is 162.67 Hz.
Limitations
- Static axial stiffness only; torsional wind-up of the screw adds lag in dynamic moves.
- The preload torque coefficient is the usual catalogue estimate.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check a machine axis for positioning stiffness
- Choose a preload class for a ball nut
- Compare one-end and both-end thrust bearing layouts
Who it is for
- Machine designers
- Automation engineers
- Maintenance engineers
What you get
- Screw, nut and bearing stiffness combined
- Deflection at the largest load
- Preload checks, drag torque and axial frequency
Interpreting the result
- Thrust bearings at both ends quadruple the weakest screw stiffness; the nut and bearings then often govern.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 2561–2568. Topic reference; no source tables or recommended settings are reproduced.
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Open the Ball-Screw Axis Stiffness, Preload and Axial Frequency calculator
