BIZOMAT 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. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Nominal screw diameter
- What it is
- The nominal size of the ball screw, the diameter of the circle through the ball centres. It sets the lead angle used in the preload torque estimate.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 300 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 32 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Used only in the lead angle for preload drag torque; stiffness comes from the root diameter.
- Get it right
- Use the catalogue nominal diameter, not the outside diameter of the thread lands or the root diameter.
What this means: Screw root diameter
- What it is
- The diameter at the bottom of the screw thread, the solid core that actually stretches and compresses under axial load.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 300 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 27 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Screw stiffness rises with the square of the root diameter.
- Get it right
- Take the root diameter from the catalogue; using the nominal diameter overstates stiffness.
What this means: Lead
- What it is
- How far the nut travels along the screw for one revolution of the screw.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A longer lead raises the preload drag torque.
- Get it right
- Lead equals pitch only for a single-start screw; multi-start screws have a lead of starts times pitch.
What this means: Thrust bearing layout
- What it is
- Whether the axial thrust is taken by bearings at both ends of the screw or at one end only. Fixing both ends shares the load between two lengths of screw.
- Units and format
- Choose one of 2 options from the list: Thrust bearings at both ends (fixed–fixed); Thrust bearing at one end (fixed–free or fixed–supported).
- Example
- The form opens with “Thrust bearings at both ends (fixed–fixed)” selected as a worked example. Choose the option that matches your case.
- Effect on the result
- Thrust at both ends gives four times the screw stiffness at mid-span.
- Get it right
- A floating support bearing that carries no thrust still counts as one-end thrust.
What this means: Span (between thrust bearings, or thrust bearin…
- What it is
- The length of screw that stretches under load: between the thrust bearings for both-end mounting, or from the thrust bearing to the farthest nut position for one end.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 10000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1000 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Screw stiffness is inversely proportional to this length.
- Get it right
- For one-end mounting use the nut’s farthest travel position, where the axis is least stiff.
What this means: Elastic modulus
- What it is
- The stiffness of the screw steel in tension and compression, from the material data or ball screw catalogue.
- Units and format
- Enter a number in GPa: gigapascals (1 GPa = 1000 MPa). The form accepts 0.000001 GPa to 300 GPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 206 GPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Screw stiffness is directly proportional to the modulus.
- Get it right
- Enter GPa, not MPa; 1 GPa is 1000 MPa.
What this means: Nut stiffness (catalogue, at the preload)
- What it is
- The axial stiffness of the ball nut at the preload chosen, from the ball screw catalogue.
- Units and format
- Enter a number in N/µm: N/µm. The form accepts 0.000001 N/µm to 100000 N/µm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 600 N/µm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Acts in series with the screw and bearings; the weakest member dominates the axis stiffness.
- Get it right
- Use the value for the preload actually ordered; catalogues list nut stiffness at a stated preload.
What this means: Support bearing stiffness
- What it is
- The combined axial stiffness of the support bearings and their housing, which act in series with the screw and nut.
- Units and format
- Enter a number in N/µm: N/µm. The form accepts 0.000001 N/µm to 100000 N/µm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1000 N/µm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Acts in series; a low bearing stiffness limits the whole axis.
- Get it right
- Include housing and mounting compliance if known; the bearing catalogue figure alone overstates it.
What this means: Largest axial load
- What it is
- The largest axial force on the screw in service: cutting or process force plus the force to accelerate the moving mass.
- Units and format
- Enter a number in N: newtons. The form accepts 0.000001 N to 1000000 N. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 3000 N as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Deflection and the least preload needed rise in direct proportion; stiffness does not change.
- Get it right
- Include acceleration force; at fast rapids it can exceed the cutting force.
What this means: Allowable elastic deflection
- What it is
- The elastic movement of the axis under the largest load that you can accept for the accuracy you need.
- Units and format
- Enter a number in µm: micrometres (µm, microns). The form accepts 0.000001 µm to 1000 µm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20 µm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- It does not change the deflection, only the pass or fail check.
- Get it right
- This is your accuracy budget for elastic deflection only, not total positioning accuracy.
What this means: Nut preload
- What it is
- The internal force set between the two halves of the preloaded nut to remove backlash and raise stiffness.
- Units and format
- Enter a number in N: newtons. The form accepts 0 N to 1000000 N. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1200 N as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Preload drag torque and the share of the dynamic rating rise in proportion.
- Get it right
- Enter the preload force in newtons, not as a catalogue preload class or a percentage.
What this means: Dynamic load rating
- What it is
- The basic dynamic load rating of the ball nut from the catalogue, used to express the preload as a share.
- Units and format
- Enter a number in kN: kilonewtons (1 kN = 1000 N). The form accepts 0.000001 kN to 10000 kN. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 30 kN as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A higher rating lowers the preload share; stiffness does not change.
- Get it right
- Enter kN, not N; catalogue tables often list it in newtons.
What this means: Largest preload as a share of the dynamic rating
- What it is
- The highest preload you allow, as a share of the dynamic load rating, to limit heat and loss of life.
- Units and format
- Enter a number in %: percent: the number is read as a percentage, so type 8 for 8 %. The form accepts 0.000001% to 20%. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10% as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- It does not change any calculated value, only the preload level check.
- Get it right
- Take the limit from the ball screw maker; it is your design rule, not a calculated value.
What this means: Moving mass (table, workpiece, fixture)
- What it is
- The total mass the axis moves: table, workpiece, fixtures and any moving parts of the nut and carriage.
- Units and format
- Enter a number in kg: kilograms. The form accepts 0.000001 kg to 100000 kg. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 200 kg as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The axial natural frequency falls with the square root of mass.
- Get it right
- Use the heaviest load the axis will carry, since it gives the lowest frequency.
What this means: Least axial natural frequency wanted
- What it is
- The lowest axial natural frequency you need for good servo tuning and response, from the control or machine builder.
- Units and format
- Enter a number in Hz: hertz (cycles per second). The form accepts 0 Hz to 1000 Hz. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 50 Hz as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- It does not change the frequency, only the check.
- Get it right
- Zero switches the check off in practice, since any positive frequency passes.
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