Ball screw stiffness affects positioning accuracy, machining stability and the response of a linear motion system under axial load. However, the value measured on an assembled axis is not determined by the ball nut alone. The screw shaft, support bearings, bearing housings, nut bracket and test fixture can all contribute to the measured displacement.
For this reason, measuring ball screw axial stiffness requires more than applying a force and recording a single displacement value. The loading direction, preload, mounting condition, temperature and deformation of the test equipment must also be controlled.
What Is Ball Screw Axial Stiffness?
Axial stiffness describes the resistance of a ball screw assembly to elastic displacement under an axial load. Within a specified load range, it can be expressed as:
k = ΔF / Δδ
Where k is axial stiffness, ΔF is the change in axial load, and Δδ is the corresponding change in axial displacement.
Ball screw stiffness is commonly expressed in N/μm or N/mm. A higher value means that less elastic displacement occurs under the same axial load.
The load–displacement relationship of the ball-to-groove contact is not perfectly linear. Therefore, stiffness should normally be reported for a defined load range rather than as one universal value. Depending on the purpose of the test, engineers may use either:
- Secant stiffness: the load change divided by the total displacement change over a specified interval.
- Tangent stiffness: the local slope of the load–displacement curve at a particular operating load.
Components That Affect the Measured Stiffness
When a complete ball screw axis is tested, the measured result represents the combined stiffness of several components:
- The ball screw nut and ball-to-raceway contacts
- Axial extension or compression of the screw shaft
- The fixed-side and supported-side bearing arrangement
- The nut housing and mounting bracket
- The bearing housing and machine structure
- The test fixture, load cell and sensor mounting arrangement
These elements behave approximately like springs connected in series. The combined axial stiffness can be represented by:
1 / ktotal = 1 / knut + 1 / kshaft + 1 / kbearing + 1 / kbracket + …
This explains why an assembled axis may have substantially lower stiffness than the theoretical rigidity value listed for the ball nut alone.
Static Load Testing
Static axial loading is the most direct method for measuring ball screw stiffness. The ball screw is mounted in a test fixture or in its actual machine arrangement, and a controlled axial force is applied while the resulting displacement is recorded.
Equipment Required
- A rigid test frame
- A hydraulic, pneumatic, screw-driven or electromechanical loading device
- A calibrated load cell
- An LVDT, laser displacement sensor or high-resolution dial indicator
- Temperature sensors when thermal effects may influence the result
Recommended Test Procedure
- Mount the ball screw in the intended support arrangement. Record whether it uses fixed–fixed, fixed–supported or fixed–free mounting.
- Align the loading axis. The applied force should be concentric with the screw axis to prevent bending from being included in the displacement reading.
- Install the displacement sensor close to the measured interface. Measuring across unnecessary fixture components will add their deformation to the result.
- Apply a small seating load. This helps settle the contacts and reduces errors caused by incomplete seating or mounting clearance.
- Load the assembly in controlled increments. Record force and displacement at each step without exceeding the specified operating load or permissible axial load.
- Unload the assembly in the same increments. Comparing loading and unloading curves can reveal backlash, hysteresis, friction or movement in the fixture.
- Repeat the test in the opposite axial direction. Bidirectional results are important for evaluating preload and axial clearance.
- Calculate stiffness over the relevant load range. The selected range should correspond to the actual working load of the machine.
How to Interpret the Load–Displacement Curve
A preloaded ball screw should normally produce a smooth load–displacement curve with relatively small displacement near the load reversal point. A visible dead zone around zero load may indicate axial clearance, loose mounting parts or movement between the nut and its bracket.
If the loading and unloading curves do not follow the same path, the difference is hysteresis. Possible causes include seal friction, lubricant resistance, slight misalignment, contact settling or deformation within the mounting structure.
Because stiffness changes with load, the test report should include the complete load–displacement curve whenever possible. At minimum, it should state:
The measured stiffness and unit
The load range used for calculation
The direction of loading
The ball screw model and preload condition
The screw length and nut position
The support bearing arrangement
The test temperature and lubrication condition
Separating Ball Screw Stiffness From Fixture Deformation
A common measurement error is treating the total movement of the test frame as deformation of the ball screw. Even a rigid-looking fixture can deflect under a large axial load.
The compliance of the fixture can be determined through calibration or by testing a reference component with much higher stiffness. The fixture displacement at each load can then be subtracted from the total measured displacement:
δball screw = δmeasured − δfixture
For an assembled machine axis, sensors can also be positioned at different interfaces to identify where displacement occurs. For example, measuring between the screw shaft and nut housing focuses more closely on the nut assembly, while measuring table movement relative to the machine base includes the entire feed structure.
Dynamic Stiffness Testing
Static testing evaluates deformation under slowly applied loads. Dynamic testing is used when vibration, resonance and servo response are important. The assembly may be excited using an impact hammer, shaker or controlled motor input while force, acceleration or displacement is measured.
Modal analysis can identify natural frequencies, damping and mode shapes. However, natural frequency alone does not provide the stiffness of the ball screw unless the effective moving mass and boundary conditions are also known. For a simplified single-degree-of-freedom system:
k = m(2πfn)²
Here, m is the effective mass and fn is the natural frequency. Actual ball screw feed systems usually have several vibration modes, so dynamic results should be interpreted using an appropriate system model.
Analytical Estimation
Ball screw stiffness may also be estimated before physical testing. The calculation can include Hertzian contact deformation between the balls and raceways, axial deformation of the screw shaft, support bearing rigidity and deformation of the mounting structure.
Finite element analysis can be useful for evaluating complex housings and brackets, but its accuracy depends on realistic contact conditions, material properties, preload and boundary conditions. Analytical values should therefore be verified against test data when positioning accuracy is critical.
Factors That Must Be Controlled During Testing
Preload and Axial Clearance
Preload reduces axial clearance and increases nut rigidity by maintaining contact between the balls and raceways. However, excessive preload increases friction, heat generation and internal load, which may shorten service life. The preload condition used during testing should match the intended operating condition.
Screw Length and Nut Position
The screw shaft stretches or compresses under axial load. Its contribution to total displacement increases with the effective shaft length and decreases as the root diameter increases. Consequently, stiffness measured with the nut near one end of the screw may differ from stiffness measured at another position.
Support Bearings
The type, arrangement and preload of the support bearings directly affect system stiffness. A fixed–fixed arrangement generally behaves differently from fixed–supported or fixed–free mounting. Bearing housing rigidity and shaft-end fit must also be considered.
DLY supplies matching ball screw support units for different shaft-end and mounting arrangements.
Temperature
Temperature changes can alter preload, screw length and bearing fit. Allow the assembly and measuring equipment to reach a stable temperature before testing. The temperature should also be recorded when comparing measurements taken at different times.
Lubrication
Lubrication has a greater direct influence on friction, temperature and hysteresis than on the basic elastic stiffness of the steel components. To obtain repeatable results, use the specified lubricant and keep the lubricant type, amount and distribution consistent between tests.
Conclusion
The most practical way to measure ball screw axial stiffness is to apply a controlled axial load and record the corresponding displacement in both directions. Accurate results require careful alignment, calibrated load and displacement sensors, a clearly defined load range and compensation for fixture deformation.
When the complete feed axis is measured, the result represents the combined stiffness of the ball nut, screw shaft, support bearings, brackets and machine structure. These components should be evaluated separately when the measured stiffness is lower than expected.
For ball screw selection, preload options or shaft-end machining requirements, view DLY's ball screw range or send us your screw diameter, lead, stroke, load and mounting arrangement.
