How to Calculate Ball Screw Axial Stiffness and Positioning Error Under Load

A ball screw may have adequate load capacity and a sufficiently high accuracy grade but still produce measurable positioning error under axial load. This happens because the screw shaft, ball nut, support bearings and mounting structure all deform elastically when force passes through the feed system.

Ball screw axial stiffness describes how strongly the complete system resists this deformation. It is commonly expressed in N/μm or N/mm. A higher stiffness means less elastic displacement under the same axial force.

This article explains how to calculate screw-shaft stiffness, combine the stiffness of multiple components and estimate the resulting positioning error. It also explains why the nut position and end-support arrangement must be included in the calculation.

Axial Stiffness and Elastic Displacement

For a component operating within its elastic range, axial stiffness can be expressed as:

K = F ÷ δ

The corresponding elastic displacement is:

δ = F ÷ K

Where:

• K = axial stiffness
• F = applied axial force
• δ = elastic axial displacement

If stiffness is entered in N/μm and force is entered in newtons, the resulting displacement is in micrometres.

For example, a component with a stiffness of 200 N/μm subjected to a 2,000 N axial force has an ideal elastic displacement of:

δ = 2,000 ÷ 200 = 10 μm

This displacement is elastic: it should substantially recover when the load is removed, provided the component remains within its permissible load range. It is different from permanent deformation caused by overload.

The Complete Feed System Determines Axial Stiffness

The axial force in a ball screw axis normally passes through several deformable components:

1. Screw shaft
2. Ball nut and ball-to-raceway contacts
3. Fixed-end support bearings
4. Nut bracket and bearing housings
5. Machine frame and moving table

If these components are treated as springs connected in series, their individual elastic displacements add together:

δ_system = δ_screw + δ_nut + δ_bearing + δ_mount

The equivalent system stiffness is:

1/K_system = 1/K_screw + 1/K_nut + 1/K_bearing + 1/K_mount

This relationship shows why installing one very stiff component does not automatically create a stiff axis. The component with the lowest stiffness can dominate the total deformation.

How to Calculate Screw-Shaft Axial Stiffness

The axial stiffness of a steel screw-shaft section is calculated from:

K_screw = A × E ÷ L

Where:

• A = effective shaft cross-sectional area in mm²
• E = Young's modulus of the shaft material in N/mm²
• L = effective loaded length in mm
• K_screw = shaft stiffness in N/mm

For steel, a commonly used value is:

E ≈ 206,000 N/mm²

To express the result in N/μm:

K (N/μm) = K (N/mm) ÷ 1,000

Use the Root Diameter, Not the Nominal Diameter

A ball screw is not a solid cylinder with a diameter equal to its nominal model size. The helical raceway reduces the effective cross-sectional area. For an initial calculation, the thread minor diameter or root diameter should therefore be used.

The approximate effective cross-sectional area is:

A = πd_r² ÷ 4

Where:

• d_r = screw-shaft root or thread minor diameter
• A = effective cross-sectional area

Using the nominal outside diameter will overestimate the screw-shaft stiffness. The root diameter should come from the manufacturer's drawing or technical data whenever possible.

Fixed-Supported Ball Screw Stiffness

In a fixed-supported arrangement, the fixed-end bearing normally establishes the axial position of the screw. The supported end mainly provides radial support and accommodates axial expansion.

For axial stiffness, the effective loaded length is generally the distance between the fixed-end bearing reference position and the ball nut:

K_screw = A × E ÷ L_n

Here, L_n changes as the nut moves. Therefore, screw-shaft stiffness is not constant over the stroke.

When the nut is close to the fixed bearing, the loaded shaft section is short and the stiffness is high. As the nut moves farther away, the loaded length increases and the stiffness decreases.

This position-dependent deformation can create a systematic positioning difference along the axis, particularly on long vertical axes or machines carrying a substantial process load.

Fixed-Fixed Ball Screw Stiffness

In an ideal fixed-fixed arrangement, both ends provide axial restraint. The nut load can be resisted by the shaft sections on both sides of the nut.

If:

• L₁ = distance from the nut to the first fixed bearing
• L₂ = distance from the nut to the second fixed bearing

the two shaft sections act approximately as parallel axial springs:

K_screw = AE/L₁ + AE/L₂

This can also be written as:

K_screw = AE(L₁ + L₂) ÷ (L₁L₂)

The formula assumes that both ends provide effective axial restraint and that the bearings, housings and machine structure are sufficiently rigid. A system that only appears fixed at both ends mechanically may not behave as an ideal fixed-fixed arrangement.

Fixed-fixed mounting can improve axial rigidity and critical-speed performance, but bearing preload, thermal expansion and assembly accuracy require more careful control.

Ball Nut Axial Stiffness

Ball nut stiffness comes mainly from the elastic contact between the balls and raceways. It cannot be calculated accurately from the nut's outside dimensions alone.

The nut manufacturer's catalogue or test data should be used because ball nut stiffness depends on:

• Ball diameter
• Number of loaded balls
• Raceway geometry
• Contact angle
• Single-nut or double-nut structure
• Preload level
• Applied axial load

Ball-to-raceway contact is not perfectly linear. The stiffness value may change with preload and applied load. A catalogue stiffness value should therefore be used under the conditions defined by the manufacturer rather than treated as a constant valid at every load.

Preload can reduce axial clearance and increase nut rigidity, but excessive preload also increases running torque, heat generation and internal fatigue loading.

For the structural difference between nut types, see Single-Nut vs Double-Nut Ball Screws.

Support Bearing Axial Stiffness

The fixed-end bearing set transfers axial force from the screw shaft into the bearing housing and machine structure. Its deformation must be included in a high-accuracy feed system.

Support-bearing stiffness depends on:

• Bearing type and size
• Contact angle
• Number and arrangement of bearings
• Bearing preload
• Fit between the bearing, shaft and housing
• Locknut installation

A generic bearing stiffness should not be assigned based only on the support-unit code. Use the bearing manufacturer's axial rigidity data or obtain the value from the support-unit supplier.

The fixed-end bearing and housing must also be installed correctly. A nominally rigid bearing set can still move if the locknut, shoulder, housing or mounting bolts are not properly controlled.

Nut Bracket and Mounting Structure Stiffness

Catalogue ball nut stiffness normally describes the ball screw nut itself. It does not automatically include the deformation of:

• Nut housing
• Nut mounting flange
• Connecting bolts
• Moving table
• Fixed-end bearing housing
• Machine base

A thin nut bracket or flexible mounting plate may contribute more deformation than the ball screw itself. This is why the measured stiffness of the assembled axis is often lower than a value calculated from the screw and nut data alone.

When structural stiffness is unknown, it can be estimated by finite element analysis or measured by applying a known axial force and recording displacement at the working point.

Worked Example: Complete Axial Stiffness Calculation

The following example illustrates the calculation method. The values are engineering examples rather than specifications for a particular DLY ball screw model.

• Axial working force: 2,000 N
• Screw-shaft root diameter: 20 mm
• Distance from fixed bearing to nut: 800 mm
• Steel modulus: 206,000 N/mm²
• Ball nut stiffness: 300 N/μm
• Support-bearing stiffness: 200 N/μm
• Mounting-structure stiffness: 500 N/μm

Step 1: Calculate the Screw-Shaft Area

A = π × 20² ÷ 4

A ≈ 314.16 mm²

Step 2: Calculate Screw-Shaft Stiffness

K_screw = 314.16 × 206,000 ÷ 800

K_screw ≈ 80,896 N/mm

K_screw ≈ 80.9 N/μm

Step 3: Calculate Each Component's Displacement

Step 4: Add the Elastic Displacements

δ_system = 24.72 + 6.67 + 10.00 + 4.00

δ_system ≈ 45.39 μm

Step 5: Calculate Equivalent System Stiffness

1/K_system = 1/80.9 + 1/300 + 1/200 + 1/500

K_system ≈ 44.1 N/μm

The result can be verified:

δ = 2,000 ÷ 44.1 ≈ 45.35 μm

The small difference is caused by rounding. Both methods show that the complete system deforms by approximately 45 μm under the example load.

Although the ball nut itself has a stiffness of 300 N/μm, the complete feed system stiffness is only about 44 N/μm. In this example, the long screw-shaft section is the largest source of elastic displacement.

Position-Dependent Error Along the Stroke

For a fixed-supported ball screw, screw-shaft stiffness decreases as the nut moves away from the fixed bearing. Under a constant axial force:

δ_screw = F × L ÷ (A × E)

Because displacement is proportional to the loaded length, the same force produces different elastic errors at different positions.

Using the screw shaft from the previous example:

The difference in shaft deformation between the 200 mm and 800 mm positions is approximately:

24.72 − 6.18 = 18.54 μm

This difference can appear as a position-dependent mechanical error when the axial working force remains in the same direction.

Elastic Error Is Not Backlash or Lead Error

Several different errors may affect a ball screw axis, but they should not be combined without understanding their direction and behaviour.

A high-resolution encoder cannot eliminate these mechanical effects unless the feedback system measures the final table position and the controller can compensate for the error.

For the distinction between commanded resolution and actual mechanical accuracy, see Ball Screw Positioning Resolution Calculation.

Thermal Expansion Can Exceed Elastic Error

Axial stiffness calculation does not include thermal growth. For an unconstrained steel screw shaft, thermal length change can be estimated as:

ΔL = α × ΔT × L

Where:

• α = thermal expansion coefficient
• ΔT = temperature change
• L = effective shaft length

A commonly used thermal expansion coefficient for steel is approximately:

α ≈ 12 × 10⁻⁶/°C

A one-metre screw rising by 1°C can therefore lengthen by approximately 12 μm. Thermal displacement must be evaluated separately from load-induced elastic deformation.

How to Improve Ball Screw System Stiffness

Increase the Screw Root Diameter

Screw-shaft area is proportional to the square of the root diameter. A larger root diameter can substantially improve shaft stiffness, although it also increases rotating inertia and required installation space.

Reduce the Effective Loaded Length

A shorter distance between the fixed bearing and nut increases shaft stiffness directly. Axis layout and fixed-end position should therefore be considered early in the machine design.

Select an Appropriate Nut and Preload

A suitably preloaded nut can improve reversal response and contact stiffness. The preload should be selected according to the real axial load, accuracy requirement, operating speed and duty cycle.

Increase Support-Bearing Rigidity

Use a fixed-end bearing arrangement with adequate axial rigidity and ensure that the bearing preload, locknut, shaft shoulder and housing fit are correctly controlled.

Strengthen the Nut Bracket and Machine Structure

Increasing ball screw stiffness provides limited benefit if the nut housing or bearing mounting plate remains flexible. Reinforcing the complete load path may be more effective than changing only the screw specification.

Consider Fixed-Fixed Mounting Carefully

A correctly designed fixed-fixed system can improve shaft rigidity, especially on long or high-speed axes. However, it requires appropriate bearing arrangements and careful control of thermal expansion and assembly preload.

Practical Calculation Checklist

1. Determine the maximum and normal axial working forces.
2. Obtain the screw-shaft root diameter from the actual drawing.
3. Confirm whether the end arrangement is fixed-supported or fixed-fixed.
4. Calculate shaft stiffness at the most important nut positions.
5. Obtain ball nut stiffness under the relevant preload and load conditions.
6. Obtain axial stiffness data for the fixed-end bearing set.
7. Estimate or measure nut-bracket and housing stiffness.
8. Combine the component stiffness values using the series-stiffness equation.
9. Calculate elastic displacement under each operating load.
10. Evaluate backlash, lead error and thermal displacement separately.
11. Verify the assembled axis using suitable force and displacement measurements.

Conclusion

Ball screw axial stiffness is a system property rather than a single catalogue number. The screw shaft, ball nut, support bearings and mounting structure all deform under axial force, and their displacements combine to produce the final positioning error.

For fixed-supported assemblies, screw-shaft stiffness changes with the distance between the fixed bearing and nut. For fixed-fixed assemblies, the shaft sections on both sides of the nut can contribute to stiffness when both ends provide effective axial restraint.

A reliable calculation should use the screw root diameter, actual nut position, manufacturer-provided nut and bearing stiffness data, and the rigidity of the complete mounting structure. Lead accuracy, backlash and thermal expansion should then be evaluated separately.

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