How to Calculate Ball Screw Thermal Expansion and Positioning Error

A ball screw does not remain at exactly the same length while a machine is operating. Heat generated by the ball nut, support bearings, seals and surrounding machine structure raises the screw-shaft temperature, causing the steel shaft to expand.

Even a small temperature rise can create a positioning change larger than the theoretical resolution of the motor and controller. On long or high-speed axes, thermal growth may also be comparable to or greater than the permitted lead error of the selected ball screw.

This article explains how to calculate ball screw thermal expansion, how that expansion becomes positioning error and how the appropriate response changes between fixed-supported and fixed-fixed bearing arrangements.

Ball Screw Thermal Expansion Formula

The free thermal expansion of a screw shaft can be estimated using:

ΔL = α × ΔT × L

Where:

• ΔL = change in screw length
• α = coefficient of linear thermal expansion
• ΔT = change in screw-shaft temperature
• L = effective heated length

For a steel screw shaft, a commonly used coefficient is:

α ≈ 12 × 10⁻⁶/°C

When the screw length is entered in millimetres, the result is also in millimetres. To calculate the expansion directly in micrometres:

ΔL (μm) = 0.012 × ΔT (°C) × L (mm)

This equation estimates the free expansion of a uniformly heated steel shaft. The actual table-position error also depends on the bearing arrangement, feedback location, nut position and temperature distribution.

Worked Example: 1,500 mm Screw with a 5°C Temperature Rise

Consider a steel ball screw with:

• Effective heated length: 1,500 mm
• Initial temperature: 20°C
• Operating temperature: 25°C
• Temperature rise: 5°C
• Thermal expansion coefficient: 12 × 10⁻⁶/°C

Substitute these values into the formula:

ΔL = 12 × 10⁻⁶ × 5 × 1,500

ΔL = 0.09 mm

Convert the result into micrometres:

0.09 mm = 90 μm

A 5°C uniform temperature rise therefore produces approximately 90 μm of free thermal growth over 1,500 mm.

The calculation can be checked using the direct micrometre formula:

ΔL = 0.012 × 5 × 1,500 = 90 μm

Both methods produce the same result.

Thermal Expansion at Different Lengths and Temperatures

The following table shows the estimated free expansion of uniformly heated steel screw shafts.

The table shows that thermal growth increases in direct proportion to both screw length and temperature rise. Doubling the heated length or temperature rise doubles the calculated expansion.

Why a Ball Screw Becomes Hot

The screw shaft is heated by several components and operating conditions rather than by one source alone.

Ball-to-Raceway Contact

Rolling contact is highly efficient, but it is not friction-free. Repeated ball circulation, contact deformation and sliding within parts of the return path generate heat.

Ball Nut Preload

Preload reduces axial clearance and can improve rigidity, but it also increases internal contact force and running torque. Excessive preload can produce unnecessary heat during continuous high-speed operation.

Support Bearings

Fixed-end angular-contact bearings generate heat from preload, rotational speed, lubricant resistance and installation conditions. Part of this heat can enter the screw shaft.

Seals and Lubrication

Seals add friction, while excessive or unsuitable grease can increase churning resistance. Insufficient lubrication can also increase friction and wear. The correct lubricant type and quantity are both important.

Operating Speed and Duty Cycle

Higher rotational speed increases the number of contact cycles per unit of time. Long periods of continuous motion may produce a different thermal condition from intermittent operation, even if the maximum speed is the same.

Heat from the Surrounding Machine

The motor, coupling, spindle, gearbox, cutting process and machine enclosure may change the temperature around the ball screw. The machine bed and measurement system may also expand, so the final positioning error cannot always be predicted from screw temperature alone.

Free Expansion Is Not Always Equal to Positioning Error

The calculated thermal growth describes the change in length of the heated screw section. The actual positioning error at the table depends on where the screw is axially located and where the position is measured.

Important factors include:

• Fixed-supported or fixed-fixed bearing arrangement
• Distance between the fixed bearing and nut
• Temperature distribution along the screw
• Motor encoder or direct linear feedback
• Thermal growth of the machine bed and table
• Controller compensation strategy

For a uniformly heated fixed-supported screw with a rotary encoder on the motor, the approximate uncompensated position change at the nut can be estimated from the heated distance between the fixed bearing and the nut:

Δx ≈ α × ΔT × x

Where:

• Δx = estimated thermal position change at the nut
• x = distance from the fixed axial reference to the nut

This means that the positioning change is normally smaller near the fixed end and larger as the nut moves toward the floating end.

Fixed-Supported Ball Screw Thermal Behavior

In a conventional fixed-supported arrangement:

• The fixed-end bearing determines the axial reference position.
• The supported or floating end mainly provides radial support.
• The screw is allowed to expand away from the fixed end.

This arrangement reduces the risk of generating a large thermally induced axial force because the floating end does not rigidly restrain the screw's length change.

However, allowing the screw to expand does not automatically eliminate positioning error. A motor encoder measures motor or screw rotation, not the thermally shifted table position. The error therefore tends to increase with the nut's distance from the fixed axial reference.

Fixed-Supported Position Example

Assume a uniformly heated screw has a 5°C temperature rise.

At a nut position 300 mm from the fixed end:

Δx = 0.012 × 5 × 300

Δx = 18 μm

At a nut position 1,200 mm from the fixed end:

Δx = 0.012 × 5 × 1,200

Δx = 72 μm

Under the simplified uniform-temperature assumption, the difference in thermal position change between the two locations is approximately 54 μm.

Fixed-Fixed Ball Screw Thermal Behavior

In a fixed-fixed arrangement, both ends provide axial restraint. This can improve shaft rigidity and permissible critical speed, but thermal expansion must be handled more carefully.

If an un-tensioned screw were held by perfectly rigid supports and prevented from expanding, heating would create compressive thermal force rather than free length growth.

For an ideal fully restrained shaft:

F_thermal = A × E × α × ΔT

Where:

• F_thermal = ideal thermally induced axial force
• A = effective screw-shaft cross-sectional area
• E = Young's modulus of the screw material

This equation represents a theoretical upper-bound condition with perfect restraint. Actual bearings, housings and machine structures have finite stiffness, so part of the thermal strain may appear as structural displacement instead of axial force.

Illustrative Thermal Force Example

Assume:

• Screw root diameter: 20 mm
• Effective area: 314.16 mm²
• Young's modulus: 206,000 N/mm²
• Temperature rise: 5°C

The ideal fully restrained thermal force is:

F_thermal = 314.16 × 206,000 × 12 × 10⁻⁶ × 5

F_thermal ≈ 3,883 N

This example shows why a screw should not simply be clamped rigidly at both ends without calculating bearing load, shaft tension and structural compliance.

Ball Nut Preload and Screw-Shaft Pre-Tension Are Different

The terms preload and pre-tension are sometimes used as though they describe the same operation. In a ball screw system, they refer to two different design measures.

A double-nut arrangement creates preload within the ball screw nut assembly. It does not, by itself, pre-tension the complete screw shaft between the support bearings.

Screw-shaft pre-tension requires a suitable fixed-fixed bearing arrangement and a defined assembly procedure. The required tension must be calculated together with support-bearing capacity, shaft stress, expected temperature rise and machine-structure stiffness.

Why a Higher Accuracy Grade Does Not Eliminate Thermal Error

Ball screw lead accuracy describes the difference between actual and specified travel under defined measurement conditions. Thermal expansion is a separate operating effect.

For example, a 1,000 mm steel screw rising by 5°C expands approximately:

0.012 × 5 × 1,000 = 60 μm

Installing a higher-accuracy ball screw does not prevent this thermal growth. The machine must also control temperature, manage the support arrangement or compensate for the resulting position change.

For the distinction between lead error and other sources of positioning error, see What Is Ball Screw Lead Accuracy? C3, C5 and C7 Explained.

Thermal Expansion, Elastic Deformation and Backlash

These three effects can all change the final table position, but they behave differently.

For the relationship between axial load, system stiffness and elastic positioning error, see Ball Screw Axial Stiffness Calculation.

Practical Methods for Reducing Thermal Positioning Error

Control Ball Nut and Bearing Preload

Use enough preload to meet rigidity and backlash requirements, but avoid specifying more preload than the application needs. Higher preload is not automatically better for a high-speed axis.

Use the Correct Lubricant and Quantity

Lubricant viscosity, quantity and replenishment interval should match the screw size, speed, load and environment. Excess grease can increase resistance, while insufficient lubrication can raise friction and accelerate wear.

Reduce Screw Rotational Speed Where Practical

A larger lead can provide the same linear speed at a lower screw rpm. However, changing the lead also affects motor torque, positioning resolution and acceleration, so it must be evaluated as part of the complete axis design.

Allow Warm-Up Before Precision Operation

Some machines perform a repeatable warm-up cycle before high-accuracy production. The purpose is to bring the feed system closer to a stable thermal condition before calibration or machining begins.

Use Air, Oil or Other Cooling Where Necessary

Cooling can reduce temperature rise, but the cooling method must produce a controlled and sufficiently uniform thermal condition. Simply cooling one local area may create a temperature gradient and uneven dimensional change.

Use Temperature-Based Compensation

A controller can apply a position correction based on measured screw or machine temperature. A simple uniform-temperature model may use:

Correction ≈ −α × ΔT × x

The negative sign indicates that the command correction is applied opposite to the predicted thermal growth.

Real machines may require an experimentally identified model because the screw temperature is often not uniform and other structures also expand.

Use Direct Linear Feedback

A linear encoder measures the position of the moving table more directly than a motor-mounted rotary encoder. In a closed-loop control system, it can help correct errors occurring between the motor and final table position.

It does not remove heat or mechanical stress, so temperature control and correct bearing design remain necessary.

Apply Screw-Shaft Pre-Tension When the System Is Designed for It

Pre-tensioning may be used in suitable fixed-fixed high-precision systems. It should be specified as an engineering value rather than applied by arbitrarily tightening the support bearings.

The initial tension, expected temperature rise, bearing preload, support stiffness and permissible screw-shaft stress must be evaluated together.

Why One Temperature Sensor May Not Be Enough

The basic expansion formula assumes that the entire effective length has the same temperature rise. A real screw may have:

• Higher temperature near the ball nut
• Heat entering from the fixed-end bearings
• Different temperatures at opposite ends
• Cooling air affecting only part of the shaft
• Changing heat generation as the nut moves

When the temperature varies along the screw, a more general calculation is:

ΔL = ∫ α × ΔT(x) dx

For practical calculations, the screw can be divided into several sections:

ΔL ≈ α[ΔT₁L₁ + ΔT₂L₂ + ... + ΔT_nL_n]

This sectional approach is more useful when measured temperatures show a clear gradient along a long screw shaft.

Sectional Temperature Example

Assume a 1,500 mm screw is divided into three 500 mm sections with measured temperature rises of:

• Section 1: 3°C
• Section 2: 6°C
• Section 3: 4°C

The estimated total free expansion is:

ΔL = 0.012 × [(3 × 500) + (6 × 500) + (4 × 500)]

ΔL = 0.012 × (1,500 + 3,000 + 2,000)

ΔL = 78 μm

Using only the highest measured temperature rise of 6°C for the entire 1,500 mm length would predict 108 μm, while using a single low-temperature measurement could underestimate the expansion. Multiple measurements can therefore improve the compensation model.

How to Measure Thermal Behavior on the Actual Machine

1. Record the ambient, screw, nut and fixed-bearing temperatures before operation.
2. Run the normal production motion cycle rather than an unrealistic no-load test.
3. Measure temperature at consistent intervals until the system approaches a stable condition.
4. Record table-position error at several locations along the stroke.
5. Repeat the test from a similar initial temperature to check reproducibility.
6. Compare the measured displacement with the theoretical thermal-growth model.
7. Adjust the compensation model using measured machine data where necessary.

The measurement method should match the required accuracy. Precision applications may require a laser interferometer, calibrated linear encoder or another traceable displacement-measurement system.

Practical Design Checklist

1. Define the required positioning accuracy after the machine reaches its operating temperature.
2. Estimate the likely screw-shaft temperature rise under the real duty cycle.
3. Calculate free thermal growth using the effective heated length.
4. Confirm the fixed-supported or fixed-fixed bearing arrangement.
5. Identify the axial reference point and feedback location.
6. Separate nut preload from screw-shaft pre-tension.
7. Check thermally induced bearing and shaft loads in restrained systems.
8. Evaluate lubrication, cooling and warm-up requirements.
9. Determine whether rotary feedback is sufficient or direct linear feedback is required.
10. Verify the model with temperature and position measurements on the assembled machine.

Conclusion

Ball screw thermal expansion can be estimated from the material expansion coefficient, temperature rise and effective heated length. For steel, each metre of screw length expands by approximately 12 μm for every 1°C uniform temperature increase.

The resulting machine-position error is not always equal to the total free expansion. It depends on the nut position, bearing arrangement, feedback location, machine-structure temperature and whether the screw is free to expand or axially restrained.

Fixed-supported systems normally allow expansion away from the fixed end but may develop position-dependent thermal error. Fixed-fixed systems can provide higher rigidity but require the thermally induced change in shaft tension and bearing load to be evaluated.

Reliable compensation therefore begins with calculation but should end with measurement. Temperature control, correct preload, appropriate lubrication, cooling, direct position feedback and a verified compensation model may all contribute to maintaining accuracy after the machine reaches its operating condition.

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