Thermal growth

Thermal expansion formula

ΔL = α × L × ΔT

ΔL

Expansion (mm)

α

Coefficient of linear expansion (1/°C)

L

Length (mm)

ΔT

Temperature change (°C)

NSK writes ΔLθ = ρ·θ·L, and THK writes Δℓ = ρ·Δt·ℓ (only the symbols differ; it is the same formula).^[1][2]

Coefficient of linear expansion of ball screw shafts and guide values for expansion

^[1][2]

Worked example (NSK's example)

For a screw shaft length of 1,300 mm and a temperature rise of 3 °C, ΔL = 12.0×10⁻⁶ × 3 × 1,300 ≈ 0.047 mm. In the example, pretension (tensile) is applied for this amount, and the target value for the reference travel is set to −0.047/1,300 mm.^[1](1 source, for reference)

Countermeasures for ball screw temperature rise

Guide values for pretension and the target value differ by source. NSK says pretension equivalent to a 2–3 °C temperature rise is normal, and THK says it is common to allow for a 2–5 °C rise and set the reference travel to −0.02 to −0.06 mm/m. In NSK's table, the target values for an NC lathe are X axis −0.02 to −0.05 and Z axis −0.02 to −0.03 (per 1 m).^[1][2]

Causes of machine tool thermal displacement

Machining accuracy varies greatly with "temperature changes around the machine," "heat from the machine," and "heat from machining." Spindle thermal displacement changes greatly not only while it rotates but also when it stops, and it is also affected by machining that changes speed frequently and by the cutting fluid temperature. For structural thermal displacement, how it shows up in the machined dimensions depends on the workpiece position on the table for machining centers and on the workpiece size for lathes.^[3](1 source, for reference)

Examples of machine tool makers' thermal displacement countermeasures

Okuma makes its structures expand and contract "cleanly" with heat (without twisting or tilting) by using a thermally symmetric (left-right symmetric) design, a stacked-box structure, and an arrangement that equalizes heat transfer paths. It then estimates thermal displacement from the spindle temperature, rotation state, temperature sensors at various machine parts, and axis positions, and compensates through the control. This is an introduction to the maker's own technology, not a general rule.^[3](1 source, for reference)

Reduction of interference fit due to bearing temperature difference (reference)

Δdt = (0.10〜0.15) × Δt × α × d ≒ 0.0015 × Δt × d × 10⁻³

Δdt

Reduction of interference fit (mm)

Δt

Temperature difference between bearing interior and housing surroundings (°C)

α

Coefficient of linear expansion of bearing steel ≈ 12.5×10⁻⁶ /°C

d

Bearing nominal bore diameter (mm)

^[4](1 source, for reference)

Coefficients of linear expansion (representative values used in maintenance)

The value for "steel" varies by source, from 10.2 to 12.5 × 10⁻⁶. The calculator lets you choose the material and shows the source's value as is.^{[1][2][4][5][6]}