Assembly Simulation Considering Thermal Deformation
Theory: expand with heat, grip as it cools
Overview
A shrink-fitted gear slipped on its shaft in service, though the interference was as calculated. Why?
A shrink fit expands the bore with heat during assembly and grips as it cools. But the same expansion acts in service: if the gear runs hotter than the shaft, from mesh friction for example, only its bore grows and interference shrinks. Interference is only tens of micrometres, so a few tens of kelvin can almost erase it. Dissimilar materials are worse, loosening from expansion mismatch even at uniform temperature. Shrink-fit design therefore uses thermal analysis to get operating temperature distributions, then structural analysis to check remaining interference and contact pressure.
Pressure, torque and assembly temperature
For a solid shaft and hub of the same material. E is Young's modulus (steel 206 GPa), δ diametral interference, D shaft diameter (50 mm), D_o hub outer diameter (100 mm), L fit length (40 mm), μ friction coefficient (0.15), c assembly clearance and α expansion coefficient (steel 11.5×10⁻⁶ /K).
Interference loss from temperature
Railway wheels and gun barrels
Shrink fitting is ancient: heating an iron tyre and fitting it to a wooden wheel was blacksmith's work. In the 19th century gun barrels were built of several shrink-fitted layers to leave compressive stress inside and resist bursting. Railway wheel tyres were long shrink-fitted too, and accidents where brake heat expanded tyres until they loosened were known, so the link between temperature and interference has long been a safety issue.
Worked examples
Example 1: steel-on-steel shrink fit (shaft 50 mm, hub OD 100 mm, length 40 mm)
| Interference | Contact pressure | Torque capacity | Hub bore hoop stress |
|---|---|---|---|
| 0.02 mm | 30.9 MPa | 728 N·m | 52 MPa |
| 0.03 mm | 46.3 MPa | 1,092 N·m | 77 MPa |
| 0.05 mm | 77.2 MPa | 1,820 N·m | 129 MPa |
Assembling 0.03 mm interference with 0.03 mm clearance requires heating the hub about 104 K. Cooling the shaft in liquid nitrogen (20 to −196 °C) shrinks it about 0.097 mm.
Example 2: interference lost to operating temperature
| Condition | Interference lost | Against 0.03 mm interference |
|---|---|---|
| Steel/steel, hub 50 K hotter | 0.029 mm | nearly all |
| Al hub/steel shaft, uniform +50 K | 0.029 mm | nearly all |
| Al hub/steel shaft, uniform +100 K | 0.058 mm | comes loose |
An aluminium hub (70 GPa) on a steel shaft at 0.03 mm interference gives only 18.8 MPa, about 40% of steel/steel.
If 50 K nearly loosens it, why not just use more interference?
More interference raises pressure but also raises stress around the bore proportionally: at 0.05 mm the bore hoop stress reaches 129 MPa, and in thin hubs or gears with bores near the tooth roots, operating stresses add on and cause fatigue or cracks. Required heating temperature also rises, running into limits such as tempering of the material. Interference is therefore chosen so that torque is still transmitted in the worst loosening case (hub hottest, shaft coldest, centrifugal expansion of the hub) and stresses stay acceptable in the tightest case in assembly and operation. Finding that window is the job of coupled thermal and structural analysis.
Assembly simulation workflow
- Bound interference by hand from required torque and allowable stress.
- Check assembly heating/cooling temperatures and times by thermal analysis.
- Compute operating temperature distributions (heating, cooling, ambient).
- Map temperatures to structural contact analysis for remaining interference and pressure.
- Add interference losses from centrifugal expansion, tolerances and surface roughness.
“A coupling that slipped only just after start-up”
A coupling in one installation left slight slip marks on its shaft only for a while after start-up. At steady operating temperature interference had margin and calculations showed no issue. Transient thermal analysis revealed that just after start-up the coupling's outside warmed first from friction while the shaft was still cold, briefly creating a hub–shaft difference near 40 K, at which interference nearly vanished. Revising the minimum interference and ramping load gradually at start-up solved it.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Assembly temperature only | Loosens in service | Couple operating temperatures |
| Steady state only | Slip at start-up | Check transients |
| Ignoring dissimilar expansion | Loosens even uniformly | Use each material's α |
| Excessive interference | Hub cracking | Limit by stress |
| Ignoring centrifugal expansion | Loosens at speed | Include rotation |
I'd like to learn related topics.
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