A bearing coefficient needs an operating-point passport.
A stiffness or damping value is not a portable material property. It is a local description of a bearing, at a defined equilibrium state, in a defined coordinate system, produced by a defined method.
1. Three quantities that must not be confused
For a scalar load–displacement relation F(x), three useful quantities answer three different questions:
Secant stiffness
The ratio from the selected origin to the operating point. It can describe an overall load-to-deflection relation, but it is not automatically the local perturbation coefficient.
Tangent stiffness
The local force derivative at equilibrium. This is the linearised stiffness required for small-motion equations around that state.
Dynamic coefficient
A frequency- and speed-conditioned stiffness/damping representation. It may be a matrix and can include cross-coupled terms.
2. A constructed example: the wrong stiffness changes the rotor screen
Consider the deliberately simple nonlinear illustration F(x) = 2.0×10⁶x + 0.8×10¹²x², with x in metres. At x0 = 10 µm, the load is 100 N:
If each value is inserted into the same single-degree-of-freedom support-only illustration with m = 0.64 kg, n = 60/(2π)√(K/m) gives 37,747 rpm from Ksec and 50,643 rpm from Kt. The 34% frequency shift is created by coefficient choice, not by a change in hardware.
3. The operating-point passport
A usable coefficient record should be sufficient for another engineer to decide whether the number can enter a particular model. At minimum, preserve:
| Passport field | What must be recorded | Why it matters |
|---|---|---|
| Identity | Case ID, version, coefficient type and lifecycle state | Prevents a screening value from being mistaken for a released value. |
| Bearing state | Geometry, clearance, restrictor/feed architecture, gas, temperature, supply and ambient pressure with pressure basis | Defines the physical state that produced the coefficient. |
| Equilibrium | Load vector, displacement/eccentricity, attitude and zero-reference method | Defines where the local derivative was taken. |
| Dynamic state | Rotor speed Ω, perturbation frequency ν and motion amplitude | Dynamic K/C values can change with speed and excitation frequency. |
| Coordinates | Axis definitions, signs, direct/cross-coupled matrix terms and any transformation | Prevents a valid number from entering the wrong equation or direction. |
| Method | Analytical derivative, numerical perturbation, identification or test procedure; fit window and solver version | Defines what the number actually represents. |
| Uncertainty | Measurement resolution, repeatability, tolerance/parameter variation and confidence interval where available | Separates nominal precision from supported confidence. |
| Evidence boundary | Benchmark/test anchor, residuals, applicability limits and allowed decision | States what the coefficient is ready to prove. |
4. From a scalar to the rotor model
For two lateral directions, the small-motion bearing force is generally written as a matrix relation around equilibrium. Direct and cross-coupled terms must retain their signs and coordinate basis:
Al-Bender treats stiffness and damping as force derivatives and expresses dynamic coefficients through complex stiffness at a defined steady eccentricity, rotational speed and perturbation frequency. Rowe develops the small-displacement stiffness matrix around equilibrium. Childs likewise introduces linear bearing coefficients through a Taylor expansion about the equilibrium position. Across these sources, the common principle is local linearisation—not a free-floating constant.
AURA handoff rule: the coefficient object, operating-point passport and transformation record enter the rotor workflow together. If the state is missing, the result remains REVIEW or SCREENING ONLY.
5. Manufacturing variation belongs in the same record
The passport is also the natural place to add uncertainty. Vainio and co-authors measured load capacity, air-film stiffness, pressure distribution and flow across porous aerostatic-bearing samples and found meaningful within- and between-manufacturer variation. The engineering implication is not that every coefficient is unreliable; it is that a nominal model value and a manufactured population are different evidence objects.
A mature record can therefore progress from a nominal coefficient to a tolerance-conditioned distribution or envelope, then attach the test evidence used to narrow it. This is the uncertainty-aware extension now being considered for AURA.
6. Source basis
- Al-Bender, F. Air Bearings: Theory, Design and Applications. Wiley, 2021. Chapter 9: linearised stiffness/damping matrices and complex dynamic stiffness.
- Rowe, W. B. Hydrostatic, Aerostatic, and Hybrid Bearing Design. Butterworth-Heinemann, 2012. Chapter 14: small-displacement stiffness coefficients about equilibrium.
- Childs, D. Turbomachinery Rotordynamics: Phenomena, Modeling, and Analysis. Wiley, 1993. Chapter 3: linear bearing coefficients obtained by expansion about equilibrium.
- Vainio, V.; Miettinen, M.; Leutonen, O.; Majuri, J.; Theska, R.; Viitala, R. “Statistical variation of porous aerostatic bearings.” Precision Engineering 96 (2025), 840–850.