Aerostatic Shallow-Pocket and Slim Linear Bearing Calculator

Equations used

All pressures are absolute and all calculations use SI units internally. R₀, R₁, and R₂ are the bearing, pocket, and inlet radii; h₀ is land film thickness and h₂ is added pocket depth.

Definitions: pᵢ and pₒ are the inner- and outer-boundary absolute pressures for each radial region, so p(Rᵢ) = pᵢ and p(Rₒ) = pₒ; pᵣ is pocket-edge pressure (Pa); β and F* are dimensionless. In shallow-pocket mode, the displayed pᵣ/pₐ uses the 0.1% perturbed clearance from the stiffness step; β, F*, load, and flow remain evaluated at the entered clearance. F is positive compressive load (N); µ is dynamic viscosity (Pa·s); R = 287 J/(kg·K) for air; T is absolute temperature (K); ṁ is mass flow (kg/s); Q is ambient volume flow (m³/s); and S = −dF/dh is inherent stiffness (N/m). The flow gap h_f is h₂ for shallow-pocket mode and h₀ for slim mode, matching each published model.

Assumptions and limits

The method assumes steady, isothermal, laminar, axisymmetric ideal-gas flow, parallel rigid surfaces, constant viscosity, and uniform absolute boundary pressures. The two radial integrals use 100 midpoint annuli. It omits inlet inertia, choking, edge leakage, groove-detail effects, deformation, waviness, roughness, contact, thermal gradients, and dynamic stability. Geometry must satisfy 0 < R₂ < R₁ < R₀. It is unsuitable for compressible-flow choking, noncircular pads, contact operation, transient motion, or safety-critical bearing qualification.

Reference

Anton van Beek, Advanced Engineering Design: Lifetime Performance and Reliability, aerostatic bearing design chapter and design charts.

How to use

1. Select Shallow pocket for entered 2R₁, 2R₂, and h₂ geometry, or Slim profile for 2R₁ = 0.8(2R₀), 2R₂ = 1 mm, and h₂ = 8 µm.
2. Enter bearing diameter in mm. In shallow mode, enter pocket and inlet diameters in mm so 0 < 2R₂ < 2R₁ < 2R₀, plus positive pocket depth in µm.
3. Enter positive film thickness h₀ in µm. It represents uniform parallel clearance and must remain large enough to prevent contact under deflection and flatness errors.
4. Enter absolute supply and ambient pressures in MPa; pₛ must exceed pₐ. Do not enter gauge pressure.
5. Enter dynamic air viscosity in Pa·s and absolute temperature in K. The defaults represent the model’s nominal dry-air condition.
6. Select Calculate to update load, stiffness, flow, pressure metrics, and the film-thickness sensitivity plot. Correct any red warning before interpreting results.
7. Read F as predicted compressive capacity above ambient, S as local clearance sensitivity, and Q as volume flow referenced to ambient pressure. F* and β support dimensionless comparison.
8. Select Reset example to restore the shallow-pocket demonstration. Do not apply the model when surfaces contact, geometry is noncircular, inlet inertia/choking is important, or structural/thermal distortion controls clearance.

Worked example

1. Select shallow pocket. 2. Enter 2R₀ = 40 mm, 2R₁ = 32 mm, 2R₂ = 1 mm, h₂ = 5 µm, h₀ = 5 µm, pₛ = 0.5 MPa absolute, pₐ = 0.1 MPa absolute, µ = 18×10⁻⁶ Pa·s, and T = 293 K. 3. Calculate. Expected results are 240.7 N load, 24.0 MN/m stiffness, 0.399 L/min ambient flow, F* = 0.479, and β = 0.507. The local stiffness indicates that a 1 µm clearance increase would reduce load by about 24 N near this point; verify deformation and supply capacity separately.

Theory and method

Pressurized air enters at R₂, spreads through the recess to R₁, and then flows across the land to ambient at R₀. Isothermal compressible Reynolds flow makes pressure-squared vary linearly with logarithmic radius in each constant-gap region. Continuity between regions gives Πᵣ. Numerical radial integration of pressure above ambient gives load. A 0.1% film perturbation gives local inherent stiffness. The model preserves separate shallow-pocket and slim-profile flow-gap conventions and their public display rounding.

Larger pockets generally improve potential stiffness and self-alignment, while a small inlet can improve self-alignment but may add unmodeled inertia losses. A deeper recess raises load potential, whereas a shallow recess contributes stiffness. Smaller running films tend to increase stiffness and reduce flow but demand tighter flatness and deflection control; use detailed design charts and measured performance for final selection.