Four-Pocket Hydrostatic Bearing Suite Calculator

Screen four-pocket hydrostatic journal-bearing stiffness and flow, or size an orifice-fed annular collar bearing.

Engineering screening tool—verify inputs and results independently. Use at your own risk.

Journal geometry and supply

Journal screening result

Load at ε = 0.5—N

Equations used

Journal pressure input is Δp=p_s−p_a. The positive radial load shown opposes an eccentricity e=0.5c. Collar pressures are absolute; positive axial load acts away from the bearing face.

Journal I: Aₑ/A = 0.9(1−Lₐ/L)[π/4−(Lₜ/L)(L/D)]

Journal I: C_f=(2/3){[π/(4L/D)−Lₜ/L]/(Lₐ/L)+(1−Lₐ/L)/(Lₜ/L)}

Journal II: γ=4(Lₐ/L)(1−Lₐ/L)(L/D)/[π(Lₜ/L)]; Aₑ/A=0.45√2(1−Lₐ/L); C_f=π/[6(L/D)(Lₐ/L)]

Capillary: S′=6(Aₑ/A)β(1−β)/[1+γ(1−β)]

Orifice: S′=12(Aₑ/A)β(1−β)/[2−β+2γ(1−β)]; fixed flow: S′=6(Aₑ/A)/(1+γ), β=1

c=(c/R)D/2; S=S′(L/D)D²Δp/c; F=S(0.5c); Q=c³ΔpβC_f/η; P=QΔp

Collar: pᵣ=pₐ+β(pₛ−pₐ); R₁ₒ=R₁+δ/2; R₁ᵢ=R₁−δ/2

F₀=π(pᵣ−pₐ){(R₀²−R₁ₒ²)/ln(R₀/R₁ₒ)−2R₁ₒ²}/2

F₁=π(R₁ₒ²−R₁ᵢ²)(pᵣ−pₐ); F₂=−π(pᵣ−pₐ){(R₁ᵢ²−R₂²)/ln(R₁ᵢ/R₂)−2R₁ᵢ²}/2; F=F₀+F₁+F₂

C_F0=π/[6ln(R₀/R₁ₒ)]; C_F2=π/[6ln(R₁ᵢ/R₂)]; Q=h₀³(pᵣ−pₐ)(C_F0+C_F2)/η

Aₒ=(Q/Cᴅ)√[ρ/{2(pₛ−pᵣ)}]; d=√[4Aₒ/(πnₒ)]; F′=F/[π(R₀²−R₂²)(pₛ−pₐ)]; S′=6(1−β)F′/(2−β)

Symbols and units. D and L are journal diameter and length; Lₐ and Lₜ are axial and tangential ridge dimensions; c is one-sided radial clearance; β is recess pressure factor; η is Pa·s. R₀, R₁ and R₂ are collar radii; δ is groove width; h₀ is land film; ρ is kg/m³; Cᴅ and nₒ describe equal orifices. SI units are used internally.

Assumptions and limits

Both models assume steady, laminar, isoviscous, incompressible flow, rigid aligned surfaces, and negligible inertia, turbulence, cavitation, thermal gradients, deformation, edge leakage, supply-line loss, and dynamics. Journal load is a linearized ε=0.5 result, not a full eccentric pressure solution. The collar requires R₀>R₁ₒ>R₁ᵢ>R₂>0, uniform h₀, and a deep groove with negligible pressure drop. Do not apply the orifice relation to compressible gas or use either model for contact, stability, transient response, or safety qualification.

How to use

Select Journal bearing or Collar bearing. Journal mode also requires configuration I/II and capillary, orifice, or fixed-flow compensation.
For the journal, enter D in mm, geometry ratios, c/R, pressure difference in bar, β, and η in Pa·s. Configuration II includes pocket cross-flow; fixed flow sets β=1.
For the collar, enter the three diameters and groove width/depth in mm, h₀ in µm, absolute pₛ and pₐ in MPa, β, η, density, orifice count, and Cᴅ.
Select Calculate. Correct any red geometry, range, or pressure warning before interpreting results.
Read the single primary load result, then stiffness, flow, hydraulic power, and mode-specific dimensionless metrics. The journal load applies only at ε=0.5.
Use Reset example to restore the active mode’s demonstration values and recalculate.
Repeat with manufacturing extremes for clearance/film, viscosity, pressure, and dimensions. Flow scales with the cube of clearance or film thickness.
Do not use these relations when compressibility, thermal/elastic distortion, turbulence, cavitation, rotation-driven hydrodynamics, groove pressure loss, contact, or transient stability dominates.

Worked example

1. In Journal mode use configuration I, capillary, D=60 mm, L/D=1, Lₐ/L=Lₜ/L=0.25, c/R=0.001, Δp=50 bar, β=0.5, and η=0.04 Pa·s. 2. Calculate. Expected load is 4,878.816 N, stiffness 325.254 MN/m, flow 20.8235 L/h, and ideal hydraulic power 28.921 W. 3. This is the ideal linearized load at e=0.5c.

Collar check. Reset Collar mode to 40/29/21 mm diameters, δ=1 mm, h₂=1 mm, h₀=5 µm, pₛ=0.5 MPa abs, pₐ=0.1 MPa abs, β=0.6, η=0.02 Pa·s, ρ=870 kg/m³, three orifices and Cᴅ=0.7. Expected load is 116.957 N, flow 19.6567 mL/h, and each orifice diameter 0.01314 mm.

Theory and method

A hydrostatic bearing carries load with externally supplied pressure rather than requiring relative motion. In the four-pocket journal model, ridge geometry determines effective pressure area and flow conductance. Compensation controls how recess pressure changes with displacement. Configuration I treats pockets as circumferentially isolated; configuration II allows cross-flow from a higher-pressure pocket to a lower-pressure pocket, which reduces the predicted stiffness and supply flow.

The collar model solves axisymmetric radial Poiseuille flow on each side of a deep annular groove. Integrating the logarithmic pressure fields gives outer- and inner-land load contributions; the groove itself contributes uniform recess pressure over its area. Flow balance and the incompressible discharge relation then give the required equal-orifice diameter. Because Q scales with h₀³ and journal Q with c³, realistic tolerance and viscosity sweeps are essential.

Applicability. Use the suite for preliminary liquid-lubricated geometry, feed, stiffness, and power screening. Refine recess pressure distribution, restrictor network losses, groove pressure drop, thermal viscosity, elastic deformation, rotor dynamics, manufacturing tolerances, and stability before design release.

References

A. van Beek, Advanced Engineering Design: Lifetime Performance and Reliability, bearing calculation material at tribology-abc.com.
B. J. Hamrock, S. R. Schmid, and B. O. Jacobson, Fundamentals of Fluid Film Lubrication, 2nd ed., doi:10.1201/9780203021187.

Continue in TriboSolver

Use TriboSolver to refine pressure fields, restrictor networks, thermal and elastic effects, and coupled bearing behavior beyond these screening relations.