The calculator analyses a hydrostatic journal bearing with Z recesses, axial drainage grooves and capillary restrictors to ISO 12167-1: from geometry, clearance, feed pressure, capillary dimensions and oil viscosity the approximation method of Annex A yields the recess pressures, the eccentricity the journal takes under the load with the minimum film thickness, the stiffness, the oil demand and the pumping and friction power. Capillary and bearing temperatures are estimated per Annex B and iterated with the power ratio.
Capillary restrictors (linear resistance law with inertia correction), 3 to 12 recesses, load on recess or land centre, laminar flow, steady state; optimisation target P* = Pf/Pp ≈ 1 … 3 per clause 6.5.
Define bearing, oil supply and lubricant. The defaults are example B.1 of ISO 12167-1.
The calculator determines the recess pressures, the journal displacement under load, film thickness, stiffness, oil flow and pumping and friction power of a hydrostatic journal bearing with capillary restrictors using the approximation method of ISO 12167-1 Annex A.
Bearing load, speed, diameter D and width B, axial and circumferential land lengths lax and lc, groove width bG, number of recesses Z, recess depth, relative clearance ψ, load direction (recess or land centre), feed pressure pen, capillary diameter and length, oil temperature in front of the bearing and the dynamic viscosity at two temperatures, density and heat capacity of the oil.
For an assumed journal position (ε, β) the pressure of each recess follows from continuity: inflow through the capillary (Hagen-Poiseuille, resistance Rcp with inertia factor a) equals the axial outflow across the lands plus the circumferential outflow minus the circumferential inflow (Poiseuille and Couette shares) – ISO 12167-1 A.3.2. The pressures give the force components per A.3.3; β is set so that the film force acts along the load, ε so that it balances the load. Then hmin = CR·(1 − ε), stiffness c = F/(ε·CR), flow Q = Σ(pen − pi)/Rcp (A.3.4), pumping power Pp = Q·pen, friction power from land and recess shares (A.3.5) and the temperature estimate per B.1.5, iterated with P* = Pf/Pp and the inertia factor.
Example B.1 of ISO 12167-1: D = B = 120 mm, lax = lc = 12 mm, bG = 6 mm, Z = 4, ψ = 1.5 ‰ (CR = 90 µm), capillaries 3.25 mm × 1,140 mm, pen = 6 MPa, ISO VG 46 (41.4/26.58 mPa·s at 40/50 °C), Ten = 45 °C, F = 20 kN, N = 16.66 s⁻¹. The calculator gives κ = 1.416, ξ = 1.01, p₀/pen = 0.50, πf = 0.227, F* = 0.231, ε = 0.350 with hmin = 58.5 µm, Q = 45.1 l/min, Pp = 4,510 W, Pf = 279 W, P* = 0.062, Rep = 695, Recp = 2,098, a = 0.20, TB = 47.8 °C. The standard quotes κ = 1.417, ξ = 1.0006, ε = 0.355 (chart reading, hmin = 58 µm), Q = 45.4 l/min, Pp = 4,542 W, Pf = 281 W, P* = 0.062, Rep = 687, Recp = 2,092, TB = 47.55 °C.
Sources and limits: ISO 12167-1:2023 (clause 6.2 eq. 1–5, 6.3 eq. 6–8, 6.4 eq. 9–10, 6.5 eq. 11–13; Annex A.3.1–A.3.5 approximation method; Annex B.1 worked example as reference test). Limits: capillary restrictors with a linear resistance law (orifices with a square-law characteristic are not included), laminar flow in gap, recess and capillary, equal recesses with pressureless drainage grooves, rigid sliding faces, steady state; the temperatures are estimates per Annex B without heat conduction through the housing. The chart functions of ISO 12167-2 are replaced by the direct continuity calculation.
The calculator determines the recess pressures, the journal displacement under load, film thickness, stiffness, oil flow and pumping and friction power of a hydrostatic journal bearing with capillary restrictors using the approximation method of ISO 12167-1 Annex A.
A hydrostatic bearing carries the shaft with oil that a pump delivers at constant pressure into several recesses – the pressure is generated outside the bearing, not by rotation as in a hydrodynamic bearing. Each recess is connected to the supply through a restrictor (here a capillary) and separated from its neighbours by axial drainage grooves. When the shaft displaces under load, the land on the loaded side narrows: less oil leaks away and the recess pressure rises; on the opposite side it falls. The pressure difference carries the load, and the restrictor is the reason it works – without it all recesses would settle at the supply pressure. ISO 12167-1 describes these relations with characteristic values for load capacity, flow and friction and gives in Annex A an approximation method based on Hagen-Poiseuille and Couette flow, which the calculator executes in full. It belongs to the design step in which number of recesses, land dimensions, clearance, feed pressure and capillary are matched – typical for machine-tool spindles, large bearings with jacking at start-up or test rigs where stiffness and wear-free starting matter.
Hydrostatic bearing design starts with the basic geometry – ISO 12167-2 quotes B/D = 1, Z = 4, lax/B = 0.1, lc/D = 0.1, bG/D = 0.05 for optimised bearings – and with the feed pressure, which sets the load capacity through F* = F/(B·D·pen). Then clearance and capillary are matched so that ξ ≈ 1 and P* lies between 1 and 3. Exactly this matching is what the calculator checks: for every combination it shows recess pressures, film thickness, stiffness, oil demand and power losses, so clearance, capillary length and feed pressure can be iterated in a few steps. Viscosities come from the oil data sheet, if necessary via dynamic/kinematic viscosity conversion. Whether a hydrodynamic bearing would do instead is shown by the hydrodynamic journal bearing to DIN 31652 calculator; for hydrostatically jacked large bearings this calculator gives the start-up state with N = 0.
Geometry: D, B and Z from the design; land lengths and groove width per the ISO 12167-2 guide values or the drawing. Clearance: ψ = 2·CR/D; 1.5 ‰ in the example of the standard, the optimum follows from the power balance. Supply: pump feed pressure, capillary dimensions from the catalogue – long, thin capillaries keep Recp below 1,000 to 1,500. Oil: viscosity at two temperatures, from which the exponential relation of Annex B is formed; ρ·cp ≈ 1.75 MJ/(m³·K) for mineral oil. Temperature: the oil temperature in front of the bearing after cooling.
(pen − pi)/Rcp = Qax,i + Qex,i − Qen,i → pi → F = Σ Fi, c = F/(ε·CR)
Rcp = 128·ηcp·lcp/(π·dcp⁴)·(1 + a), a = 1.08/32·Recp·dcp/lcp (A.3.2.2)Qax,i = CR³·D·αi·pi/(12·ηB·lax), αi = ∫(1 + ε·cos φ)³ dφ over the recess (A.3.2.3)Qen,i = bc·(U·hen/2 − pi·hen³/(12·ηB·lc)), Qex,i = bc·(U·hex/2 + pi·hex³/(12·ηB·lc)) (A.3.2.4)Fi = D·bc·pi·sin(π/Z − φG), F = √(Fh² + Fv²) (A.3.3) · Q = Σ(pen − pi)/Rcp, Pp = Q·pen (A.3.4)Pf = ηB·U²·Alan/(CR·√(1 − ε²)) + 4·ηB·U²·Ap/hp (A.3.5) · P* = Pf/Ppξ = Rcp/RP,0, RP,0 = 6·ηB·lax/(bax·CR³·(1 + κ)), κ = lax·bc/(lc·bax), πf = ηB·ω/(pen·ψ²)Tcp = Ten + ΔTcp/2, TB = Ten + ΔTcp + ΔTB/2 with ΔTcp = pen/(ρ·cp)·ξ/(1 + ξ), ΔTB = pen/(ρ·cp)·(1/(1 + ξ) + P*) (B.1.5)| Symbol / input | Meaning |
|---|---|
| F, N, U | Bearing load, rotational frequency of the shaft and circumferential speed U = ω·D/2. |
| D, B, lax, lc, bG, Z | Bearing diameter, width, axial and circumferential land lengths, groove width and number of recesses; from these the recess length bax = π·D/Z − (lc + bG) and the angle φG = (lc + bG)/D. |
| ψ, CR, hp | Relative clearance, radial clearance CR = ψ·D/2 and recess depth. |
| pen, pi, p₀ | Feed pressure, pressure in recess i and recess pressure with a concentric shaft. |
| Rcp, RP,0, ξ, a | Capillary resistance, recess resistance at ε = 0, restrictor ratio and capillary inertia factor. |
| ε, β, hmin, c | Relative eccentricity, attitude angle, minimum film thickness and stiffness. |
| Q, Q*, Pp, Pf, P* | Oil flow, flow characteristic, pumping power, friction power and their ratio. |
| ηB, ηcp, TB, Tcp | Viscosities and mean oil temperatures in the bearing and in the capillary. |
| Rep, Recp | Reynolds numbers of the secondary flow in the recess and of the capillary flow. |
Bearing load, speed, diameter D and width B, axial and circumferential land lengths lax and lc, groove width bG, number of recesses Z, recess depth, relative clearance ψ, load direction (recess or land centre), feed pressure pen, capillary diameter and length, oil temperature in front of the bearing and the dynamic viscosity at two temperatures, density and heat capacity of the oil.
Enter bearing load and speed, D, B, land lengths, groove width, number of recesses and recess depth, and set clearance and load direction. Give feed pressure, capillary dimensions and oil temperature in front of the bearing, plus the two viscosity points of the oil. After calculating, first check ε and hmin (reserve against overload), then ξ (near 1) and P* (1 to 3). If P* is well below, reduce the clearance or choose a thicker oil; if ξ is too high, shorten the capillary or enlarge its bore. The flow is the design value for the pump, the sum of pumping and friction power the heat input into the oil.
Example B.1 of ISO 12167-1: D = B = 120 mm, lax = lc = 12 mm, bG = 6 mm, Z = 4, ψ = 1.5 ‰ (CR = 90 µm), capillaries 3.25 mm × 1,140 mm, pen = 6 MPa, ISO VG 46 (41.4/26.58 mPa·s at 40/50 °C), Ten = 45 °C, F = 20 kN, N = 16.66 s⁻¹. The calculator gives κ = 1.416, ξ = 1.01, p₀/pen = 0.50, πf = 0.227, F* = 0.231, ε = 0.350 with hmin = 58.5 µm, Q = 45.1 l/min, Pp = 4,510 W, Pf = 279 W, P* = 0.062, Rep = 695, Recp = 2,098, a = 0.20, TB = 47.8 °C. The standard quotes κ = 1.417, ξ = 1.0006, ε = 0.355 (chart reading, hmin = 58 µm), Q = 45.4 l/min, Pp = 4,542 W, Pf = 281 W, P* = 0.062, Rep = 687, Recp = 2,092, TB = 47.55 °C.
The minimum film thickness follows directly from the eccentricity; hydrostatic bearings are designed for ε ≤ 0.5, usually 0.4, so that reserve remains under overload. The stiffness c = F/e is the actual design quantity: it is maximal when the capillary resistance roughly equals the recess resistance (ξ ≈ 1), i.e. when the recess pressure with a concentric shaft is half the feed pressure. A ξ well below 1 wastes stiffness, a ξ well above 1 wastes load capacity. Oil flow and pumping power grow with the cube of the clearance, friction power falls with it – their ratio P* = Pf/Pp should lie at 1 to 3 per clause 6.5 for minimum total loss; the example of the standard sits far off at P* = 0.06 and recommends halving the clearance from 1.5 to 0.75 ‰. The Reynolds numbers in capillary and recess must stay laminar, otherwise the resistance law and friction model no longer hold.
Force in N, speed in min⁻¹, lengths in mm, clearance in ‰, feed pressure in MPa, viscosity in mPa·s, temperatures in °C, heat capacity in MJ/(m³·K); outputs in µm, N/µm, MPa, l/min and W. Internally the core works in SI units (m, Pa, Pa·s, s⁻¹).
The capillary acts as a resistance in series with the recess: with a concentric shaft the feed pressure divides in the ratio of the resistances, p₀/pen = 1/(1 + ξ). When the shaft displaces, the resistance of the loaded recess rises and its pressure approaches pen; the opposite recess unloads towards zero. The maximum possible pressure swing between the two recesses is largest at ξ = 1, which is why that ratio gives the highest stiffness. ISO 12167-1 assumes ξ = 1 as the standard case; the calculator determines the actual ξ from capillary dimensions, viscosity and geometry and warns when it lies outside 0.5 to 2.
Shaft rotation superimposes a hydrodynamic share on the hydrostatic pressure build-up: the Couette flow across the circumferential lands carries oil from the recess with the wider gap into the recess with the narrower one, raises the pressure there and rotates the displacement by the angle β. The ratio of the two shares is described by the relative friction pressure πf or the parameter Krot = ξ·κ·πf·lc/D; in the example of the standard Krot = 0.03 and the influence is negligible, at high speeds and tight clearance it becomes noticeable.
Spindle bearings of machine and measuring tools, bearings of large rotors with hydrostatic jacking at start-up, test-rig and telescope bearings, slow-running heavy-duty bearings without hydrodynamic pressure build-up; matching feed pressure, clearance and capillary for stiffness and minimum power loss; sizing pump and cooler.
ISO 12167-1:2023 (clause 6.2 eq. 1–5, 6.3 eq. 6–8, 6.4 eq. 9–10, 6.5 eq. 11–13; Annex A.3.1–A.3.5 approximation method; Annex B.1 worked example as reference test). Limits: capillary restrictors with a linear resistance law (orifices with a square-law characteristic are not included), laminar flow in gap, recess and capillary, equal recesses with pressureless drainage grooves, rigid sliding faces, steady state; the temperatures are estimates per Annex B without heat conduction through the housing. The chart functions of ISO 12167-2 are replaced by the direct continuity calculation.
Common mistake: Do not confuse the land lengths with the recess length – lax and lc are the narrow sealing lands, the recess is the rest. Enter the clearance as relative clearance ψ = 2·CR/D, not as a diametral clearance in millimetres. Do not choose the capillary too short and thick: Recp then becomes turbulent, the inertia factor large and the linear restrictor law invalid. Do not design for maximum load capacity but for ε ≈ 0.4 with ξ ≈ 1 – stiffness is the real goal. And take the pumping power seriously: it grows with the cube of the clearance and is often the largest loss.
The capillary has a linear resistance law (Hagen-Poiseuille), the orifice a square law. ISO 12167-1 assumes capillaries with ξ = 1 in its characteristic values; orifices give somewhat higher stiffness for the same design but are more sensitive to contamination and viscosity changes. The calculator models capillaries only and corrects their resistance for the inertia share of the inlet.
ISO 12167-2 provides characteristic functions as charts for fixed geometry ratios; the calculator replaces them by the direct per-recess continuity calculation of Annex A, which holds for arbitrary land and recess dimensions. In the example of the standard it reads ε = 0.355 from the chart, the calculation gives 0.350.
Up to about ε = 0.5 the load capacity is nearly linear in the displacement, and the bearing behaves like a linear spring with stiffness c. Above that the load capacity keeps rising, but the film becomes thin and the reserve against overload and form errors disappears.
P* is the ratio of friction to pumping power. Pumping power grows with CR³, friction power falls with 1/CR; the sum has a minimum that lies at P* ≈ 1 to 3 per ISO 12167-1 clause 6.5. A very small P* as in the example of the standard indicates too much clearance or too thin an oil.
Approximately per Annex B.1.5: the throttling losses in the capillary and the throttling plus friction power in the bearing heat the oil; capillary and bearing temperature are the mean values of this heating above the inlet temperature. Heat conduction through the housing is not applied, which tends to overestimate the temperature.
Yes, with zero speed: the Couette share vanishes, β becomes zero, and the calculation gives recess pressures, lift and oil demand of the purely hydrostatic state. For operation at speed the hydrodynamic bearing to DIN 31652 then has to be verified.