Precision Machine-Tool Tables & Observatory Azimuth Bearings, Hydrostatic Thrust Pad
Hydrostatic Circular Step Thrust-Pad Load & h³ Flow Formula
Calculate Exact Radial-Log Pad Load Capacity W (kN), Viscous Sill Leakage Flow Q (L/min), Hydraulic Sill Pumping Power (W and kW), and Cubic Film-Gap Sensitivity.
Doubling a hydrostatic bearing’s oil-film gap from 55 to 110 um leaves load lift unchanged—while Poiseuille’s h^3 law multiplies oil leakage and pump heating 8-FOLD!
Unlike *hydrodynamic* tilting-pad bearings that need sliding velocity to build an oil wedge, **Externally Pressurized Hydrostatic Circular Step-Recess Thrust Pads** (`Outer Sill Radius R_o` mm, `Central Pocket Radius R_i` mm, `Recess Supply Pocket Pressure P_r` MPa, `Oil Dynamic Viscosity mu` Pa·s) float `200`-ton vertical boring-mill tables and `30`-m optical telescope mounts at **zero RPM with zero stick-slip friction**! Integrating radial Poiseuille laminar flow `p(r) = P_r * ln(R_o / r) / ln(R_o / R_i)` across the annular land yields **Exact Vertical Lift Load `W = ((pi * P_r * (R_o^2 - R_i^2)) / (2 * ln(R_o / R_i))) / 1,000`** (`kN`), **Sill Oil Flow `Q = ((pi * (P_r * 10⁶) * ((h * 10⁻⁶)^3)) / (6 * mu * ln(R_o / R_i))) * 60,000`** (`L/min` at Film Clearance `h` um), and **Pocket Hydraulic Pumping Power `P_{pump} = (P_r * 1,000 * Q) / 60`** (`W`, `/ 1,000` kW)—scaling strictly with **`h^3`**!
Verify on a Heavy Vertical Turning Lathe Hydrostatic Table Pad (`ISO VG 46` Oil at `40°C`)—Outer_Pad_Radius `R_o` = 160 mm (B2), Recess_Pocket_Radius `R_i` = 85.0 mm (B3), Pocket_Pressure `P_r` = 4.80 MPa (`48` bar, B4), Dynamic_Viscosity `mu` = 0.046 Pa·s (`46` mPa·s, B5), and Film_Clearance `h` = 55.0 um (B6): Vertical Pad Load Capacity equals **`219.03` kN** (`22.3` metric tons per pad! E2), Sill Oil Flow equals **`0.862` L/min** (E3), consuming just **`69.0` W** (`0.069` kW in E5) of hydraulic pocket power!
| Hydrostatic Circular Step Pad KPI (Ro=160 mm, Ri=85 mm, Pr=4.8 MPa, mu=0.046 Pa·s) | Formula | Verified Output |
|---|---|---|
| Pad Lift Load Capacity W (kN) | ((PI() * Pr * (Ro^2 - Ri^2)) / (2 * LN(Ro/Ri))) / 1000 | 219.03 kN (independent of gap h @ fixed Pr) |
| Viscous Sill Flow Q (L/min) | ((PI * Pr_Pa * h_m^3) / (6 * mu * LN(Ro/Ri))) * 60000 | 0.862 L/min @ 55 um (6.898 L/min @ 110 um!) |
| Pocket Pumping Power (W) | (Pocket_Pr_MPa * 1000 * Flow_L_min) / 60 | 69.0 W cool (551.9 W = 8.0x thermal load!) |
| Pocket Pumping Power (kW) | Pocket_Pumping_Power_W / 1000 | 0.069 kW (0.552 kW chiller load) |
Radial Poiseuille Hydrostatic Step-Pad Load & Cubic h³ Flow Formula
Validate `Inner_Radius R_i >= 5` mm, `Outer_Radius R_o` in `(1.15 * R_i, 2,500]` mm, `Pocket_Pressure P_r` in `[0.2, 50.0]` MPa, `Oil_Viscosity mu` in `[0.002, 1.000]` Pa·s, and `Film_Gap h` in `[5, 500]` um!
=LET(ro_m, B2, ri_m, B3, pr_m, B4, mu_p, B5, hf_u, B6, ql_m, ((PI() * (pr_m * 1000000) * ((hf_u * 0.000001) ^ 3)) / (6 * mu_p * LN(ro_m / ri_m))) * 60000, pw_w, (pr_m * 1000 * ql_m) / 60, IF(OR(ri_m < 5, ro_m <= 1.15 * ri_m, ro_m > 2500, pr_m < 0.2, pr_m > 50, mu_p < 0.002, mu_p > 1, hf_u < 5, hf_u > 500), "Invalid Pad Input", ROUND(pw_w / 1000, 3)))Doubled Film Clearance (`h = 110` um) 8x Oil-Chiller Thermal-Distortion Check
Open capillary/constant-pressure restrictor clearance to `B6 = 110.0` um (`2.0x` baseline) at fixed `4.80` MPa pocket pressure: Pad Load E2 stays `219.03` kN, YET Sill Leakage E3 jumps **`2^3 = 8.00x` to `6.898` L/min** and Pumping Heat E4 jumps **`8.00x` to `551.9` W** (`0.552` kW per pad in E5)—forcing oil chillers onto precision CNC beds so warm return oil never thermally bows the machine table!
=IF(ISTEXT(E5), E5, ROUND(((PI() * (B4 * 1000000) * ((B6 * 0.000001) ^ 3)) / (6 * B5 * LN(B2 / B3))) * 60000, 3))Try the example
Use the original values or change them to test your understanding.