# Static Load Characteristics of Hydrostatic Journal Bearings: Measurements and Predictions

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{s}) into the test bearings. The test results demonstrate notable effects of the test fluids and their temperatures, as well as P

_{s}, on the bearing performance. In general, the measured bearing flow rate, rotor displacement, and stiffness of the test bearings steadily increase with P

_{s}. The static load bearing characteristics predictions considering flow turbulence and compressibility matched well with the experimental results. The work with independent test data and engineering computational programs will further the implementation of hydrostatic bearings in high-performance turbopump shaft systems with improved efficiency and enhanced reusability of liquid rocket engine sub-systems.

## 1. Introduction

_{s}) into the test bearings at a non-rotating condition prior to extensive rotordynamic tests of the hydrostatic journal bearing supported rotor system. The non-rotating operating condition is intended to eliminate the contribution of hydrodynamic pressure within the test bearings on the bearing performance and characteristics. That is, the current work only shows the bearing static load performance due to external pressurization of a test fluid into the test bearings. Air (bearing inlet temperature 25 °C, which is a controlled room temperature), water (bearing inlet temperature 6 °C, 25 °C, 48 °C, and 70 °C), and liquid nitrogen (bearing inlet temperature −197 °C) are used as test fluids to demonstrate the effects of lubricant properties and conditions on the bearing performance. The feed pressure condition of each test fluid is manipulated to identify its effect on the static load characteristics of the test bearings. In addition, the measurements are compared with the predictions.

## 2. Experimental Facility

_{t}) and polar moment of inertia (I

_{p}) of the test rotor are 2.12 × 10

^{−1}kg·m

^{2}and 4.61 × 10

^{−3}kg·m

^{2}, respectively. Hard chrome plating (0.12 mm in thickness) is applied to the rotor outer surfaces at the bearing location. The center of mass of the test rotor is 293 mm from the rotor free end. The fraction of rotor weight acting on the free end and the drive end bearings equals ~55 N and ~47 N, respectively. The test bearings, made of SUS 630, are orifice-compensated-type hydrostatic bearings with nine square recesses (0.823 mm in depth). The inner diameter and axial length of both test bearings are ~60 mm and 25 mm, respectively. The bearing inner surfaces are coated with Ag (i.e., ~0.02 mm thickness silver plating). The axial and the circumferential lengths of the recesses fabricated on the bearing inner surfaces are both ~9.86 mm. The outer diameter of the test bearings has circumferential grooves as a fluid path for pressurized lubricant. The bearing outer surfaces also have circumferential grooves for insertion of O-rings. Note that the current bearing recess and orifice configurations follow the design approach and dimensions detailed in Refs. [19,20,21] for rocket engine turbopump applications. A separate and independent parametric study of the bearing recess and orifice dimensions (not shown here for brevity) confirms that the current bearings are capable of supporting a shaft system in rocket engine cryogenic turbopumps. Rotor displacements are recorded using two pairs of displacement sensors located at the free end rotor and drive end rotor. The displacement sensors are orthogonally affixed on each side cover.

## 3. Test Cases and Experimental Methods

_{s}increases from 0.1 bar(g) to 1.6 bar(g) in 0.1 bar(g) increments. The measured parameters are P

_{s}, flow rate, rotor displacement, and torque. From the measured data, bearing orifice discharge coefficients and bearing stiffness coefficients along horizontal direction are estimated. Note that when air (i.e., test case #1) or liquid nitrogen (i.e., test case #3) is fed into the test bearings, pneumatic hammer instability occurs even with P

_{s}=1 bar(g). For test cases #1 and #3, pneumatic hammer instability becomes more distinctive as P

_{S}increases. Therefore, to prevent damage to the test bearings and the test rotor, measurements are conducted up to P

_{s}=1.6 bar(g) for test case #1 and up to P

_{s}= 4 bar(g) for test case #3. Note, for test cases #1 and #2, flow rate and P

_{s}are measured for both the bearing-only (i.e., without rotor in the test bearings) and the rotor-bearing conditions (i.e., rotor inserted in the test bearings). However, in test case #3, flow rate and P

_{s}are measured for the bearing-only condition. Note that the test fluid temperature is monitored at the inlet locations of the test bearing.

_{s}. The flow rate through an orifice is calculated using

_{S}for complete lift-off (i.e., no physical contact) of the test rotor from the bearing inner surfaces. The force is measured while pulling the sting connected between the test rotor and the force gauge. The bearing torque is measured by multiplying the force applied to the rotor by the radius of the rotor at the bearing locations. Each torque measurement result uses the average value of the data measured thrice. A series of rap tests is conducted while supplying a test fluid to the test bearings to estimate bearing stiffnesses. The bearing stiffness (K) is identified by the measured acceleration response obtained from the rap test. The accelerometer is attached to one end of the test rotor. Note that $K={\omega}_{n}^{2}M$, where K is a bearing stiffness, ${\omega}_{n}$ is a measured natural frequency, and M is the rotor weight. In Figure 4b, m

_{1}and m

_{2}represent the static load acting on the free end bearing and the drive end bearing, respectively. In addition, K

_{1}and K

_{2}indicate the stiffness of each test bearing. Presently, all measurements are conducted thrice under static steady-state conditions and the average value of three results is shown used for the test result.

## 4. Experimental Results

#### 4.1. Flow Rate and Orifice Discharge Coefficient of Test Bearings

_{s}into the test bearings for each test fluid. The bearing flow rates are measured for both the bearing-only (i.e., without rotor in the test bearings) and the rotor-bearing (i.e., rotor inserted in the test bearings) conditions. Recall that the loads acting on the free end bearing and drive end bearing are 55 N and 47 N, respectively. When 25 °C air is used as a test fluid (i.e., test case #1), the flow rate increases linearly as P

_{s}increases for both the bearing-only and rotor-bearing conditions. Obviously, at the same P

_{s}, the measured bearing flow rates at the bearing-only condition are larger than those measured at the rotor-bearing condition. The flow rate difference between the rotor-bearing and the bearing-only conditions is nearly constant (~16 L/min) with increasing P

_{S}. Note that there is no notable difference in flow rate between the free end bearing and the drive end bearing. Recall that pneumatic hammer instability occurs from P

_{s}= 1 bar(g) and the vibration amplitude caused by pneumatic hammer instability increases with P

_{s}.

_{s}for both the bearing-only and rotor-bearing conditions. Note that for test case #2, at the same P

_{s}, the measured bearing flow rates increase with temperature of the test fluid for the rotor-bearing condition due to the changes in bearing eccentricity. On the other hand, there is no noticeable difference in flow rate with increasing temperature of the test fluid for the bearing-only condition.

_{s}, the bearing flow rates for the air-lubricated condition (i.e., test case #1) are larger than those for the water-lubricated (i.e., test case #2) and liquid-nitrogen-lubricated (i.e., test case #3) conditions. The recorded flow rates of the free end bearing and the drive end bearing under the bearing-only condition show similar values. Note that for the rotor-bearing condition under the same P

_{s}, the flow rates of the free end bearing are always slightly higher than those of the drive end bearing.

_{d}) versus P

_{s}. The orifice discharge coefficients are estimated using Equation (1) for water and liquid nitrogen and Equation (2a) for air. Note that the physical properties of air, water, and liquid nitrogen are taken from Refs. [22,23,24,25,26]. When air is used as a test fluid (i.e., test case #1), C

_{d}gradually increases while P

_{s}increases from 0.1 bar(g) to 1 bar(g), then shows an almost constant value of ~0.72 above 1 bar(g). When water is used as a test fluid (i.e., test case #2), C

_{d}shows a nearly invariant value of ~0.74. In addition, for test case #2, the difference in C

_{d}with increasing temperature of the test fluid is not notable. For test case #3, C

_{d}ranges from 0.6 to 0.7. The differences in C

_{d}for the free end bearing and the drive end bearing are not noticeable.

#### 4.2. Rotor Centerline Motions and Bearing Eccentricity Ratio

_{s}. The initial position of the test rotor (i.e., the coordinates (0, 0) in Figure 7) denotes the rotor position within the test bearing when P

_{s}= 0 bar(g). The centerline of the test rotor increases along the vertical plane as P

_{s}increases. For test case #1 (i.e., tests with air), the test rotor is lifted off from the bottom of the bearing inner surface to ~25% of the (room temperature assembly) bearing diametrical clearance for P

_{s}= 1 bar(g). For test case #3 (i.e., tests with liquid nitrogen), the test rotor is lifted off from the bottom of the bearing inner surface to ~23% of the (room temperature assembly) bearing diametrical clearance for P

_{s}= 1 bar(g). For test case #2 (i.e., tests with water) with P

_{s}= 1 bar(g), the test rotor is lifted off from the bottom of the bearing inner surface to ~31%, ~28%, ~26%, and ~25% of the (room temperature assembly) bearing diametrical clearance for at 6 °C water, 25 °C water, 48 °C water, and 70 °C water, respectively.

_{s}for the free end bearing. The bearing eccentricity ratios rapidly decrease as P

_{s}increases when P

_{s}ranges from P

_{s}= 0 bar(g) to P

_{s}= ~5 bar(g), then becomes nearly invariant to P

_{s}when P

_{s}> 5 bar(g). For test case #2 (i.e., tests with water), at the same P

_{s}, the eccentricity ratios decrease as the test fluid temperature increases. Note that the eccentricity ratios for test case #3 (i.e., tests with liquid nitrogen) are larger than test case #2 (i.e., tests with water).

#### 4.3. Bearing Torque

_{s}for test cases #1 and #2. Note that (nearly) zero bearing torque represents complete separation (i.e., lift-off) of the rotor surface from the bearing surfaces due to fluid external pressurization. For test case #1 (i.e., tests with air), the measured bearing torques at P

_{s}> 0.8 bar(g) are ~0 N-m. For test case #2 (i.e., tests with water), the measured bearing torques at P

_{s}> 1 bar(g) are ~0 N-m. Interestingly, for test case #1, when P

_{s}< 0.6 bar(g), the bearing torque linearly decreases with P

_{s}.

#### 4.4. Bearing Stiffness

_{s}. The figure also includes the identified bearing stiffnesses from the excited frequencies due to pneumatic hammer instability for test cases #1 and #3. For test case #1 (i.e., tests with air), pneumatic hammer instability occurs when P

_{s}> 1 bar(g). Therefore, when P

_{s}< 1 bar(g), the natural frequencies are identified from the rap test, while those are identified from the frequencies excited by pneumatic hammer instability when P

_{s}> 1 bar(g). The bearing stiffnesses for test case #2 (i.e., tests with water) are estimated by the rap test. For test case #3 (i.e., tests with liquid nitrogen), when P

_{s}< 4 bar(g), it is important to note that due to insufficient thermal insulation around the test bearings in the bearing housing and low P

_{s}for external pressurization into the bearings, the test fluid (i.e., liquid nitrogen) experiences a phase transition from all-liquid to two-phase (liquid–gas) flow in the thin bearing films. This two-phase fluid condition in cryogenic bearings for turbopump applications is not uncommon, see Refs. [27,28]. The test results clearly show significant effects of test fluids and bearing inlet fluid temperature (T

_{s}) on the measured bearing stiffnesses.

_{s}. For test case #1, K for P

_{s}= 1.6 bar(g) ≈ ~2 × K for P

_{s}= 0.8 bar(g). For test case #3 (i.e., tests with liquid nitrogen), the K rapidly increases with P

_{s}when P

_{s}< 2 bar(g) while the K slightly increases with P

_{s}when 2 bar(g) < P

_{s}< 4 bar(g). That is, for test case #3, K for P

_{s}= 2 bar(g) ≈ ~2 × K for P

_{s}= 1 bar(g) while K for P

_{s}= 4 bar(g) ≈ ~1.1 × K for P

_{s}= 2 bar(g). For test case #2 (i.e., tests with water), the K gradually increases with P

_{s}and tends to increase as the temperature of the test fluid decreases. For tests with 70 °C water, K for P

_{s}= 5 bar(g) ≈ ~2.5 × K for P

_{s}= 2.5 bar(g) and K for P

_{s}= 10 bar(g) ≈ ~2.75 × K for P

_{s}= 5 bar(g). For tests with 6 °C water, K for P

_{s}= 5 bar(g) ≈ ~2.6 × K for P

_{s}= 2.5 bar(g) and K for P

_{s}= 10 bar(g) ≈ ~1.5 × K for P

_{s}= 5 bar(g). Note that, K for T

_{s}= 6 °C ≈ ~3.2 × K for T

_{s}= 70 °C when P

_{s}= 5 bar(g) and K for T

_{s}= 6 °C ≈ ~1.8 × K for T

_{s}= 70 °C when P

_{s}= 10 bar(g).

## 5. Predictions and Comparison to Measurements

_{s}and then flows out of the recesses. The steady-state Reynolds equation for an isothermal and isoviscous fluid film is written as

_{r}), recall Equations (1) and (2a–d). That is, P

_{s}, Q

_{r}, and C

_{d}mainly determine P

_{r}. See Refs. [14,30] for further details on the fundamental model of multi-recess hydrostatic bearings which is employed in the present work.

_{s}. The predicted bearing flow rates are in good agreement with the test results.

_{s}. For test case #1 (i.e., tests with air), the predicted eccentricity ratios agree well with the test data. For test case #2 (i.e., tests with water), correlations between measurements and predictions become less favorable as the water temperature increases while those for 6 °C water are remarkable. For test case #3 (i.e., tests with liquid nitrogen), the predictions show a good agreement with the measurements even though the predicted eccentricity ratios are slightly larger than the test data.

_{s}, and the agreement between predictions and measurements is remarkable. For test case 2 (i.e., tests with water), in general, the predicted stiffnesses agree reasonably with the measurements. For test case #3 (i.e., tests with liquid nitrogen), the trends between measurements and predictions appear quite similar as P

_{s}increases. However, the comparisons between predicted and measured stiffnesses for test case #3 are less favorable than those for test cases #1 and #2. As discussed in the previous chapter (i.e., 4. Experimental Results), this is mainly due to the phase transition of liquid nitrogen from all-liquid to two-phase flow in the test bearings.

## 6. Conclusions

_{s}) into the test bearings. In addition, measurements are compared to predictions for validation of the bearing prediction model. The test results show that the static load characteristics of the test hydrostatic bearing strongly rely on test fluids, as well as their bearing inlet temperatures, and static load conditions. The measured bearing flow rates for the tests with air are much larger than those for the tests with water and liquid nitrogen. For the tests with water fed into the bearings, the measured bearing flow rates increase as the water temperature increases when the rotor is installed within the bearings (i.e., the rotor-bearing condition). However, when the rotor is removed from the test rig and measurements are conducted only with the bearings (i.e., the bearing-only condition), the measured bearing flow rates do not notably change with water temperature. Interestingly, the measured bearing flow rates for the tests with water are not quite different from those for the tests with liquid nitrogen. When water is used as a test fluid, as the water gets warmer, the measured bearing eccentricity ratios increase. The measured bearing eccentricity ratios for the tests with liquid nitrogen are higher than those for the tests with water. Bearing stiffnesses are identified by the rap test, as well as the excited vibration frequencies by pneumatic hammer instability for the tests with air and liquid nitrogen. The measured bearing stiffnesses for the tests with water gradually increase with fluid supply pressure into the bearings and decreases with the water temperatures. Bearing performance predictions are benchmarked against the comprehensive measurement data tested with air, water, and liquid nitrogen. The predicted bearing flow rate, eccentricity ratio, and stiffness are in notable agreement with the test data for various supply pressure conditions. Note that for the tests with liquid nitrogen, the comparisons between predictions and measurements clearly infer a phase transition of liquid nitrogen in the test bearings due to a large thermal gradient from the outside of the bearing housing to the thin film of the test bearings and relatively low fluid (i.e., liquid nitrogen) supply pressure into the test bearings. This evidence a need for employing a thermo-hydrodynamic model considering a two-phase fluid condition for more improved and accurate bearing characteristics predictions. The present work provides an extensive database on the static load characteristics of hydrostatic bearings lubricated with compressible (air), incompressible (water), and cryogenic (liquid nitrogen) fluids. Currently, comprehensive rotordynamic testing is underway to measure shaft motions of the present test rig for various fluid supply conditions while increasing rotor speed.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

Nomenclature | |

A_{o} | Orifice area (m^{2}) |

C_{d} | Orifice discharge coefficients (-) |

C | Bearing radial clearance (m) |

D | Bearing diameter (m) |

D_{rotor} | Rotor diameter (m) |

d_{orifice} | Orifice diameter (m) |

f | Frequency (Hz) |

$g$ | Flow function (-) |

G_{x}, G_{z} | Turbulence parameters (-) |

h | Film thickness at recess edge (m) |

h_{recess} | Recess depth (m) |

I_{p} | Polar moment of inertia (kg-m^{2}) |

I_{t} | Transverse moment of inertia (kg-m^{2}) |

k_{1} | Bearing stiffness of free end bearing (N/m) |

k_{2} | Bearing stiffness of drive end bearing (N/m) |

K | Bearing stiffness |

L | Bearing length (m) |

l | Recess length (m) |

m_{1} | Mass acting on free end bearing (kg) |

m_{2} | Mass acting on drive end bearing (kg) |

M | Rotor mass (kg) |

P | Fluid film pressure (Pa) |

P_{s} | Supply pressure (Pa) |

P_{r} | Recess pressure (Pa) |

Q | Flow rate (kg/s) |

$\Re $ | Gas constant |

T_{s} | Supply fluid temperature (°C) |

U | Rotor surface speed (m/s^{2}) |

X, Y, Z | Inertial coordinate system (m) |

κ | Specific heat ratio of air (-) |

ρ | Density (kg/m^{3}) |

µ | Viscosity (Pa-s) |

υ | Normal fluid velocity to recess edge (m/s) |

Φ | Flow equation |

ω | Measured natural frequency (rad/s) |

Acronyms | |

FEB | Free end bearing |

DEB | Drive end bearing |

Exp. | Experiment |

Pred. | Prediction |

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**Figure 3.**Schematic view (not to scale) of test fluid supply systems. (

**a**) System configuration for air-lubricated bearing tests. (

**b**) System configuration for water-lubricated bearing tests. (

**c**) System configuration for liquid-nitrogen-lubricated bearing tests.

**Figure 4.**Schematic view (not to scale) of (

**a**) torque measurement and (

**b**) rap test for stiffness estimation.

**Figure 5.**Test cases #1 through #3: recorded flow rate versus supply pressure. (

**a**) Test case #1. DEB: drive end bearing. FEB: free end bearing. (

**b**) Test cases #2 and #3. Free end bearing. (

**c**) Test cases #2 and #3. Drive end bearing.

**Figure 6.**Test cases #1 through #3: estimated orifice discharge coefficient (C

_{d}) versus supply pressure (P

_{s}). (

**a**) Free end bearing. (

**b**) Drive end bearing. Note: The lowest value of the Y-axis is not 0 but 0.5.

**Figure 7.**Test cases #1 through #3: measured rotor centerline motions for increasing P

_{s}. Free end bearing. Static load on the bearing: ~55 N. (

**a**) Test cases #1 and #3. (

**b**) Test case #2.

**Figure 8.**Test cases #1 through #3: measured bearing eccentricity ratio versus supply pressure. Free end bearing. Static load on the bearing: ~55 N.

**Figure 13.**Predictions versus measurements. Test cases #2 and #3: bearing flow rate of free end bearing (FEB). (

**a**) Bearing-only condition. (

**b**) Rotor-bearing condition.

**Figure 14.**Predictions versus measurements. Test cases #1, #2, and #3: eccentricity ratio of free end bearing. (

**a**) Test cases #1 and #3. (

**b**) Test case #2.

**Figure 15.**Predictions versus measurements. Test cases #1, #2, and #3: stiffnesses of free end bearing. (

**a**) Test cases #1 and #3. (

**b**) Test case #2.

Test Rotor | Value |

Material | SUS630 |

Outer diameter at bearing locations | 59.896 (±0.002) mm |

Length | 590 mm |

Mass center from rotor free end | 293 mm |

Polar moment of inertia (I_{p}) | 4.61 × 10^{−3} kg·m^{2} |

Transverse moment of inertia (I_{t}) | 2.12 × 10^{−1} kg·m^{2} |

Mass | 10.40 kg |

Test Bearings | Value |

Material | SUS630 |

Outer diameter | 100.000 (±0.002) mm |

Inner diameter | 60.000 (±0.002) mm |

Axial length | 25.00 (±0.002) mm |

Radial clearance | 0.052 (±0.002) mm |

Orifice diameter | 0.82 (±0.005) mm |

Number of recesses | 9 |

Axial and circumferential lengths of recess | 9.86 mm (±0.003) |

Test Case # | Test Fluid | Controlled Bearing Inlet Fluid Temperature, T _{s}[°C] | Measured or Estimated Parameters |
---|---|---|---|

1 | Air | 25 | Supply pressure, flow rate, orifice discharge coefficient, rotor centerline motion, torque, and stiffness |

2 | Water | 6, 25, 48, and 70 | Supply pressure, flow rate, orifice discharge coefficient, rotor centerline motion, torque, and stiffness |

3 | Liquid nitrogen | −197 | Supply pressure, flow rate, orifice discharge coefficient, rotor displacement, and stiffness |

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**MDPI and ACS Style**

Yi, H.; Jung, H.; Kim, K.; Ryu, K.
Static Load Characteristics of Hydrostatic Journal Bearings: Measurements and Predictions. *Sensors* **2022**, *22*, 7466.
https://doi.org/10.3390/s22197466

**AMA Style**

Yi H, Jung H, Kim K, Ryu K.
Static Load Characteristics of Hydrostatic Journal Bearings: Measurements and Predictions. *Sensors*. 2022; 22(19):7466.
https://doi.org/10.3390/s22197466

**Chicago/Turabian Style**

Yi, Howon, Hyunsung Jung, Kyuman Kim, and Keun Ryu.
2022. "Static Load Characteristics of Hydrostatic Journal Bearings: Measurements and Predictions" *Sensors* 22, no. 19: 7466.
https://doi.org/10.3390/s22197466