The Impact of Pad Flexibility on the Rotordynamic Coefficients of Tilting-Pad Journal Bearings

2016 ◽  
Vol 138 (8) ◽  
Author(s):  
Jennifer E. Gaines ◽  
Dara W. Childs
Keyword(s):  
Dynamic Stiffness ◽  
Dynamic Performance ◽  
Element Model ◽  
Bending Stiffness ◽  
Test Fluid ◽  
Unit Load ◽  
Rotordynamic Coefficients ◽  
Damping Coefficients ◽  
Tilting Pad ◽  
The Impact

Static and dynamic load tests were performed on a three-pad, rocker-pivot, tilting-pad journal bearing (TPJB) with three interchangeable pad configurations, each with measurably different pad flexibilities. Measured dynamic-stiffness data for the bearing were readily fitted by a frequency-independent, constant-coefficient [K][C][M] model. The test-bearing had a 101.74 mm diameter with L/D = 0.6. Tests were conducted over the speed range of 6–12 krpm, with unit loads varying from 0.172 to 1.724 MPa. An ISO VG 46 lubricant was used as the test fluid. Pad flexibility was characterized as the change in the pad's bending stiffness or the change in pad thickness. A finite-element model (FEM) was created to predict the structural bending stiffness of each pad configuration, showing a significant pad flexibility increase as pad thickness decreased. To examine the effect of pad flexibility on the rotordynamic coefficients, the measured results were compared across pad configurations and showed that the pad flexibility increase reduced the direct damping coefficients by 12–20%. As pad flexibility increased, the direct-stiffness coefficients could increase or decrease, depending on the unit load. They varied from an increase of 12% at low unit loads to a decrease of 3% at high unit loads. Results show that the pad's structural bending stiffness or flexibility is important when predicting the bearing’s dynamic performance. Damping is consistently overpredicted when neglecting pad flexibility. A nondimensional pad flexibility parameter αflex was developed. It related the average deflection across the pad surface to the pad's arc length and was to relate the pad flexibility of multiple bearings of different sizes. A bearing code was used to predict the percent change in direct damping coefficients for rigid-pad/flexible-pivot and flexible-pad/flexible-pivot models for a surface speed of 54 m/s and a unit load of 783 kPa for the three-pad configuration tested here plus five additional tested bearings from the literature. For the minimum pad thickness configuration tested here, the code predicted a 20% drop in predicted Cxx (off-load axis direct damping) when comparing a model that included pad flexibility with a model that neglected pad flexibility. In terms of αflex, the two thinnest pad configurations tested here are quite flexible compared to both TPJB's pads used in industry and previously tested TPJB pads.

The Impact of Pad Flexibility on the Rotordynamic Coefficients of Tilting-Pad Journal Bearings

2015 ◽  
Author(s):  
Jennifer E. Gaines ◽  
Dara W. Childs
Keyword(s):  
Dynamic Stiffness ◽  
Dynamic Performance ◽  
Element Model ◽  
Bending Stiffness ◽  
Test Fluid ◽  
Unit Load ◽  
Rotordynamic Coefficients ◽  
Damping Coefficients ◽  
Tilting Pad ◽  
The Impact

Static and dynamic load tests were performed on a three-pad, rocker-pivot, tilting-pad journal bearing (TPJB) with three interchangeable pad configurations, each with measurably different pad flexibilities. Measured dynamic-stiffness data for the bearing were readily fitted by a frequency-independent, constant-coefficient [K][C][M] model. The test bearing had a 101 .74 mm diameter with L/D = 0.6 Tests were conducted over the speed range of 6 to 12 krpm, with unit loads varying from .172 to 1.724 MPa. An ISO VG 46 lubricant was used as the test fluid. Pad flexibility was characterized as the change in the pad’s bending stiffness or the change in pad thickness. A finite-element model was created to predict the structural bending stiffness of each pad configuration, showing a significant pad-flexibility increase as pad thickness decreased. To examine the effect of pad flexibility on the rotordynamic coefficients, the measured results were compared across pad configurations and showed that the pad-flexibility increase reduced the direct damping coefficients by 12–20%. As pad flexibility increased, the direct stiffness coefficients could increase or decrease, depending on the unit load. They varied from an increase of 12% at low unit loads to a decrease of 3% at high unit loads. Results show that the pad’s structural bending stiffness or flexibility is important when predicting the bearing’s dynamic performance. Damping is consistently over-predicted when neglecting pad flexibility. A non-dimensional pad flexibility parameter αflex was developed. It related the average deflection across the pad surface to the pad’s arc length and was to relate the pad flexibility of multiple bearings of different sizes. A bearing code was used to predict the percent change in direct damping coefficients for rigid-pad/flexible-pivot and flexible-pad/flexible-pivot models for a surface speed of 54 m/s and a unit load of 783 kPa for the three pad configurations tested here plus five additional tested bearings from the literature. For the minimum-pad-thickness configuration tested here, the code predicted a 20% drop in predicted Cxx (off-load-axis direct damping) when comparing a model that included pad flexibility with a model that neglected pad flexibility. In terms of αflex, the two thinnest pad configurations tested here are quite flexible compared to both TPJB’s pads used in industry and previously-tested TPJB pads.

Effects of Clearance on the Performance of a Labyrinth Seal Under Wet-Gas Conditions

2020 ◽  
Author(s):  
Min Zhang ◽  
Dara W. Childs ◽  
Dung L. Tran ◽  
Hari Shrestha
Keyword(s):  
Dynamic Stiffness ◽  
Labyrinth Seal ◽  
Test Fluid ◽  
Radial Clearance ◽  
Liquid Volume ◽  
Rotordynamic Coefficients ◽  
Whirling Motion ◽  
Wet Gas ◽  
Different Diameters ◽  
The Impact

Abstract The labyrinth seal is one of the most popular non-contact annular seals used in centrifugal compressors to improve machine efficiency by reducing the secondary flow leakage. Reducing the radial clearance Cr can effectively decrease the seal’s leakage and therefore increase the machine efficiency. However, reducing Cr can also introduce undesired effects on the machine’s vibration behaviors. This paper experimentally studies the impact of reducing Cr on the leakage and rotordynamic coefficients of a 16-tooth see-through labyrinth seal under wet-gas conditions. The test seal’s inner diameter is 89.256 mm. Two rotors with different diameters are used to obtain two radial clearances (0.102 mm and 0.178 mm). Tests are carried out at a supply pressure of 62 bars, three speeds from 10krpm to 20 krpm, three pressure ratios from 0.21 to 0.46, and six inlet liquid volume fractions (LVFs) from zero to 15%. The test fluid is a mixture comprised of air and silicon oil. Test results show that, for all pure-air and mainly-air conditions, decreasing Cr decreases (as expected) the test seal’s leakage mass flow rate. For all test cases, direct dynamic stiffness KΩ is negative, producing a negative centering force on the associated rotor. For inlet LVF ≤ 8%, the effects of decreasing Cr on KΩ are negligible. When inlet LVF = 12% and 15%, decreasing Cr increases KΩ (decreases the magnitude). In other words, when inlet LVF = 12% and 15%, decreasing Cr reduces the test seal’s negative centering force on the rotor, and would increase the critical speeds of the rotor. The value of the effective damping Ceff near 0.5ω represents the seal’s capability to suppress the rotor’s potential whirling motion at about 0.5ω. For all pure-air and mainly-air conditions, decreasing Cr generally increases the Ceff value near 0.5ω; i.e., decreasing Cr improves the test seal’s stabilizing capability against the rotor’s potential whirling motion at about 0.5ω.

Rotordynamic Characteristics of a Five Pad, Rocker-Pivot, Tilting Pad Bearing in a Load-on-Pad Configuration; Comparisons to Predictions and Load-Between-Pad Results

2011 ◽  
Vol 133 (8) ◽  
Author(s):  
Dara W. Childs ◽  
Clint R. Carter
Keyword(s):  
Dynamic Stiffness ◽  
Excitation Frequency ◽  
Bulk Flow ◽  
Quadratic Functions ◽  
Dynamic Force ◽  
Unit Load ◽  
Damping Coefficients ◽  
Stiffness Coefficients ◽  
Computational Fluid Dynamics Analysis ◽  
Tilting Pad

Rotordynamic data are presented for a rocker-pivot tilting pad bearing in load-on-pad (LOP) configuration for (345–3101 kPa) unit loads and speeds from 4000 rpm to 13,000 rpm. The bearing was directly lubricated through a leading edge groove with five pads, 0.282 preload, 60% offset, 57.87 deg pad arc angle, 101.587 mm (3.9995 in.) rotor diameter, 0.1575 mm (0.0062 in.) diametral clearance, and 60.325 mm (2.375 in.) pad length. Measured results were reported for this bearing by Carter and Childs (2008, “Measurements Versus Predictions for the Rotordynamic Characteristics of a 5-Pad, Rocker-Pivot, Tilting-Pad Bearing in Load Between Pad Configuration,” ASME Paper No. GT2008-50069) in the load-between-pad (LBP) configuration. Results for the LOP are compared with predictions from a bulk-flow Navier–Stokes model (as utilized by San Andres (1991, “Effect of Eccentricity on the Force Response of a Hybrid Bearing,” STLE Tribol. Trans., 34, pp. 537–544)) and to the prior LBP results. Frequency effects on the dynamic-stiffness coefficients were investigated by applying dynamic-force excitation over a range of excitation frequencies. Generally, the direct real parts of the dynamic-stiffness coefficients could be modeled as quadratic functions of the excitation frequency, and accounted for by adding a mass matrix to the conventional [K][C] model to produce a frequency-independent [K][C][M] model. Measured added-mass terms in the loaded direction approached 60 kg. The static load direction in the tests was y. The direct stiffness coefficients Kyy and Kxx depend strongly on the applied unit load, more so than speed. They generally increased linearly with load, shifting to a quadratic dependence at higher unit loads. At lower unit loads, Kyy and Kxx increase monotonically with running speed. The experimental results were compared with predictions from a bulk-flow computational fluid dynamics analysis. Stiffness orthotropy was apparent in test results, significantly more than predicted, and it became more pronounced at the heavier unit loads. Measured Kyy values were consistently higher than predicted, and measured Kxx values were lower. Comparing the LOP results to prior measured LBP results for the same bearing, at higher loads, Kyy is significantly larger for the LOP configuration than LBP. Measured values for Kxx are about the same for LOP and LBP. At low unit loads, stiffness orthotropy defined as Kyy/Kxx is the same for LOP and LBP, progressively increasing with increasing unit loads. At the highest unit load, Kyy/Kxx=2.1 for LOP and 1.7 for LBP. Measured direct damping coefficients Cxx and Cyy were insensitive to changes in either load or speed, in contrast to predictions of marked Cyy sensitivity for changes in the load. Only at the highest test speed of 13,000 rpm were the direct damping coefficients adequately predicted. No frequency dependency was observed for the direct damping coefficients.

Effects of Clearance on the Performance of a Labyrinth Seal Under Wet-Gas Conditions

2020 ◽  
Vol 142 (11) ◽  
Author(s):  
Min Zhang ◽  
Dara W. Childs ◽  
Dung L. Tran ◽  
Hari Shresth
Keyword(s):  
Dynamic Stiffness ◽  
Labyrinth Seal ◽  
Test Fluid ◽  
Radial Clearance ◽  
Liquid Volume ◽  
Rotordynamic Coefficients ◽  
Whirling Motion ◽  
Wet Gas ◽  
Different Diameters ◽  
The Impact

Abstract The labyrinth seal is one of the most popular noncontact annular seals used in centrifugal compressors to improve machine efficiency by reducing the secondary flow leakage. Reducing the radial clearance Cr can effectively decrease the seal's leakage and therefore increase the machine efficiency. However, reducing Cr can also introduce undesired effects on the machine's vibration behaviors. This paper experimentally studies the impact of reducing Cr on the leakage and rotordynamic coefficients of a 16-tooth see-through labyrinth seal under wet-gas conditions. The test seal's inner diameter is 89.256 mm. Two rotors with different diameters are used to obtain two radial clearances (0.102 mm and 0.178 mm). Tests are carried out at a supply pressure of 62 bars, three speeds from 10 krpm to 20 krpm, three pressure ratios from 0.21 to 0.46, and six inlet liquid volume fractions (LVFs) from zero to 15%. The test fluid is a mixture comprised of air and silicon oil. Test results show that, for all pure-air and mainly air conditions, decreasing Cr decreases (as expected) the test seal's leakage mass flowrate. For all test cases, direct dynamic stiffness KΩ is negative, producing a negative centering force on the associated rotor. For inlet LVF ≤ 8%, the effects of decreasing Cr on KΩ are negligible. When inlet LVF = 12% and 15%, decreasing Cr increases KΩ (decreases the magnitude). In other words, when inlet LVF = 12% and 15%, decreasing Cr reduces the test seal's negative centering force on the rotor, and would increase the critical speeds of the rotor. The value of the effective damping Ceff near 0.5ω represents the seal's capability to suppress the rotor's potential whirling motion at about 0.5ω. For all pure-air and mainly air conditions, decreasing Cr generally increases the Ceff value near 0.5ω; i.e., decreasing Cr improves the test seal's stabilizing capability against the rotor's potential whirling motion at about 0.5ω.

Rotordynamic Characteristics of a 5 Pad, Rocker-Pivot, Tilting Pad Bearing in a Load-on-Pad Configuration: Comparisons to Predictions and Load-Between-Pad Results

2009 ◽  
Author(s):  
Dara W. Childs ◽  
Clint R. Carter
Keyword(s):  
Dynamic Stiffness ◽  
Mass Matrix ◽  
Excitation Frequency ◽  
Bulk Flow ◽  
Quadratic Functions ◽  
Dynamic Force ◽  
Unit Load ◽  
Damping Coefficients ◽  
Stiffness Coefficients ◽  
Tilting Pad

Rotordynamic data are presented for a rocker-pivot tilting-pad bearing in load-on-pad (LOP) configuration for (345–3101 kPa) unit loads and speeds from 4k to 13k rpm. The bearing was direct lubricated through a leading-edge groove with 5 pads, .282 preload, 60% offset, 57.87° pad arc angle, 101.587 mm (3.9995 in) rotor diameter, 0.1575 mm (.0062 in) diametral clearance, and 60.325 mm (2.375 in) pad length. Measured results were reported for this bearing by Carter and Childs in 2008 in the load-between-pad (LBP) configuration. Results for the LOP are compared to predictions from a bulk-flow Navier-Stokes model (as utilized by San Andres in 1991) and to the prior LBP results. Frequency effects on the dynamic-stiffness coefficients were investigated by applying dynamic-force excitation over a range of excitation frequencies. Generally, the direct real parts of the dynamic-stiffness coefficients could be modeled as quadratic functions of the excitation frequency and accounted for by adding a mass matrix to the conventional [K][C] model to produce a frequency-independent [K][C][M] model. Measured added mass terms in the loaded direction approached 60 kg. The static load direction in the tests was y. The direct-stiffness coefficients Kyy and Kxx depend strongly on the applied unit load, more so than speed. They generally increased linearly with load, shifting to a quadratic dependence at higher unit loads. At lower unit loads, Kyy and Kxx increase monotonically with running speed. The experimental results were compared to predictions from a bulk-flow CFD analysis. Stiffness orthotropy was apparent in test results, significantly more than predicted, and it became more pronounced at the heavier unit loads. Measured Kyy values were consistently higher than predicted, and measured Kxx values were lower. Comparing the LOP results to prior measured LBP results for the same bearing, at higher loads, Kyy is significantly larger for the LOP configuration than LBP. Measured values for Kxx are about the same for LOP and LBP. At low unit loads, stiffness orthotropy defined as Kyy / Kxx is the same for LOP and LBP, progressively increasing with increasing unit loads. At the highest unit load, Kyy / Kxx = 2.1 for LOP and 1.7 for LBP. Measured direct damping coefficients Cxx and Cyy were insensitive to changes in either load or speed in contrast to predictions of marked Cyy sensitivity for changes in the load. Only at the highest test speed of 13 krpm were the direct damping coefficients adequately predicted. No frequency dependency was observed for the direct damping coefficients.

Measured Static and Rotordynamic Coefficient Results for a Rocker-Pivot, Tilting-Pad Bearing With 50 and 60% Offsets

2012 ◽  
Vol 134 (5) ◽  
Author(s):  
Chris D. Kulhanek ◽  
Dara W. Childs
Keyword(s):  
Dynamic Stiffness ◽  
Mass Matrix ◽  
Excitation Frequency ◽  
Quadratic Function ◽  
Added Mass ◽  
Damping Coefficients ◽  
Stiffness Coefficients ◽  
Tilting Pad ◽  
Frequency Independent ◽  
Direct Stiffness

Static and rotordynamic coefficients are measured for a rocker-pivot, tilting-pad journal bearing (TPJB) with 50 and 60% offset pads in a load-between-pad (LBP) configuration. The bearing uses leading-edge-groove direct lubrication and has the following characteristics: 5-pads, 101.6 mm (4.0 in) nominal diameter,0.0814 -0.0837 mm (0.0032–0.0033 in) radial bearing clearance, 0.25 to 0.27 preload, and 60.325 mm (2.375 in) axial pad length. Tests were performed on a floating bearing test rig with unit loads from 0 to 3101 kPa (450 psi) and speeds from 7 to 16 krpm. Dynamic tests were conducted over a range of frequencies (20 to 320 Hz) to obtain complex dynamic stiffness coefficients as functions of excitation frequency. For most test conditions, the real dynamic stiffness functions were well fitted with a quadratic function with respect to frequency. This curve fit allowed for the stiffness frequency dependency to be captured by including an added mass matrix [M] to a conventional [K][C] model, yielding a frequency independent [K][C][M] model. The imaginary dynamic stiffness coefficients increased linearly with frequency, producing frequency-independent direct damping coefficients. Direct stiffness coefficients were larger for the 60% offset bearing at light unit loads. At high loads, the 50% offset configuration had a larger stiffness in the loaded direction, while the unloaded direct stiffness was approximately the same for both pivot offsets. Cross-coupled stiffness coefficients were positive and significantly smaller than direct stiffness coefficients. Negative direct added-mass coefficients were obtained for both offsets, especially in the unloaded direction. Cross-coupled added-mass coefficients are generally positive and of the same sign. Direct damping coefficients were mostly independent of load and speed, showing no appreciable difference between pivot offsets. Cross-coupled damping coefficients had the same sign and were much smaller than direct coefficients. Measured static eccentricities suggested cross coupling stiffness exists for both pivot offsets, agreeing with dynamic measurements. Static stiffness measurements showed good agreement with the loaded, direct dynamic stiffness coefficients.

The Surface Roughness Effect on the Dynamic Stiffness and Damping Characteristics of the Hydrostatic Thrust Spherical Bearings: Part 5 of 5 — Clearance Type of Bearings

2007 ◽  
Author(s):  
Ahmad W. Yacout
Keyword(s):  
Surface Roughness ◽  
Capillary Tube ◽  
Dynamic Stiffness ◽  
Dynamic Performance ◽  
Major Effect ◽  
Design Parameters ◽  
Compressibility Effect ◽  
Damping Coefficients ◽  
Stiffness And Damping ◽  
Fluid Compressibility

This study has theoretically analyzed the surface roughness, centripetal inertia and recess volume fluid compressibility effects on the dynamic behavior of a restrictor compensated hydrostatic thrust spherical clearance type of bearing. The stochastic Reynolds equation, with centripetal inertia effect, and the recess flow continuity equation with recess volume fluid compressibility effect have been derived to take into account the presence of roughness on the bearing surfaces. On the basis of a small perturbations method, the dynamic stiffness and damping coefficients have been evaluated. In addition to the usual bearing design parameters the results for the dynamic stiffness and damping coefficients have been calculated for various frequencies of vibrations or squeeze parameter (frequency parameter) and recess volume fluid compressibility parameter. The study shows that both of the surface roughness and the centripetal inertia have slight effects on the stiffness coefficient and remarkable effects on the damping coefficient while the recess volume fluid compressibility parameter has the major effect on the bearing dynamic characteristics. The cross dynamic stiffness showed the bearing self-aligning property and the ability to oppose whirl movements. The orifice restrictor showed better dynamic performance than that of the capillary tube.

Numerical analysis of the dynamic performance of aerostatic thrust bearings with different restrictors

2018 ◽  
Vol 233 (3) ◽  
pp. 406-423 ◽  
Author(s):  
Hailong Cui ◽  
Yang Wang ◽  
Xiaobin Yue ◽  
Yifei Li ◽  
Zhengyi Jiang
Keyword(s):  
Load Capacity ◽  
Dynamic Stiffness ◽  
Dynamic Performance ◽  
Dynamic Mesh ◽  
Eccentricity Ratio ◽  
Thrust Bearings ◽  
Stiffness And Damping Coefficient ◽  
Stiffness And Damping ◽  
The Impact ◽  
The Relationship

This study utilizes a dynamic mesh technology to investigate the dynamic performance of aerostatic thrust bearings with orifice restrictor, multiple restrictors, and porous restrictor. An experiment, which investigates the bearing static load capacity, was carried out to verify the calculation accuracy of dynamic mesh technology. Further, the impact of incentive amplitude, incentive frequency, axial eccentricity ratio, and non-flatness on the bearing dynamic performance was also studied. The results show incentive amplitude effect can be ignored at the condition of amplitude less than 5% film thickness, while the relationship between dynamic characteristics and incentive frequency presented a strong nonlinear relationship in the whole frequency range. The change law of dynamic stiffness and damping coefficient for porous restrictor was quite different from orifice restrictor and multiple restrictors. The bearing dynamic performance increased significantly with the growth of axial eccentricity ratio, and the surface non-flatness enhanced dynamic performance of aerostatic thrust bearings.

Effects of Increased Tooth Clearance on the Performance of a Labyrinth Seal With Oil-Rich Bubbly Laminar Flow

2021 ◽  
Author(s):  
Min Zhang ◽  
Dara W. Childs
Keyword(s):  
Smooth Surface ◽  
Volume Fraction ◽  
Silicone Oil ◽  
Bubbly Flow ◽  
Labyrinth Seal ◽  
Test Fluid ◽  
Axial Thrust ◽  
Rotordynamic Coefficients ◽  
Piston Seal ◽  
The Impact

Abstract In recent years, multiphase pumps have become more and more popular because of the capability to simplify the process, reduce the footprint, and lower the cost. To compensate for the axial thrust force, an annular seal is normally used as a balance piston seal, and the labyrinth seal is one of the choices. A typical labyrinth seal consists of a surface with teeth and a smooth surface. The teeth are either on the rotor or the stator. To protect the machine, one side (either the teeth or the smooth surface) is made of a material that can be safely sacrificed during a rub. After the rub, the teeth clearance is increased. This paper studies the impact of the increased teeth clearance on the performance of the labyrinth seal under oil-rich bubbly flow conditions. The test fluid is a mixture of silicone oil (PSF 5cSt) and air with inlet Gas Volume Fraction GVF up to 9%. Tests are conducted with pressure drop PD = 34.5 bars, rotor speed ω = 5 krpm, and radial tooth clearance Cr = 0.102 mm and 0.178 mm. Test results show that, for all test conditions (before and after injecting air bubbles into the oil flow), increasing Cr from 0.102 mm to 0.178 mm increases the mass flow rate by about 40% but barely changes the test seal’s rotordynamic coefficients; i.e., the increased tooth clearance would not change the pump vibration performance.

A Study on the Leakage and Rotordynamic Performance of a Long Labyrinth Seal Under Mainly Air Conditions

2019 ◽  
Vol 141 (12) ◽  
Author(s):  
Min Zhang ◽  
Dara W. Childs
Keyword(s):  
Silicone Oil ◽  
Dynamic Stiffness ◽  
Excitation Frequency ◽  
Labyrinth Seal ◽  
System Stability ◽  
Test Fluid ◽  
Liquid Volume ◽  
Frequency Dependent ◽  
Frequency Independent ◽  
The Impact

Abstract This paper investigates the impact of liquid presence in air on the leakage and rotordynamic coefficients of a long (length-to-diameter ratio L/D = 0.747) teeth-on-stator labyrinth seal. The test fluid is a mixture of air and silicone oil (PSF-5cSt). Tests are carried out at inlet pressure Pi = 62.1 bars, three pressure ratios from 0.21 to 0.46, three speeds from 10 to 20 krpm, and six inlet liquid volume fractions (LVFs) from 0% to 15%. Complex dynamic-stiffness coefficients Hij are measured. The real parts of Hij are too frequency dependent to be fitted by frequency-independent stiffness and virtual-mass coefficients. Therefore, this paper presents frequency-dependent direct stiffness KΩ and cross-coupled stiffness kΩ. The imaginary parts of Hij produce frequency-independent direct damping C. Test results show that, under both pure- and mainly air conditions, the leakage mass flowrate m˙ of the test seal steadily increases as inlet LVF increases. KΩ is negative under all test conditions, and the magnitude of KΩ increases as inlet LVF increases, leading to a larger negative centering force on the associated compressor rotor. Under pure-air conditions, kΩ is a small negative value. Injecting oil into the air increases kΩ slightly and make the magnitude of kΩ closer to zero. Under mainly air conditions, increasing inlet LVF from 2% to 15% has little impact on kΩ. C normally increases as inlet LVF increases. The value of the effective damping Ceff = C − kΩ/Ω near 0.5ω is of significant interest to the system stability since an unstable centrifugal compressor may precess at approximately 0.5ω. Ω denotes the excitation frequency. The oil presence in the air has little impact on the value of Ceff near 0.5ω. Also, the liquid presence does not change the insensitiveness of m˙, KΩ, kΩ, C, and Ceff to change in ω; i.e., under both pure- and mainly air conditions, changes in ω has little impact on m˙, KΩ, kΩ, C, and Ceff.

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