Measurements Versus Predictions for Rotordynamic and Leakage Characteristics of a Convergent-Tapered, Honeycomb-Stator/Smooth-Rotor Annular Gas Seal

2008 ◽  
Author(s):  
Daniel E. van der Velde ◽  
Dara W. Childs
Keyword(s):  
Control Volume ◽  
Excitation Frequency ◽  
Test Fluid ◽  
Low Frequencies ◽  
Rotordynamic Coefficients ◽  
Stiffness Coefficients ◽  
Direct Stiffness ◽  
Honeycomb Seals ◽  
Excitation Frequencies ◽  
The Cross

Measured results are presented for rotordynamic coefficients and leakage rates for two honeycomb-stator seal geometries, a convergent-tapered honeycomb seals (CTHC) and a constant-clearance honeycomb seals (CCHC) tested by Sprowl and Childs in 2007. The rotor diameter was 114.3 mm (4.500 in). The CTHC seals had inlet and exit clearances of 0.334 and 0.204 mm, respectively. The CCHC seal had a constant clearance of 0.204 mm. Honeycomb cells had depths of 3.175 mm (0.125 in) and widths of 0.79 mm (0.031 in). Measurements are reported with air as the test fluid, zero preswirl, ω = 20,200 rpm, a supply pressure of 69 bar (1,000 psi) and supply temperature of 18°C (64.4°F) for both seal geometries. The test pressure ratios are 0.5 for the CCHC seal, and 0.46 for the CTHC seal. The tapered seal leaks about 20% more than the constant-clearance seal. Measured and predicted dynamic coefficients are strong functions of excitation frequency. The measured direct stiffness coefficient was higher for the tapered seal at all excitation frequencies, including a projection to zero frequency, where the CCHC seal was on the order of −2MN/m versus roughly +13MN/m for the tapered seal. The CTHC seal had higher cross-coupled stiffness coefficients than the CCHC seal at all excitation frequencies. The CCHC and CTHC seals had comparable direct damping out to ∼80 Hz. For higher excitation frequencies, the CTHC seal had larger direct damping values. The effective damping Ceff combines the positive effect of direct damping and the destabilizing effect of cross-coupled-stiffness coefficients. It is negative at low frequencies and becomes positive for higher frequencies. The frequency at which it changes sign is called the cross-over frequency. The CCHC had a lower cross-over frequency (better from a stability viewpoint) and higher Ceff values out to ∼80 Hz. At higher excitation frequencies from ∼120Hz onward, the tapered seal has higher effective damping values. Kleynhans and Childs’ 1997 two-control-volume model did a generally good job of predicting the direct stiffness coefficients of both seals. It closely predicted the cross-coupled stiffness coefficients for the CCHC seal but substantially under predicted the values for the CTHC seal. It under predicted the direct damping for the CCHC seal at frequencies below ∼120Hz, but did a good job for higher frequencies. It under predicted direct damping for the CTHC seal at all frequencies. For the CCHC seal, the model did a good job of predicting Ceff at all frequencies and also accurately predicted the cross-over frequency. For the CTHC seal, the model accurately predicted the cross-over frequency but over predicted Ceff below the cross-over frequency (the seal was more destabilizing than predicted) and under predicted Ceff at higher frequencies.

Measurement Versus Predictions of Rotordynamic Coefficients of a Hole-Pattern Gas Seal With Negative Preswirl

2012 ◽  
Author(s):  
Philip D. Brown ◽  
Dara W. Childs
Keyword(s):  
Control Volume ◽  
Pressure Ratio ◽  
Excitation Frequency ◽  
Ideal Gas ◽  
Rotor Speed ◽  
Rotordynamic Coefficients ◽  
Damping Coefficients ◽  
The Cross ◽  
Gas Seals ◽  
Stiffness And Damping

Test results are presented for rotordynamic coefficients of a hole-pattern annular gas seals at supply pressures to 84 bar and running speeds to 20,200 RPM. The principal test variable of interest was negative preswirl. Preswirl signifies the circumferential fluid flow entering a seal, and negative preswirl indicates a fluid swirl in a direction opposite to rotor rotation. The influences of pressure ratio and rotor speed were also investigated. Measured results produce direct and cross-coupled stiffness and damping coefficients that are a function of excitation frequency Ω. Changes in pressure ratio had only small effects on most rotordynamic coefficients. Cross-coupled stiffness showed slightly different profiles through the mid-range of Ω values. Increasing rotor speed significantly increased cross-coupled stiffness and cross-coupled damping. At 10,200 RPM, high negative inlet preswirl produced negative cross-coupled stiffness over an excitation frequency range of 200–250 Hz. Negative preswirl did not affect direct stiffness and damping coefficients. Effective damping combines the stabilizing effect of direct damping and the destabilizing effect of cross-coupled stiffness. The cross-over frequency is the precession frequency where effective damping transitions from a negative value to a positive value with increasing frequency. At 20,200 RPM with a pressure ratio of 50%, peak effective damping was increased by 50%, and the cross-over frequency was reduced by 50% for high-negative preswirl versus zero preswirl. Hence, reverse swirl can greatly enhance the stabilizing capacity of hole-pattern balance-piston or division-wall seals for compressors. A two-control-volume model that uses the ideal gas law at constant temperature was used to predict rotordynamic coefficients. The model predicted direct rotordynamic coefficients well, but substantially under predicted cross-coupled rotordynamic coefficients especially at high negative preswirls.

Experimental Study of the Static and Dynamic Characteristics of a Long Smooth Seal With Two-Phase, Mainly-Air Mixtures

2017 ◽  
Author(s):  
Min Zhang ◽  
James E. Mclean ◽  
Dara W. Childs
Keyword(s):  
Dynamic Stiffness ◽  
Excitation Frequency ◽  
Test Fluid ◽  
Radial Clearance ◽  
Test Cases ◽  
Test Results ◽  
Rotordynamic Coefficients ◽  
Stiffness Coefficients ◽  
Rate Of Increase ◽  
Annular Seal

A 2-phase annular seal stand (2PASS) has been developed at the Turbomachinery Laboratory of Texas A&M University to measure the leakage and rotordynamic coefficients of division wall or balance-piston annular seals in centrifugal compressors. 2PASS was modified from an existing pure-air annular seal test rig. A special mixer has been designed to inject the oil into the compressed air, aiming to make a homogenous air-rich mixture. Test results are presented for a smooth seal with an inner diameter D of 89.306 mm, a radial clearance Cr of 0.188 mm, and a length-to-diameter ratio L/D of 0.65. The test fluid is a mixture of air and Silicone oil (PSF-5cSt). Tests are conducted with inlet LVF = 0%, 2%, 5%, and 8%, shaft speed ω = 10, 15, and 20 krpm, and pressure ratio PR = 0.43, 0.5, and 0.57. The test seal is concentric with the shaft (centered), and the inlet pressure is 62.1 bars. Complex dynamic stiffness coefficients are measured for the seal. The real parts are generally too dependent on excitation frequency Ω to be modeled by constant stiffness and virtual mass coefficients. The direct real dynamic stiffness coefficients are denoted as KΩ; the cross-coupled real dynamic stiffness coefficients are denoted as kΩ. The imaginary parts of the dynamic stiffness coefficients are modeled by frequency-independent direct C and cross-coupled c damping coefficients. Test results show that the leakage and rotordynamic coefficients are remarkable impacted by changes in inlet LVF. Leakage mass flow rate ṁ drops slightly as inlet LVF increases from zero to 2%, and then increases with further increasing inlet LVF to 8%. As inlet LVF increases from zero to 8%, KΩ generally decreases except it increases as inlet LVF increases from zero to 2% when PR = 0.43. kΩ increases virtually with increasing inlet LVF from zero to 2%. As inlet LVF further increases to 8%, kΩ decreases or remains unchanged. C increases as inlet LVF increases; however, its rate of increase drops significantly at inlet LVF = 2%. Effective damping Ceff combines the stabilizing impact of C and the destabilizing impact of kΩ. Ceff is negative (destabilizing) for lower Ω values and becomes more destabilizing as inlet LVF increases from zero to 2%. It then becomes less destabilizing as inlet LVF is further increased to 8%. Measured ṁ and rotordynamic coefficients are compared with predictions from XLHseal_mix, a program developed by San Andrés [1] based on a bulk-flow model, using the Moody wall-friction model while assuming constant temperature and a homogenous mixture. Predicted ṁ values are close to measurements when inlet LVF = 0 and 2%, and are larger than measured values when inlet LVF = 5% and 8%. As with measurements, predicted ṁ drops slightly as inlet LVF increases from zero to 2%, and then increases with increasing inlet LVF further to 8%. However, in the inlet LVF range of 2∼8%, the predicted effects of inlet LVF on ṁ are weaker than measurements. XLHseal_mix poorly predicts KΩ in most test cases. For all test cases, predicted KΩ decreases as inlet LVF increases from zero to 8%. The increase of KΩ induced by increasing inlet LVF from zero to 2% at PR = 0.43 is not predicted. C is reasonably predicted, and predicted C values are consistently smaller than measured results by 14∼34%. Both predicted and measured C increase as inlet LVF increases. kΩ and Ceff are predicted adequately at pure-air conditions, but not at most mainly-air conditions. The significant increase of kΩ induced by changing inlet LVF from zero to 2% is predicted. As inlet LVF increases 2% to 8%, predicted kΩ continue increasing versus that measured kΩ typically decreases. As with measurements, increasing inlet LVF from zero to 2% decreases the predicted negative values of Ceff, making the test seal more destabilizing. However, as inlet LVF increases further to 8%, the predicted negative values of Ceff drops versus measured values increase. For high inlet LVF values (5% and 8%), the predicted negative values of Ceff are smaller than measurements. So, the seal is actually more stable than predicted for high inlet LVF cases.

Rotordynamic Performance of a Negative-Swirl Brake for a Tooth-on-Stator Labyrinth Seal

2014 ◽  
Author(s):  
Dara W. Childs ◽  
James E. Mclean ◽  
Min Zhang ◽  
Stephen P. Arthur
Keyword(s):  
Labyrinth Seal ◽  
Test Results ◽  
Labyrinth Seals ◽  
Rotordynamic Coefficients ◽  
Stiffness Coefficients ◽  
Shaft Rotation ◽  
Direct Stiffness ◽  
The Cross ◽  
Inlet Conditions ◽  
The Stability

In the late 1970’s, Benckert and Wachter (Technical University Stuttgart) tested labyrinth seals using air as the test media and measured direct and cross-coupled stiffness coefficients. They reported the following results: (1) Fluid pre-swirl in the direction of shaft rotation creates destabilizing cross-coupled stiffness coefficients, and (2) Effective swirl brakes at the inlet to the seal can markedly reduce the cross-coupled stiffness coefficients, in many cases reducing them to zero. In recent years, “negative-swirl” swirl brakes have been employed that attempt to reverse the circumferential direction of inlet flow, changing the sign of the cross-coupled stiffness coefficients and creating stabilizing stiffness forces. This study presents test results for a 16-tooth labyrinth seal with positive inlet preswirl (in the direction of shaft rotation) for the following inlet conditions: (1) No swirl brakes, (2) Straight, conventional swirl brakes, and (3) Negative-swirl swirl brakes. The negative-swirl swirl-brake designs were developed based on CFD predictions. Tests were conducted at 10.2, 15.35, and 20.2 krpm with 70 bars of inlet pressure for pressure ratios of 0.3, 0.4, 0.5. Test results include leakage and rotordynamic coefficients. In terms of leakage, the negative-swirl brake configuration leaked the least, followed by the conventional brake, followed by the no-brake design. Normalized to the negative-swirl brake configuration, the conventional-brake and no-brake configurations mass flow rate were greater, respectively, by factors of 1.04 and 1.09. The direct stiffness coefficients are negative but small, consistent with past experience. The conventional swirl brake drops the destabilizing cross-coupled stiffness coefficients k by a factor of about 0.8 as compared to the no-brake results. The negative-swirl brake produces a change in sign of k with an appreciable magnitude; hence, the stability of forwardly-precessing modes would be enhanced. In descending order, the direct damping coefficients C are: no-swirl, negative-swirl, conventional-swirl. Normalized in terms of the no-swirl case, C for the negative and conventional brake designs are, respectively, 0.7 and 0.6 smaller. The effective damping Ceff combines the effect of k and C. Ceff is large and positive for the negative-swirl configuration and near zero for the no-brake and conventional-brake designs. The present results for a negative-brake design are very encouraging for both eye-packing seals (where conventional swirl brakes have been previously employed) and division-wall and balance-piston seals where negative shunt injection has been employed.

Prediction of Rotordynamic Performance of Smooth Stator-Grooved Rotor Liquid Annular Seals Utilizing Computational Fluid Dynamics

2017 ◽  
Vol 140 (3) ◽  
Author(s):  
Farzam Mortazavi ◽  
Alan Palazzolo
Keyword(s):  
Fluid Dynamics ◽  
Computational Fluid Dynamics ◽  
Excitation Frequency ◽  
Computationally Efficient ◽  
Operational Conditions ◽  
Rotordynamic Coefficients ◽  
Stiffness Coefficients ◽  
Liquid Seal ◽  
Frequency Ratios ◽  
Annular Seals

Circumferentially grooved, annular liquid seals typically exhibit good whirl frequency ratios (WFRs) and leakage reduction, yet their low effective damping can lead to instability. The current study investigates the rotordynamic behavior of a 15-step groove-on-rotor annular liquid seal by means of computational fluid dynamics (CFD), in contrast to the previous studies which focused on a groove-on-stator geometry. The seal dimensions and working conditions have been selected based on experiments of Moreland and Childs (2016, “Influence of Pre-Swirl and Eccentricity in Smooth Stator/Grooved Rotor Liquid Annular Seals, Measured Static and Rotordynamic Characteristics,” M.Sc. thesis, Texas A&M University, College Station, TX). The frequency ratios as high as four have been studied. Implementation of pressure-pressure inlet and outlet conditions make the need for loss coefficients at the entrance and exit of the seal redundant. A computationally efficient quasi-steady approach is used to obtain impedance curves as functions of the excitation frequency. The effectiveness of steady-state CFD approach is validated by comparison with the experimental results of Moreland and Childs. Results show good agreement in terms of leakage, preswirl ratio (PSR), and rotordynamic coefficients. It was found that PSR will be about 0.3–0.4 at the entrance of the seal in the case of radial injection, and outlet swirl ratio (OSR) always converges to values near 0.5 for current seal and operational conditions. The negative value of direct stiffness coefficients, large cross-coupled stiffness coefficients, and small direct damping coefficients explains the destabilizing nature of these seals. Finally, the influence of surface roughness on leakage, PSR, OSR, and stiffness coefficients is discussed.

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.

scholarly journals Annular Honeycomb Seals: Test Results for Leakage and Rotordynamic Coefficients; Comparisons to Labyrinth and Smooth Configurations

1989 ◽  
Vol 111 (2) ◽  
pp. 293-300 ◽  
Author(s):  
D. Childs ◽  
D. Elrod ◽  
K. Hale
Keyword(s):  
Labyrinth Seal ◽  
Test Results ◽  
Rotordynamic Coefficients ◽  
Damping Coefficients ◽  
Stiffness Coefficients ◽  
Shaft Rotation ◽  
Honeycomb Seal ◽  
Cell Dimensions ◽  
Honeycomb Seals ◽  
Rotordynamic Coefficient

Test results are presented for leakage and rotordynamic coefficients for seven honeycomb seals. All seals have the same radius, length, and clearance; however, the cell depths and diameters are varied. Rotordynamic data, which are presented, consist of the direct and cross-coupled stiffness coefficients and the direct damping coefficients. The rotordynamic-coefficient data show a considerable sensitivity to changes in cell dimensions; however, no clear trends are identifiable. Comparisons of test data for the honeycomb seals with labyrinth and smooth annular seals shows the honeycomb seal had the best sealing (minimum leakage) performance, followed in order by the labyrinth and smooth seals. For prerotated fluids entering the seal, in the direction of shaft rotation, the honeycomb seal has the best rotordynamic stability followed in order by the labyrinth and smooth. For no prerotation, or fluid prerotation against shaft rotation, the labyrinth seal has the best rotordynamic stability followed in order by the smooth and honeycomb seals.

An Extended Three-Control-Volume Theory for Circumferentially-Grooved Liquid Seals

1996 ◽  
Vol 118 (2) ◽  
pp. 276-285 ◽  
Author(s):  
O. R. Marquette ◽  
D. W. Childs
Keyword(s):  
Control Volume ◽  
Current Theory ◽  
Flow Section ◽  
Leakage Flow ◽  
Centrifugal Pumps ◽  
Rotordynamic Coefficients ◽  
Stiffness Coefficients ◽  
Through Flow ◽  
Control Volumes ◽  
Theoretical Results

Circumferentially-grooved seals are used in centrifugal pumps to reduce leakage flow. They can also have a significant impact on pump rotordynamic characteristics. Florjancic (1990) developed an analysis for leakage and rotordynamic coefficients, using a partition of the seal into three control volumes. This paper presents a new theory, based on an extension of Florjancic’s work (1990) for circumferentially-grooved liquid seals. The current theory differs from Florjancic’s analysis in the retention of transfer momentum terms and the introduction of diverging flow in the through-flow section within a seal groove. Validation of the new analysis is achieved through a comparison with existing experimental data taken from Kilgore (1988), and Florjancic (1990). Theoretical results are reasonable and consistent; i.e., a modification in the seal parameters induces a correct evolution of the rotordynamic coefficients. Direct and cross-coupled stiffness coefficients are slightly underpredicted, whereas the direct damping coefficient is underpredicted within 40 percent. Leakage flow predictions are very good.

scholarly journals Static and Rotordynamic Analysis of a Plain Annular (Liquid) Seal in the Laminar Regime With a Swirl Brake for Three Clearances

2019 ◽  
Vol 141 (8) ◽  
Author(s):  
Ovais Ahmed Bin Najeeb ◽  
Dara W. Childs
Keyword(s):  
Dynamic Force ◽  
Laminar Regime ◽  
Dynamic Measurements ◽  
Stiffness Coefficients ◽  
Stationary Coordinate System ◽  
Liquid Seal ◽  
Direct Stiffness ◽  
The Cross ◽  
Positive Direct ◽  
San Andres

Tests are reported for a smooth seal with radial clearances 127 μm, 254 μm, 381 μm (1×, 2×, and 3×); length 45.72 mm, diameter 101.6 mm. An insert induced upstream preswirl. Swirl brakes (SBs), comprising 36 square cuts with axial depth 5.08 mm, radial height 6.35 mm, and circumferential width 6.35 mm each. Static and rotordynamic data were produced at ω = 2, 4, 6, 8 krpm, ΔP = 2.07, 4.14, 6.21, 8.27 bar, and eccentricity ratios ε0 = e0/Cr = 0.00, 0.27, 0.53, and 0.80. ISO VG 46 oil at a range of 46–49 °C was used, netting laminar flow (total Re ≤ 650). Dynamic measurements included components of the following vectors: (a) stator–rotor relative displacements, (b) acceleration, and (c) applied dynamic force in a stationary coordinate system. SBs were effective at the 3× clearance only. With the 3× seal, the cross-coupled stiffness coefficients have the same sign (not destabilizing). However, the seal has a negative direct stiffness K that could potentially “suck” the rotor into contact with the stator wall, along with dropping the pump rotor's natural frequency, further reducing its dynamic stability. Measurements were compared to predictions from a code by Zirkelback and San Andrés. Most predictions agree well with test data. Notable exceptions are the direct and cross-coupled stiffness coefficients for the 3× clearance. Predictions showed positive direct stiffness and opposite signs for the cross-coupled stiffness coefficients.

Experimental Analysis of Angled Injection Aerostatic Hybrid Bearings

2017 ◽  
Vol 140 (2) ◽  
Author(s):  
Julian Le Rouzic ◽  
Mihai Arghir
Keyword(s):  
Equations Of Motion ◽  
Cross Coupling ◽  
Impulse Turbine ◽  
Rotordynamic Coefficients ◽  
Counter Rotation ◽  
Theoretical Investigations ◽  
Coupling Stiffness ◽  
Direct Stiffness ◽  
The Cross ◽  
Dynamic Excitations

Counter-rotation angled injection employed for aerostatic hybrid bearings reduces the cross coupling stiffness that may lead to whirl–whip instabilities at high rotation speeds. The benefits of counter-rotation injection have been known for years. Theoretical investigations were performed for water or air fed hybrid bearings but experiments were conducted only for water fed bearings. The present work is the first effort dedicated to angled injection in air fed hybrid bearings. The tests were performed for a simple rotor supported by two identical hybrid bearings. The hybrid bearings are provided with small size, shallow pockets and are fed with air via counter-rotation-oriented orifice type restrictors. An impulse turbine fed with air entrains the rotor. An impact gun applies dynamic excitations and the rotordynamic coefficients are identified from the equations of motion of the rotor. Different air feeding pressures are tested as well as high rotational speeds. Compared to the dynamic characteristics of radial injection hybrid bearings, the direct stiffness of counter-rotation injection bearings has slightly lower values and the direct damping is higher but the main impact is the drastic reduction of the cross-coupling stiffness that may have even negative values.

Non-Axisymmetric Flows and Rotordynamic Forces in an Eccentric Shrouded Centrifugal Compressor: Part 1 — Measurement

2019 ◽  
Author(s):  
Jieun Song ◽  
Suyong Kim ◽  
Tae Choon Park ◽  
Bong-Jun Cha ◽  
Dong Hun Lim ◽  
...  
Keyword(s):  
Centrifugal Compressor ◽  
System Level ◽  
Labyrinth Seals ◽  
Stiffness Coefficients ◽  
Pressure Distributions ◽  
Order Of Magnitude ◽  
Direct Stiffness ◽  
The Cross ◽  
Rotordynamic Forces ◽  
Pressure Perturbations

Abstract Centrifugal compressors can suffer from rotordynamic instability. While individual components (e.g., seals, shrouds) have been previously investigated, an integrated experimental or analytical study at the compressor system level is scarce. For the first time, non-axisymmetric pressure distributions in a statically eccentric shrouded centrifugal compressor with eye-labyrinth seals have been measured for various eccentricities. From the pressure measurements, direct and cross-coupled stiffness coefficients in the shrouded centrifugal compressor have been determined. Thus, the contributions of the pressure perturbations in the shroud cavity and labyrinth seals have been simultaneously investigated. The cross-coupled stiffness coefficients in the shroud and labyrinth seals are both positive and one order of magnitude larger than the direct stiffness coefficients. Furthermore, in the tested compressor, contrary to the common assumption, the cross-coupled stiffness in the shroud is 2.5 times larger than that in the labyrinth seals. Thus, the shroud contributes more to rotordynamic instability than the eye-labyrinth seals.

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