Cross-coupled Stiffness Research Articles

Sensitivity Analysis of Scallop Damper Seal Design Parameters for Leakage and Static Performance

The leakage characteristics and static stiffness of scallop damper seals have a significant impact on rotor vibration and stability. A parameter sensitivity analysis model for geometrical parameters in scallop damper seals was developed using a design of experiments (DOE) approach. The method employed a central composite design, integrating factorial, axial, and center points to assess non-linear effects efficiently. And the effects of radial clearance, cavity depth, and length–diameter ratios on leakage performance and rotor stability were investigated. The leakage rate, flow-induced force, and static stiffness coefficient for 15 different combinations of geometric parameters at eccentricities of 0.2 and 0.4 were numerically calculated. The results show that eccentricity has little effect on leakage and its parameter sensitivity. Larger cavity depths and length–diameter ratios are beneficial for seal leakage performance. The tangential force increases with increasing eccentricity but decreases with increasing radial clearance, while it first decreases and then increases with the increase in the cavity depth and length–diameter ratios. Additionally, the radial force decreases with the increase in the length-to-diameter ratio and increases first and then decreases with the increase in radial clearance. The parameter level in this study is defined as the ratio of the actual parameter value to the maximum parameter value. Static direct stiffness reaches its maximum value at a radial clearance level of 30.2%. It remains positive within a cavity depth range of 92.3~100%, as well as a length–diameter ratio range of 0~20.3%. The static cross-coupled stiffness gradually decreases with the increase in radial clearance but first decreases and then increases with the increase in the cavity depth or length–diameter ratio levels. The research results presented in this paper can serve as a reference for the analysis of the performance and design of scallop damper seals.

Nonlinear Dynamics of Whirling Rotor with Asymmetrically Supported Snubber Ring

Rotor–stator whirling is a critical malfunction frequently encountered in rotating machinery, often resulting in severe damages. This study investigates the nonlinear dynamics of a whirling rotor interacting with a snubber ring through numerical simulations that account for the stiffness asymmetries of the snubber ring. A two-degrees-of-freedom (DOF) model is employed to analyse the contact interactions that occurred between the rotor and the snubber ring, assuming a linear elastic contact model. The analysis also incorporates the static offset between the centers of the rotor and the snubber ring. The dynamic behaviour of the whirling system is characterised by pronounced nonlinearity due to transitions between contact and non-contact states. The model is first validated against our prior theoretical and experimental studies. The nonlinear responses of the rotor are analysed to evaluate the effects of stator asymmetry through various techniques, including time-domain waveforms, frequency spectra, rotor orbits, and bifurcation diagrams. Furthermore, the influence of varying system parameters, such as rotational speed and the damping ratio, both with and without stator asymmetry, are systematically analysed. The results demonstrate that the rubbing response is highly sensitive to small variations in system parameters, with stator asymmetry significantly affecting system behaviour, even at low asymmetry levels. Direct stiffness asymmetry is shown to have a more pronounced effect than cross-coupling stiffness. The system exhibits a range of dynamics, including periodic, quasi-periodic, and chaotic responses, with regions of periodic orbits coexisting with chaotic ones. Complex phenomena such as period doubling, period halving, and jump bifurcations are identified, alongside quasi-periodic and period doubling routes to chaos. These findings contribute to a deeper understanding of the nonlinear phenomena associated with rotor–stator whirling and provide valuable insights into the unique characteristics of rubbing faults, which could facilitate fault diagnosis.

Study on Dynamic Characteristics of Single-Span Rotor Under Cross-Coupling Stiffness Control on Non-drive End

Abstract The sealing-induced cross-coupling stiffness (CCS) often decreases the rotordynamic stability of turbomachinery. In our previous study, an active negative CCS control strategy applied between two bearings in the middle of the rotor was validated to enhance the stability of the rotor system. In this paper, a different approach was taken by applying both positive and negative CCS control strategies to the non-drive end, aiming to investigate their effects on rotordynamic stability using a rotordynamic model. The model represented a single-span centrifugal compressor rotor with CCS at the seal node. The logarithmic decrements and unbalance response of the rotor system were compared under different CCS applications at the non-drive end and in the span. The results demonstrate that applying both positive and negative CCS at the non-drive end can improve system stability, but the positive CCS strategy has limitations, and its applicability is not as extensive as Negative CCS. A “critical stiffness” phenomenon emerges when the rotor system is actively stabilized at the non-drive end, resembling critical resonance. Furthermore, when components like seals introduce cross-coupling stiffness out of the span, they contribute to system stabilization. These findings provide valuable insights into actively enhancing the stability of rotor systems.

Parametric instability analysis of rotors under anisotropic boundary conditions

Ensuring rotor stability is a major concern in engineering, as instabilities can lead to catastrophic failures. Existing literature shows that anisotropic boundary conditions significantly affect the parametric instability characteristics of rotors under periodical axial loads. However, there is little literature systematically analyzing the formation mechanism of parametric resonance under these boundary conditions or providing a detailed classification of the parametric instability regions. Therefore, this paper presents a comprehensive parametric instability analysis of a rotor subjected to periodic axial loads under anisotropic boundary conditions. A novel approach based on the multiple scales method is proposed to address anisotropy in the boundary conditions. Using this approach, the analytical boundaries of the parametric instability regions are derived, and a proof regarding the absence of certain parametric resonances is presented. These analytical solutions are validated by numerical results obtained from the discrete transition matrix method, which form the basis for systematically investigating the effects of anisotropy in direct or cross-coupling stiffness/damping coefficients on the rotor instability. The key scientific contributions of this work include: Deriving analytical instability boundaries, providing a more efficient alternative to purely numerical methods while maintaining high accuracy; Demonstrating the absence of parametric resonance of difference type under both isotropic or anisotropic boundary conditions; Discovering that anisotropy in stiffness coefficients can induce self-interaction within a given forward or backward whirl mode, as well as interaction between two forward or two backward whirl modes, leading to additional instability regions; Reducing anisotropy in direct damping coefficients may increase critical dynamic load coefficients, potentially enhancing rotor safety; If the cross-coupling stiffness coefficients exceed the threshold for triggering intrinsic instability, the rotor may become unstable in all operating conditions. All these findings offer insights into the stability management of rotors under various operating conditions and provide valuable guidance for designing and operating safer, more efficient rotor systems.

Analytical Determination of Stick–Slip Whirling Vibrations and Bifurcations in Rotating Machinery

In rotating machinery, aerodynamic forces and oil film forces often lead to cross-coupling stiffness. This paper is aimed at studying the stick-slip whirling vibrations induced by the piecewise smooth rotor/stator frictions in a modified flexible rotor subjected to cross-coupling stiffness. Governing equations determining the sliding region and boundaries of piecewise discontinuous friction are defined. This analytical study was conducted to discuss the complex vibrations and bifurcations. Various types of sliding motions (continuous pure rolling, continuous crossing, and grazing–sliding) were observed in this research. Further, as for discussing the impacts of the parameters on nonlinear sliding vibrations, a parametric study was conducted. The obtained results reveal that with an increase in the cross-coupling stiffness coefficient, continuous pure rolling occurs earlier, and the disk vibration time around the contact regime becomes shorter. For studying the self-excited backward whirling vibration of stick–slip nonlinear motions, analytical formulations are established. Detailed vibration amplitude and frequency studies of friction-induced backward whirling vibrations were carried out. Numerical simulations were performed to compare them with the analytical solutions and to validate the results as well. The proposed theory and results provide fresh perspectives for predicting friction-induced whirlings and creating proper designs for turbo machinery.

A comparative study on rotordynamics characteristics of hole-pattern damping seals with different cavity shapes

PurposeThe purpose of this paper is to improve the seal performance by proper design of the cavity shape of the damping holes, especially the rotordynamics characteristics of the hole-pattern damped seal (HPDS).Design/methodology/approachA new damping seal structure that comprises a circle-shaped cavity and two directional leaf-shaped cavities with a dovetail-shaped diversion groove is proposed. The comparative study on the sealing characteristics of dovetail-shape, leaf-shape and classical circular HPDSs was carried out using ANSYS CFX.FindingsThe dovetail-shaped HPDS significantly outperformed two other damping seal designs in leakage and rotordynamic performance. At a rotating speed of 7,500 rpm, it showed a 25% reduction in leakage, a 23% increase in average effective damping and a 119% increase in average effective stiffness. The cross-coupled stiffness Kxy shifted from positive to negative, reducing circumferential flow. The dovetail's inclined leaf-shaped grooves create a double vortex that slows jet velocity in the seal clearance and alters spiral flow direction, resulting in a uniform pressure distribution and enhanced rotor stability at low frequencies.Originality/valueThis study proposes a novel HPDS with dovetail-shaped diversion grooves. The seal can realize the simultaneous improvement of rotordynamics and leakage characteristics compared to the current seal structure.Peer reviewThe peer review history for this article is available at: https://publons.com/publon/10.1108/ILT-04-2024-0127/

Dynamic Performance of Liquid Smooth Annular Seal Operating in the Transition Regime

Abstract This paper presents the experimental leakage and rotordynamic performance for a liquid smooth annular seal operating in the transition regime. The test conditions include pressure differentials up to 64 bars with 1∼2 bar increments for 6 rotor speeds (2.5, 3.8, 5, 7.5, 8.8, and 10 krpm), as well as nonrotating rotor case under zero preswirl condition. The rotordynamic coefficients for all the test conditions are obtained by pseudo-random excitation of the seal at multiple subsynchronous frequencies. By considering the transition Reynolds number (1000 < Re < 3000) and the Taylor Number (Ta) versus Axial Reynolds Number (Rez), the variations in the direct stiffness coefficients (K) can used as an indicator of the flow regime transition boundaries. The direct stiffness K resulting from the Lomakin and hydrodynamic effects significantly drops until Rez reaches ∼1500. For higher Rez, K increases mainly due to hydrodynamic effects. When K drops, the cross-coupled stiffness k, the direct damping C and the cross-coupled virtual mass m increase while the cross-coupled damping c and virtual mass M decrease. None of predictions based on either laminar or turbulent flow show the variations in rotordynamic coefficients obtained from experimental results. The leakage is not highly influenced by rotor speeds for low speed cases crossing laminar boundary as ΔP increases, however, results for higher speeds in the superlaminar region show reduced leakage rates as rotor speed increases.

Enhanced rotordynamic performance under various structural arrangements of hole diaphragm labyrinth seal

In this study, we study the influence of five drilling hole arrangements on the rotordynamic and leakage performance of hole diaphragm labyrinth seal (HDLS) type seals with precise CFD analysis. Key findings reveal that hole distance significantly affects direct and cross-coupled stiffness, thereby impacting stability and leakage. Notably, the smallest hole in rectangle four holes(RF)-HDLS does not yield the minimum leakage. Vortex structure analysis, employing the Q-criterion, elucidate horizontal two holes(HT)-HDLS's superior leakage reduction owing to enhanced turbulence eddy dissipation. Seals with larger hole diameters, down single hole(DS)-HDLS and up single hole(US)-HDLS, demonstrate stable rotordynamic performance, with DS-HDLS exhibiting a notable 106.3% improvement in cross-coupled stiffness. In contrast, seals with holes positioned farther away, DS-HDLS, US-HDLS and vertical two holes(VT)-HDLS, experienced increased leakage with whirl frequency. HT-HDLS and RF-HDLS displayed a decrease, attributed to the stronger centrifugal effect. Comparing to center single hole(CS)-HDLS, VT-HDLS and US-HDLS achieved a significant 159.3% direct stiffness improvement and 432.7% reduction in cross-coupled stiffness. Despite a maximum 4.2% leakage increase, this substantial enhancement in rotor stability should not be overlooked. Moreover, all seals exhibited lower leakage rates than the traditional labyrinth seal.

Rotordynamic characteristics of a novel labyrinth seal with partition walls and helical groove teeth

To improve the stability of the conventional labyrinth seal (LS), in this paper, four novel fully partitioned helically labyrinth seals (FPHGLS) were designed and they were in comparison with one FPPGLS. The influences of the preswirl ratio and the helical groove pitch on the leakage flow and rotordynamic characteristics were numerically investigated, using the transient computational fluid dynamics (CFD) method based on the multi-frequency elliptical whirling orbit model. The accuracy and availability of the present transient numerical method were demonstrated based on the experiment data. The results show that the partition walls design can significantly increase the direct damping and cross-coupling stiffness for labyrinth seals and the helical teeth design can significantly decrease the cross-coupling stiffness and tangential force. When the helical groove pitch is equal to the seal length, the leakage nearly remains unchanged. Compared to the baseline design (LS), the FPPGLS and the FPHGLS have similar and significantly larger direct stiffness and direct damping. The two designs possess positive direct stiffness throughout the frequency range. The FPPGLS and FPHGLS possess significantly higher direct damping (∼323.7% larger than LS). But the FPPGLS possesses the largest cross-coupling stiffness among three seals at two preswirl ratios. From preswirl ratio = 0.13–0.84, the cross-coupling stiffness of the FPHGLS decreases by 53.4-310.1% compared with the FPPGLS. Increasing the helical groove pitch increases both direct stiffness and direct damping and reduces cross-coupling stiffness, but it also leads to greater leakage losses. In general, the novel FPHGLS whose helical groove pitch is equal to seal length possesses superior rotordynamic characteristics and similar leakage characteristic. This work provides the reference of the seal design and safety operation for the turbomachinery.

Effect of Air-Liquid Homogeneous Mixture on the Linear Dynamic Characteristics of a Hydrostatic Journal Bearing

Abstract Water lubricated hydrostatic journal bearings are widely used to support radial load of pumps in nuclear power plants. In some situations, these pumps may be required to operate with a mixture of water and air. This compressible air–water mixture will alter the dynamic characteristics of the hydrostatic journal bearings and can have a significant impact on the vibration behavior of the entire pump shaft line. This work is a parametrical investigation of the rotordynamic characteristics of these bearings relative to the volume fraction of the ingested air. The compressibility, the density, and the viscosity of the air–water mixture depend on this volume fraction. In order to evaluate this influence, a homogeneous air-liquid mixture model were developed. Moreover, for low viscosity lubricants, inertia effects can also have an important impact on bearing rotordynamic coefficients. Therefore, the flow is modeled first with the Reynolds equation and then with the bulk flow system of equations to evaluate this impact. The results show the variation of the direct and cross-coupling stiffness and of the direct damping with the evolution of inertial effects and of the level of air contamination.

Computational Fluid Dynamics Analysis and Experimental Results for the Dynamic Performance of Two Long Smooth Surface Annular Seals Operating With a Liquid in Air Mixture

Abstract Compressors in subsea oil and gas production must handle wet gases to reliably operate for extended periods of time. Annular clearance seals contribute to compressor performance and do affect system rotordynamic stability. Prior experimental work with two smooth surfaces, uniform clearance seals supplied with a light oil in air mixture and undergoing similar operating conditions produced direct stiffnesses (K) with distinct trends as the liquid content increased to 8% in volume. Both seals differ in length and diameter albeit having similar radial clearance. Other force coefficients for both seals, namely, cross-coupled stiffness (k) and direct damping (C) increase as the inlet liquid volume fraction (LVF) grows. Rationale for the peculiar differences in centering stiffness (K) is missing. Hence, a computational fluid dynamics (CFD) model and its predictions, the thrust of this paper, unveil flow field details (pressure, velocity fields, and liquid content evolution) for the oil in air mixture. Besides the CFD model, an enhanced bulk-flow model (BFM) also predicts the seals' leakage and dynamic force coefficients. Both models predict through flows agreeing well with the measured ones, the maximum difference is less than 16%. The BFM direct stiffness (K) does reproduce closely the experimental K whereas the direct damping coefficient (C) is up to ∼41% lower than the test result. The CFD model captures the variation trend of K versus inlet LVF for the first seal, albeit its magnitude is thrice the experimental stiffness. The CFD C agrees well with the test data for both seals, the largest difference is less than 10%. In spite of the complexity of the CFD model, significant differences with the experimental results persist, in particular for K. When considering the seal inlet corner as round, the CFD model produces a significant reduction in K to better approach the test result for a seal supplied with air. Attention to the seal geometry is paramount to produce accurate predictions.

Computational Fluid Dynamics Prediction of Rotordynamic Coefficients for Grooved Seals: Model and Numerical Sensitivity

Abstract Computational fluid dynamics (CFD) studies for annular seals continues to gain popularity as a powerful tool for seal rotordynamic analysis, but the wide variety in setup and modeling choices without sufficient justification prevent more widespread use of these methods. This study applies the quasi-steady (QS) method for rotordynamic coefficient prediction to an incompressible grooved seal model in order to rigorously quantify the influence of various prominent modeling choices on prediction results. Variation of the upstream region configuration confirms the dependence of the cross-coupled stiffness on the development of circumferential velocity, which is a strong function of upstream geometry. An axial inlet upstream region is shown to be a sufficient approximation to a radial inlet with axial symmetry, representative of a back-to-back seal configuration. Significant stiffness variation is observed for particular downstream region configurations, though the additional pressure perturbation is mostly confined to the downstream region itself. When no downstream region is included, sensitivity to the amount of pressure profile blending enforced at the seal outlet plane is demonstrated, further underscoring the need for an experimentally accurate downstream geometry. This is the first paper dedicated to a quantitative sensitivity investigation of this nature specific to incompressible grooved seals that examines upstream and downstream region configuration, rotor centrifugal growth effects, and modeling whirl amplitude. Use of these results will foster more accurate application of the QS method for rotordynamic predictions of incompressible grooved seals, ultimately enabling more widespread use of CFD methods for general seals predictions.

Numerical Investigation on the Leakage and Rotordynamic Characteristics for a Hole-Pattern Seal in Wet-Gas Conditions

Abstract For turbomachinery working in wet-gas conditions, liquid-phase fluid may worsen the rotordynamic characteristic of an annular seal, which induces a subsynchronous vibration problem and destabilizes the rotor-bearing system. The hole-pattern seal, demonstrated as effective to eliminate synchronous or subsynchronous vibrations for gas turbomachinery, is an ideal seal scheme to increase rotor stability and liquid tolerance capability of wet-gas turbomachinery. In this paper, the leakage and rotordynamic characteristics of a hole-pattern seal are numerically investigated under wet-gas conditions, using a three-dimensional transient CFD-based perturbation method. The accuracy and reliability of the present numerical method are demonstrated based on published experimental data. The rotordynamic force coefficients are presented and compared for the wet-gas hole-pattern seal with various inlet liquid volume fractions (LVF = 0%–20%), rotor speeds (ω = 0–20 krpm), inlet preswirl ratios (Sr = −0.2–0.5), and pressure ratios (Pr = 0.3–0.7). Numerical results show that the hole-pattern seal possesses desired tolerance capability for high inlet liquid volume fraction (LVF) of up to 20%. With inlet LVF increasing from 0 to 20%, the effective damping of the hole-pattern seal increases by about 50%, suggesting an improvement in rotor stability. The leakage flow rate of the oil-air mixture increases by 97.5%, combined with the sharply increasing oil leakage flow rate (by 636%) and decreasing air leakage flow rate (by 40%). The increasing rotor speed and inlet preswirl ratio both result in an obvious increase (by 50%) in the cross-coupled stiffness, yielding a smaller effective damping and worse rotor stability. With the increase in pressure ratio, all the rotordynamic force coefficients show a weaker frequency dependency and smaller magnitudes. The swirl velocity in the seal clearance can cause an accumulation of the liquid component in the hole cavities. With the increase of swirl velocity, more liquid component accumulates in the hole cavities, and the main accumulation position gradually moves upstream.

Experimental Force Coefficients for a Fully-Partitioned Pocket Damper Seal and Comparison to Other Two Seal Types

Abstract Labyrinth seals (LSs), honeycomb seals (HCSs) and hole-pattern seals, pocket damper seals (PDSs), and fully-partitioned damper seals (FPDSs) serve as balance pistons on the discharge side of barrel-type centrifugal compressors. These seals, facing large pressure differentials along with significant changes in gas density, produce sizeable lateral forces that impact compressor stability and rotordynamic performance. There is extensive archival data on the dynamic forced performance of LS and textured surface seals. However, the experimental database for a FPDS is insufficient, thus a comprehensive comparison of their dynamic force performance against that of textured surface seals is missing. The paper presents experimentally derived rotordynamic force coefficients for a FPDS along with a direct comparison to published data for a HCS and a LS, both similar in size and in operating conditions. The dynamic load tests with the FPDS include operation at a shaft speed (Ω) of 10 krpm (rotor surface speed of 60 m/s) while supplied with air at Pin = 70 bar and Tin =10 °C, and a discharge at Pout = 0.25, 0.5, and 0.65 of Pin. The preswirl circumferential velocity at the seal inlet is low, approximately 6% of rotor surface speed. The FPDS having eight pockets × eight axial blades produces a direct stiffness (K) increasing with excitation frequency (ω), though K < 0 for the lowest pressure ratio PR = (Pout/Pin) = 0.25. The cross-coupled stiffness (k) is invariant to frequency whereas the direct damping (C) steadily decreases as ω/Ω → 1, and then increases for larger frequencies. Hence, the effective damping coefficient (Ceff = C − k/ω) is approximately constant for ω < Ω, while increasing for higher frequencies. For the three PRs, the FPDS leakage is roughly up to 26% larger than the HCS leakage, while the 20-blade LS leaks in between both. The comparison of results shows the LS has insignificant forces compared to those from the HCS and FPDS, both producing a comparable Ceff and similar crossover frequencies (Ceff > 0). The HCS produces a large direct stiffness (K), three to four times that of the FPDS. Comparison of computational fluid dynamics (CFD) predictions for the FPDS against the experimental force coefficients shows significant differences, in particular an overestimation of direct stiffness as the frequency grows along with a lower direct damping for frequencies equal and above the shaft speed. Besides manufacturing considerations, a FPDS is a better alternative to a HCS whose large centering stiffness (K) may affect the compressor's critical speed, hence shortening the separation margin with respect to the operating speed.

A Review of Turbine and Compressor Aerodynamic Forces in Turbomachinery

Aerodynamic forces due to blade-tip clearance eccentricity are a known destabilizing source in rotating machinery with unshrouded impellers. Dynamic forces also appear in shrouded impellers, due to changes in the pressure in the gap between the impeller casing and its shroud. These are load-dependent forces typically characterized by a cross-coupled stiffness coefficient (k > 0). This paper reviews the archival literature for quantification of blade-tip clearance induced forces and impeller-casing forces in both unshrouded and shrouded turbines and compressors. Most distinctive are the lack of experimental results and the indiscriminate application of simple formulas to predict k, including Alford’s and Wachel’s equations. The disparity in estimations of the destabilizing k extends to recent CFD models and results. Hence, rotordynamic predictions vary widely. This review reveals that engineering practice ignores accurate physical models that could bridge the gap between practice and theory. As the energy market shifts toward carbon capture and hydrogen compression, accurate knowledge of aerodynamic forces from unshrouded compressors and open impellers will become necessary in multi-stage rotors.

Cross-coupling stiffness for natural goal-directed robot motion

The capability of humans to use their natural dynamics for generating explosive motions in a highly-coordinated sequence is a feat that has yet to be reached in robotics. With the introduction of intrinsically elastic joints, great progress towards this goal has been made. However, there are still some challenges associated with this type of actuation, which limits its application. Generating goal-directed sequences has proven difficult as optimal control solutions tend to result in uncoordinated swing-up motions. This can be explained when viewing the structure of the stiffness matrix: If the elastic elements are placed in series with the motor, a diagonal stiffness matrix is generated. This in turn leads to a multitude of frequencies at which the system can oscillate. By adding off-diagonal elements, a dominant primary resonance behaviour can be achieved. Leveraging this cross-coupling stiffness, we show that robots can produce natural goal-directed oscillatory and explosive movements that closely resemble human throwing.

Nonlinear dynamics of asymmetrically supported supercritical rotor equipped with a dry friction damper

To suppress the excessive transcritical vibration of a supercritical rotor, dry friction damper is a popular choice especially in rotorcraft and aero-engine design. In the current work, we focus on the nonlinear dynamics of asymmetrically supported supercritical rotor equipped with a dry friction damper and the influence of support asymmetry on damper’s performance. Governing equations of the rotor/damper system, which fully consider the nonlinear rub-impact and side dry friction effects, are derived. Typical results of symmetrically supported rotor/damper systems are firstly demonstrated and quantitatively compared with experimental counterparts to confirm the predicting capability of the nonlinear dry friction damper model. Afterward, influences of asymmetric direct stiffness and skew-symmetric cross-coupling stiffness are discussed. It is found that direct stiffness asymmetry deteriorates the transcritical vibration attenuation capability of the damper but reduces the width of transcritical region. For system with a severer direct stiffness asymmetry, a larger pre-tightening force is suggested to be imposed. Besides, unstable self-excited vibration induced by skew-symmetric cross-coupling stiffness is found to be well-limited by the dry friction damper within entire interested rotational speed range. Moreover, during critical speed transition, the self-excited frequency component is completely suppressed.

Measured Leakage and Rotordynamic Force Coefficients for Two Liquid Annular Seal Configurations: Smooth-Rotor/Grooved-Stator Versus Grooved-Rotor/Smooth-Stator

Abstract This paper reports and compares the experimental results of leakage and dynamic force coefficients for two liquid annular pressure seals, one having a smooth-rotor/circumferentially grooved stator (SR/GS), the other one with a circumferentially grooved rotor/smooth-stator (GR/SS). Differing only in the grooves’ location, the GR/SS seal’s geometry and operating conditions are representative of those in electrical submersible pumps (ESPs) used for oil recovery. Supplied with an ISO VG2 oil at 46 °C, both seals have the same diameter D = 102 mm, length-to-diameter ratio L/D = 0.5, and nominal land clearance Cr = 0.203 mm. The seals have 15 circumferential grooves with grooves and land lengths equal to 1.52 mm. Test variable ranges include (a) shaft speeds (ω) ranging from 2 to 8 krpm (shaft surface speed ∼ 43 m/s), and (b) pressure differences (ΔP) from 2 to 8 bar. Upstream of the test seals, three separate prerotation rings generate a range of inlet circumferential velocities (entrance swirl). Under all conditions, the GR/SS seal leaks about 10% less than the SR/GS seal. For both seals, the direct stiffnesses (KXX, KYY) have low magnitudes that drop with increasing ω; in some cases, they turn negative at 6 krpm. The GR/SS seal produces cross-coupled stiffnesses (KXY, KYX) that are ∼1.5 times larger than those for the SR/GS seal. Under the same conditions, the SR/GS seal is more stabilizing as its direct damping, and added mass coefficients are ∼ 20% larger than those for the GR/SS seal. Instability issues are likely to arise with either seal geometry because negative KXX and KYY drop a pump critical speed, aggravating the well-known destabilizing coefficients KXY and KYX. The whirl frequency ratio (WFR) combines the effects of the cross-coupled stiffness, direct damping and cross-coupled mass terms, thus providing a good basis for comparing two seals’ stability characteristics. Overall, the WFR magnitudes for the GR/SS seal are about three times higher than those for the SR/GS seal. Note that, irrespective of the inlet swirl condition, the GR/SS approaches a WFR ∼ 0.50 for operating shaft speeds greater than 4 krpm. At the lowest shaft speed (2 krpm), the WFR ≪ 0.5 for the low inlet preswirl ring, whereas WFR > 0.8 for the medium and high preswirl rings. For the SR/GS seal, the WFR ∼ 0.2 at the highest shaft speed of 6 krpm, and not affected by the inlet preswirl condition. On the other hand, at the lowest shaft speed, the WFR ranges from 0.5 to 0.75 for the SR/GS seal with medium and high preswirl rings. At this speed, the WFR = 0 with the low inlet preswirl ring. Hence, to enhance the operational stability, an effective swirl brake that could drop the inlet preswirl ratio upstream of a seal is helpful for the GR/SS seal out to 4 krpm and for the SR/GS seal out to 6 krpm.

Abdominal Aortic Wall Cross-coupled Stiffness Could Potentially Contribute to Aortic Length Remodeling

BackgroundWall stiffness of the abdominal aorta is an important factor in the cardiovascular risk assessment. We investigated abdominal aortic wall stiffness divided in direct and cross-coupled stiffness components with respect to sex and age.MethodsThirty healthy adult males (n = 15) and females were recruited and divided into three age groups: young, middle aged and elderly. Pulsatile diameter changes were determined noninvasively by an echo-tracking system, and intra-aortic pressure was measured simultaneously. A mechanical model was used to compute stress and stiffness in circumferential and longitudinal directions.ResultsCircumferential stretch had a higher impact on longitudinal wall stress than longitudinal stretch had on circumferential wall stress. Furthermore, there were an age-related and sex-independent increase in circumferential and longitudinal direct and cross-coupled stiffnesses and a decrease in circumferential and longitudinal stretch of the abdominal aortic wall. For the young group, females had a stiffer wall compared to males, while the male aortic wall grew stiffer with age at a higher rate, reaching a similar level to that of the females in the elderly group.ConclusionTemporal changes in aortic stiffness suggest an age-related change in wall constituents that is expressed in terms of circumferential remodeling impacting longitudinal stress. These mechanisms may be active in the development of aortic tortuosity. We observed an age-dependent increase in circumferential and longitudinal stiffnesses as well as decrease in stretch. A possible mechanism related to the observed changes could act via chemical alterations of wall constituents and changes in the physical distribution of fibers. Furthermore, modeling of force distribution in the wall of the human abdominal aorta may contribute to a better understanding of elastin–collagen interactions during remodeling of the aortic wall.

Contribution to the estimation of force coefficients of plain gas seals with high preswirl considering rotor-foundation dynamics

Rotor dynamic force coefficients of gas seals strongly depend on the machine operational conditions. These force coefficients influence the overall dynamical response and modal properties of machines, consequently defining the machine vibration levels. Accurate estimations of the rotor dynamic coefficients are required for designing machines with low vibration amplifications and well-defined stability margins throughout the operational range. Experimental methods applied to test benches are used to validate such force coefficients and they normally rely on (i) the quality of the measurements and (ii) the assumption that the mathematical model is able to capture the whole system dynamics. If relevant dynamical contributions in a system are neglected by the mathematical model, the contribution will erroneously be concluded to originate from the seal being tested. The theoretical and experimental investigation in this paper focuses on quantifying and qualifying the effect of neglected system dynamics modelling on the estimation of seals force coefficients and stability margins. The in-situ identification of seal forces shows that the direct stiffness, cross-coupling stiffness, and direct damping coefficient estimations for a gas seal with high preswirl are statistically significantly affected by the baseline model. Nevertheless, the baseline model leads to small deviations of the seal force coefficient estimations. The prediction accuracy of stability margins is found to be more influenced by the baseline model describing the system dynamics than by the deviations between the seal force coefficient estimations.