Cross-coupled Stiffness Coefficients Research Articles

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.

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.

Theoretical Comparison of Characteristics between Elliptical and Conventional Hydrodynamic Journal Bearing

In this research, two subjects are presented. The first one was, studying the effect of the ellipticity ratio on the bearing performance characteristics (flow rate, load number, power absorbed, and stiffness coefficients), where three different values of the ellipticity ratio (0.5, 1, and 1.5) were studied and the conventional bearing was considered as a reference for comparison with the elliptical bearing of different values of ellipticity ratio to demonstrate the effect of ellipticity ratio on the bearing performance, while the other aspect was, studying the effect of aspect ratio on the elliptical bearing performance. Three different values of aspect ratio (0.5, 1, and 1.5) were studied. This was achieved within the range (0.1 - 0.8) for the eccentricity ratio. The finite difference method was used to solve Reynold’s equation numerically to obtain the pressure distribution on the bearing surface and then the bearing characteristics were computed. The computer program (Matlab R2015a) was used for solving the equations used in this study. From the results, it was observed that increasing the ellipticity ratio gives an increase in the flow rate values and a decrease in the (load number, power losses, Krr, Kss, |Krs| and Ksr) values, while, increasing the aspect ratio gives an increase in the (flow rate, power losses, Krr, Kss, |Krs| and Ksr) values and a decrease in the load number values. It was also observed that the elliptical bearing has a (higher flow rate, lower load capacity, and less power losses), than the conventional bearing. In addition, the elliptical bearing has higher principle stiffness coefficients (Krr and Kss) in the region (n<0.41) and (n<0.66), respectively, and lower cross-coupling stiffness coefficients (Krs and Ksr), than the conventional bearing.

Investigation of stiffness and damping coefficients in fluid film bearing with bio-oils and mineral-based oil

Rotating machinery utilizes fluid film bearings because of their good load carrying capacity and inherent damping properties. Fluid film bearings offer stiffness and damping parameters to ensure dynamic stability. The damping and stiffness characteristics of oil film significantly change the critical speed and unbalance response of the rotor. The damping and stiffness coefficients (direct, cross-coupled) may lead to system instability, also known as self-excited vibrations. Bio-oils have good tribological and physiochemical properties. They are also environment-friendly and bio-degradable. In this work, bio-oils (rapeseed, palm olein, and soybean) are evaluated in terms of their dynamic characteristics to that of mineral-based SAE40 lubricant for journal-bearing applications. The analytical extraction of damping and stiffness coefficients is performed based on the experimental dynamic viscosity data at different temperatures. In comparison to SAE40, the bio-oils have shown improved responsiveness. Bio-oils showed lower cross-coupled stiffness coefficients than SAE40, the primary source of rotor instability. The main contribution of this study is that it provides damping and stiffness coefficients for various bio-oils at different eccentricity ratios and viscosities for different types of oils. This may be used to characterize the dynamic properties of the fluid film bearing under various operating circumstances. It is concluded that the difference in cross-coupled stiffness favors bio-oils. Whereas, the difference in cross-coupled damping favors SAE40 by around 6%. This discrepancy is decreased to less than 2% for all oil’s final eccentricity ratio values (bio-oils and SAE40).

Computational Fluid Dynamics Analysis of the Influence of Gas Content on the Rotordynamic Force Coefficients for a Circumferentially Grooved Annular Seal for Multiple Phase Pumps

Abstract Pumping efficiency and rotordynamic stability are paramount to subsea multiphase pump operation since, during the life of a well, the process fluid transitions from a pure liquid to a mixture of gas in the liquid to just gas. Circumferentially grooved seals commonly serve as balance pistons in pumps while also restricting secondary flow. Prior experimental results obtained with a grooved seal operating with a mixture of air and mineral oil show the seal rotordynamic force coefficients vary significantly with the gas volume fraction (GVF). This paper, complementing an exhaustive experimental program, presents a computational fluid dynamics (CFD) analysis to predict the leakage and dynamic force coefficients of a circumferentially grooved seal supplied with air in an oil mixture with a GVF varying from 0 to 0.7. The test seal has 14 grooves, an overall axial length of 43.6 mm, and radial clearance of 0.211 mm. The 127 mm diameter rotor spins at constant angular speed (Ω = 3500 rpm). The mixture enters the seal at a supply pressure (Pin) of 2.9 bar(a), and the seal exit pressure (Pout) is 1 bar(a). The CFD two-phase flow simulations utilize the Euler–Euler multiphase model to predict the mass flowrate and the pressure field as a function of the operating conditions. Using a multi-frequency shaft orbit motion method, the CFD simulations deliver the variations of reaction force on the rotor with respect to the excitation frequency. For a pure liquid condition (GVF = 0), both the CFD and experimental results produce constant stiffness, damping, and added mass coefficients. The experimental and CFD results demonstrate the seal rotordynamic force coefficients are quite sensitive to the GVF. When introducing a small amount of air into the oil (GVF = 0.1), the direct damping coefficient increases by approximately 10%. For operation with a mixture with inlet GVF > 0.1, the cross-coupled stiffness coefficients develop strong frequency-dependent characteristics. In contrast, the direct damping coefficient has a negligible variation with excitation frequency. The CFD predictions, as well as the experimental results, evidence that air injection in a liquid stream can significantly change the seal rotordynamic characteristics, and thus can affect the rotordynamic stability of a pump. An accurate CFD analysis provides engineers to design reliable grooved seals operating under two-phase flow conditions.

Evaluation of sand p–y curves by predicting both monopile lateral response and OWT natural frequency

Abstract Extending the use of the p–y curves included in the regulation codes API and DNV to design large-diameter monopiles supporting offshore wind turbines (OWTs) was unsuccessful as it resulted in an inaccurate estimation of the monopile behavior. This had prompted many investigators to propose formulations to enhance the performances of Winkler model. In this paper, two case studies are considered. A case consisting of an OWT at Horns Rev (Denmark) supported by a monopile in a sandy soil was studied first. Taking the FEA using ABAQUS as reference, results of WILDOWER 1.0 (a Winkler computer code) using the recently proposed p–y curves giving design parameters were plotted and evaluated. In order to see the ability of proposed p–y curves to predict the monopile head movements, and consequently the first natural frequency (1st NF), a second case study consisting of a monopile supporting an OWT at North Hoyle (UK) was selected. The monopile head stiffness in terms of lateral, rocking, and cross-coupling stiffness coefficients, necessary for the 1st NF, were computed using both ABAQUS and WILDPOWER 1.0. Comparisons with the measured 1st NF showed that with the exception of one p–y model, none of other proposed Winkler methods is able to predict accurately this parameter.

Leakage and rotordynamic characteristics of labyrinth seal and hole-pattern damping seal with special-shaped 3D cavity

Purpose The purpose of this paper is to enhance sealing and rotordynamic performance of hole-pattern damping seal (HPDS) and labyrinth seal (LS) by structural innovation and geometrical optimization of special-shaped hole or annular-groove cavity. Design/methodology/approach The unsteady flow was transformed into steady one using moving reference frame method. The full period numerical models of LS and HPDS were established. The influence of special-shaped hole or annular-groove cavity at axial inclined angle on leakage rate and rotordynamic coefficient of these two seals at different whirl angular speed were investigated. Findings The results show that dynamic characteristics of straight-tooth LS are better than that of slanted-tooth LS. Compared to typical straight-hole damping seal, HPDS with windward oblique-hole when axial inclined angle ranges from 50 to 60° has superiority in both leakage and rotordynamic characteristics by considering smaller cross-coupled stiffness coefficient and whirl frequency ratio, larger direct damping coefficient and effective damping coefficient. Originality/value A novel HPDS with special-shaped three-dimensional hole cavity was proposed to enhance leakage and rotordynamic performance. The optimized geometrical structures of HPDS for excellent sealing and rotordynamic characteristics were obtained. Peer review The peer review history for this article is available at: https://publons.com/publon/10.1108/ILT-07-2020-0262/

Comparative Study on Dynamic Characteristics of Double-Pad Inwardly and Outwardly Pumping Water-Lubricated Spiral-Groove Thrust Bearings

AbstractThe dynamic characteristics of the double-pad inwardly pumping water-lubricated spiral-groove thrust bearing (SGTB) is different from that of the outwardly pumping version, especially under a high-speed condition. However, no literature on this topic can be found up to now. In this study, a comparative study on the dynamic characteristics of the two types of SGTBs is conducted. First, a turbulent lubrication model considering centrifugal and cavitating effects for the double-pad SGTB is developed based on gas–liquid two-phase flow. The steady-state and perturbed Reynolds equations of liquid phase are derived using linear perturbation method. Subsequently, the boundary fitted coordinate system and control volume method are applied to solve the Reynolds equations. Then, the axial stiffness coefficients of high-speed water-lubricated bearings are verified by an experimental study. Finally, the influence of operating conditions and configuration parameters on the dynamic behaviors for the two types of SGTBs is investigated. The results show that the double-pad outwardly pumping bearing is superior to the inwardly pumping version in all the dynamic coefficients when the radius ratio exceeds 0.65. When the tilt angle exceeds 1.2 × 10−4 as well as when the eccentricity ratio exceeds 0.7, the dynamic damping, angular and cross-coupled stiffness coefficients of the outwardly pumping bearing may be larger than those of the inwardly pumping one.

Study the Foundation Effect on the Hydrodynamic Bearing Performance.

The hydrodynamic bearings are important components of most of the rotary parts in different applications. The current study examines the performance of the traditional type, which is the most common because they are easy to the manufactured, relatively low cost, but they have low stability compared with other type because they have large values for the cross coupling stiffness coefficients, which are responsible for the whirl phenomenon. The mathematical model was constructed and the equations of the thickness change oil film were presented. In addition to the mathematical formula of Reynolds equation, the numerical solution was adopted to solve it using a (five-nodes) scheme method and it is solved using the (MATLAB) software. The oil film and the foundation were considered as springs that were connected to each other in series and the equivalent stiffness coefficient are calculated. The results showed that taking the foundation effect into account reduces the overall values of the stiffness coefficient. The results showed that the use of different materials in the covering leads to a decrease in the equivalent stiffness coefficient value depending on the layer thickness of the covering. The method of partial covering caused an important of some of stiffness coefficient and decrease..

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

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 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, not only eye-labyrinth seals but also shrouds need to be considered in rotordynamic analysis.

Application of a Novel Rotordynamic Identification Method for Annular Seals With Arbitrary Elliptical Orbits and Eccentricities

The identification method using infinitesimal theory is proposed to predict rotordynamic coefficients of annular gas seals. The transient solution combined with moving grid method was unitized to obtain the fluid reaction force at a specific position under different whirling frequencies. The infinitesimal method is then applied to obtain the rotordynamic coefficients, which agrees well with published experimental results for both labyrinth seals and eccentric smooth annular seals. Particularly, the stability parameter of the effective damping coefficient can be solved precisely. Results show that the whirling frequency has little influence on direct damping coefficient, effective damping coefficient, and cross-coupled stiffness coefficient for the labyrinth seal. And the effective damping coefficients decrease as the eccentricity ratio increases. A higher eccentricity ratio tends to destabilize the seal system, especially at a low whirling frequency. Results also show that the fluid velocity in the maximum clearance in the seal leakage path is less than that in the minimum clearance. The inertial effect dominates the flow field. Then it results in higher pressure appearing in maximum clearances. The pressure difference aggravates the eccentricity of rotor and results in static instabilities of the seal system.

Numerical and Experimental Research on the Fluid-Induced Forces of Clearance Flow in Canned Motor Reactor Coolant Pump

Reactor coolant pump (RCP) is one of the most important equipment of the coolant loop in a pressurized water reactor system. Its safety relies on the characteristics of the rotordynamic system. For a canned motor RCP, the liquid coolant fills up the clearance between the metal shields of the rotor and stator inside the canned motor, forming a long clearance flow. The fluid-induced forces of the clearance flow in canned motor RCP and their effects on the rotordynamic characteristics of the pump are numerically and experimentally analyzed in this work. A transient computational fluid dynamics (CFD) method has been used to investigate the fluid-induced force of the clearance. A vertical experiment rig has also been established for the purpose of measuring the fluid-induced forces. Fluid-induced forces of clearance flow with various whirl frequencies and various boundary conditions are obtained through the CFD method and the experiment. Results show that clearance flow brings large mass coefficient into the rotordynamic system and the direct stiffness coefficient is negative under the normal operating condition. The rotordynamic stability of canned motor RCP does not deteriorate despite the existence of significant cross-coupled stiffness coefficient from the fluid-induced forces of the clearance flow.

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

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.

Stability Analysis of Various Lengths Conical Hydrodynamic Bearing for Variable Load Conditions

Conical hydrodynamic journal bearing is used to sustain radial load and observed to adjust the effect of axial load on rotating elements. Small compact size conical hydrodynamic fluid film bearings are used in many applications, such as compact hard disc drive. Hence need arises to explore the performance of these bearing for variable load condition and various sizes in terms of aspect ratio. These conical bearings also emerge with better feature to replace two separate bearings (journal and thrust bearings) in various applications. The present study proposed the stability analysis of conical hydrodynamic journal bearing in terms of stiffness and damping coefficients for bearing of semi cone angle (γ = 30o) and bearing size in terms of aspect ratio (λ = 0.5, 0.8, 1.0) for a wide range of radial load ((W_r ) =0.1-0.9) on rotating journal at a speed of 500 rpm. Finite element method has been used to solve the modified Reynolds equation in spherical coordinate system to investigate the flow of lubricant in the clearance space. Stability parameters for various configuration of conical hydrodynamic bearing have been presented and discussed in terms of direct and cross coupled stiffness coefficients, damping coefficients and threshold speed margin. Results show that threshold speed (ω _th) of conical journal bearing is improved for short bearings when aspect ratio changes from 1.0 to 0.5.

A novel negative dislocated seal and influential parameter analyses of static/rotordynamic characteristics

This paper proposes a new type of negative dislocated seal (NDS) based on the dislocated bearing theory to investigate the influential parameters of static rotor eccentricity and dislocated ratio on the static and rotordynamic characteristics of an example seal solved with computational fluid dynamics (CFD) the flow field. The rotordynamic characteristics of the NDS were investigated in respect of the effects of rotor whirling frequency on response force, stiffness coefficients, damping coefficients and rotor system stability, by multifrequency elliptical orbit rotor whirling model. Based on the studies, we reached the following conclusions. The circumferential pressure distribution of NDS and conventional labyrinth seal (LS) presents sine periodic variation approximately, while relative to the LS, the NDS has two divergent wedge gaps and reduces the hydrodynamic pressure effects, then the circumferential pressure difference and tangential force on rotor surface decreases by about 40 %~190 %. The leakage of the NDS linearly increases with the rising eccentricity ratio and dislocation ratio approximately, and is larger (about 0.9 %~1.5 %) than the LS. The direct stiffness coefficients of the two seals increase with the rise of rotor whirling frequency, while the absolute values of both the cross-coupled stiffness coefficients and the damping coefficients decrease with raising rotor whirling frequency. Compared with the LS, the NDS has smaller (about 28.8 %~206.2 %) cross-coupled stiffness coefficient, larger (about 26.15 %~60.39 %) effective damping coefficient, and good stability of rotor system. This study has developed a novel seal structure to improve the seal performance of turbomachinery such as aeroengine.

A Numerical Investigation of the Effect of Inlet Preswirl Ratio on Rotordynamic Characteristics of Labyrinth Seal

Full three-dimensional (3D) computational fluid dynamics (CFD) simulations are carried out using ANSYS cfx to obtain the detailed flow field and to estimate the rotordynamic coefficients of a labyrinth seal for various inlet swirl ratios. Flow fields in the labyrinth seal with the eccentricity of the rotor are observed in detail and the detailed mechanisms that increase the destabilizing forces at high inlet swirl ratios are discussed based on the fluid governing equations associated with the flow fields. By evaluating the contributions from each term of the governing equation to cross-coupled force, it is found that circumferential velocity and circumferential distribution of axial mass flow rate play key roles in generating cross-coupled forces. In the case that circumferential velocity is high and decreases along the axial direction, all contributions from each term are positive cross-coupled force. On the other hand, in the case that circumferential velocity is low and increases along the axial direction, one contribution is positive but the other is negative. Therefore, cross-coupled force can be negative in the local chamber depending on the balance even if circumferential velocity is positive. CFD predictions of cross-coupled stiffness coefficients and direct damping coefficients show better agreement with experimental results than a bulk flow model does by considering the force on the rotor in the inlet region. Cross-coupled stiffness coefficients derived from the force on the rotor in the seal section agree well with those of the bulk flow model.

Numerical and Experimental Analyses of Dynamic Characteristics for Liquid Annular Seals With Helical Grooves in Seal Stator

Numerical and experimental analyses were carried out to investigate the dynamic characteristics of liquid annular seals with helical grooves in the seal stator. In the numerical analysis, the governing equations were the momentum equations with turbulent coefficients and the continuity equation, all averaged across the film thickness and expressed using an oblique coordinate system in which the directions of coordinate axes coincided with the circumferential direction and the direction along the helical grooves. These governing equations were solved numerically to obtain the dynamic characteristics, such as the dynamic fluid-film forces, dynamic coefficients, and whirl-frequency ratio (WFR). The numerical analysis included the effect of both fluid inertia and energy loss at the steps between the helical groove and the land sections. In the experiments, the dynamic fluid-film pressure distributions, which were induced by a small whirling motion of the rotor about the seal center, were measured to obtain the dynamic characteristics. The equivalent numerical results reasonably agree with the experimental results, demonstrating the validity of the numerical analysis. The value of the tangential dynamic fluid force induced by the rotor whirling motion decreased with increasing the helix angle γ. Consequently, the values of the cross-coupled stiffness coefficient and WFR decreased with increasing γ and became negative for large γ. In general, pump rotors rotate with a forward whirling motion under normal operating conditions. Hence, the negative value of WFR for helically grooved seals contributes to rotor stability by suppressing the forward whirling motion of the rotor.

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

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.

Numerical Investigation on the Leakage and Static Stability Characteristics of Pocket Damper Seals at High Eccentricity Ratios

Annular gas seals for compressors and turbines are designed to operate in a nominally centered position in which the rotor and stator are at concentric condition, but due to the rotor–stator misalignment or flexible rotor deflection, many seals usually are suffering from high eccentricity. The centering force (represented by static stiffness) of an annular gas seal at eccentricity plays a pronounced effect on the rotordynamic and static stability behavior of rotating machines. The paper deals with the leakage and static stability behavior of a fully partitioned pocket damper seal (FPDS) at high eccentricity ratios. The present work introduces a novel mesh generation method for the full 360 deg mesh of annular gas seals with eccentric rotor, based on the mesh deformation technique. The leakage flow rates, static fluid-induced response forces, and static stiffness coefficients were solved for the FPDS at high eccentricity ratios, using the steady Reynolds-averaged Navier–Stokes solution approach. The calculations were performed at typical operating conditions including seven rotor eccentricity ratios up to 0.9 for four rotational speeds (0 rpm, 7000 rpm, 11,000 rpm, and 15,000 rpm) including the nonrotating condition, three pressure ratios (0.17, 0.35, and 0.50) including the choked exit flow condition, two inlet preswirl velocities (0 m/s, 60 m/s). The numerical method was validated by comparisons to the experiment data of static stiffness coefficients at choked exit flow conditions. The static direct and cross-coupling stiffness coefficients are in reasonable agreement with the experiment data. An interesting observation stemming from these numerical results is that the FPDS has a positive direct stiffness as long as it operates at subsonic exit flow conditions; no matter the eccentricity ratio and rotational speed are high or low. For the choked exit condition, the FPDS shows negative direct stiffness at low eccentricity ratio and then crosses over to positive value at the crossover eccentricity ratio (0.5–0.7) following a trend indicative of a parabola. Therefore, the negative static direct stiffness is limited to the specific operating conditions: choked exit flow condition and low eccentricity ratio less than the crossover eccentricity ratio, where the pocket damper seal (PDS) would be statically unstable.

Numerical and Experimental Analyses of the Dynamic Characteristics of Journal Bearings With Square Dimples

Numerous previous numerical studies have investigated the effect of surface texturing upon the static characteristics of journal bearings, including their load-carrying capacity and friction torque. In general, the dynamic characteristics of journal bearings are also important, since they are essential factors in predicting the vibration behavior of actual rotors supported by journal bearings. However, the effects of surface texture upon these dynamic characteristics have not been investigated through either numerical or experimental analysis. Thus, in the present study, such analyses were conducted to investigate the dynamic characteristics of textured journal bearings, such as their dynamic coefficients of oil film and the stability-threshold shaft speed supported by the bearings. Numerical analysis was done using a model that included inertial effects and energy loss; this model agreed well with experimental results concerning static characteristics from our previous study. Dynamic testing based on a sinusoidal-excitation method was also performed using textured journal bearings with uniform square dimples to verify the numerical results, which agreed qualitatively with those of experiment, confirming the validity of the numerical analysis. These results suggest that under the same operating conditions, the main effect of texturing upon the dynamic coefficients is to yield the cross-coupled stiffness coefficients with lower absolute values than the conventional ones with a smooth surface. The linear stability-threshold shaft speed of the rotor supported by the textured journal bearings became higher than that of a smooth bearing, mainly due to the reduction of cross-coupled stiffness coefficients. This tendency became more pronounced for high Reynolds number operating conditions and textured bearings with a large number of dimples.