Application of Dry-Friction Whip and Whirl Solution Methods to Systems With Support Asymmetry

Dry-friction whip and whirl occurs when a rotor contacts a stator across a clearance annulus. In a general sense, the relative motion between the two bodies is described by a circular precessing motion. While this problem is generally well understood, the author is unaware of any papers that discuss the problem for systems having asymmetric rotor or stator supports. The current work will investigate a general model to describe dry-friction whip and whirl for the case of continuous contact between a rotor and stator in the presence of asymmetry. This paper will show that for light asymmetry, the rotor and stator motions are elliptical; however, the relative motion between the two bodies remains circular.

Nonlinear Analysis of Rotordynamic Fluid Forces in the Annular Plain Seal by Using Extended Bulk-Flow Analysis: Influence of Static Eccentricity and Whirling Amplitude

Rotor-dynamic fluid force (RD fluid force) of turbo-machinery is one of the causes of the shaft vibration problem. Bulk flow theory is the method for analyzing this RD fluid force, and it has been widely used in the design stage of machine. Conventional bulk flow theory has been carried out under the assumption of concentric circular shaft’s orbit with small amplitude. However, actual rotating machinery’s operating condition often does not hold this assumption, for example, existence of static load on the machinery causes static eccentricity. In particular, when such a static eccentricity is significant, the nonlinearity of RD fluid force may increase and become non-negligible. Therefore, conventional bulk flow theory is not applicable for the analysis of RD fluid force in such situation. In this paper, RD fluid force of the annular plain seal in the case of circular whirling orbit with static eccentricity is investigated. The case with both the significant static eccentricity and the moderate whirling amplitude is considered, and the perturbation analysis of the bulk-flow theory is extended to investigate RD fluid force in such cases. In this analysis, the assumption of the perturbation solution is extended to both static terms and whirling terms up to the third order. Then, the additional terms are caused by the coupling of these terms through nonlinearity, and these three kinds of terms are considered in the extended perturbation analysis of the bulk flow theory. As a result, a set of nonlinear analytical equations of the extended perturbation analysis of the bulk flow theory, for the case with both the significant static eccentricity and the moderate whirling amplitude, is deduced. RD fluid force for such cases are analyzed, and the occurrence of constant component, backward synchronous component and super-harmonic components in RD fluid force is observed in addition to the forward synchronous component. The representation of RD fluid force coefficients (RD coefficients) are modified for the case with significant static eccentricity, and the variation of RD fluid force coefficients for the magnitude of static eccentricity is analyzed. These analytical results of RD fluid force and its RD coefficients are compared with the numerical results using finite difference analysis and experimental results. As a result, the validity of the extended perturbation analysis of the bulk-flow theory for the case with both the significant static eccentricity and the moderate whirling amplitude is confirmed.

Predictions for Non-Contacting Mechanical Face Seal Vibration With External Excitation From Pump Vibration: Part I — Flexibly Mounted Stator

Stability and response predictions are presented for a Flexibly Mounted Stator (FMS) mechanical seal ring using the model developed by Childs in 2018. The seal ring is excited by external vibration from the rotor/housing. The model includes a frequency dependent stiffness and damping model for the O-ring and a frequency independent model for the fluid film. The dynamic coefficients depend on both speed and excitation frequency. Data used in defining the model are representative of a typical FMS mechanical seal. Parameters for radius and O-Ring placement are varied. The predictions show an insignificant dependency on speed. The predictions are strongly frequency dependent with a critical speed of 90 kRPM. The FMS is predicted to be stable to frequencies below 140 kRPM. The distance between the O-Ring and seal ring inertia center doz couples lateral and pitch-yaw motion of the seal ring. Overall, if doz is kept small, the seal ring is predicted to not have any stability or response issues.

Rotordynamic Characterization of a Staggered Labyrinth Seal: Experimental Test Data and Comparison With Predictions

As well known, the stability assessment of turbomachines is strongly related to internal sealing components. For instance, labyrinth seals are widely used in compressors, steam and gas turbines and pumps to control the clearance leakage between rotating and stationary parts, owing to their simplicity, reliability and tolerance to large thermal and pressure variations. Labyrinth seals working principle consists in reducing the leakage by imposing tortuous passages to the fluid that are effective on dissipating the kinetic energy of the fluid from high-pressure regions to low-pressure regions. Conversely, labyrinth seals could lead to dynamics issues. Therefore, an accurate estimation of their dynamic behavior is very important. In this paper, the experimental results of a long-staggered labyrinth seal will be presented. The results in terms of rotordynamic coefficients and leakage will be discussed as well as the critical assessment of the experimental measurements. Eventually, the experimental data are compared to numerical results obtained with the new bulk-flow model (BFM) introduced in this paper.

Evaluation of Coated Top Foil Bearings: Dry Friction, Drag Torque, and Dynamic Force Coefficients

Despite their many advantages, bump-type foil bearings (BFBs) have issues of dry-friction during sliding contact at rotor start/stop cycles. To prevent premature wear of both shaft and the BFB, the proper selection and application of a coating on the top foil is of importance to ensure bearing long life. This thesis presents measurements characterizing the static and dynamic load performance of a Generation I BFB having uncoated and coated (VN, TiSiN, MoS2) top foils. The bearing, with length L and diameter D = 38 mm, integrates a 360° 0.127 mm thick top foil made of Inconel X-750, and a 27 bumps strip layer, 0.47 mm in height, made of the same stock as for the top foils. The VN and TiSiN coating, 0.005 mm thick, applies to the front and back surfaces of a top foil. The MoS2 coating, 0.020 mm thick, is sacrificial. The tests were conducted at room temperature (21°C), determined by the existing test facility. The dry-sliding torque (T) increases linearly with an increase in applied static load, max W/(LD) = 25.6 kPa. The bearing with a VN coated top foil shows the largest turning torque. The dry-sliding friction factor f = T/(½WD) decreases as the specific load (W/(LD)) increases. As expected, journal rotation towards the top foil free end (clockwise) produces a larger f than for rotations in reverse. A test-rig records the BFB drag torque during rotor acceleration and deceleration procedures to/from 70 krpm (138 m/s). The vertical load applied into a bearing equals W/(LD) = −8.0 kPa, 0 kPa and 8.0 kPa. In general, the bearing with a coated top foil shows a lesser drag torque than that of the uncoated top foil bearing. Among the coated foil bearings, the one with VN coating shows the highest drag torque, whereas another with MoS2 coating shows the lowest. When the rotor starts up, the dry-sliding friction coefficient (f) of the bearing with VN coating is ∼0.4 while f for the bearing with TiSiN coating is 0.3∼0.4. The uncoated bearing shows the largest f ∼0.6, and the MoS2 coated one has the lowest f = 0.2∼0.3. The drag torque, increasing with an increase in applied static load, is small when the rotor is airborne (lesser than ∼10% of peak torque). Dynamic load tests spanning excitation frequencies (ω) from 200 Hz to 400 Hz serve to identify force coefficients for the test BFBs with a specific load of 16 kPa and operating with shaft speed at 50 krpm (833 Hz). Baseline measurements correspond to a null applied load and no shaft rotation. The test bearings show a remarkable behavior with nearly isotropic direct coefficients and very small cross-coupled ones. The bearing direct stiffnesses (K) increase with frequency whereas the direct damping coefficients (C) quickly decrease. The bearing material loss factor, γ = ωC/K, represents best the BFB ability to dissipate mechanical energy. Over the excitation frequency range, γ = 0.34, 0.28, and 0.12 for the uncoated top foil, VN coated and TiSiN coated bearings. The test data show the bearing loss factor correlates with the dry friction coefficient as γ ∼ (0.71 × f) at a rotor speed of 50 krpm (95 m/s). Since the top foils with VN or TiSiN are coated on both sides, kinetic friction between the back of a top foil and the bumps’ crests likely lessens during sustained contact.

Turbulence Input Parameters Correction Methodology in Water Lubricated Thrust Bearings

Oil-lubricated bearings are widely used in high speed rotating machines such as those found in the aerospace and automotive industries. However, environmental issues and risk-averse operations are resulting in the removal of oil and the replacement of all sealed oil bearings with reliable water-lubricated bearings. Due to the different fluid properties between oil and water, the low viscosity of water increases Reynolds numbers drastically and therefore makes water-lubricated bearings prone to turbulence effects. This requires finer meshes when compared to oil-lubricated bearings as the low-viscosity fluid produces a very thin lubricant film. Analyzing water-lubricated bearings can also produce convergence and accuracy issues in traditional oil-based analysis codes. Thermal deformation largely affects oil-lubricated bearings, while having limited effects on water lubrication; mechanical deformation largely affects water lubrication, while its effects are typically lower than thermal deformation with oil. One common turbulence model used in these analysis tools is the eddy-viscosity model. Eddy-viscosity depends on the wall shear stress, therefore effective wall shear stress modeling is necessary in determining an appropriate turbulence model. Improving the accuracy and efficiency of modeling approaches for eddy-viscosity in turbulence models is of great importance. Therefore, the purpose of this study is to perform mesh refinement for water-lubricated bearings based on methodologies of eddy-viscosity modeling to improve their accuracy. According to Szeri [1], εm/v for the Boussinesq hypothesis is given by Reichardt’s formula. Fitting the velocity profile with experiments having a y+ in the range of 0–1,000 results in Ng-optimized Reichardt’s constants k = 0.4 and δ+ = 10.7. He clearly states that for y+ > 1000 theoretical predictions and experiments have a greater variance. Armentrout and others [2] developed an equation for δ+ as a function of the pivot Reynolds number, which they validated with CFD simulations. The definition of y+ can be used to approximate the first layer thickness calculated for a uniform mesh. Together with Armentrout’s equation, the number of required elements across the film thickness can be obtained. For typical turbulence models, the y+ must be within a certain range to be accurate. On the condition that the y+ is fixed to that of a standard oil bearing for which an oil bearing code was validated, the number of elements across the film thickness and coefficients used in the eddy-viscosity equation can be adjusted to allow for convergence with other fluids other than that which the traditional oil bearing code was designed for. In this study, the number of required elements across the film for improved prediction quality was calculated based on the proposed eddy-viscosity model mesh correction from the known literature. A comparison between water lubrication using the parameter correction and oil lubrication was also made. The results of this study could aid in improving future designs and models of water-lubricated bearings.

Leakage and Dynamic Force Coefficients for Two Labyrinth Gas Seals: Teeth-on-Stator and Interlocking Teeth Configurations — A CFD Approach to Their Performance

Labyrinth gas seals (LS) commonly used in turbomachines reduce secondary flow leakage. Conventional see-through labyrinth seal designs include either all Teeth-On-Stator (TOS) or all Teeth-On-Rotor (TOR). Experience shows that an interlocking labyrinth seal (ILS), with teeth on both stator and rotor, reduces gas leakage by up to 30% compared to the conventional see-through designs. However, field data for ILS rotordynamic characteristics is still vague and scarce in the literature. This work presents flow predictions for an ILS and a TOS LS, both seals share identical design features, namely radial clearance Cr = 0.2 mm, rotor diameter D = 150 mm, tooth pitch Li = 3.75 mm, and tooth height B = 3 mm. Air enters the seal at supply pressure Pin = 3.8, 6.9 bar (absolute) and temperature of 25 °C. The ratio of gas exit pressure to supply pressure ranges from 0.5 to 0.8, and the rotor speed is fixed at 10 krpm (surface speed of 79 m/s). The analysis implements a computational fluid dynamics (CFD) method with a multi-frequency-orbit rotor whirl model. The CFD predicted mass flow rate for the ILS is ∼21% lower than that of the TOS LS, thus making the ILS a more efficient choice. Integration of the dynamic pressure fields in the seal cavities, obtained for excitation frequency (ω) ranging from 12% to 168% of rotor speed (sub and super synchronous whirl), allows an accurate estimation of the seal dynamic force coefficients. For all the considered operating conditions, at low frequency range the TOS LS shows a negative direct stiffness (K < 0), frequency independent; whereas the ILS has K > 0 that increases with both frequency and supply pressure. For both seals, the magnitude of K decreases when the exit pressure/inlet pressure ratio increases. On the other hand, the cross-coupled stiffness (k) from both seals is frequency dependent, its magnitude increases with gas supply pressure, and the k for the ILS is more sensitive to a change in the exit/inlet pressure ratio. Notably, k turns negative for subsynchronous frequencies below rotor speed (Ω) for both the TOS LS and ILS. The direct damping (C) for the TOS LS remains constant for ω > ½ Ω and has a larger magnitude than the damping for the ILS over the frequency range up to 1.5Ω. An increase in exit/inlet pressure ratio decreases the direct damping for both seals. The effective damping coefficient, Ceff = (C-k/ω) whenever positive aids to damp vibrations, whereas Ceff < 0 is a potential source for an instability. For frequencies ω /Ω < 1.3, Ceff for the TOS LS is higher in magnitude than that for the ILS. From a rotordynamics point of view, the ILS is not a sound selection albeit it reduces leakage. Comparison of the CFD predicted force coefficients against those from a bulk flow model demonstrate the later simple model delivers poor results, often contradictory and largely indifferent to the type of seal, ILS or TOS LS. In addition, CFD model predictions are benchmarked against experimental dynamic force coefficients for two TOS LSs published by Ertas et al. (2012) and Vannini et al. (2014).

Theoretical and Experimental Analyses of Directly Lubricated Tilting-Pad Journal Bearings With Leading Edge Groove

Flooded lubrication of tilting-pad journal bearings provides safe and robust operation for many applications due to a completely filled gap at the leading edge of each pad. While flooded conditions can be ensured by restrictive seals on the lateral bearing ends for any conventional bearing design, direct lubrication by leading edge grooves (LEG) placed on the pads represents an alternative to produce completely filled gaps at the entrance to the convergent lubricant film. Moreover, this design is flexible to apply different axial sealing baffles in order to influence the thermal equilibrium within the entire bearing. A theoretical model is presented that describes the specific influences of LEG design on the operating characteristics. First, in opposite to conventional tilting-pad journal bearing designs the LEG is a self-contained lube oil pocket which is generally connected to an outer annular oil supply channel. Consequently, each leading edge groove can feature a specific speed and load dependent effective pocket pressure and flow rate. As a consequence of this and the fact that the LEG is part of the pad, it directly influences its tilting angle. Secondly, the thermal inlet mixing model must consider the specific flow conditions depending on the main flow direction within the film as well as the one between outer annular channel and pocket. The novel LEG model is integrated into a comprehensive bearing code validated earlier for other bearing designs. The code is based on an extended Reynolds equation and a three-dimensional energy equation. The entire theoretical model is validated with test data from high performance journal bearing test rig for a four tilting-pad bearing in load between pivot orientation. The bearing is described by the following specifications: 0.5 nominal preload, 60% offset, 70° pad arc angle, 120 mm inner diameter, 72 mm pad length and 1.7 per mille relative bearing clearance. Measurements are conducted for rotational speeds between 4000 and 15000 rpm and specific bearing loads between 0.5 and 2.5 MPa. Within the investigated operating range good agreement between theoretical and experimental data is achieved if all boundary conditions are accurately considered. Additionally, the impact of single simplifications within the model are studied and evaluated. Finally, the test data is compared to results from the same test bearing with modified lubricant oil supply conditions in order to identify specific properties of LEG design. Here, the leading groove edge elements are replaced by conventional spray-bars. It is shown that an assessment of the comparison depends on the definition of reference conditions as the bearings require different oil flow rates for nominal operation due to their design.

Dynamic Characterization of a Novel Externally Pressurized Compliantly Damped Gas-Lubricated Bearing With Hermetically Sealed Squeeze Film Damper Modules

Ever-increasing demand for cleaner energy is driving the need for higher power density turbomachinery while reducing cost and simplifying design. Gas lubricated bearings, representing one of the enabling technologies that can help maximize these benefits and have been successfully implemented into turbomachinery applications with rotors weights in the order few kg’s. However, load capacity and damping limitations of existing gas bearing technologies prevents the development of larger size oil-free drive trains in the MW power output range. Compliantly damped hybrid gas bearings (CHGB) were introduced as an alternative design to overcome these limitations by providing external pressurization to discrete tilting pads while retaining flexibility in the bearing support to help tolerate misalignment and rotor-pad geometry changes. Additionally, the CHGB concept addresses damping entitlement through the application of bearing support dampers such a metal mesh. An alternative CHGB design, featuring a novel hermetically seal squeeze film damper (HSFD) in the bearing support, was introduced as alternative approach to metal mesh dampers (MMD) to further improve bearing damping. This paper details the rotordynamic characterization of a CHGB with modular HSFD for various operating conditions. Direct and cross-coupled stiffness and damping coefficients are presented for different rotor speeds up to 12,500 rpm, frequencies of excitation between 20–200 Hz, bearing loads between 200–400 1bf, and external hydrostatic pressures reaching 180psi. Direct comparisons to experimental results for a CHGB using (MMD) shows 3X increase in direct damping levels when using HSFD in the compliant bearing support. In addition to the experimental results, an analytical model is presented based on the implementation of the isothermal compressible Reynolds equation coupled to a flexible support possessing a pad with 3 degrees of freedom. The numerical results capture the direct stiffness and frequency dependency but underpredict the absolute values for both case when compared to experimental data.

Potentials and Limitations of an Extended Approximation Method for Nonlinear Dynamic Journal and Thrust Bearing Forces

Nonlinear dynamic journal bearing modeling within rotordynamic analyses requires the calculation of the nonlinear bearing forces particularly depending on shaft eccentricity and velocity. The bearing forces can be calculated properly using Reynolds differential equation and mass conserving cavitation algorithms, based for example on Elrod’s cavitation algorithm. This approach achieves high model accuracy and allows the consideration of additional effects like misalignment, variable viscosity and transient local oil distribution in the lubricant film. However, despite rising calculating capacity dynamic bearing analyses are still very CPU-time consuming and, consequently, approximation methods are commonly applied in multibody or rotordynamic analyses, especially in day-to-day business. While many approximation procedures are limited to special bearing geometries Glienicke et al. describe a method which is flexible to model different journal bearing geometries, as well as to consider many additional effects like oil supply pressure or starved lubrication conditions in a time averaged manner. It can be applied for both fixed-pad and tilting-pad journal bearings and its characteristic data is included in an a priori calculated map enabling a time-efficient call up of characteristic parameters of the bearing forces from a look-up table in dynamic simulations. Further, the data can be transferred to any other bearing if the requirements of the theory of similarity are supposed to be valid. In this investigation, the method is first successfully extended by the authors to consider misalignment. Secondly, the general idea of the procedure is transferred and applied to thrust bearings in order to enable a six degree of freedom rotordynamic modeling. In case of a simply lateral movement and rotation-symmetric bearing design the procedure is simple, though, in case of tilting movements it becomes more complicated. A misaligned thrust bearing provides tilting and cross-coupling moments. Cross coupling moments are smaller than the main moments, but have similar orders of magnitude and should therefore be considered. Strategies are investigated for a proper approximation of the nonlinear thrust bearing main and cross-coupling forces and moments. All steps are verified using a direct solution of Reynolds differential equation based on an extended mass conserving algorithm adapted from Elrod’s numerical implementation for the stationary case. Finally, the whole procedure and its application to rotordynamic analysis is verified by comparisons with results gained using direct online solution of Reynolds equation in rotordynamic simulation. While good simulation quality of this approximation approach is documented for selected rotor-bearing-systems in literature the range of validity is not clearly defined. Here, the influences of different parameters on the simulation error are investigated conducting different variation calculations for an overhung rotor with documented vibrational behavior from literature. It is shown that the simulation quality depends on the cavitation zone and decreases with rising vibrational velocity. The root cause for this upcoming error and a possible modification for the elimination of this limitation are presented.