Hydrodynamic flexure pivot tilting pad gas bearings (FPTPGBs) can enable successful operation of oil-free microturbomachinery. This work presents the experimental and analytical study of such bearings with different damper configurations. A test rig was constructed that could safely operate a ~28.6 mm, 0.8 kg rotor beyond 120 krpm. A time domain orbit simulation, which integrates nonlinear equations of motion for the rotor-bearing elements, was implemented as the primary analysis tool to predict rotor-bearing responses to imbalance, the presence and location of critical speeds, etc. Complementary analyses were also performed with a model that uses linear bearing impedance coefficients to predict system natural frequencies. Imbalance response testing verified that the rotor-bearing system behaved linearly in the region above the critical speed, and orbit simulations predicted the response to a calibrated imbalance with notable agreement. Viscoelastic dampers added behind the FPTPGB pads delayed the onset of subsynchronous vibrations (from 43 krpm without damper to above 50 krpm with damper) of the system with bearing clearance increased by shims. Midrange subsynchronous vibrations initiated at ~20 krpm were eventually suppressed by ~25 krpm due to the stabilizing effect of rotor centrifugal growth. The viscoelastic dampers had a negligible effect on suppressing these midrange subsynchronous vibrations in experiments, but this was not demonstrated in simulations, presumably due to much lower stiffness contribution of the damper at lower frequencies. The ideal, perfectly aligned models in the simulations were able to tolerate shims up to only 10% of nominal clearance, but the test rig exhibited surprising stability with shims as much as 200% of nominal clearance; this increase may be caused by imposed eccentricities due to misalignments in the test rig. FPTPGBs supported by compliant bump foils can have the ability to tolerate rotor misalignments and shock loading like foil gas bearings. Simulation studies on imbalance response characteristics for several bearing shell mass and support stiffness configurations present initial design guidelines for the application. Namely, results showed that FPTPGBs favored large bearing shell mass and large support stiffness, while FPTPGBs with radial compliance favored small bearing shell mass with large support stiffness.

This thesis provides experimental frequency dependent stiffness and damping coefficient results for a high-speed, lightly loaded, flexible-pivot tilting-pad bearing, with a load-on-pad configuration. Test conditions include four shaft speeds (6000, 9000, 13000 and 16000 rpm), and bearing unit loads from 172 kPa to 690 kPa. The results show that the bearing stiffness is a quadratic function of the frequency of vibration; hence their frequency dependency can be modeled by added-mass terms. The additional degrees of freedom introduced by the pads and the influence of the inertial forces generated in the fluid film account for this frequency dependency. The conventional frequency-dependent stiffness and damping model for tilting-pad bearings is extended with an added-mass matrix to account for the frequency dependency. This approach allows the description of the bearing dynamic characteristics with frequency-independent stiffness, damping and added-mass matrices. Experimental results are compared with predictions from the Reynolds equation and from a bulk-flow Navier-Stokes model. Both models produce good predictions of the stiffness and damping coefficients. However, results show that the bulk-flow model is more adequate for predicting the direct added-mass terms because it accounts for the fluid inertial forces. A bulk-flow solution of the Navier-Stokes equations that includes the effects of fluid inertia should be used to calculate the rotordynamic coefficients of a flexible-pivot tilting-bearing. Static performance measurement results are also detailed. Results include pad metal temperatures, eccentricity-ratios and attitude-angle as a function of bearing load, and estimated power losses.

In the theoretical analysis of high speed rotor bearing systems, it is common to use four displacement and four velocity based coefficients, which characterise the behaviour of the lubricating fluid film. Although a great deal of work has been published establishing theoretical models of all types of hydrodynamic journal bearings, the large amount of experimental work has centred on relatively low speed conditions. This work presents a contribution to the experimental study of the static and dynamic characteristics of oil films in journal bearings used in high-speed rotating machinery. The main objectives of the work are: • To devise new experimental techniques for the measurement of dynamic coefficients suitable for use at high rotational speeds • To design, manufacture, assemble and commission a test facility to measure the static and dynamic characteristics of journal bearings at speeds up to 30000 rpm • To determine the static and dynamic characteristics of a 5 Pad Tilting Pad Journal Bearing Unit of 80 mm diameter at speeds up to 25 000 rpm using the said test facility. New techniques are particularly necessary for the measurement of velocity coefficients because these invoke the necessity of imposing a velocity on to the bearing housing and previous techniques have utilised synchronous motion of the bearing. Consequently a new experimental procedure for measuring the four velocity or damping coefficients of an oil film journal bearing from imposed dynamic "orbits" has been devised called the "double pulse" technique. All four velocity coefficients are derived from one imposed journal centre dynamic orbit and, therefore may be regarded as being obtained at the same time. The method requires the production of a "cross- over" point similar to that of a "figure of eight" shaped orbit and utilises the "cross-over" point therein. Coefficients are initially evaluated in a co-ordinate system, which is chosen to align with the designated parts of the measured orbit. Each coefficient is then evaluated from single values of instantaneous imposed force and resulting journal centre velocity. Coefficients are them converted into any other desired axes system. The result is a simpler experimental procedure, with reduced uncertainty compared to hitherto existing methods. The use of non-sinusoidal excitation of the oil film was explored, in the form of applying a step-pulse train load pattern to produce a cross-over pattern in the journal displacement ·orbit'. Experimental tests were completed on a tilting pad bearing at speeds up to 15 000 rpm inclusive. At speeds above this, the bearing exhibited a vibrational response, which precluded the accurate measurement of journal centre displacement.

A novel tilting pad journal bearing model including pivot flexibility as well as temporal fluid inertia effects on the thin film fluid flow aims to accurately predict the bearing forced performance. The predictive model also accounts for the thermal energy transport effects in a TPJB. A Fortran program with an Excel GUI models TPJBs and delivers predictions of the bearing static and dynamic forced performance. The calculation algorithm uses a Newton-Raphson procedure for successful iterations on the equilibrium pad radial and transverse displacements and journal center displacements, even for bearings pads with very soft pivots. The predictive model accounts for the effect of film temperature on the operating bearing and pad clearances by calculating the thermal expansion of the journal and pad surfaces. The pad inlet thermal mixing coefficient (lambda) influences moderately the predicted fluid film temperature field. Pad pivot flexibility decreases significantly and dominates the bearing stiffness and damping coefficients when the pivot stiffness is lower than 10% of the fluid film stiffness coefficients (with rigid pivots). Pivot flexibility has a more pronounced effect on reducing the bearing damping coefficients than the stiffness coefficients. Pad pivot flexibility may still affect the bearing behavior at a light load condition for a bearing with a large pad preload. Pad pivot flexibility, as well as the fluid inertia and the pads’ mass and mass moment of inertia, could influence the bearing impedance coefficients, in particular at high whirl frequencies. The stiffness and damping coefficients of a TPJB increase with a reduction in the operating bearing and pad clearances. The work delivers a predictive tool benchmarked against a number of experimental results for test bearings available in the recent literature. The static and dynamic forced performance characteristics of actual TPJBs can not be accurately predicted unless their pad flexibility and pivot flexibility, fluid film temperature, pad inlet thermal mixing coefficient, operating bearing and pad clearances, among others are well known in advance. However, the extensive archival literature showcasing test procedures and experimental results for TPJBs does not report the above parameters. Thus, reasonable assumptions on the magnitude of certain elusive parameters for use in the predictive TPJB model are necessary.

This dissertation deals with three research topics; i) the catcher bearings life prediction method, ii) the Morton effect, and iii) the two dimensional modified Reynolds equation. Firstly, catcher bearings (CB) are an essential component for rotating machine with active magnetic bearings (AMBs) suspensions. The CB's role is to protect the magnetic bearing and other close clearance component in the event of an AMB failure. The contact load, the Hertzian stress, and the sub/surface shear stress between rotor, races, and balls are calculated, using a nonlinear ball bearing model with thermal growth, during the rotor drop event. Fatigue life of the CB in terms of the number of drop occurrences prior to failure is calculated by applying the Rainflow Counting Algorithm to the sub/surface shear stress-time history. Numerical simulations including high fidelity bearing models and a Timoshenko beam finite element rotor model show that CB life is dramatically reduced when high-speed backward whirl occurs. Secondly, the theoretical models and simulation results about the synchronous thermal instability phenomenon known as Morton Effect is presented in this dissertation. A transient analysis of the rotor supported by tilting pad journal bearing is performed to obtain asymmetric temperature distribution of the journal by solving variable viscosity Reynolds equation, energy equation, heat conduction equation, and equations of motion for rotor. The tilting pad bearing is fully nonlinear model. In addition, thermal mode approach and staggered integration scheme are utilized in order to reduce computation time. The simulation results indicate that the temperature of the journal varies sinusoidally along the circumferential direction and linearly across the diameter, and the vibration envelope increased and decreased, which considers as a limit cycle that is stable oscillation of the envelope of the amplitude of synchronous vibration. Thirdly, the Reynolds equation plays an important role to predict pressure distribution in the fluid film for the fluid film bearing analysis. One of the assumptions on the Reynolds equation is that the viscosity is independent of pressure. This assumption is still valid for most fluid film bearing applications, in which the maximum pressure is less than 1 GPa. In elastohydrodynamic lubrication (EHL) which the lubricant is subjected to extremely high pressure, however, the pressure independent viscosity assumption should be reconsidered. With considering pressure-dependent viscosity, the 2D modified Reynolds equation is derived in this study. The solutions of 2D modified Reynolds equation is compared with that of the classical Reynolds equation for the plain journal bearing and ball bearing cases. The pressure distribution obtained from modified equation is slightly higher pressures than the classical Reynolds equations. / PDF file replaced 10-21-2012 at the request of the Thesis Office.

Measured and predicted static and dynamic characteristics are provided for a four-pad, rocker-pivot, tilting-pad journal bearing in the load-on-pad and load-between-pad orientations. The bearing has the following characteristics: 4 pads, .57 pad pivot offset, 0.6 L/D ratio, 60.33 mm (2.375in) pad axial length, 0.08255 mm (0.00325 in) radial clearance in the load-on-pad orientation, and 0.1189 mm (0.00468 in) radial clearance in the load-between-pad orientation. Tests were conducted on a floating test bearing design with unit loads ranging from 0 to 2903 kPa (421.1 psi) and speeds from 6.8 to 13.2 krpm. For all rotor speeds, hot-clearance measurements were taken to show the reduction in bearing clearance due to thermal expansion of the shaft and pads during testing. As the testing conditions get hotter, the rotor, pads, and bearing expand, decreasing radial bearing clearance. Hot-clearance measurements showed a 16-25% decrease in clearance compared to a clearance measurement at room temperature. For all test conditions, dynamic tests were performed over a range of excitation frequencies to obtain complex dynamic stiffness coefficients as a function of frequency. The direct real dynamic stiffness coefficients were then fitted with a quadratic function with respect to frequency. From the curve fit, the frequency dependence was captured by including a virtual-mass matrix [M] to produce a frequency independent [K][C][M] model. The direct dynamic stiffness coefficients for the load-on-pad orientation showed significant orthotropy, while the load-between-pad did not. The load-between-pad showed slight orthotropy as load increased. Experimental cross-coupled stiffness coefficients were measured in both load orientations, but were of the same sign and significantly less than direct stiffness coefficients. In both orientations the imaginary part of the measured dynamic stiffness increased linearly with increasing frequency, allowing for frequency independent direct damping coefficients. Rotordynamic coefficients presented were compared to predictions from two different Reynolds-based models. Both models showed the importance of taking into account pivot flexibility and different pad geometries (due to the reduction in bearing clearance during testing) in predicting rotordynamic coefficients. If either of these two inputs were incorrect, then predictions for the bearings impedance coefficients were very inaccurate. The main difference between prediction codes is that one of the codes incorporates pad flexibility in predicting the impedance coefficients for a tilting-pad journal bearing. To look at the effects that pad flexibility has on predicting the impedance coefficients, a series of predictions were created by changing the magnitude of the pad's bending stiffness. Increasing the bending stiffness used in predictions by a factor of 10 typically caused a 3-11% increase in predicted Kxx and Kyy, and a 10-24% increase in predicted Cxx and Cyy. In all cases, increasing the calculated bending stiffness from ten to a hundred times the calculated value caused slight if any change in Kxx, Kyy, Cxx, and Cyy. For a flexible pad an increase in bending stiffness can have a large effect on predictions; however, for a more rigid pad an increase in pad bending stiffness will have a much lesser effect. Results showed that the pad's structural bending stiffness can be an important factor in predicting impedance coefficients. Even though the pads tested in this thesis are extremely stiff, changes are still seen in predictions when the magnitude of the pad?s bending stiffness is increased, especially in Cxx, and Cyy. The code without pad flexibility predicted Kxx and Kyy much more accurately than the code with pad flexibility. The code with pad flexibility predicts Cxx more accurately, while the code without pad flexibility predicted Cyy more accurately. Regardless of prediction Code used, the Kxx and Kyy were over-predicted at low loads, but predicted more accurately as load increased. Cxx, and Cyy were modeled very well in the load-on-pad orientation, while slightly overpredicted in the load-between-pad orientation. For solid pads, like the ones tested here, both codes do a decent job at predicting impedance coefficients

This thesis presents the dynamic and static forced performance of a flexure-pivot tilting-pad bearing load between pad (LBP) configuration for different rotor speeds and bearing unit loadings. The bearing has the following design parameters: 4 pads with pad arc angle 72o and 50% pivot offset, pad axial length 0.0762 m (3 in), pad radial clearance 0.254 mm (0.010 in), bearing radial clearance 0.1905 mm (0.0075 in), preload 0.25 and shaft nominal diameter of 0.11684 m (4.600 in). The dynamic coefficients and the static performance parameters of the FPB have been compared with the theoretical predictions using the isothermal analysis from the rotordynamic software suite XLTRC2-XLTFPBrg. The bearing shows a small attitude angle, about 10o, which indicates small crosscoupling stiffnesses. The pad temperatures increase in the circumferential direction of rotation with speed and load. The pads maximum temperature was measured near the trailing edge. The dependency of the stiffness and damping coefficients on the excitation frequency has been studied. The frequency dependency in the dynamic coefficients was removed by introducing an added mass coefficient to the bearing model. The direct added mass coefficients were around 32 kg. The direct stiffness and damping coefficients increase with load, while increasing and decreasing with rotor speed, respectively. A small whirl frequency ratio (WFR) was found of about 0.15, and it decreases with load and increases with speed. A comparison between the dynamic stiffnesses using a Reynolds equation and the bulk-flow Navier-Stokes models with the experimental dynamic stiffnesses shows that the Reynolds model (even for laminar flows) is not adequate, and that the bulk-flow model should be used for rotordynamic coefficients prediction. The bulk-flow model in general predicts well the static performance parameters and the direct dynamic coefficients, and underpredicts the cross-coupled coefficients (overpredicts the stability).

Many researchers have compared predicted stiffness and damping coefficients for tilting-pad journal bearings (TPJBs) to measurements. Most have found that direct damping is consistently overpredicted. The thrust of this research is to explain the difference between measured and predicted stiffness and damping coefficients for TPJBs, and to provide some confidence to designers that TPJB dynamic coefficients can be accurately predicted. Most analytical models for TPJBs are based on the assumption that explicit dependence on pad motion can be eliminated by assuming harmonic rotor motion such that the amplitude and phase of pad motions resulting from radial and transverse rotor motions are predicted by rotor-pad transfer functions. In short, these transfer functions specify the amplitude and phase of pad motion (angular, radial, translational, etc.) in response to an input rotor motion. A new pad perturbation model is developed including the effects of angular, radial, and circumferential pad motion and changes in pad clearance due to pad bending compliance. Though all of these pad variables have previously been included in different analyses, there are no publications containing perturbations of all four variables. In addition, previous researchers have only perturbed the journal, while both the bearing and journal motions are perturbed in the present analysis, and the applicability of comparing rotor-perturbed bearing impedance predictions to impedances measured on a bearing-perturbed test rig is discussed. This perturbation model was implemented in a Reynolds-based TPJB code to predict the frequency-dependent bearing impedances and rotor-pad transfer functions. Direct measurements of pad motion during test excitation were recorded to produce measured transfer functions between rotor and pad motion, and a comparison between these measurements and predictions is given. Motion probes were added to the loaded pad (having the static load vector directed through its pivot) of a 5-pad TPJB to obtain accurate measurement of pad radial and tangential motion, as well as tilt, yaw, and pitch. Strain gages were attached to the side of the loaded pad to measure static and dynamic bending strains, which were then used to determine static and dynamic changes in pad curvature (pad clearance). Good agreement was found between the amplitude of the measured and predicted transfer functions concerning radial and transverse pad motions throughout the range of speeds and loads tested, while pad tilt was moderately underpredicted. For the bearing investigated, radial pad motions resulting from pivot compliance were as large as 60% of the radial component of shaft motion when operating at 4400 rpm under heavily loaded conditions. Hence, if a dynamic load applied to the shaft resulted in a shaft displacement of 25 microns (1 mil), the pad would displace radially 15 microns (0.6 mils), and the fluid film height would only decrease by 10 microns (0.4 mils). The consequence of this pad motion is that fluid film stiffness and damping forces produced by relative rotor-pad motions are significantly reduced, resulting in a bearing having significantly less direct stiffness and damping than predicted. A similar effect occurs when shaft motions produce significant changes in pad clearance due to pad compliance. For the pad tested here, the measurements show that predicting TPJB stiffness and damping coefficients without accounting for pad and pivot compliance will produce large errors, and is not advised. Transverse pad motion was predicted and observed. Based on phase measurements, this motion is lightly damped, and appears to be caused by pivot deflection instead of slipping. Despite observing a lightly damped phase change, an increase in magnitude at this natural frequency was not observed. Predicted direct stiffness and damping for unit loads from 0-3200 kPa (0-450 psi) fit through 1.5× running speed are within 18% of measurements at 4400 rpm, while predictions at 10200 rpm are within 10% of measurements. This is a significant improvement on the accuracy of predictions cited in literature. Comparisons between predictions from the developed bearing model neglecting pad, pivot, and pad and pivot flexibility show that predicted direct stiffness and damping coefficients for a model having a rigid pad and pivot are overestimated, respectively, by 202% and 811% at low speeds and large loads, by 176% and 513% at high speeds and high loads, and by 51% and 182% at high speeds and light loads. While the reader is likely questioning the degree to which these predictions are overestimated in regard to previous comparisons, these predictions are based on measured operating bearing clearances, which are 20-30% smaller than the cold bearing clearances that previous comparisons were based on. The effect of employing a full bearing model (retaining all of the pad degrees of freedom) versus a reduced bearing model (where only journal degrees of freedom are retained) in a stability calculation for a realistic rotor-bearing system is assessed. For the bearing tested, the bearing coefficients reduced at the frequency of the unstable eigenvalue (subsynchronously reduced) predicted a destabilizing cross-coupled stiffness coefficient at the onset of instability within 1% of the full model, while synchronously reduced coefficients for the lightly loaded bearing required 25% more destabilizing cross-coupled stiffness than the full model to cause system instability. This overestimation of stability is due to an increase in predicted direct damping at the synchronous frequency over the subsynchronously reduced value. This increase in direct damping with excitation frequency was also seen in highly loaded test data at frequencies below approximately 2×running speed, after which direct damping decreased with increasing excitation frequency. This effect was more pronounced in predictions, occurring at all load and speed combinations. The same stability calculation was performed using measured stiffness and damping coefficients at synchronous and subsynchronous frequencies at 10200 rpm. It was found that both the synchronously measured stiffness and damping and predictions using the full bearing model were more conservative than the model using subsynchronously measured stiffness and damping. This outcome contrasts with the comparison between models using synchronously and subsynchronously reduced impedance predictions, which showed the subsynchronously reduced model to be the most conservative. This contrast results from a predicted increase in damping with increasing excitation frequency at all speeds and loads, while this increase in damping with increasing excitation frequency was only measured at the most heavily loaded conditions.

Swedish Hydropower Centre - SVC

Rotordynamic and thermal analyses of compliant flexure pivot tilting pad gas bearings were performed. First, compliant flexure pivot tilting pad gas bearings with pad radial compliance (CFTPBs) were introduced and designed for high-speed oil-free micro turbomachinery. The pad radial compliance was for accommodation of large rotor growth at high speeds. Parametric studies on pivot offset, preload, and tilting stiffness were performed using non-linear orbit simulations and coast-down simulations for an optimum design. Second, coast-down tests for imbalance response and stability of typical rotor-bearing system with a rigid rotor and two CFTPBs designed from the above design studies were conducted over operating speeds up to 55 krpm. Prediction of synchronous rotordynamic responses was made in terms of critical speed for various imbalance modes by using a rotordynamic analysis software (XLTRC), combined with dynamic force coefficients from the perturbation analysis. For stability analyses, a generalized orbit simulation program was developed considering both the translational and angular rotor motions with two different bearings. Linear stability analyses for the conical vibration mode were also performed by using XLTRC and the perturbation analysis based on the Lund method. Predictions of whirl speed showed good agreement to the tests, but the estimated onset speed of instability appeared lower than the measured instability. Finally, a new thermo-hydrodynamic analysis model of a typical rotor-bearing system with CFTPBs was presented, accompanying linear perturbation analyses to investigate thermal effects on the rotordynamic performance. A numerical procedure was established for solving the generalized Reynolds equation, the 3-D energy equation, and the associated boundary conditions at the pad inlet flow and solid walls (rotor and pad) simultaneously. Parametric studies were conducted on nominal clearance and external load. Nominal clearance showed significant influence on temperature fields, and external load had uneven thermal effects among pads. Case studies with heat flux and temperature boundary conditions on the rotor end surface were performed to simulate various working conditions of the bearing. Large rotor thermal growth due to the high rotor temperature showed noticeable influence on rotordynamic performance by increasing direct stiffness and damping coefficients.