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.

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

Static characteristics and rotordynamic coefficients were experimentally determined for a four-pad tilting-pad journal bearing with ball-in-socket pivots in loadbetween- pad configuration. A frequency-independent [M]-[C]-[K] model fit the measurements reasonably well, except for the cross-coupled damping coefficients. Test conditions included speeds from 4,000 to 12,000 rpm and unit loads from 0 to 1896 kPa (0 to 275 psi). The test bearing was manufactured by Rotating Machinery Technology (RMT), Inc. Though it has a nominal diameter of 101.78 mm (4.0070 in.), measurements indicated significant bearing crush with radial bearing clearances of 99.6 μm (3.92 mils) and 54.6 μm (2.15 mils) in the axes 45º counterclockwise and 45º clockwise from the loaded axis, respectively. The pad length is 101.6 mm (4.00 in.), giving L/D = 1.00. The pad arc angle is 73º, and the pivot offset ratio is 65%. The preloads of the loaded and unloaded pads are 0.37 and 0.58, respectively. A bulk-flow Navier-Stokes model was used for predictions, using adiabatic conditions for the bearing fluid. Because the model assumes constant nominal clearances at all pads, the average of the measured clearances was used as an estimate. Eccentricities and attitude angles were markedly under predicted while power loss was under predicted at low speeds and very well predicted at high speeds. The maximum detected pad temperature was 71ºC (160ºF) and the rise from inlet to maximum bearing temperature was over predicted by 10-40%. Multiple-frequency force inputs were used to excite the bearing. Direct stiffness and damping coefficients were significantly over predicted, but addition of a simple stiffness-in-series model substantially improved the agreement between theory and experiment. Direct added masses were zero or negative at low speeds and increased with speed up to a maximum of about 50 kg; they were normally greater in the unloaded direction. Although significant cross-coupled stiffness terms were present, they always had the same sign. The bearing had zero whirl frequency ratio netting unconditional stability over all test conditions. Static stiffness in the y direction (obtained from steadystate loading) matched the rotordynamic stiffness Kyy (obtained from multiple-frequency excitation) reasonably at low loads but poorly at the maximum test load.