A theoretical basis for static and dynamic operation of tilting pad journal bearings (TPJBs)
has evolved over the last 50 years. Originally demonstrated by Lund using the pad assembly
method and a classic Reynolds equation solution, the current state of the art includes full
thermoelastohydrodynamic solutions of the generalized Reynolds equation that include fluid
convective inertia effects, pad motions; and thermal and mechanical deformations of the
pads and shaft. The development of TPJB theory is reviewed, emphasizing dynamic modeling. The paper begins with the early analyses of fixed geometry bearings and continues to
modern analyses that include pad motion and stiffness and damping effects. The development
of thermohydrodynamic, thermoelastohydrodynamic, and bulk-flow analyses is reviewed. The
theories of TPJB dynamics, including synchronous and nonsynchronous models, are reviewed.
A discussion of temporal inertia effects in tilting pad bearing is considered. Future
trends are discussed, and a path for experimental verification is proposed.

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A Review of Tilting Pad Bearing Theory

This paper presents the identification of the rotordynamic force coefficients for a direct lubrication five-pad and four-pad tilting pad bearing. The bearing is 110 mm in diameter with an L/D of 0.4. The experiments include load-on-pad (LOP) and load-between-pad (LBP) configurations, with a 0.5 and 0.6 pivot offset, for rotor speeds ranging from 7500 rpm to 15000 rpm. The bearing force coefficients are identified from multiple frequency excitations (20 to 300 Hz) exerted on the bearing housing by a pair of hydraulic shakers, and are presented as a function of the excitation frequency, rotor speed, for a 300 kPa unit load. The experimental results also include temperatures at the trailing edge of three pads. The direct force coefficients, identified from curve-fits of the complex dynamic stiffness, are frequency independent if considering an added mass term much smaller than the test device modal mass. The force coefficients from the four-pad bearing load-between-pad configuration show similar coefficients in the loaded and orthogonal direction. On the other hand, as expected, the five-pad bearing load-on-pad shows larger coefficients (∼25%) in the loaded direction. The maximum pad temperature recorded for the 0.5 pivot offset configurations are up to 20° C higher than those associated to the 0.6 offset configuration. Results from a predictive code are within 50% of the experimental results for the direct stiffness coefficients and within 30% for the direct damping coefficients.Copyright © 2010 by ASME

Identification and Prediction of Force Coefficients in a Five-Pad and Four-Pad Tilting Pad Bearing for Load-on-Pad and Load-Between-Pad Configurations

Recent comprehensive experimental data showcasing the force coefficients of commercial size tilting pad journal bearings has brought to rest the long standing issue on the adequacy of the [K,C,M] physical model to represent frequency independent bearing force coefficients, in particular viscous damping. Most experimental works test TPJBs with large preloads, operating over a large wide range of rotor speeds, and with null to beyond normal specific loads. Predictions from apparently simple fluid film bearing models stand poor against the test data which invariably signals to theory missing pivot and pad flexibility effects, and most importantly, ignoring significant differences in bearing and pad clearances due to actual operation, poor installation and test procedures, or simply errors in manufacturing and assembly. Presently, a conventional thermo hydrodynamic bulk flow model for prediction of the pressure and temperature fields in TPJBs is detailed. The model accounts for various pivot stiffness types, all load dependent and best when known empirically, and allows for dissimilar pad and bearing clearances. The algorithm, reliable even for very soft pad-pivots, predicts frequency reduced bearing impedance coefficients and, over a certain frequency range, delivers the bearing stiffness, damping and virtual mass force coefficients. Good correlation of predictions against a number of experimental results available in the literature bridges the gap between a theoretical model and the applications. Predicted pad reaction loads reveal large pivot deflections, in particular for a bearing with large preloaded pads, with significant differences in pivot stiffness as a function of specific load and operating speed. The question on how pivot stiffness acts to increase (or decrease) the bearing force coefficients, in particular the dynamic stiffness vs. frequency, remains since the various experimental data show contradictory results. A predictive study with one of the test bearings varies its pivot stiffness from 10% of the fluid film stiffness to an almost rigid one, 100 times larger. With certainty, bearings with nearly rigid pivot stiffness show frequency independent force coefficients. However, for a range of pad pivot stiffness, 1/10 to ten times the fluid film stiffness, TPJB impedances vary dramatically with frequency, in particular as the excitation frequency grows above synchronous speed. The bearing virtual mass coefficients become negative, thus stiffening the bearing for most excitation frequencies.© 2013 ASME

The Role of Pivot Stiffness on the Dynamic Force Coefficients of Tilting Pad Journal Bearings

For several years, researchers have presented predictions showing that using a full tilting-pad journal bearing (TPJB) model (retaining all of the pad degrees of freedom) is necessary to accurately perform stability calculations for a shaft operating on TPJBs. This paper will discuss this issue, discuss the importance of pad and pivot flexibility in predicting impedance coefficients for the tilting-pad journal bearing, present measured changes in bearing clearance with operating temperature, and summarize the differences between measured and predicted frequency dependence of dynamic impedance coefficients. The current work presents recent test data for a 100 mm (4 in.) five-pad TPJB tested in load on pad (LOP) configuration. Measured results include bearing clearance as a function of operating temperature, pad clearance and radial displacement of the loaded pad (the pad having the static load vector directed through its pivot), and frequency-dependent stiffness and damping. Measured hot-bearing clearances are approximately 30% smaller than measured cold-bearing clearances and are inversely proportional to pad surface temperature; predicting bearing impedances with a rigid pad and pivot model using these reduced clearances results in overpredicted stiffness and damping coefficients that are several times larger than previous comparisons. The effect of employing a full bearing model 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. The same stability calculation was performed using measured stiffness and damping coefficients at synchronous and subsynchronous frequencies. These predictions showed 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, an outcome that is completely opposite from conclusions reached by comparing different prediction models. This contrasting outcome results from a predicted increase in damping with increasing excitation frequency at all speeds and loads; however, this increase in damping with increasing excitation frequency was only measured at the most heavily loaded conditions.

Tilting Pad Journal Bearings—A Discussion on Stability Calculation, Frequency Dependence, and Pad and Pivot

Measured and predicted static and dynamic characteristics are provided for a four-pad, rocker-pivot, tilting-pad journal bearing (TPJB) in the load-on-pad (LOP) and load-between-pad (LBP) orientations. The bearing has the following characteristics: pad-pivot offset = 0.57, LID = 0.6, pad length = 60.33 mm. Unit loads ranged from 0 to 2903 kPa, and speeds ranged from 6.8 to 13.2 krpm. Nonrotating tests were carried out using a small rotating load to precess the test-bearing stator around the rotor shaft while measuring the clearances. These tests produced "clearance rectangles" for the LOP case and "clearance rhombuses" for the LBP cases. These tests defined the bearing clearances for facing bearing pads that were significantly different with a ratio between the larger and smaller clearances at approximately 1.6. Clearances were measured at room temperatures and immediately following tests to obtain room temperature and "hot" clearances. Hot-clearance measurements showed a 16%-25% decrease as compared to room-temperature clearances. Static load-deflection tests were carried out to determine the pad's flexibility characteristics with respect to the housing (pad-pivot flexibility). Detailed circumferential temperature measurements were made on the loaded pad(s) with only leading and trailing temperatures for the unloaded pads. The radial thermal gradient was examined in the loaded pad via embedded thermocouples on the rotor and outside of the pads. Results showed a 5-25 °C decrease from the rotor side of the pad to housing side. An FEM analysis predicted that the radial and circumferential temperature gradients caused an uneven thermal deflection in the pad, changing the pads' radii of curvature. (However, the changes made scant differences in predictions.) Dynamic-excitation tests were performed over a range of excitation frequencies Ω to obtain 2 × 2 complex dynamic-stiffness matrices [H ij ] as a function of Ω. The Re(H ij ) coefficients were readily fitted as a linear function of Ω 2 , producing frequency-independent stiffness and virtual-mass coefficients. The Im(H ij ) coefficients were readily fitted as a linear function of Ω, producing frequency-independent damping coefficients and supporting the adequacy of a constant-frequency MCK model for bearings out to running speed. Measured (separate) pad clearances, pad-contact flexibility characteristics, and input temperatures were used as input for a recently-developed code to predict the static and dynamic characteristics of the bearing. The code used a Reynolds equation model plus an adiabatic energy equation. It also accounts for pad-contact flexibility. Measurements versus predictions were made for the temperature distributions, the dynamic-stiffness coefficients, and the direct rotordynamic coefficients (stiffness, damping, and virtual-mass). The measured cross-coupled stiffness and damping coefficients were insignificant, and are not presented. Generally, the code predicts the trends of the circumferential temperature distributions well; however, it predicted a continuing increase in temperature from leading to trailing edge, while the tests show an increase through the next-to-last temperature probe and then a drop to the last probe nearest the trailing edge. Generally speaking, the code does an adequate job of predicting rotordynamic coefficients for both LOP and LBP conditions. The input data (clearances, pad-flexibility, etc.) and output results (temperatures, dynamic stiffness coefficients, rotordynamic coefficients) presented allow other researchers to directly make predictions for these bearings using alternate models and codes.

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Measurements Versus Predictions for the Static and Dynamic Characteristics of a Four-Pad, Rocker-Pivot, Tilting-Pad Journal Bearing

Static performance characteristics and rotordynamic coefficients were experimentally determined for a four-pad, spherical-seat, tilting-pad journal bearing in load-between-pad configuration. Measured static characteristics include journal static equilibrium position, estimated power loss, and trailing-edge pad temperatures. Rotordynamic coefficients were determined from curve fits of measured complex dynamic-stiffness coefficients as a functions of the excitation frequency. A frequency-independent [M]-[C]-[K] model did a good job of fitting the measurements. Test conditions included speeds from 4 to 12 krpm and unit loads from 0 to 1896 kPa (0 to 275 psi). The bearing uses cool inlet oil to decrease the pad operating temperatures and increase the bearing’s load and speed capacity. The bearing has a nominal diameter of 101.78 mm (4.0070 in). Measurements indicated significant bearing crush with a radial bearing clearance of 99.63 μm (3.92 mils) in the axis 45° counterclockwise from the loaded axis and 54.60 μm (2.15 mils) in the axis 45° clockwise from the loaded axis. The pad length is 101.60 mm (4.00 in), giving L/D = 1.00. The pad arc angle is 73°, and the pivot offset ratio is 65%. Testing was performed using a test rig described by Kaul [1], and rotordynamic coefficients were extracted using a procedure adapted from Childs and Hale [2]. A bulk-flow Navier-Stokes model was used for predictions, using adiabatic conditions for the fluid in the bearings. However, the model assumes constant nominal clearances at all pads, and an average clearance was used based on measured clearances. Measured static eccentricities and attitude angles were significantly lower than predicted. Attitude angles varied from 6° to 39° and decreased with load. Power loss was well-predicted, with a maximum value of 25 kW (34 hp). The maximum detected pad temperature was 71°C (160°C) while the temperature rise from inlet to exit was over-predicted by 8°C (14°F). Direct stiffness and damping coefficients were significantly over-predicted, but the addition of a simple pivot-stiffness in series with the measured stiffness and damping values vastly improved the agreement between theory and experiment. Direct added masses were negative to a higher degree for Myy (y load direction) at low speeds and increased with speed. With the exception of Myy at zero load, they became positive before reaching 8,000 rpm. Although significant cross-coupled stiffness terms were present, they always had the same sign, producing a whirl frequency ratio of zero and netting unconditional stability over all test conditions.© 2008 ASME

Joel Harris

Papers

Static Performance Characteristics and Rotordynamic Coefficients for a Four-Pad Ball-in-Socket Tilting Pad Journal Bearing