1. Computational Model for Tilting Pad Journal Bearings Yujiao Tao Research Assistant Dr. Luis San Andres Mast-Childs Professor TRC project 2010-2011 TRC 32514/15196B/ME START DATE: September 1, 2010

2. Justification and Objective The accurate prediction of tilting pad journal bearing (TPJB) static and dynamic forced performance is vital to the successful design and operation of high-speed rotating machinery. Pivot flexibility reduces bearing force coefficients for operation with heavy loads. XLTRC 2 TFP BRG code shows poor predictions for dynamic force coefficients when compared to test data. Research objective: To develop an advanced computational program, benchmarked by test data, to predict the static and dynamic forced performance of modern TPJBs accounting for thermal effects and the (nonlinear) effects of pivot flexibility.

3. Work to date (a)Reviewed literature on TPJBs (b)Developed analysis for effect of pivot flexibility on TPJBs load response. (c)Took XL PRESSDAM ® code and began modifications (d)Obtained initial predictions for a near-rigid TPJB Comprehensive table summing 46 papers

4. Literature review Reviewed 46 papers on TPJBs (1964-2011) and prepared a table that includes analysis methods, test methods and force coefficient identification, lubricant feeding arrangements, etc. Reviewed oil feed arrangements and other conditions to improve TPJBs’ performance. Views of leading edge groove in TPJB (Ball, J. H., and Byrne, T. R., 1998) Single externally adjustable pad fluid film bearing (Shenoy B. S. and Pai R.2009)

5. Literature review 46 papers on TPJBs (1964-2011)

6. Literature review 46 papers on TPJBs (1964-2011)

7. Work to date (a)Reviewed literature on TPJBs (b)Developed analysis for effect of pivot flexibility on force coefficients of TPJBs. (c)Took XL PRESSDAM ® code and began modifications (d)Obtained initial predictions for near-rigid TPJB Physical model and equations follow

8. Major assumptions : Laminar flow Includes temporal fluid inertia effects Average viscosity across the film On k th pad h : fluid film thickness P : hydrodynamic pressure μ : lubricant viscosity  : journal speed R J : journal radius Reynolds equation for thin film bearing

9. Thermal energy transport in thin film flows Nomenclature T : film temperature h : film thickness U,W : circ. & axial flow velocities  C v  : viscosity & density, specific heat h B, h J : heat convection coefficients T B, T J : bearing and journal temperatures  : journal speed Major assumptions : Neglect temperature variations across- film. Use bulk-flow velocities and temperature CONVECTION + DIFFUSION= DISSIPATION (Energy Disposed) = (Energy Generated)

10.   Film thickness in a pad C p : Pad radial clearance R d = R p +t : pad thickness r p : pad dimensional preload  p : pad tilt angle  piv  piv  pivot radial and transverse deflections Y θpθp h e WXWX Pivot Fluid film Journal OBOB RPRP WYWY OP’OP’  θ X P’ OPOP  piv  piv pp X Y   P Pad

11. Journal static equilibrium in a TPJB Fluid film moment on pad k=1,…Npad Journal X Y Pad WYWY WXWX   ’ P’P’ X Y   OpOp Pad equations of motion about pivot point P is pad mass matrix

12. Perturbation analysis Consider small journal motion perturbations with frequency (  ) about the equilibrium position, the journal displacements are: Journal motions induce changes in the rotation  of the k th pad and its pivot displacements  with the same frequency (  ) And, journal and pad motions induce changes in the film thickness and pressure fields

13. Reduced force coefficients 25 force impedances for the k th pad  X, Y,  The reduced force impedances are

14. Reduced force coefficients (in pad coordinates) Alternatively, reduced impedances (Z R ) are also obtained in pad local coordinates. According to the perturbation analysis, the reduced impedances obtained by two methods are identical: X Y  

15. Work to date (a)Reviewed literature on TPJBs (b)Developed analysis for effect of pivot flexibility on force coefficients of TPJBs. (c)Took XL PRESSDAM ® code and began modifications (d)Obtained initial predictions for near-rigid TPJB Fortran program and Excel GUI

16. Modified Fortran program and Excel GUI Uses finite element method to solve Reynolds equation (hydrodynamic pressure) Uses control volume method to solve energy transport equation Program updated for ideal TPJB with pivot flexibility. At this time, it works only for a near-rigid pivot (Difficulties in convergence).

17. Work to date (a)Reviewed literature on TPJBs (b)Developed analysis for effect of pivot flexibility on force coefficients of TPJBs. (c)Took XL PRESSDAM ® code and began modifications (d)Obtained initial predictions for near- rigid TPJB Comparison with other predictions and some experimental results

18. Predictions for a (near rigid) TPJB bearing *Someya, T., 1988, Journal-Bearing Databook, Springer-Verlag, Berlin, pp. 227-229. Number of Pads, N 5 ConfigurationLoad on Pad L/D0.5 Dimensionless Preload, r p 0.5 Pad Arc Angle,  p 60º Rotor Diameter, D0.06 m (2.36 inch ) Bearing Axial Length, L0.03 m (1.18 inch ) Pad radial Clearance, C p 120 μm (0.004724 inch ) Lubricant Viscosity,   0.028 Pa.s Rotor Speed 6000 rpm Offset 0.5 (Someya*) Five pad, tilting pad bearing (LOP) Isothermal flow, isoviscous Synchronous speed reduced force coefficients 1 RIGID pivot (Someya’s data) 2 RIGID pivot (My code) 3 Pivot stiffness Kp =3 GN/m (almost rigid) Comparison of results for W Y X

19. Predictions for static load versus journal eccentricity TPJB model with flexible pivot predicts a larger eccentricity than that with rigid pivot, especially at heavy loads (small S). W Y X Near rigid pivot Rigid pivot

20. Predicted stiffness coefficients K XX K YY W Y X Pivot flexibility lowers the direct stiffness coefficient K XX (along load direction), in particular for large loads. KPKP

21. Predicted damping coefficients C XX C YY W Y X Pivot flexibility lowers the direct damping coefficient C XX (along load direction), in particular for large loads.

22. Comparison with recent test data Number of Pads, N5 Load ConfigurationLoad on Pad Pad Arc Angle,  P 60º Offset0.5 Rotor Diameter, D101.59mm (4.0 in) Bearing Axial Length, L55.88 mm (2.20 in) Pad Radial Clearance, C P 120.65 μm (4.75 mil) Bearing Radial Clearance, C b 68 μm (2.67mil) Bearing Preload,0.44 Pad Mass, m p 0.44kg (0.97 lb) Pad Inertia, I G 2.49 kg-cm 2 ( 0.851 lb-in 2 ) Pad thickness, t19.05mm (3.228inch) Bearing pivot stiffness, K p nonlinear, ~0.5GN/m Bearing LubricantDTE 797, ISO VG-32 Wilkes* five pad, rocker-back pivot, tilting pad bearing (LOP) *Proceedings of ASME Turbo Expo 2011, Paper No. GT2011-46510 Operating condition Journal speed : 4,400 rpm Unit load: 1566 kPa (227 psi) Lubricant supply temperature :25 o C Used pivot stiffness: Pivot radial stiffness: 2 GN/m W Y X

23. Predicted & Test impedances versus frequency MeasuredPredicted X0.0090.006 Y-0.381-0.306 Dimensionless Eccentricity K-C model: Z=K + iωC Stiffness: K=Re (Z) Damping:C=Im (Z)/ ω W Y X Real part of impedances Re (Z YY )-prediction Re (Z YY )-measurement Re (Z XX )-measurement Re (Z XX )-prediction Dynamic stiffness K YY over predicted

24. Predicted & Test impedances versus frequency Imaginary part of impedances W Y X Im (Z YY )-measurement Im (Z YY )-prediction Im (Z XX )-measurement Im (Z XX )-prediction Both damping coefficients are underpredicted.

25. Conclusions Updated XL TRC 2 XL PRESSDAM code works for TPJBs with a near rigid pivot stiffness Predictions agree with published predictions for ideal, rigid pivot, TPJB. Comparisons with recent TPJB impedance data vs frequency, show damping coefficients are largely underpredicted while the off-load stiffness coefficients is over predicted. Test results at odds with prior test data. Current code used pivot stiffness ~ 4 times magnitude of that in test bearing.

26. Proposed work for 2 nd year 1.Complete analysis of reduced frequency force coefficients for TPJBs for NONLINEAR pivot stiffness depending on the type of contact. 2. Derivation of iterative search scheme to update the pad radial and transverse deformations and ensure reliable convergence to an equilibrium solution. 3. Implementation of various oil feed arrangements in the FE model to model TPJBs with leading edge groove supply systems and scrapers. 4. Comparison of predictions from the enhanced TPJB code to test data for various bearing geometries tested by Childs and students and preparation of a technical report (MS. Thesis).

27. TRC Budget Year II Support for graduate student (20 h/week) x $ 1,800 x 12 months$ 21,600 Fringe benefits (0.6%) and medical insurance ($191/month)$ 2,419 Travel to (US) technical conference$ 1,200 Tuition three semesters ($3,802 x 9 ch)$ 10,132 Office (PC & HD storage)$ 200 (2011-12) Year II$ 35,558 (2010-11) Year I$ 34,863 End product (code) will enable TRC members to model modern TPB configurations and to improve predictions of dynamic forced response (K-C-M model) Code for Tilting Pad Bearings

28. Questions (?)