Computational Model for Tilting Pad Journal Bearings

1. TRC project 2010-2011 TRC 32514/15196B/ME Computational Model for Tilting Pad Journal Bearings Yujiao Tao Research Assistant Dr. Luis San Andres Mast-Childs Professor 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. XLTRC2 TFPBRG 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 • Reviewed literature on TPJBs • Developed analysis for effect of pivot flexibility on TPJBs load response. • Took XLPRESSDAM® code and began modifications • Obtained initial predictions for a near-rigid TPJB Comprehensive table summing 46 papers

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

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

7. Work to date • Reviewed literature on TPJBs • Developed analysis for effect of pivot flexibility on force coefficients of TPJBs. • Took XLPRESSDAM® code and began modifications • 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 Reynoldsequation for thin film bearing On kth pad h: fluid film thickness P: hydrodynamic pressure μ: lubricant viscosity  : journal speed RJ: journal radius

9. Thermal energy transport in thin film flows • Nomenclature • T: film temperature • h : film thickness • U,W: circ. & axial flow velocities • m, r, Cv:viscosity & density, specific heat • hB, hJ : heat convection coefficients • TB, TJ: bearing and journal temperatures W: 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 Y X h x Pad Y Cp : Pad radial clearance Rd = Rp+t : pad thickness rp: pad dimensional preload dp : pad tilt angle xpiv, hpiv: pivot radial and transverse deflections Journal WY OP OP’ Fluid film WX OB X e dp θ h  θp RP P xpiv P’ Pivot h x hpiv

11. Y X h x Journal static equilibrium in a TPJB Pad equations of motion about pivot point P k=1,…Npad Y Journal WY Op X WX is pad mass matrix Fluid film moment on pad ’ x P’ Pad h

12. Consider small journal motion perturbations with frequency (w) about the equilibrium position , the journal displacements are: Perturbation analysis • Journal motions induce changes in the rotation (d) of the kthpad and its pivot displacements (z,h) with the same frequency (w) • And, journal and pad motions induce changes in the film thickness and pressure fields

13. Reduced force coefficients • 25 force impedances for the kth pad a, b=X, Y, x, h, d • The reduced force impedances are

14. Reduced force coefficients (in pad coordinates) Alternatively, reduced impedances(ZR)are also obtainedin pad local coordinates. Y X h x According to the perturbation analysis, the reduced impedances obtained by two methods are identical:

15. Work to date • Reviewed literature on TPJBs • Developed analysis for effect of pivot flexibility on force coefficients of TPJBs. • Took XLPRESSDAM® code and began modifications • 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 • Reviewed literature on TPJBs • Developed analysis for effect of pivot flexibility on force coefficients of TPJBs. • Took XLPRESSDAM® code and began modifications • Obtained initial predictions for near-rigid TPJB Comparison with other predictions and some experimental results

18. W Y X Predictions for a (near rigid) TPJB bearing (Someya*) Five pad, tilting pad bearing (LOP) • Isothermal flow, isoviscous • Synchronous speed reduced force coefficients *Someya, T., 1988, Journal-Bearing Databook, Springer-Verlag, Berlin ,pp. 227-229. Comparison of results for

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

20. W Y X Predicted stiffness coefficients KXX KP Pivot flexibility lowers the direct stiffness coefficient KXX (along load direction), in particular for large loads. KYY

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

22. Y W X Comparison with recent test data Wilkes* five pad, rocker-back pivot, tilting pad bearing (LOP) Operating condition Journal speed : 4,400 rpm Unit load: 1566 kPa (227 psi) Lubricant supply temperature :25 oC Used pivot stiffness: Pivot radial stiffness: 2 GN/m *Proceedings of ASME Turbo Expo 2011, Paper No. GT2011-46510

23. Y X W Predicted & Test impedances versus frequency Dimensionless Eccentricity Real part of impedances K-C model: Z=K + iωC Stiffness: K=Re (Z) Damping:C=Im (Z)/ ω Re (ZYY)-prediction Re (ZYY)-measurement Dynamic stiffness KYY over predicted Re (ZXX)-prediction Re (ZXX)-measurement

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

25. Updated XLTRC2 XLPRESSDAM 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. Conclusions

26. Proposed work for 2nd year • Complete analysis of reduced frequency force coefficients for TPJBs for NONLINEAR pivot stiffness depending on the type of contact. • Derivation of iterative search scheme to update the pad radial and transverse deformations and ensure reliable convergence to an equilibrium solution. • Implementation of various oil feed arrangements in the FE model to model TPJBs with leading edge groove supply systems and scrapers. • 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 Code for Tilting Pad Bearings End product (code) will enable TRC members to model modern TPB configurations and to improve predictions of dynamic forced response (K-C-M model)

28. Questions (?)