1. 1 TRC 2008 The Effect of (Nonlinear) Pivot Stiffness on Tilting Pad Bearing Dynamic Force Coefficients – Analysis Jared Goldsmith Research Assistant Dr. Luis San Andres Mast-Childs Professor TRC Project 32513/1519 T3

2. 2 XLTRC 2 Project Goals Enhance tilting pad bearing model by including nonlinear pivot flexibility for rocker, spherical, and flexure type pivots

3. 3 Tilting Pad Journal Bearing Pivot Types Tilting Pad Bearings Y X PAD ROTATION Y X Y X FLEXURE WEB

4. 4 Film Thickness Tilting pad journal bearing & coordinates rotational stiffness radial stiffness journal speed film thickness Film thickness: Pad clearance ( ) and preload ( ) and journal eccentricity ( ) Pad angular rotation ( ), radial ( ) and transverse displacement ( ) for

5. 5 Perturbation Analysis Small amplitude journal motions about an equilibrium position Applying an external static load with components ( ) to the journal determines its static equilibrium position ( ) with fluid static pressure field, film thickness, and corresponding equilibrium pad rotation and deflections ( ) Consider small amplitude journal center motions ( ) of frequency about the static equilibrium point. Hence Consider small amplitude journal and pad motions about static equilibrium position (SEP) and for

6. 6 Load and Pad EOMs Bearing forces and Pad Equations of Motion The sum of the pad fluid film forces balance the external load applied on the journal, i.e., Force and Moment EOMs for pad: Matrix representing pad inertia and mass Y X Journal Rotation Bearing Fluid film

7. 7 Nonlinear Pad Pivot Typical Nonlinear Pivot Radial Force Radial Deflection The assumption of small amplitude motions about an equilibrium position allows the pivot reaction radial force to be expressed as where is the static load on the pivot and is the force due to radial displacement Consider a typical nonlinear force ( ) versus pivot radial deflection ( ) in a bearing pivot

8. 8 Pad Forces and Moment Forces and moments acting on a pad Pad Fluid Film Forces = integration of hydrodynamic pressure fields on pad Moment on Pad: Substitution of zeroth and first order pressure fields gives Fluid film impedances:

9. 9 Reduced Force Coefficients Frequency reduced force coefficients for tilting pad bearing where and Assuming the pads move with the same excitation frequency ω as the journal whirl frequency, the frequency reduced coefficients are Matrices representing pivot stiffness and damping coefficients

10. 10 Progress Tilting pad bearing pivot Modified tilting pad bearing model now accounts for spherical and rocker nonlinear pivot stiffness Spherical pivot stiffness versus load Pivot Pivot housing Assumptions: Spherical pivot – point contact Rocker pivot – line contact

11. 11 Test Bearing Test bearing description Y X PAD ROTATION Carter and Childs* five pad, rocker pivot, tilting pad bearing (LBP) and (LOP) Bearing ParametersValues Rotor diameter101.59 mm Pad axial length60.33 mm Pivot offset60% Pad number (arc length)5 (57.87) Radial pad clearance.1105 mm Pad inertia2.49E-4 kg- Preload0.282 Radial bearing clearance.0792 mm Mobile DTE ISO 32 31 cSt 5.5 cSt Viscosity @ 40° C Viscosity @ 100° C Density @ 15°C Specific heat 850 kg/m 3 1951 J/(kg-K) Fluid Properties * ASME Paper No. GT2008-5069

12. 12 (LBP) Rotordynamic Force Coefficients Experimental and predicted direct stiffness and damping coefficients Carter and Childs measured and predicted nonsynchronous direct stiffness Rotor speed = 4000 RPM Bearing loaded in –Y direction (LBP) The TPB model (rigid pivot) generally over predicts stiffness and damping coefficients

13. 13 Static Results Direct static stiffnesses versus load Original XLTRC 2 (rigid pivot) and modified XLTRC 2 (flexible pivot) predicted direct static stiffnesses versus static load Flexible pivot Rigid pivot Rotor speed = 4000 RPM Bearing loaded in –Y direction (LBP)

14. 14 Static Results Journal eccentricity versus load Original XLTRC 2 (rigid pivot) and modified XLTRC 2 (flexible pivot) predicted journal eccentricity versus static load Flexible pivot Rigid pivot Rotor speed = 4000 RPM Bearing loaded in –Y direction (LBP)

15. 15 Future work Perform extensive comparisons between predictions and Childs et al. experimental TPB stiffness and damping coefficients Include pivot friction for spherical pivots

16. 16 Statement of work Future work 1.Account for pad clearance variations due to thermal and mechanical deformation effects 2. Improve I/O operations for Excel interface 3. Implement informed eccentricity “guess” for starting calculations