1. This material is supported by the TAMU Turbomachinery Research Consortium. Parts of the investigation were conducted under NASA NRA on Subsonic Rotary Wing, SSRW2-1.3 Oil-Free Engine Technology (Foil Gas Bearing Modeling). Grant Cooperative Agreement NNX07P98A. Luis San Andrés Mast-Childs Professor Fellow ASME Texas A&M University Keun Ryu Sr. Development Engineer BorgWarner Turbo Systems ASME Turbo Expo 2011: Power for Land, Sea and Air June 6-10, 2011, Vancouver, BC GT2011-45763 On the Nonlinear Dynamics of Rotor- Foil Bearing Systems : Effects of Shaft Acceleration, Mass Imbalance and Bearing Mechanical Energy Dissipation Presentation available athttp://rotorlab.tamu.edu

2. Series of corrugated foil structures (bumps) assembled within a bearing sleeve. Integrate a hydrodynamic gas film in series with one or more structural layers. Applications: ACMs, micro gas turbines, turbo expanders, blowers, etc Reliable with adequate load capacity and high temperature capability Tolerant to misalignment and debris Need coatings to reduce friction at start-up & shutdown Damping from dry-friction and operation with limit cycles Gas Foil Bearings – Bump type OIL-FREE Systems! reduce overall system weight, complexity, and maintenance cost increase system efficiency due to low power losses extend maintenance intervals.

3. Gas Foil Bearings Issues Endurance: performance at start up & shut down Little test data for rotordynamic force coefficients Thermal management for high temperature applications (gas turbines, turbochargers) Prone to subsynchronous whirl and limit cycle operation – Forced nonlinearity! NOT rotordynamic instability (San Andrés, 2007) AIAA2007-5094 San Andrés, L. and Kim, T. H., 2008, “Forced Nonlinear Response of Gas Foil Bearing Supported Rotors,” Tribol. Int., 41(8)

4. TAMU research on foil bearings yearTopic 2008-11 Metal Mesh Foil Bearings: construction, verification of lift off performance and load capacity, identification of structural stiffness and damping coefficients, identification of rotordynamic force coefficients 2008-10 Extend nonlinear rotordynamic analysis Performance at high temperatures, temperature and rotordynamic measurements 2007-09 Thermoelastohydrodynamic model for prediction of GFB static and dynamic forced performance at high temperatures 2005-07 Integration of Finite Element structure model for prediction of GFB static and dynamic forced performance Effect of feed pressure and preload (shims) on stability of FBS. Measurements of rotordynamic response. 2005-07 Rotordynamic measurements: instability vs. forced nonlinearity? 2005-06 Model for ultimate load capacity, Isothermal model for prediction of GFB static and dynamic forced performance 2004-09 Measurement of static load capacity, Identification of structural stiffness and damping coefficients. Ambient and high temperatures

5. Overview – Subsynchronous motions Heshmat (2000): Operation of a flexible rotor-GFB system at super critical bending mode rotor speeds. Large amplitude subsynchronous motions suddenly appear while crossing system bending critical speed. Heshmat (1994): GFB operates at max speed of 132 krpm, i.e. 4.61 ×10 6 DN, showing stable limit cycle operation with large amplitude subsynchronous motions at frequency = rigid body mode natural frequency. Lee, et al. (2004, 2003): GFBs with viscoelastic layer eliminate large subsynchronous whirl motions appearing in flexible rotor-GFB system (2004) and a two stage centrifugal compressor (2003). San Andrés et al. (2006): Small imbalances lead to mainly synchronous rotor motions. Large mass imbalances cause sub harmonic motions at rotor speeds > 2 x system natural frequency (whirl frequency ratio ~ 50%) => nonlinear forced rotor responses San Andrés et al. (2007): Introduce simple GFB model as a nonlinear structure. rotor-GFB performs as a Duffing oscillator with multiple frequency response. Agreement between predictions and test data. 1/2 and 1/3 WFRs due to nonlinearity. (First paper predicting NL forced response of rotor-GFB systems with validations to reliable test data)

6. Example 1 – Subsynchronous motions Heshmat (1994) - Maximum speed 132 krpm, i.e. 4.61 ×10 6 DN. - Stable limit cycle operation with large amplitude sub harmonic motions at whirl frequency = rigid body mode natural frequency. Subsynchronous amplitude at 350 Hz Synchronous, 2,200 Hz (132 krpm) Supersynchronous amplitude at 3,300 Hz (bending mode)

7. Subsynchronous amplitude recorded during rotor speed coastdown from 132 krpm (2,200 Hz) Whirl amplitude remains ~ constant as subsynchronous frequency drops from 350 Hz to 180 Hz Heshmat (1994) - Maximum speed 132 krpm, i.e. 4.61 ×10 6 DN. - Stable limit cycle operation but with large amplitude subsynchronous motions. Whirl frequency tracks rotor speed Example 1 – Subsynchronous motions

8. Heshmat (2000) Flexible rotor- GFB system operation to 85 krpm (1.4 kHz): 1 st bending critical speed:34 krpm (560 Hz) Waterfall plot recorded during rotor speed coastdown test from 45 krpm (750 Hz) Rotor orbit shape at 45k rpm Large amplitude limit cycle motions above bending critical speed, whirl frequency = natural frequency (rigid body) Example 2 – Subsynchronous motions

9. Lee, et al. (2003, 2004) Flexible rotor supported on GFBs with viscoelastic layer Viscoelastic layer eliminates large motions at natural frequency & appearing above 1 st bending critical speed. 50 kRPM (833 Hz) Bump type GFB Viscoelastic GFB Synchronous vibration 1st bending mode Rigid body mode Bump type GFB Viscoelastic GFB Synchronous vibration Example 3 – Subsynchronous motions

10. San Andrés, L., et al., 2011, “Identification of Rotordynamic Force Coefficients of a Metal Mesh Foil Bearing Using Impact Load Excitations,” ASME J. Eng. Gas Turbines Power, Vol. 133 Example 4 – Subsynchronous motions Metal Mesh Foil Bearing

11. RudDloff, L., Arghir, M., et al., 2011, “Experimental Analysis of a First generation foil Bearing. Start-Up Torque and Dynamic Coefficients,” ASME GT2010-22966 Example 5 – Subsynchronous motions Unloaded FB: “Self-Excited” whirl motions at speed 30 krpm (500 Hz) with whirl frequency=165 Hz (WFR=0.33)

12. Kim, D., Shetty P., Lee. D., 2011, “Imbalance Response of a Rotor Supported by Hybrid Air Foil Bearings,” ASME GT2011- 45576 Example 6a – Subsynchronous motions Loaded hybrid FB (vertical): 2.67 bar gauge supply pressure Sub sync whirl motions start at 20 krpm with (nat) freq 5900 rpm (WFR=0.30). Too large amplitudes at 30 krpm, test stopped

13. Kim, D., Shetty P., Lee. D., 2011, “Imbalance Response of a Rotor Supported by Hybrid Air Foil Bearings,” ASME GT2011- 45576 Example 6b – Subsynchronous motions Loaded hybrid FB (vertical): 4 bar gauge supply pressure Large amplitude whirl motions start at 34 krpm (567 Hz) with whirl frequency~natural frequency 7200 rpm (WFR=0.21)

14. Amplitudes of subsynchronous motions INCREASE as imbalance increases ( forced nonlinearity! ) Rotor speed + Imbalance + San Andrés, L., Rubio, D., and Kim, T.H, 2007, “Rotordynamic Performance of a Rotor Supported on Bump Type Foil Gas Bearings: Experiments and Predictions,” ASME J. Eng. Gas Turbines Power, 129 Example 7 – Subsynchronous motions Gen II foil bearings

15. Large amplitudes locked at natural frequency (50 krpm to 27 krpm) …… but stable limit cycle! Kim, T.H., and San Andrés, L., 2009, “Effects of a Mechanical Preload on the Dynamic Force Response of Gas Foil Bearings - Measurements and Model Predictions,” STLE Tribol. Trans., 52 Rotor speed decrea ses Example 7 – Subsynchronous motions Gen II foil bearings

16. Analysis vs. test data Subsynchronous whirl frequencies concentrate in a narrow band enclosing natural frequency (132 Hz) of test system Amplitude vs. frequency Frequency vs. rotor speed Test data Predictions Test data Predictions Rotor speed (krpm) Frequency (Hz) San Andrés, L. and Kim, T. H., 2008, “Forced Nonlinear Response of Gas Foil Bearing Supported Rotors,” Tribol. Int., 41(8)

17. AIR SUPPLY Cooling flow/feed pressure on FB motions Typically foil bearings DO not require pressurization. Cooling flow is for thermal management : to remove heat from drag or to reduce thermal gradients in hot/cold engine sections Side effect: Axial flow retards evolution of circumferential flow velocity San Andres et al, ASME JGT, 209, v31

18. Effect of side flow on rotordynamics (a) 0.35 bar (b) 1.4 bar (c) 2.8 bar Whirl frequency locks at RBS natural frequency ( not affected by level of feed pressure For Ps ≥ 2.8 bar rotor subsync. whirl motions disappear; (stable rotor response) ω sub = 132 Hz ω sub = 147 Hz ω sub = 127 Hz Subsynchronous ω syn = 508 Hz Synchronous FFT of shaft motions at 30 krpm San Andres et al, ASME JGT, 209, v31

19. Onset of subsynchronous whirl motions (a) 0.35 bar (b) 1.4 bar (c) 2.8 bar Synchronous Subsynchronous N OS : 25 krpm N OS : 30.5 krpm N OS : 27 krpm Delay of large amplitude subsynchro nous rotor motions with increase in axial cooling flow (feed pressure) Effect of side flow on rotordynamics San Andres et al, ASME JGT, 209, v31

20. Objectives  To extend earlier analysis to predict the forced response of a rigid rotor supported on FBs modeled as nonlinear structure with material damping.  To determine the effects of rotor acceleration, imbalance mass, and the FB structural loss factor on the dynamic forced response of simple RBS. Most GFB analyses are complex; coupling top foil & under spring models with gas film flow model. but GFB forced performance depends mainly on the material properties of the support elastic structure Dynamic Stiffness & Damping Mechanism for Foil Bearing Dynamic Stiffness & Damping Mechanism for Foil Bearing GT2011-45763 Fast accelerations are typical in MTM due to small rotor mass moment of inertia. This work provides design and operation considerations for the appropriate selection and use of GFBs to avoid the build up of excessive nonlinear RBS response.

21. FB load–deflection structural test Nonlinear bearing forced deflection. Hysteresis loop shows energy dissipation Loading Unloading Stiffness hardening is likely to induce internal resonances at rotor speeds greater than the RBS natural frequency Kim and San Andrés (2007): Eight cyclic load - unload structural tests on Gen II foil bearing

22. Load–Deflection Structural tests Nonlinear bearing forced deflection: test data, polynomial fit & model prediction F = r (0.0675 -0.002 r + 0.0001 r 2 ) Test data Prediction Kim and San Andrés (2007): Eight cyclic load - unload structural tests on Gen II foil bearing r F FB FB load–deflection structural test

23. FB Structural Loss Factor  Loss factor (γ) represents structural damping and is obtained from load- deflection hysteresis loop TYP, loss factor is large at small displacements BUT decreases for large displacements. Typical of structural system with dry-friction where local stiffness coefficient r 

24. Natural frequency for small amplitude motions about SEP: = 130 Hz (7.8 krpm) Rotor-GFB system Rotor mass, M =1.02 kg Gen II foil bearings FB structure (static): Static equilibrium X E =37.5  m X Y Replicates laboratory set-up Shaft length (L) = 209.5 mm Shaft diameter (D) = 38.1 mm ( 25 bumps) L B =38.1 mm Hollow rotor L D

25. Equations of Motion EOMs: rigid rotor & in-phase imbalance Assumption: minute gas film with infinite film stiffness FB dynamic reaction force Varying rotor speed  (t) Rotor speed Rotor angle

26. Use Runge-Kutta scheme Rotor speeds: 2 - 36 krpm (600 Hz) Numerical solution Rotor speed ramp rate: α=±35 Hz/s, ±71 Hz/s, and ±283 Hz/s (2 krpm ↔ 36 krpm) Sampling rate: 12 k/s, 24 k/s, and 96 k/s Time step: 0.0833 ms, 0.04167 ms, and 0.01042 ms # integration points = 192,000 Simple MATHLAB or MATHCAD code: numerical integration and post- processing of results in frequency domain The fast rotor acceleration requires of a smaller time step (faster acquisition rate) since the speed changes quickly.

27. Typical rotor response Rotor acceleration +283 Hz/s Rotor deceleration -283 Hz/s u=8 µm; FB γ=0.14 Solutions obtained in a few seconds. Post-processing filters responses and finds synchronous and subsynchronous components of motion

28. Effect of rotor acceleration/deceleration on RBS forced response

29. 29 Effect of rotor acceleration (+) α= +283 Hz/s (FAST) α= +35 Hz/s (SLOW) u=8 µm. FB γ=0.14 WFR=ω/Ω Subsynchronous motion amplitudes more severe for the slowest rotor acceleration: More elapsed time for the whirl motions to build up! Ramp rate

30. 30 Effect of rotor deceleration u=8 µm. FB γ=0.14 α= -283 Hz/s (FAST DECEL)α= -35 Hz/s (SLOW DECEL) Notable differences in the onset speed and persistence of whirl motions show the RBS has a marked mechanical hysteresis Ramp rate

31. 31 Whirl frequencies: +α α= +35 Hz/s (SLOW acceleration) u=8 µm. FB γ=0.14 WFR=ω/Ω Subsynchronous whirl motions from 11 to 20 krpm with WFR=½ at first, and later from 20 to 36 krpm jump to WFR=⅓. Above 28 krpm, more complex WFRs ranging from 0.31 to 0.37, slightly above and below ⅓. Once a subsynchronous frequency motion appears, its amplitude rapidly increases with rotor speed. Significant motion amplitudes with WFR=½ and WFR=⅓ appear at ~twice and ~three times the system natural frequency.

32. 32 Whirl frequencies: - α α= -35 Hz/s (SLOW deceleration) u=8 µm. FB γ=0.14 WFR=ω/Ω For rotor speeds > ~20 krpm (333 Hz), motions with WFRs ranging from 0.27 to 0.41, i.e., a chaotic regime, are apparent. The motions with a 50% WFR are not as severe in amplitude as when the rotor accelerates, occurring over a shorter rotor speed span.

33. 33 Effect of rotor acceleration u=8 µm. FB γ=0.14 The peak amplitude during rotor deceleration is ~10 µm smaller than the one predicted during rotor acceleration x L =u/γ = 8 µm/0.14 = 57.1 µm Synchronous response  ± 71 Hz/s Linearized model =7800 rpm

34. Effect of mass imbalance on RBS forced response

35. 35 Effect of rotor mass imbalance α= +283 Hz/s, FB γ=0.14 u=4 μm u=20 μm Imbalance Mass imbalance exacerbates the bearings’ nonlinearity and showcases a distinctive jump phenomenon

36. 36 Effect of rotor mass imbalance u=20 µm. FB γ=0.14 α= -283 Hz/s α= +283 Hz/s Synchronous amplitude shows stiffness hardening effect with jump phenomenon while rotor accelerates Synchronous response

37. 37 Whirl frequencies: Imbalance ↑ u= 20 µm α= +283 Hz/s, FB γ=0.14 WFR=ω/Ω As imbalance mass increases, and for rotor speeds above (3×f n ), the rotor motion has a broader whirl motions at ⅓ WFR. ½ frequency whirl disappears for u>20 µm!

38. Effect of FB loss factor (material damping) on RBS forced response

39. 39 Effect of FB loss factor γ=0.07γ=0.28 α= +283 Hz/s. FB u=8 µm Bearing loss factor affects the onset and persistence of rotor sub harmonic motions FB loss factor

40. Whirl frequency: FB loss factor ↑ WFR=ω/Ω ½WFR motions are apparent from 8 ~ 15 krpm (1×f n ~ 2× f n ) For γ>0.2, ⅓WFR frequency components disappear! γ=0.28 α= +283 Hz/s. u=8 µm

41. 41 Effect of FB loss factor u=8 µm. γ=0.07 α= -283 Hz/s α= +283 Hz/s As rotor accelerates, the FB hardening stiffness with little damping (  low) affects more the response (multi-frequency). On deceleration, the rotor synchronous response appears free of nonlinearities Synchronous response

42. 42 Other NL response: small u & γ u=1 µm. γ=0.015 For speeds above RBS natural frequency (f n =130 Hz), whirl motions of large amplitudes with whirl frequency locked at the natural frequency! α=+283 Hz/s System natural frequency (f n =130 Hz) Bearing with little damping! (poor mechanical energy dissipation) Example TEST DATA (not same FBs)

43. Conclusions For operation above the RBS system critical speed and as the rotor accelerates, large amplitude whirl motions appear with a main subsynchronous frequency tracking rotor speed, first at 50% speed and later bifurcating into at 33% whirl frequency. Large rotor imbalances produce both jump phenomenon and a stronger hysteresis during slow acceleration and deceleration cases. Slow rotor accelerations result in responses with more abundant subsynchronous whirl patterns, increased amplitudes of whirl, and accompanied by a pronounced mechanical hysteresis when the rotor decelerates. Material damping (dry friction) in the FB aids to reduce and delay the nonlinear response, eventually eliminating the multiple frequency behavior. GT2011-45763

44. Recommendation GT2011-45763 Fast rotor start up and coast down procedures Reductions in rotor inertia or larger drive torque To ameliorate subsynchronous rotor motions resulting from nonlinear effect of hardening support structure Minimize rotor imbalance (a forcing function) Well-balanced rotor is a must! Large FB material damping Improvements in GFB design and materials

45. Acknowledgments Thanks support of TAMU TRC (2004-2008) NASA GRC (2007-09) & Dr. Samuel Howard NSF (2003-06), Foster-Miller (FBs) Learn more http://rotorlab.tamu.edu Questions (?) © 2011 Luis San Andres