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Research Papers: Gas Turbines: Structures and Dynamics

Orbit-Model Force Coefficients for Fluid Film Bearings: A Step Beyond Linearization

[+] Author and Article Information
Luis San Andrés

Mast-Childs Chair Professor
Fellow ASME
Mechanical Engineering Department,
Texas A&M University,
College Station, TX 77843
e-mail: Lsanandres@tamu.edu

Sung-Hwa Jeung

Mechanical Engineering Department,
Texas A&M University,
College Station, TX 77843
e-mail: sean.jeung@gmail.com

1Corresponding author.

Contributed by the Structures and Dynamics Committee of ASME for publication in the JOURNAL OF ENGINEERING FOR GAS TURBINES AND POWER. Manuscript received June 23, 2015; final manuscript received July 21, 2015; published online September 1, 2015. Editor: David Wisler.

J. Eng. Gas Turbines Power 138(2), 022502 (Sep 01, 2015) (11 pages) Paper No: GTP-15-1217; doi: 10.1115/1.4031237 History: Received June 23, 2015

High-performance turbomachinery demands high shaft speeds, increased rotor flexibility, tighter clearances in the flow passages, advanced materials, and increased tolerance to imbalances. Operation at high speeds induces severe dynamic loading with large amplitude shaft motions at the bearing supports. Typical rotordynamic models rely on linearized force coefficients (stiffness K, damping C, and inertia M) to model the reaction forces from fluid film bearings and seal elements. These true linear force coefficients are derived from infinitesimally small amplitude motions about an equilibrium position. Often, however, a rotor–bearing system does not reach an equilibrium position and displaces with motions amounting to a sizable portion of the film clearance; the most notable example being a squeeze film damper (SFD). Clearly, linearized force coefficients cannot be used in situations exceeding its basic formulation. Conversely, the current speed of computing permits to evaluate fluid film element reaction forces in real time for ready numerical integration of the transient response of complex rotor–bearing systems. This approach albeit fast does not help to gauge the importance of individual effects on system response. Presently, an orbit analysis method estimates force coefficients from numerical simulations of specified journal motions and predicted fluid film reaction forces. For identical operating conditions in static eccentricity and whirl amplitude and frequency as those in measurements, the computational physics model calculates instantaneous damper reaction forces during one full period of motion and performs a Fourier analysis to characterize the fundamental components of the dynamic forces. The procedure is repeated over a range of frequencies to accumulate sets of forces and displacements building mechanical transfer functions from which force coefficients are identified. These coefficients thus represent best fits to the motion over a frequency range and dissipate the same mechanical energy as the nonlinear mechanical element. More accurate than the true linearized coefficients, force coefficients from the orbit analysis correlate best with SFD test data, in particular for large amplitude motions, statically off-centered. The comparisons also reveal the fallacy in representing nonlinear systems as simple K–C–M models impervious to the kinematics of motion.

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References

Zeidan, F. , San Andrés, L. , and Vance, J. , 1996, “ Design and Application of Squeeze Film Dampers in Rotating Machinery,” 25th Turbomachinery Symposium, Texas A&M University, Houston, TX, Sept. 16–19, pp. 169–188.
Vance, J. , 1988, Rotordynamics of Turbomachinery, Wiley, New York, pp. 234–246.
Tiwari, R. , Lees, A. W. , and Friswell, M. I. , 2004, “ Identification of Dynamic Bearing Parameters: A Review,” Shock Vib. Dig., 36(2), pp. 99–124. [CrossRef]
Choy, F. K. , Braun, M. J. , and Hu, Y. , 1991, “ Nonlinear Effects in a Plain Journal Bearing: Part 1—Analytical Study,” ASME J. Tribol., 113(3), pp. 555–561. [CrossRef]
El-Shafei, A. , and Eranki, R. , 1994, “ Dynamic Analysis of Squeeze Film Damper Supported Rotors Using Equivalent Linearization,” ASME J. Eng. Gas Turbines Power, 116(3), pp. 682–691. [CrossRef]
Müller-Karger, C. , and Granados, A. , 1997, “ Derivation of Hydrodynamic Bearing Coefficients Using the Minimum Square Method,” ASME J. Tribol., 119(4), pp. 802–807. [CrossRef]
Czolczynski, K. , 1999, Rotordynamics of Gas-Lubricated Journal Bearing Systems, Springer-Verlag, New York, Chap. 2.4.
Sawicki, J. , and Rao, T. , 2001, “ Nonlinear Prediction of Rotordynamic Coefficients for a Hydrodynamic Journal Bearing,” Tribol. Trans., 44(3), pp. 367–374. [CrossRef]
Sawicki, J. , and Rao, T. , 2004, “ Nonlinear Prediction of Rotordynamic Coefficients for a Hydrodynamic Journal Bearing,” Int. J. Rotating Mach., 10(6), pp. 507–513. [CrossRef]
Diaz, S. , and San Andrés, L. , 2000, “ Orbit-Based Identification of Damping Coefficients for a Rotor Mounted on Off-Centered Squeeze Film Dampers and Including Support Flexibility,” ASME Paper No. 2000-GT-0394.
San Andrés, L. , and De Santiago, O. , 2004, “ Forced Response of a Squeeze Film Damper and Identification of Force Coefficients From Large Orbital Motions,” ASME J. Tribol., 126(2), pp. 292–300. [CrossRef]
San Andrés, L. , 2012, “ Experimental Identification of Bearing Force Coefficients,” Modern Lubrication Theory, Notes 14, Texas A&M University Digital Libraries, College Station, TX.
San Andrés, L. , and Jeung, S.-H. , 2015, “ Experimental Performance of an Open Ends, Centrally Grooved, Squeeze Film Damper Operating With Large Amplitude Orbital Motions,” ASME J. Eng. Gas Turbines Power, 137(3), p. 032508. [CrossRef]
Bradley, G. , 2013, “ Performance of a Short Open-End Squeeze Film Damper With Feed Holes: Experimental Analysis of Dynamic Force Coefficients,” M.S. thesis, Texas A&M University, College Station, TX.
San Andrés, L. , 2012, “ Extended Finite Element Analysis of Journal Bearing Dynamic Forced Performance to Include Fluid Inertia Force Coefficients,” ASME Paper No. IMECE2012-87713.
San Andrés, L. , and Delgado, A. , 2012, “ A Novel Bulk-Flow Model for Improved Predictions of Force Coefficients in Grooved Oil Seals Operating Eccentrically,” ASME J. Eng. Gas Turbines Power, 134(5), p. 052509. [CrossRef]
San Andrés, L. , 2012, “ Damping and Inertia Coefficients for Two Open Ends Squeeze Film Dampers With a Central Groove: Measurements and Predictions,” ASME J. Eng. Gas Turbines Power, 134(10), p. 102506. [CrossRef]
San Andrés, L. , and Seshagiri, S. , 2013, “ Damping and Inertia Coefficients for Two End Sealed Squeeze Film Dampers With a Central Groove: Measurements and Predictions,” ASME J. Eng. Gas Turbines Power, 135(12), p. 112503. [CrossRef]
San Andrés, L. , 2014, “ Force Coefficients for a Large Clearance Open Ends Squeeze Film Damper With a Central Groove: Experiments and Predictions,” Tribol. Int., 71, pp. 17–25. [CrossRef]
San Andrés, L. , Jeung, S.-H. , and Bradley, G. , 2014, “ Dynamic Forced Performance of Short Length Open-Ends Squeeze Film Damper With End Grooves,” IFToMM International Conference on Rotordynamics, Sept. 22–25, Milan, Italy, pp. 855–866.
Jeung, S.-H. , 2013, “ Performance of an Open Ends Squeeze Film Damper Operating With Large Amplitude Orbital Motions: Experimental Analysis and Assessment of the Accuracy of the Linearized Force Coefficients Model,” M.S. thesis, Texas A&M University, College Station, TX.

Figures

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Fig. 1

Schematic view of a simple journal bearing describing small amplitude whirl motions about an equilibrium position

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Fig. 2

Depiction of small amplitude journal motions about an equilibrium position (not to scale) [15]

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Fig. 3

Example of journal describing off-centered, large amplitude elliptical motions within clearance circle

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Fig. 4

Example of analysis for an elliptical off-centered orbit: journal motion X versus Y and fluid film bearing reaction forces (FX versus FY). Dots indicate discrete points at which the numerical program predicts forces.

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Fig. 5

Cross section views of two test SFD configurations: (a) damper A—L = 25.4 mm, D = 127 mm, cA = 251 μm (nominal), and 12.7 × 9.65 mm feed central groove [13] and (b) damper B—L = 25.4 mm, D = 127 mm, cB = 127 μm (nominal), and no feed groove [14]

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Fig. 6

Open ends damper B (cB = 129 μm): real and imaginary parts of direct complex stiffnesses (H) versus excitation frequency. Predictions with circular orbits r/cB = 0.2 and e/cB = 0.5. Symbol o: H̃ constructed from force responses to various orbits; continuous line: H̃=K̃−ω2M̃+i ω C̃ with orbit-based force coefficients; and dash line: H = K − ω2M + iωC with linearized force coefficients.

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Fig. 7

Open ends damper B: forces FX versus FY determined from off-centered (e/cB = 0.5) circular orbit (r/cB = 0.2) and whirl frequency 50 Hz and 100 Hz. Symbol • : periodic force response; continuous line: F1 fundamental component of periodic force response; and dashed-line: constructed with orbit-based coefficients, F1 ≈(K̃−ω2M̃+i ω C̃) z1.

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Fig. 8

Open ends damper B (cB = 129 μm): real and imaginary parts of direct complex stiffnesses versus excitation frequency. Tests with circular orbits: (a) r/cB = 0.3 and e/cB = 0; (b)r/cB = 0.5 and e/cB = 0; and (c) r/cB = 0.3 and e/cB = 0.4. Symbol □ : test data [14]; continuous line: constructed with orbit-based coefficients; and dashed-line: constructed with linearized force coefficients, H̃=(K̃−ω2M̃+i ω C̃) and H=(K−ω2M+i ω C).

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Fig. 9

Open ends damper A (cA = 251 μm): SFD effective stiffness ( −Keff-A) and damping (Ceff-A) coefficients—(left) versus frequency and (right) versus amplitude (r/cA). Top: circular centered orbits and bottom: off-centered orbits at es/cA = 0.76. Test data from Ref. [13]. (The total uncertainty for the direct damping (C), added mass (M), and stiffness (K) coefficients are (with rounding) UC < 8%, UM < 12%, and UK < 3%, respectively. Note the uncertainties are valid exclusively for the identification frequency range (10–100 Hz) [14].)

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Fig. 10

Open ends damper B (cB = 127 μm): SFD effective stiffness (−Keff-B) and damping (Ceff-B) coefficients—(left) versus frequency and (right) versus amplitude (r/cB). Top: circular centered orbits and bottom: off-centered orbits at es/cB = 0.4. Test data from Ref. [14].

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Fig. 11

Open ends damper B (cB = 127 μm): experimental and predicted SFD direct damping and added mass coefficients versus amplitude (rX/cB). Parameters identified for elliptical (5:1) orbits with static eccentricity eS/cB = 0.1. Test data from Ref. [14].

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Fig. 12

Open ends damper A (cA = 251 μm): SFD forces (FX, FY): actual from test data and derived from orbit analysis. Operation at (a) r = 0.08cA and es/cA = 0.00, (b) r = 0.08cA and es/cA = 0.75, and (c) r = 0.71cA and es/cA = 0.0. Whirl frequency of 100 Hz. Test data from Ref. [21]. Note different scales for the force magnitudes.

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Fig. 13

Open ends damper A (cA = 251 μm): SFD mechanical work performed from the (a) experimental forces and (b) forces with coefficients from orbit model. Circular orbit amplitudes from r/cA = 0.08–0.71 and static eccentricities from es/cA = 0–0.76. Whirl frequency = 100 Hz. Test data from Ref. [21].

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Fig. 14

Open ends SFDs A [21] and B [14]: measure of energy differences (test-model) for circular orbits with amplitude r = 0.08c–0.71c and static eccentricity es = 0.0–0.76c. Motions with whirl frequency of 100 Hz.

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