XII Congreso y Exposición Latinoamericana de Turbomaquinaria, Queretaro, Mexico, February 24 2011

1. Turbomachinery Laboratory, Mechanical Engineering Department Texas A&M University Identification of Force Coefficients in Mechanical Components: Bearings and Seals A guide to a frequency domain technique Dr. Luis San Andres Mast-Childs Tribology Professor ASME Fellow, STLE Fellow Lsanandres@tamu.edu XII Congreso y Exposición Latinoamericana de Turbomaquinaria, Queretaro, Mexico, February 24 2011

2. Turbomachinery A turbomachinery is a rotating structure where the load or the driver handles a process fluid from which power is extracted or delivered to. Fluid film bearings (typically oil lubricated) support rotating machinery, providing stiffness and damping for vibration control and stability.In a pump, neck ring seals and inter stage seals and balance pistons also react with dynamic forces. Pump impellers also act to impose static and dynamic hydraulic forces.

3. Turbomachinery Acceptable rotordynamic operation of turbomachinery Ability to tolerate normal (even abnormal transient) vibrations levels without affecting TM overall performance (reliability and efficiency)

4. Rotordynamics primer (2) Model structure (shaft and disks) and find free-free mode natural frequencies Model bearings and seals: predict or IDENTIFY mechanical impedances (stiffness, damping and inertia force coefficients) Eigenvalue analysis: find damped natural frequencies and damping ratios for various (rigid & elastic) modes of vibration as rotor speed increases (typically 2 x operating speed) Synchronous response analysis: predict amplitude of 1X motion, verify safe passage through critical speeds and estimate bearing loads Certify reliable performance as per engineering criteria (API 610 qualification) and give recommendations to improve system performance

5. The need for parameter identification Experimental identification of force coefficients is important • to predict, at the design stage, the dynamic response of a rotor-bearing-seal system (RBS); • to reproduce rotordynamic performance when troubleshooting RBS malfunctions or searching for instability sources, & • to validate (and calibrate) predictive tools for bearing and seal analyses. The ultimate goal is to collect a reliable data base giving confidence on bearings and/or seals operation under both normal design conditions and extreme environments due to unforeseen events

6. The physical model Y Z X • Lateral displacements (X,Y) For lateral rotor motions (x, y), a bearing or seal reaction force vector f is modeled as K,C,Mare matrices of stiffness, damping, and inertia force coefficients (4+4+4 = 16 parameters) representing a linear physical system. The (K, C, M) coefficients are determined from measurements in a test system or element undergoing small amplitude motions about an equilibrium condition.

7. Bearings: dynamic reaction forces Y Z X • Lateral displacements (X,Y) Damping coefficients Stiffness coefficients Typical of oil-lubricated bearings: No fluid inertia coefficients accounted for. Force coefficients are independent of excitation frequency for incompressible fluids (oil). Functions of speed & applied load

8. Seals: dynamic reaction forces Liquid seals: Inertia coefficients Stiffness coefficients Damping coefficients Gas seals Typically: frequency dependent force coefficients

9. The concept of force coefficients Stiffness: Damping: Inertia: i,j = X,Y Strictly valid for small amplitude motions. Derived from SEP The “physical” idealization of force coefficients in lubricated bearings and seals

10. Modern parameter identification Modern techniquesrely on frequency domain procedures, where force coefficients are estimated from transfer functions of measured displacements (or velocities or accelerations) due to external loads of a prescribed time varying structure. Frequency domain methods take advantage of high speed computing and digital signal processors, thus producing estimates of system parameters in real time and at a fraction of the cost (and effort) than with antiquated and cumbersome time domain algorithms.

11. A test system example Consider a test bearing or seal element as a point mass undergoing forced vibrations induced by external forcing functions force, fY Kh,Ch:support stiffness and damping Mh: effective mass KYY, CYY Y KXY, CXY KXX, CXX Bearing or seal (K,C,M): test element stiffness, damping & inertia force coefficients Ω Journal X force, fX KhY, ChY KYX, CYX Soft Support structure KhX, ChX

12. Equations of motion (EOMs) For small amplitudes about an equilibrium position, the EOMs of a linear mechanical system are Kh,Ch:structure stiffness and damping Mh: effective mass where (K,C,M): test element force coefficients Note:The system structural stiffness and damping coefficients, {Kh,Ch}i=X,Y, are obtained from prior shake tests results under dry conditions, i.e. without lubricant in the test element

13. Identification model (1) and measure and measure Apply two independent force excitations on the test element Step (1) Apply Step (2) Apply How to apply the forces? Use impact hammers, mass imbalances, shakers (impulse, periodic-single frequency, sine-swept, random, etc)

14. Excitations with shakers X Y

15. Identification model (2) where, Obtain the discrete Fourier transform(DFT) of the applied forces and displacements, i.e., and use the property

16. Identification model (3) The DFT operator transforms the EOMS from the time domain into the frequency domain For the assumed physical model, the EOMS become algebraic

17. Identification model (4) Define the complex impedance matrix The impedances are functions of the excitation frequency (). REAL PART = dynamic stiffness, IMAGINARY PART = (quadrature stiffness), proportional to viscous damping K - w2 M wC

18. Identification model (5) With the complex impedance The EOMS become, for the first & second tests Add these two eqns. and reorganize them as At each frequency (ωk=1,2,…n), the eqn. above denotes four independent equations with four unknowns, (HXX, HYY , HXY , HYX)

19. Identification model (6) Find H since Then where The need for linear independence of the test forces (and ensuing motions) is obvious

20. Condition number In the identification process, linear independence is MOST important to obtain reliable and repeatable results. In practice, measured displacements may not appear similar to each other albeit producing an identification matrix that is ill conditioned, i.e., the determinant of In this case, the condition number of the identification matrix tell us whether the identified coefficients are any good. Test elements that are ~isotropic or that are excited by periodic (single frequency) loads producing circular orbits usually determine an ill conditioned system

21. The estimated parameters Estimates of the system parameters {M, K, C},j=X,Y are determined by curve fitting of the test derived discrete set of impedances (HXX, HYY , HXY , HYX ) k=1,2…., one set for each frequency ωk, to the analytical formulas over a pre-selected frequency range. For example:

22. Meaning of the curve fit Analytical curve fitting of any data gives a correlation coefficient (r2)representing the goodness of the fit. A low r2<< 1, does not mean the test data or the obtained impedance are incorrect, but rather that the physical model (analytical function) chosen to represent the test system does not actually reproduce the measurements. On the other hand, a high r2~ 1 demonstrates that the physical model with stiffness, damping and inertia giving K-ω2M and ωC, DOES model well the system response with accuracy.

23. Transfer functions=flexibilities Transfer functions (displacement/force) are the system flexibilities G derived from G=H-1 where

24. The instrumental variable filter method In the experiments there are many more data sets (one at each frequency) than parameters (4 K, 4 C,4 M=16). Fritzen (1985) introduced the IVFM as an extension of a least-squares estimation method to simultaneously curve fit all four transfer functions from measured displacements due to two sets of (linearly independent) applied loads. The IVFM has the advantage of eliminating bias typically seen in an estimator due to measurement noise GH = I Recall that

25. The IVFM (1) Since G=H-1 GH = I The product However, in any measurement process there is always some noise. Introduce the error matrix (e) and set Above G is the measured flexibility matrix while H represents the (to be) estimated test system impedance matrix

26. The IVFM (2) It is more accurate to minimize the approximation errors (e) rather than directly curve fitting the impedances. Let Hence Let

27. The IVFM (3) Stack all the equations, one for each frequency k= 1,2…,n , to obtain the set where Acontains the stack of measured flexibility functions at discrete frequencies k=1,2…,n. Eqs. make an over determined set, i.e. there are more equations than unknowns. Hence, use least-squares to minimize the Euclidean norm of e

28. The IVFM (4) The minimization leads to the normal equations A first set of force coefficients (M,C,K) is determined In the IVFM, the weight function A is replaced by a new matrix function Wcreated from the analytical flexibilities resulting from the (initial) least-squares curve fit. W is free of measurement noise and contains peaks only at the resonant frequencies as determined from the first estimates of K, C, M coefficients

29. The IVFM (5) At stepm, where when m=1 use W1=A = least-squares solution. Continue iteratively until a given convergence criterion or tolerance is satisfied

30. The IVFM (6) At stepm, Substituting W for the discrete measured flexibility A(which also contains noise) improves the prediction of parameters. Note that the product ATAamplifies the noisy components and adds them. Therefore, even if the noise has a zero mean value, the addition of its squares becomes positive resulting in a bias error. On the other hand, W does not have components correlated to the measurement noise. That is, no bias error is kept in WTA. Hence, the approximation to the system parameters improves.

31. The IVFM (7) In the IVFM, the flexibility coefficients (G) work as weight functions of the errors in the minimization procedure. Whenever the flexibility coefficients are large, the error is also large. Hence, the minimization procedure is best in the neighborhood of the system resonances (natural frequencies) where the dynamic flexibilities are maxima (i.e., null dynamic stiffness, K-2M=0) External forcing functions exciting the test system resonances are more reliable because at those frequencies the system is more sensitive, and the measurements are accomplished with larger signal to noise ratios

32. An example of parameter identification

33. Texas A&M University Mechanical Engineering Dept. – Turbomachinery Laboratory Identification of force coefficients in a SFD Luis San Andrés Sanjeev Seshagiri Paola Mahecha Research Assistants Sponsor: Pratt & Whitney Engines SFD EXPERIMENTAL TESTING & ANALYTICAL METHODS DEVELOPMENT

34. P&W SFD test rig Static loader Shaker assembly (Y direction) Shaker assembly (X direction) Static loader Shaker in Y direction Shaker in X direction SFD test bearing

35. Test rig description

36. P & W SFD Test Rig – Cut Section Piston ring seal (location) Test Journal Bearing Cartridge Supply orifices (3) Circumferential groove Flexural Rod (4, 8, 12) Main support rod (4) Journal Base Pedestal in

37. Lubricant flow path Oil inlet in

38. Objective & task • Evaluate dynamic load performance of SFD with a central groove. • Dynamic load measurements: circular orbits (centered and off centered) and identification of test system and SFD force coefficients

39. Circular orbit tests Oil in, Qin Journal (D) Oil out, Qt End groove End groove c Bearing Cartridge L L L Central groove Oil out, Qb Oil collector Oil out Base Support rod • Frequency range: 5-85 Hz • Centered and off-centered, eS/c = 0.20, 0.40, 0.60 • Orbit amplitude r/c = 0.05 – 0.50

40. Typical circular orbit tests Forces (fyvs.fx) motion (yvs.x) • Frequency range: 5-85 Hz • CenteredeS=0 • Orbit amplitude r/c=0.66

41. Typical circular orbit tests Forces (fyvs.fx) motion (yvs.x) • Frequency: 85 Hz • Off-centered at eS/c= 0.31 • Orbit amplitude r=0.05 – 0.5

42. Typ system direct impedances Imaginary part Real part HXX r/c= 0.66, centered es=0 HYY

43. Typ. system direct impedances r/c= 0.66, centered es=0 HXX K - w2 M wC Excellent correlation between test data and physical model REAL PART = dynamic stiffness IMAGINARY PART proportional to viscousdamping

44. Test cross-coupled impedances Imaginary part Real part r/c= 0.66, centered es=0 HXY One order of magnitude lesser than direct impedances = Negligible cross- coupling effects HYX

45. SFD force coefficients DRY system parameters Ks = 21 klbf/in Ms = 40 lb Cs= 7 lbf-s/in Nat freq = 73-75 Hz Damping ratio = 0.04 SFD Difference between lubricated system and dry system (baseline) coefficients CSFD=Clubricated - Cs MSFD=Mlubricated - Ms KSFD=Klubricated - Ksh

46. SFD damping coefficients Damping increases mildly as static eccentricity increases CXX CYY ~ CXXfor circular orbits, independent of static eccentricity

47. SFD mass coefficients MXX MXX ~ MYYdecreases with orbit radius (r) for centered motions. Typical nonlinearity

48. Conclusions • SFD test rig: completed measurements of dynamic loads inducing small and large amplitude orbits, centered and off-centered. • Identified SFD damping and inertia coefficients behave well. IVFM delivers reliable and accurate parameters. • Comparison to predictions are a must to certify the confidence of numerical models.

49. Acknowledgments • Thanks to Pratt & Whitney Engines • Turbomachinery Research Consortium Learn more http:/rotorlab.tamu.edu Questions (?)

50. References Fritzen, C. P.,1985, “Identification of Mass, Damping, and Stiffness Matrices of Mechanical Systems,” ASME Paper 85-DET-91. Massmann, H., and R. Nordmann,1985, “Some New Results Concerning the Dynamic Behavior of Annular Turbulent Seals,” Rotordynamic Instability Problems of High Performance Turbomachinery, Proceedings of a workshop held at Texas A&M University, Dec, pp. 179-194. Diaz, S., and L. San Andrés, 1999, "A Method for Identification of Bearing Force Coefficients and its Application to a Squeeze Film Damper with a Bubbly Lubricant,” STLE Tribology Transactions, Vol. 42, 4, pp. 739-746. L. San Andrés, 2010, “identification of Squeeze Film Damper Force Coefficients for Jet Engines,” TAMU Internal Report to Sponsor (proprietary)