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

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 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 Rotordynamics primer (2)

5 The need for parameter identification 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 Experimental identification of force coefficients is important

6 The physical model For lateral rotor motions (x, y), a bearing or seal reaction force vector f is modeled as K,C,M are 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. X Y Z Lateral displacements (X,Y)

7 Bearings: dynamic reaction forces Stiffness coefficients Damping 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 X Y Z Lateral displacements (X,Y)

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

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

10 Modern parameter identification Modern techniques rely 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 K h,C h : support stiffness and damping M h : effective mass X Y K YY, C YY K XY, C XY force, f Y Bearing or seal Journal K YX, C YX K XX, C XX Ω force, f X K hX, C hX K hY, C hY Soft Support structure Consider a test bearing or seal element as a point mass undergoing forced vibrations induced by external forcing functions (K,C,M): test element stiffness, damping & inertia force coefficients

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

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

14 Excitations with shakers X Y

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

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 - 2 M C

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 (ω k =1,2,…n), the eqn. above denotes four independent equations with four unknowns, (H XX, H YY, H XY, H YX )

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

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 (H XX, H YY, H XY, H YX ) k=1,2 …., one set for each frequency ω k, 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 (r 2 ) representing the goodness of the fit. A low r 2 << 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 r 2 ~ 1 demonstrates that the physical model with stiffness, damping and inertia giving K-ω 2 M 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 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 In the experiments there are many more data sets (one at each frequency) than parameters (4 K, 4 C, 4 M =16). Recall that

25 The IVFM (1) However, in any measurement process there is always some noise. Introduce the error matrix (e) and set GH = I Since The product G=H -1 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. Hence Let

27 The IVFM (3) Stack all the equations, one for each frequency k = 1,2…,n, to obtain the set where A contains 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 W created 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 step m, where when m=1 use W 1 =A = least-squares solution. Continue iteratively until a given convergence criterion or tolerance is satisfied

30 The IVFM (6) At step m, Substituting W for the discrete measured flexibility A (which also contains noise) improves the prediction of parameters. Note that the product A T A amplifies 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 W T A. 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- 2 M=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 Sponsor: Pratt & Whitney Engines Luis San Andrés Sanjeev Seshagiri Paola Mahecha Research Assistants SFD EXPERIMENTAL TESTING & ANALYTICAL METHODS DEVELOPMENT Identification of force coefficients in a SFD Texas A&M University Mechanical Engineering Dept. – Turbomachinery Laboratory

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

35 Test rig description

36 P & W SFD Test Rig – Cut Section in Test rig main features Journal diameter: 5.0 inch Film clearance: 5.1 mil Film length: 2 x 0.5 inch Support stiffness: 22 klbf/in Bearing Cartridge Test Journal Main support rod (4) Journal Base Pedestal Piston ring seal (location) Flexural Rod (4, 8, 12) Circumferential groove Supply orifices (3)

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 Frequency range: 5-85 Hz Centered and off-centered, e S /c = 0.20, 0.40, 0.60 Orbit amplitude r/c = 0.05 – 0.50 ISO VG 2 Oil Viscosity at 73.4 o F [cPoise]2.95 Density [kg/m 3 ]784 Inlet pressure [psig]7.5 Outlet pressure [psig]0 Radial Clearance [mil]c Journal Diameter [inch]5.0 Central groove length [inch]L Land length, L [inch]L Total Length [inch]3L3L Oil out, Q b Base Support rod Bearing Cartridge Journal (D) Oil out, Q t Oil in, Q in Central groove L End groove Oil out Oil collector c

40 Typical circular orbit tests Frequency range: 5-85 Hz Centered e S =0 Orbit amplitude r/c=0.66 Forces (f y vs. f x )motion (y vs. x)

41 Typical circular orbit tests Frequency: 85 Hz Off-centered at e S /c= 0.31 Orbit amplitude r=0.05 – 0.5 Forces (f y vs. f x )motion (y vs. x)

42 Typ system direct impedances H XX H YY r/c= 0.66, centered e s =0 Imaginary partReal part

43 Typ. system direct impedances H XX r/c= 0.66, centered e s =0 Excellent correlation between test data and physical model REAL PART = dynamic stiffness IMAGINARY PART proportional to viscous damping K - 2 M C

44 Test cross-coupled impedances H XY H YX One order of magnitude lesser than direct impedances = Negligible cross- coupling effects r/c= 0.66, centered e s =0 Imaginary partReal part

45 SFD force coefficients SFD K s = 21 klbf/in M s = 40 lb C s = 7 lbf-s/in Nat freq = Hz Damping ratio = 0.04 DRY system parameters C SFD =C lubricated - C s M SFD =M lubricated - M s K SFD =K lubricated - K sh Difference between lubricated system and dry system (baseline) coefficients

46 SFD damping coefficients C XX Damping increases mildly as static eccentricity increases C YY ~ C XX for circular orbits, independent of static eccentricity

47 SFD mass coefficients M XX M XX ~ M YY decreases 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 Questions (?)

50 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 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 L. San Andrés, 2010, identification of Squeeze Film Damper Force Coefficients for Jet Engines, TAMU Internal Report to Sponsor (proprietary) References