Calibrating a Large Compressor s Rotordynamic Model: Method and Application

Transcription

1 7th IFToMM-Conference on Rotor Dynamics, Vienna, Austria, September 26 Calibrating a Large Compressor s Rotordynamic Model: Method and Application Giuseppe Vannini, Ph.D. Centrifugal Compressor NPI Department GE Oil & Gas Nuovo Pignone Via Felice Matteucci, Firenze, Italy [email protected] Anthony J. Smalley, Ph.D. Tony Smalley Consulting, LLC 454 Seasons Drive Grand Junction, CO 8153 USA [email protected] Tim J. Hattenbach Hans P. Weyermann Justin R. Hollingsworth Bechtel Corporation P. O. Box 2166 Houston, Texas [email protected] ConocoPhillips Company 6 North Dairy Ashford Houston, Texas [email protected] Mechanical & Materials Engineering Division Southwest Research Institute Post Office Drawer 2851 San Antonio, Texas [email protected] ABSTRACT This paper defines and demonstrates an effective, sequential process of rotor bearing model calibration, for a large centrifugal compressor. The calibration process involves a series of measurements and corresponding predictions, including free-free mode shapes and frequencies, casing shaker test data, amplitude and phase variation with speed, and change of vibration with known changes in unbalance. The process first uses the measured free-free data and comparison with matching predictions to calibrate the mass elastic model for the rotor alone; it then applies measured casing response to shaker excitation, measured variation of frequency and phase, and dual peak characteristics at the first critical speed, to identify the correct support characteristics. The paper demonstrates application of the calibrated model and established uncertainty in its predictions to assure integrity not readily demonstrated by testing alone. Specifically the margin between maximum continuous speed and the second critical speed is shown to be acceptable, and the amplitude of vibration at maximum continuous speed is shown to stay well below the seal clearances at all seal locations. 1 INTRODUCTION Most centrifugal compressor designs seek to keep the second critical speed (NC2) well above maximum continuous speed, and the API 617 standard [1] sets a separation margin. However, the goal to run below NC2 makes it difficult to quantify the margin directly. This paper demonstrates a process to calibrate the dynamic model of a rotor-bearing system and, thereby, to quantify likely errors in frequency and amplitude. The rotor in question has six impellers and runs in tilting pad bearings. Three identical rotors were built for a plant two to operate and one spare. The nominal power for each compressor is 3 MW. The test data from mechanical running tests and other data were available as a basis for the model calibration process. A mass-elastic rotor model was prepared from as-built drawings. The free-free mode shapes and natural frequencies were compared to measured values. The rotor model was combined with bearing and bearing support models. The predicted first critical speed (NC1) was compared with measured resonant frequencies for the three rotors. The calibration process tunes the model to reproduce important measured characteristics and quantifies its prediction uncertainty. No model can faithfully reproduce observed vibration behavior of such a complex system indeed, this behavior does not reproduce itself run to run! Recognizing and measuring this uncertainty enables quantification of potential differences between reality and predictions of integrity-related quantities, and assessment of consequences. 1 Paper 125.

2 The following paper typifies the shop test data, details the modeling and calibration process, applies the calibrated model, and, accounting for uncertainty, shows the second critical speed margin and the worst case predictions of vibration at seal locations to be acceptable. 2 COMPRESSOR DESCRIPTION Figure 1 shows one of the 5,8 Kg rotors hung from the drive end for modal testing. The compressor has two sections, back-to-back, six impellers, and 22 mm dry gas seals. Each rotor is mounted in 2-mm tilting pad bearings (5 pad) with length to diameter ratio (L/D) of.7. The maximum rotor diameter is 42 mm, and the bearing span is meters. The range from minimum to maximum continuous speed (MCS) is 4,118 to 5,87 RPM, and trip speed is 5,314 RPM. Each gas turbine driven compressor consumes up to 3 MW. Figure 1: Photograph of Rotor 3 AVAILABLE SHOP TEST VIBRATION DATA This paper presents and analyzes vibration data obtained during mechanical running tests. The test drive was a steam turbine with a light coupling of similar weight to the job coupling. The rotor for Unit 1 and the spare rotor were tested in the Unit 1 casing. The Unit 2 rotor was tested in its own casing. Figure 2 provides Bode plots for rotor 1; the left hand frame presents the rotor in balanced condition; in the right hand frame the rotor has a 75 gm.cm. couple unbalance. The two columns in each frame address first the X Probe and then the Y Probe. For each probe, the four charts in sequence going down show Drive End Phase, Drive End Amplitude, Non-Drive End Phase, and Non-Drive End Amplitude. X Y X Y DE Phase Amp DE Phase Amp 25 Micron 25 Micron NDE Phase Amp NDE 25 Micron Phase Amp 25 Micron Figure 2: Bode Plot for Rotor 1 Balanced (LH Frame) and with a 75 gm.cm. Couple Unbalance (RH Frame) Each chart shows first critical speed characteristics at 2,1 to 2,2 RPM: an amplitude peak and a sharp downwards change of phase with increasing speed. Some of the Y Probe amplitude charts exhibit a split in the first critical speed (most likely resulting from support asymmetry as will be subsequently discussed). The sharp phase change near MCS for the balanced rotor results from the vibration dropping to near zero at the same speed. Added unbalance in the second frame eliminates the phase change and zero vibration point; instead, the second critical speed vibration builds up from 4,8 RPM to trip speed. 2 Paper 125.

3 The behavior near MCS creates concern that the second critical speed, although predicted to satisfy API 617 margins, might be closer to MCS than predicted. The calibration process and its application in this paper help resolve this issue while avoiding further testing. 4 MODELING The mass-elastic model was prepared from as-built drawings. Figure 3 shows model geometry and bearing locations. This model includes rotor bending, shear flexibility, shaft mass, and rotary/polar inertia discretized at each station, with mass, polar, transverse inertias of components, such as impellers, sleeves, thrust disk, and couplings, lumped at the station corresponding to the component s center of gravity Shaft Radius, meters.8.4 Shaft Shaft Shaft Shaft Figure 3: Rotor Model The model accounts for rotor stiffening at all interference fits, by the method of Smalley, et al. (22) [2]. This method increases effective shaft diameter for second moment of area calculation, over each interference length. The effective diameter equals physical diameter multiplied by a factor (greater than one), which depends on the interference L/D ratio (component interference length divided by shaft diameter). A three-dimensional finite element analysis of typical impellers with a range of L/D ratios produces a consistent functional relationship for thick sections (e.g., impeller, balance drum, or coupling). For thin sections (e.g., a sleeve), the diameter increase is limited to 7 percent of the actual diametral thickness. The tilting pad bearings are represented conventionally as linear elements, which act on the rotor with direct stiffness and damping coefficients. The coefficients are generally unequal for vertical and horizontal directions. Bearing forces are determined from isothermal solution of the finite length Reynolds equation for each pad, accounting for cavitation. Pad and journal nominal orientations are established to satisfy equilibrium of individual pads and of the rotor. The coefficients are obtained by linear perturbation about the equilibrium condition essentially as described by Lund [3]. 5 UNDAMPED CRITICAL SPEED MAPS AND MODE SHAPES Figure 4 presents the predicted critical speed map, which shows how the rotor s synchronous undamped critical speeds vary with support stiffness. The support stiffness represents the bearings and support structure in series. The two supports have the correct axial locations but are treated as isoelastic springs, equal at each location, with neither cross-coupling nor damping. The support stiffness is varied over many decades. The shape and level of the critical speed variation compactly summarize the rotor s dynamic characteristics, and the influence of its supports. The predicted lines for bearing direct stiffnesses, and their intersections with the critical speeds, indicate approximately where the critical speeds will occur. The log-log format of Figure 4 exhibits straight-line behavior with a one-half slope in regions where the rotor acts as a rigid body (negligible bending). Fluid film-mounted rotors tend to have well damped critical speeds in this rigid body regime. In regions where rotor bending dominates, the critical speeds exhibit horizontal straightline behavior, such that further increases in bearing stiffness will not influence the critical speed value. In regions dominated by rotor bending, the critical speeds tend to be much less well damped, because the modes develop less relative motion at the bearings. For most industrial compressors, the first two critical speeds are most strongly influenced by the bearings and tend to intersect with the bearing stiffness lines between rigid body and flexible rotor regions of these critical speeds. The third and fourth critical speeds are mainly characteristic of the rotor mass-elastic system alone, with limited influence of supports. In fact, for low support stiffness, the third critical speed has an asymptote. 3 Paper 125.

4 Figure 4 displays the first three mode shapes for two different support stiffness values, which cover the range of expected support stiffness values. There is only slight variation in the mode shapes of each mode between the extremes of the range (5E8 and 8E8 N/m stiffness). All modes reveal distinct curvature. Critical Speed, cpm Undamped Critical Speed Map Kxx Kyy 1 1.E+7 1.E+8 1.E+9 1.E+1 1.E+11 Bearing Stiffness, N/m f= cpm -1 f= cpm K=5 N/m K=8 N/m f= cpm f=739.6 cpm K=5 N/m K=8 N/m f= cpm f= cpm K=5 N/m K=8 N/m Figure 4: Critical Speed Map and Mode Shapes; 5E8 and 8E8 N/m Support Stiffness 6 CALIBRATION PROCESS 6.1 Free-Free Modes Figure 1 showed one rotor suspended from a hook with a shaker at its lower end. The free rotor has very little internal damping and accelerometers at 1 points along its length respond distinctly and sharply as the shaker sweeps through each natural frequency. The displacements indicate the associated mode shapes. A matching mass elastic model was prepared with the coupling replaced by the support hook mass. Figure 5 shows measured and predicted shape and frequency for the first two free-free modes, each scaled so that measured and predicted amplitudes match exactly at the first accelerometer Measured 136 Hz Measured.4.2 Measured Hz Measured Hz Hz Hz Figure 5: First Two Mode Shapes for Free-Free Vibration Figure 6 shows predicted and measured frequencies, with a 4 to 6.6 percent discrepancy for the first three modes and almost 32 percent over-prediction for the fourth mode. The predicted and measured numbers of nodes (axis crossings) match for all four modes, but the model apparently neglects some physics that play a significant role in the prediction of the fourth mode (for example impeller flexibility) Frequency Hz_ Measured Mode Number Figure 6: Frequency Comparison and Measured Free-Free Frequencies 4 Paper 125.

5 The question arises: How many free-free modes are needed to quantify adequacy and uncertainty of the rotor-bearing system model? The rotor speed range brings into play a number of critical speeds, which reflect constraint from the bearings and whose curvatures differ from the free-free modes. An intuitive measure of normalized curvature (k) was defined to develop a rational approach to this question and to enable systematic and uniform comparison of free-free or bearing constrained mode shapes. This quantity k=(d 2 y/dz 2 ) max /(y max -y min ), where the maximum second derivative of lateral deflection, y, represents maximum curvature for the mode shape and (y max -y min ) quantifies the deflection range for the mode shape (e.g., from Figure 5, deflection range equals 1.13 for mode 1 and about 1.4 for mode 2). The normalized curvature has units of (length) -2, it could be redefined as a fully non-dimensional quantity, but was used only for comparison so this extra step was not taken. This normalized curvature is 1.67, 1.96, 4.39, and 5.56 for the first four predicted freefree modes, and 1.9, 2.63, and 4.24 for the first three predicted rotor-bearing critical speeds. Comparison suggests that coverage and uncertainty from the first three free-free modes suffices for the rotor-bearing dynamics up to the third critical speed. The asymptotic third critical speed at low bearing stiffness differs from the first free-free mode. The asymptotic third critical speed (19 Hz in Figure 4) accounts for synchronous gyroscopic effects, which do not affect the first mode of the non-rotating rotor (128.8 Hz). Also, the rotating rotor has a coupling attached, whereas the rotor for free-free tests has a lighter hook attached. The lower weight hook appears to dominate since the non-rotating frequency exceeds the rotating frequency. Accounting for the specific mass distribution under test for free-free mode calibration is clearly essential. 6.2 Tuning Bearing Support Flexibility Accurate rotor-bearing system modeling should include bearing support flexibility. Figure 7 shows the horizontally split casing, outboard bearing support structure, and a shaker in different orientations. Accelerometers on the casing, combined with applied load data, yield impedance values, as shown in Figure 8. Running frequency data from the accelerometer nearest the bearing gave 5E9 N/m (horizontal) and 2E9 N/m (vertical) impedance values; thus, the casing flange stiffens horizontally more than vertically. When included in the model, these impedance values best reproduce measured characteristics, and so represent values identified from both shaker and system data. Figure 7: Configuration Schematic and Photographs for Shaker Testing of Bearing Support Structure X Direction 1E+11 1E+1 Stiffness [N/m] 1E+9 1E+8 Acc1bk Acc13bk Acc14bk Acc15bk Acc16bk Frequency [Hz] Used 5E9 as Uniform Horizontal Stiffness Figure 8: Shaker Test Response at Several Locations on Support Structure 5 Paper 125.

6 6.3 Calibration Based on Critical Speed Characteristics Figure 9 shows predicted response to center unbalance with bearing supports set rigid then flexible. Support stiffness significantly influences peak frequency and response characteristics. Only with the inferred support stiffness values does the prediction show the split critical speed observed in Figure 2. The predicted first critical speed range is 2,12 to 2,32 RPM, and the observed range is 2,157 to 2,343 RPM. The standard deviation from multiple runs for three rotors is 67.7 and 61.6 RPM for horizontal and vertical motion, so differences in observed and predicted ranges lies well within one standard deviation. The difference between the lowest predicted value for NC1 and the lowest measured value is 37 RPM. Rotordynamic Response Plot Rotordynamic Response Plot Response, mm pk-pk Sta. No. 14: Probe Vert Amp.3.2 Excitation = 1x Rotor Speed, rpm Response, mm pk-pk Sta. No. 14: Probe Vert Amp.4 Excitation = 1x Rotor Speed, rpm Figure 9: Drive End Response to Center Unbalance Rigid and Flexible (5E9, 2E9 N/m) Supports This comparison assesses uncertainty in the rotor-bearing mass-elastic model separately from the free modal calibration, so all difference in NC1 can be attributed conservatively to support stiffness uncertainty. For extrapolation to NC2, a normalized rate of change, (K/N c )*(dn c /dk), has been defined to measure the fractional rate of change of critical speed in RPM with respect to the fractional rate of change of support stiffness, K. The critical speed map shows this quantity is twice as high for NC2 as for NC1 (specifically.24 compared to.12). Thus, with a discrepancy of 37 RPM between the low ends of the measured and predicted ranges for the first critical speed (1.75 percent), twice this uncertainty (3.5 percent) can be expected in the low end of NC2. The free-free comparison showed a maximum of 6.6 percent discrepancy in any of the first three modes resulting from rotor mass-elastic modeling, yielding a combined uncertainty of 1.1 percent ( ) in the second critical speed prediction. 6.4 Calibration Based on Influence Coefficients To calibrate sensitivity to unbalance, measured data for rotor vibration with different added unbalance levels was used to calculate influence coefficients dividing the vector change in vibration at a particular speed and location from one run to the next by the vector change in unbalance between these two runs. Figure 1 compares predicted and measured influence coefficients in the 4, to 5,1 RPM range, obtained with unbalance added at just one end. The measured influence coefficient at 5,1 RPM falls about 33 percent below the predicted value. Predicting influence coefficients to agree well with observed influence coefficients is particularly challenging. This is first because the influence coefficients depend on the difference between measured phase and amplitude for two successive runs, where thermal and other transient phenomena may significantly influence vibration, and subtracting vectors tends to amplify uncertainty. In addition, small differences in resonant frequency can translate into more significant differences in amplitude. However, based on these results, an inferred 33 percent model uncertainty in predicted unbalance response at MCS is reasonable and conservative..25 Amplitude of IC Measured Speed RPM Figure 1: Comparison of and Measured Influence Coefficients 6 Paper 125.

7 6.5 Comparing and Measured Polar Plots Figure 11 shows the qualitative and quantitative similarity between a measured polar plot with a high couple unbalance added and the polar plot predicted with a combination of 1,5 gm.cm. center unbalance and 1, gm.cm. couple unbalance. For both plots of Figure 11, the first critical speed circle diameter is about 8 microns, and the length of the vibration growth vector beyond the first critical speed circle is about 3 microns. The total unbalance for the measured data is unknown, but 1,5 gm.cm of couple unbalance was added to the rotor s initial state. While this is less precise for calibration purpose than other comparisons presented above, the sensitivity and nature of polar plot response to significant unbalance again seems acceptably and conservatively predicted Microns p-p Full Scale Figure 11: and Measured Polar Plots of Vibration for Couple Unbalance 6.6 Evaluating Uncertainty in the Second Critical Speed In Figure 12, the predicted response at Probe 1 to couple unbalance shows a well damped critical, with amplification factor (AF) below 3. The 6,-RPM peak gives a 913-RPM margin above MCS (almost 18 percent), which meets API 617 requirements for this low AF. A 1.1 percent uncertainty leaves a margin of over 3 RPM (6 percent) between MCS and the resonant peak, even if all uncertainty acts to lower the second critical speed. 6.7 Evaluating Seal Location Vibrations The preceding results add confidence in predicted response to high unbalance. Comparing measured and predicted influence coefficients indicates a 33 percent uncertainty in predicted MCS amplitude, which with the model, was used to assess seal damage potential. Rotordynamic Response Plot.12 Sta. No. 14: Probe 1 Response, mm pk-pk "Worst Case" Reduction of NC2 MCS NC2 Major Amp Horz Amp Vert Amp Excitation = 1x Rotor Speed, rpm Figure 12: Response to Couple Unbalance Flexible Supports One shop test run exhibited 48 microns at NC1 and 3 microns at MCS higher than for any other run. No intentional unbalance had been added, and the high vibrations probably resulted from transient distortion. Although abnormal, this high vibration data set provides a convincing basis for a worst case unbalance distribution. 7 Paper 125.

8 The unbalance distribution needed to cause these probe vibrations was 8,25 gm.cm at mid-rotor combined with a couple unbalance of 3, gm.cm at each end. Each vector of the couple represents four times the API reference unbalance based on the entire rotor weight, and the 8,25 gm.cm. mid-rotor vector is ten times this reference. The 48 microns at NC1 and 5 microns at MCS so predicted under test conditions was already higher than measured and to add severity, predictions of seal vibration were made with this unbalance at maximum bearing clearances, increasing to 55 microns at NC1 and 125 microns at MCS. In Figure 13, the corresponding response distribution along the rotor quantifies the seal clearance loss, at NC1 and at MCS. This loss of seal clearance is highest at NC1 (which is known to meet API 617 margins comfortably), reaching almost 3 percent. At MCS, clearance loss is less than 15 percent. Thus, response to a most severe unbalance condition under worst-case bearing clearance is readily tolerable. Even if the worst case uncertainty in predicted unbalance vibration sensitivity of 33 percent is applied, MCS seal clearance loss increases only to 2 percent. To provide additional security the predicted 125 micron vibration at the probes would cause a vibration trip at MCS, adding protection for the rotor against damaging vibrations % of Seal Clearance % at NC1 % at 51 RPM 5.. End Seal 1 Eye 1 Interstage 1 Eye 2 Interstage 2 Eye 3 Balance Drum Eye 6 Interstage 6 Eye 5 Interstage 5 Eye 4 End Seal 2 Figure 13: Response at Seals to Worst Case Unbalance As noted, clearance loss is lower at MCS than at the first critical speed in spite of the bearing vibration being substantially higher at MCS. This results from differences between the first critical mode shape (which defines the relative vibrations at NC1) and the second critical mode shape (which dominates MCS vibration). Inspection of Figure 4 shows, particularly for the softer direction, the ratio of bearing deflection to maximum rotor deflection is significantly higher for NC2 than for NC1. This causes the rotor to exhibit higher vibration along its length at NC1 than at NC2 even with lower NC1 bearing vibration. 7 CONCLUSIONS 1. Applying a calibrated rotor-bearing dynamic system model confirmed acceptability of margin and vibrations at maximum continuous speed for two critical compressors, avoiding costly, time-consuming, and difficult additional tests. 2. The following specific steps of calibration and application proved effective: Identify maximum frequency uncertainty from rotor mass-elastic model alone. Use a normalized curvature parameter to assess the required number of free-free modes. Identify added first critical frequency uncertainty associated with supports. Identify maximum uncertainty at MCS in predicted sensitivity to unbalance. Extrapolate NC1 uncertainty to NC2, using a normalized sensitivity parameter. Establish minimum residual second critical speed margin with frequency uncertainty. Identify a worst-case unbalance and apply it at MCS for most sensitive conditions. Compare seal vibrations with added amplitude uncertainty against seal clearance. 3. Effective calibration against free-free modes requires close attention to the rotor s mass distribution in the free-free tests and adjustments made to the model to account for the mounting hardware (in this case, a hook of much lower weight than the job coupling). 4. A significant difference between predicted and measured frequency for the fourth free-free mode, while not of importance in this case, seems attributable to physics not reproduced in standard rotordynamics models and needs to be better understood. 8 Paper 125.

9 REFERENCES [1] API th Edition (1995), Axial and Centrifugal Compressors and Expander-Compressors for Petroleum, Chemical and Gas Industry Services, Seventh Edition, American Petroleum Institute, Washington, D.C. [2] Smalley, A. J., Pantermuehl, J. P., Hollingsworth, J. R., and Camatti, M. (22), How Interference Fits Stiffen the Flexible Rotors of Centrifugal Compressors, Proc. IFToMM Sixth International Conference on Rotordynamics, Sydney, Australia. [3] Lund J. W. 1964, Spring and Damping Coefficients for Tilting Pad Journal Bearings, ASLE Transactions 7, pp Paper 125.