ESTIMATES OF ELECTROMAGNETIC DAMPING ACROSS AN INDUCTION MOTOR AIR GAP FOR USE IN TORSIONAL VIBRATION ANALYSIS
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1 Page 1 of 12 ESTIMATES OF ELECTROMAGNETIC DAMPING ACROSS AN INDUCTION MOTOR AIR GAP FOR USE IN TORSIONAL VIBRATION ANALYSIS ABSTRACT: Ed Hauptmann (1), Brian Howes (2), Bill Eckert (3), The effects of induction motor electromagnetic (em) damping on torsional vibration are estimated with a method previously used for em stiffness. Results indicate the significance of em damping depends on mode shape, particularly so for lower order vibratory modes. Motor parameters that may lead to drive instabilities are identified, and simple guidelines for estimating the em damping magnitudes are given. INTRODUCTION: The steady-state torque and power output of a polyphase induction motor are the result of electromagnetic fields which act across the air gap between stator and rotor. If the rotor has a torsional vibration superimposed over the steady rotation, the same electromagnetic fields across the air gap can produce an additional torque which acts in the same way as a torsional spring and damper added to a mass-elastic model as shown in Figure 1. In the past, these additional electromagnetic (em) effects were not usually included in standard torsional vibration analysis. em torque (spring) steady motor torque em torque (damper) vibratory excitation torque ROTOR COUPLING COMPRESSOR Figure 1. A conventional mass-elastic model of a motor driven compressor, with a spring and damper representing the additional em effects on the rotor due to torsional vibration. 1. E. G. Hauptmann, PhD, PEng, Director, Eng. Development, Lo-Rez Vibration Control. 2. B. C. Howes, MSc, PEng, Chief Engineer, Beta Machinery Analysis. 3. W. F. Eckert, PhD, PEng, Principal Engineer, Beta Machinery Analysis.
2 Page 2 of 12 Simple analytical methods for estimating the em stiffness and damping were presented at GMRC 2013 [1], and extended to provide estimates of current pulsation due to torsional vibration at GMRC 2014 [2]. As outlined in [1], values for the em spring, km, and em damper, dm, can be estimated from: km = (# stator poles)( TB ){ x 2 / [1 + x 2 ] }, Eq n (1), and where; dm = km / (ω 2 TL) = km (TL) / (x) 2.Eq n (2), TR = rated motor torque, [Nm] TB = breakdown torque. [Nm] sr = slip at rated load, [%], Ωs = supply frequency, [rad/s] ω = torsional vibration freq., [rad/s] TL = electrical time constant, [s], x = (ωtl), dimensionless time. The electrical time constant of the motor TL can be estimated from: TL (1/Ωs)[1/(2sR)]( TR / TB ).Eq n (3). A presentation by Knop [3] at 2012 EFRC explained that the addition of an em spring and damper adds a mode with a very low torsional natural frequency (TNF), while the second mode can be though of as a shift upward of the original first order mode as shown in Figure 2. 2 Ratio of Torsional Frequencies J M /J C A A A 1 B C k M / k C Figure 2. The effect of including an em spring to the model in Fig.1 (neglecting damping, and taking km as constant). Coupling type: A-elastomeric at different operating temperatures; B-steel spring coupling; C-disc pack. The ratios JM /JC are the total inertias on the motor and compressor side respectively (shown for reference; taken from [1]).
3 Page 3 of 12 It can be seen in Figure 2 that where a torsionally very soft (elastomeric) coupling is used, the em stiffness, km, may be many times greater than the coupling stiffness, so that the shift in the TNF can be significant. An em spring with km values estimated using Eq ns (1, 3) was added to a series of previous TVA models where field measurements of the TNF were available. Figure 3 shows the measured TNF (RPM) where the em was naturally present, versus the re-calculated TNF with em effects included in the TVA (solid markers). The proximity to the 45 o line indicates good agreement between the em model and field measurements, particularly so for soft rubber couplings where the TNF shift can be significant. The open markers show the TNF without the em effect Measured TNF (RPM) HP, 4 throw, spring cplg HP, 4 throw, spring cplg HP, 4 throw, spring cplg HP, 4 throw, rubber cplg Predicted TNF (RPM) Figure 3. Comparison of predicted versus measured TNF when em effects are included in the TVA (solid markers), and when they are not included in the TVA (open markers); Figure taken from [1]. We concluded [1, 2] that based on these measurements, the simple method presented for evaluating the em stiffness gives reasonable estimates to be used at the early design stage, and that the effect on torsional natural frequencies is particularly important for systems with very soft (elastomeric) couplings.
4 Page 4 of 12 EFFECT OF em DAMPING IN TYPICAL DRIVE TRAINS: The addition of an em spring to the model in Fig. 1 not only alters the torsional natural frequencies, but also adds a significant lower vibratory mode. The em damper however acts only on the rotor, and plays a less important role in overall system response in most cases. The em rotor damping dm is strongly dependent on torsional vibration frequency, becoming very large at lower frequencies. As an example, the em damping has been evaluated using Eq n 2 and the methods above for a 1,250 HP, 8 pole motor, 60 Hz supply frequency, running at full load with a slip of 0.89%. Figure 4 shows the damping (and stiffness) at varying torsional vibration frequencies Eq'n 1 stiffness Eq'n 2 damping Stiffness- knm/rad Damping - Nms/rad Torsional Vibration Frequency - Hz Figure 4. Electromagnetic damping and stiffness estimates made using Eq ns 1, 2, 3, for a 1,250 HP, 8 pole motor, 60 Hz electrical (line) supply, running at full load with a slip of 0.89%. Estimates of em rotor damping using Eq n 2 have also been applied to a series of previous torsional studies to better evaluate the importance of em damping. Table 1 shows results for the rotor em damping in three different drives: Drive A- system with a very low torsional stiffness (elastomeric) coupling; Drive B-system with a steel spring coupling; Drive C-rigid (disc pack) coupling [1]. The principal torsional vibration frequency, ω, is taken as the first order of run speed.
5 Page 5 of 12 Drive A Drive B Drive C Compressor Ariel JGC/6 Ariel JGK/4 Cameron MH64 speed range-rpm speed range-hz Coupling rubber-in-shear steel spring stiff: disc pack Motor 60 Hz Toshiba Reliance WEG 1 st TNF, ω 0 Hz st TNF, incl. em rated power kw 3, rated power - HP 5,000 1, k M knm / rad * d M knms / rad * * at 900 RPM Table 1. A summary of em stiffness effects on some previous torsional vibration studies. The effect is pronounced for higher power installations with ultra-soft rubber couplings, while lower power drive trains with torsionally-stiff couplings are not greatly affected [1]. The em damping, d M, in Table 1 was applied to the rotor in addition to normal system modal damping for each case. The extra em effect on system response at the compressor crankcase, coupling, and motor shaft is shown in Table 2. Drive A Drive B Drive C speed range-hz st TNF, incl. em no em incl. em no em incl. em no em incl. em d M knms / rad d M lb-ins / rad 0 28, ,166 Vibratory Torques lbf-in, 0-pk comp. crankshaft 309, , , ,132 79,949 79,991 coupling 18,781 18,677 18,622 18,384 49,397 49,417 motor shaft 9,866 9,816 15,961 15,784 45,957 45,978 effective damping -% Table 2. The effect of additional em damping added to original system damping for three types of torsional drivelines. For these systems, the effect on vibratory torques due to em damping was slight.
6 Page 6 of 12 Note that in Table 2, the operating speed range for Drive A includes the 1 st TNF. However because of the mode shape, the em damping makes a small contribution to effective damping (about 0.01%). Drive B is a situation where the compressor inertia is almost as large as the motor inertia, and the run speed (approximately 19.9 Hz) is removed from a resonant condition. Compressor crankshaft torque remains nearly constant with the inclusion of em damping, while coupling and motor shaft torques are slightly reduced. Effective system damping in Drive B is increased by about 1.56%). Drive C, a very stiff system, operates far from any interfering resonances and is very slightly affected by the addition of em damping SYSTEM RESPONSE DUE TO NEGATIVE em DAMPING: While the simplified approach above has yielded useful estimates of the em spring stiffness km, accurate damping estimates have been more elusive with Eq n 2 giving reliable results only for lower order modes (vibration frequencies). More complete studies have shown that at frequencies approaching the electrical supply frequency, the em damping becomes negative (not possible from Eq n 2). Better understanding of the complete unsteady em damping effects requires direct numerical integration of the non-linear differential equations representing stator and rotor currents and their mutually induced stator and rotor fields. Past studies have typically included startup of drive systems and estimation of the resulting transient motor torques. Jordan et al [4], [5], have developed the complete em field equations which were then solved numerically, showing dynamical effects (limit cycles and drive instabilities) for some motors which showed negative damping. Extensive numerical modeling of the electromechanical stiffness and damping effects as a result of torsional vibration have been reviewed by Holopainen et al [6]. Their results of a complete numerical simulation for the motor of Figure 4 are shown in Figure 5, which also includes the results shown in Figure 4. The full simulation results from [6] show the onset of negative damping as the torsional vibration frequency is 83% - 95% of the electrical supply frequency (60 Hz) for this motor. Except for the region of 50 Hz-60 Hz torsional vibration frequency, Eq ns (1, 2, 3) results agree well with the detailed stiffness calculation, while damping estimates using the simple results are as much as 44% lower at 25 Hz. The motor in Figure 5 was used to drive a blower in a direct, on-line application with a relatively torsionally soft coupling so that the first TNF was approximately 28.1 Hz. It should be noted that the 2 nd order run speed at 29.8 Hz, may have been responsible for the relatively large torsional amplitudes reported at the rotor. The em effects were minor and negative damping was avoided entirely.
7 Page 7 of Eq'n 1 stiffness Eq'n 2 damping Ref [6] stiffness Ref [6] damping Stiffness- knm/rad Damping - Nms/rad Torsional Vibration Frequency - Hz Figure 5. Predicted values of em damping and stiffness versus torsional excitation, using Eq ns 2 and 3, and predicted values by Holopainen et al [6]. Negative damping appears from 83 to 95% of supply frequency for this motor. Knop [3] reported on negative damping effects for a 690 HP, 6 pole motor with a 50 Hz supply frequency, running at 972 RPM (16.2 Hz). Negative damping was shown to depend critically on stator resistance, expressed as a dimensionless ratio, α = (stator resistance/ stator reactance). For the exemplar motor, having a value α = 0.01, the em damping became negative at torsional vibration frequencies from 32 to 48.5 Hz (64% to 97%) of the electrical supply frequency. The stiff coupling in the drive resulted in a 1 st TNF of 42 Hz so that the 2 nd and 3 rd order of run speed were within the region of negative damping. The resulting torque amplitudes on the coupling were reported to be as much as 5 times predicted, so that the coupling failed after a few running hours (replaced with a much softer coupling to remove the interferences). It was also noted that negative damping disappeared for values of α greater than (about) 0.05, and also for all orders of natural vibration above the supply frequency.
8 Page 8 of 12 MOTOR PARAMETERS CONDUCIVE TO NEGATIVE DAMPING: To further explore the negative damping phenomenon, we have combined the complete analytical models from [4, 5] for the em damping and adjusted the motor parameters; α, β, and σ (dimensionless stator and rotor resistances, and dimensionless leakage flux) explained in [1]) so that a virtual motor (VM) can be developed to study the importance of these parameters. As an illustration of such a virtual motor, Figure 6 shows a comparison of results from [6] for the 1,250 HP motor of Figure 5, and those of the VM motor with parameters indicated. The VM can be used to illustrate how the region of negative damping is influenced by the parameters; α, β, and σ, as well as the supply frequency (as in VFD drive arrangements). Figure 7 shows the results of the predicted em damping for the VM with a supply frequency of 30 Hz. Further parametric studies on the effect of supply frequencies for the VM of Fig.6 indicate that the negative damping range persists at all supply frequencies above approximately 30 Hz Damping- Nms/rad VM Ref [6] motor Torsional Vibration frequency - Hz Figure 6. Development of a virtual motor (VM) to match predicted results for em damping reported in [6]. VM parameters are; α = 0.01, β =0.02, σ = 0.1.
9 Page 9 of Damping- Nms/rad VM-30Hz VM-60 Hz Torsional Vibration frequency - Hz Figure 7. The em damping results for the VM of Fig. 5 at 30 Hz and 60 Hz supply frequencies, mimicking VFD outputs from 50% to 100% of nominal speed. At 30Hz, negative damping disappears over the range of frequencies shown. 400 Damping- Nms/rad HP VM Eq'n (2) Torsional Vibration frequency - Hz Figure 8. Comparison of the predicted em damping for a 1,750 HP, 8 pole VM with resistance and reactance values described above, to that of the em damping for the same motor using the approximate results using Eq n (1, 2, 3).
10 Page 10 of 12 Changing the dimensionless resistance and reactance parameters can also significantly change the shape of em damping curve for a VM. Figure 8 shows the predicted em damping for a 1,750 HP, 8 pole motor, 60 Hz supply frequency and a full load slip of 0.67%. The dimensionless parameters for this motor are; α = 0.105, β = 0.054, σ = 0.1. Figure 8 also includes the predicted em damping using the approximations made by using E qns (1, 2, 3). Negative em damping is not present for the VM in this case. Results of the simplified method applied to the same motor compare well with the VM at low vibration frequencies where the damping is significant; however not as well at vibration frequencies from 20 Hz 40 Hz, where the damping is much lower. CONCLUSIONS: 1. The addition of an em spring to the TVA alters the torsional natural frequencies, and also adds a significant lower vibratory mode. The em damper however acts only on the rotor, and plays a much less important role in overall system response according to the mode shape. 2. Motor em damping is not significant for soft coupling designs with the first TNF below run speed. Higher modes that are dominated by the compressor shaft are independent of the em effects due to the isolation provided by the soft coupling. 3. The em damping may have a more significant effect for drivelines with torsional natural frequencies in the range of 80%-100% of supply (line) frequency, with motors having ratios of stator resistance /reactance in the range of The estimates from Equations 1 and 2 are generally sufficient for use at an early design stage of the torsional vibration model. Reciprocating compressor systems typically have significant torque excitations in the critical range of % of supply frequency (usually 2, 3, or 4 times running speed). As a result, most systems are designed to have torsional natural frequencies that are well separated from the critical frequency region to avoid excessive response, even without the consideration of possible negative damping at the motor. DESIGN RECOMMENDATIONS: 1. Torsional natural frequencies within % of the electrical supply frequency should be avoided due to the potential of negative damping in the motor. API 618 requires a 10% separation margin from supply frequency, and 5% from twice supply frequency. API 514 recommends 15% separation from 1x and 2x line frequency. However, the possibility of negative em damping in the region below the supply frequency has the
11 Page 11 of 12 potential to create a torsional issue. The recommended API separation margin is meant to protect against torsional issues caused by motor torque fluctuations during electrical events (such as short circuits). A more prudent design guideline would be to use at least a 20% separation margin on the lower side of the supply frequency to avoid the potential negative em damping. 2. A mode with high motor response coupled with low, or worse, negative damping is a recipe for disaster and should be avoided. Tuning a torsional natural frequency to the % supply frequency range often means addition of significant inertia to the compressor. This can result in the motor rotor inertia being less than the total driven inertia, with the corresponding mode shape being such that the motor has higher torsional displacement than the compressor. 3. Motors with low stator resistance should be reviewed. The results of Knop [3], and of the VM show that motors with low stator resistance (α 0.01) are more likely to exhibit a region of negative em damping, and thus the 20% separation margin below the supply frequency would be more desirable. The stator resistance is a property of the electrical design of the motor, and cannot easily be changed. The torsional vibration analyst should be aware of the potential issue, especially with low stator resistance designs. 4. There may be some instances when other limits force tuning the system into the critical region. In those cases, using a more complete calculation method for the em damping might be prudent. Alternatively, a damping sensitivity analysis could demonstrate the need for more detailed modelling. 5. Variable frequency drives have additional design requirements due to the varying supply frequency. However, the tendency is for em damping to become less negative as the supply frequency decreases. This helpful behavior will make the torsional design problem with a VFD easier to solve. 6. More complete data for specific motors has to be supplied by the manufacturer. This would include data on the parameters; α, β, and σ (dimensionless stator and rotor resistances, and dimensionless leakage flux, [1]). Our understanding of these parameters is dependent on motor manufacturers at this time.
12 Page 12 of 12 ACKNOWLEDGEMENTS: Authors gratefully acknowledge the comments and support from the GMRC Research Committee Torsional Effect of Motor Magnetic Field ; Jeff Vea, for his helpful work in rerunning past analyses. REFERENCES: 1. E. Hauptmann, B. Howes, B. Eckert; The influence on torsional vibration analysis of electromagnetic effects across an induction motor air gap, GMRC, Albuquerque (2013). 2. E. Hauptmann, B. Howes, B. Eckert; Approximate method for calculating current pulsations caused by induction motors driving reciprocating compressors, GMRC, Nashville (2014). 3. G. Knop, The importance of motor dynamics in reciprocating compressor drives, EFRC, Dϋsseldorf (2012). 4. H. Jordan, J. Mϋller, H.O. Seinsch; Über elektromagnetische und mechanische Ausgleichvorgänge bei Drehstromantrieben; Wiss. Ber. AEG-TELEFUNKEN 53 (1979) H. Jordan, J. Mϋller, H.O. Seinsch; Über das Verhalten von Drehstromasynchronmotoren in drehelastischen Antrieben; Wiss. Ber. AEG- TELEFUNKEN 53 (1980) T. P. Holopainen, J. Nϋranen, P.Jöorg, D. Andreo ; Electric motors and drives in torsional vibration analysis, Forty-Second Turbomachinery Symposium, Houston TX, (2013).
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