Stability Analysis for a Flexible Rotor on Active Magnetic Bearings subject to Aerodynamic Loads

Transcription

1 Stability Analysis for a Flexible Rotor on Active Magnetic Bearings sbject to Aerodynamic Loads Simon E. Mshi 1,a, Zongli Lin 1,b, and Pal E. Allaire,c 1 Charles L. Brown Department of Electrical and Compter Engineering, University of Virginia, Charlottesville, VA , U.S.A. Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, VA , U.S.A. { a sem5t, b zl5y, c pea}@virginia.ed Keywords: Active magnetic bearing, aerodynamic cross-copling, mixed sensitivity, modeling and identification, µ-synthesis, robst control. Abstract The development of advanced control algorithms to garantee stability and high performance of a rotor-amb system sbject to ncertainties and distrbances is an active area of research. The model based control paradigm entails consideration of the dynamics of all components that will affect the stability and performance of a system. Control design for a flexible rotor spported on active magnetic bearings provides an additional challenge since ncertainties sch as the magnitdes of cross-copled stiffness forces strongly affect the stability of the closed-loop system. Recent work has highlighted the se of sensitivity fnctions in determining stability margins for AMB spported trbomachinery. The otpt sensitivity fnction provides a means of qantifying the distrbance rejection and reference command tracking properties of the closed-loop system. Or work presents an analysis of the closedloop AMB sensitivity fnctions of the rotor-amb system. We present simlations of the effects of varying mid-span aerodynamic cross-copled stiffness on stability and performance of several µ-optimal controllers designed in a mixed-sensitivity framework, measrements of experimental sensitivity fnctions from the levitated rotor, and recommendations on the selection of weighting fnctions to improve the performance. Introdction Aerodynamic cross-copled stiffness (CCS) forces in indstrial compressors and trbines are a reslt of flid-strctre interactions prodced by flow differences in clearances arond impellers, seals, and hydrodynamic bearings. Observed early on in jet engines, the sbsynchronos and self-exciting natre of these forces is of major concern to machinery designers as they create potentially nstable rotor vibrations, which can lead to machine damage in the absence of sfficient damping [1]. While empirical models exist for the factors that contribte to this aerodynamic CCS, the complex transient natre of the interactions makes it difficlt to devise experiments and develop detailed design tools []. Several athors have described experiments in which active magnetic bearings (AMBs) have been sed to excite a rotor-bearing system with CCS forces for varios prposes [3,4,5]. In this work, we discss and demonstrate the tility of sensitivity fnctions as a tool for design and analysis of controllers robst to varying levels of CCS. The se of the otpt sensitivity fnction as a tool to qantify the stability margins has been sggested in ISO

2 standard for AMB-spported trbomachinery [6], thogh it is not withot some important limitations [7]. The recommendations within this standard have recently been sed as part of a rotordynamic design adit ttorial sing a 4.8MW AMB-spported pipeline compressor [8]. System sensitivity fnctions become a valable asset in the design of control laws and respect the fndamental practical limitations on achievable stability and performance of rotor-amb systems [9,10]. Recent works have contined to highlight the se of and investment in µ-synthesis control for AMBs as a powerfl engineering tool to realize significant performance improvements over hand-tned algorithms [11,1]. Test Rig Description A rotor-amb test rig (Fig. 1a) has been constrcted at the University of Virginia to investigate the stability of AMB-spported trbomachinery sbject to aerodynamic CCS forces and other ncertain aerodynamic loads [13]. The rotor has dimensions comparable to a small indstrial centrifgal gas compressor sch as the 5M frame size from Elliott s EDGE prodct line (6,600 to 1,00 m 3 /hr flow rate at 13,100 rpm with rotor length 1.145m to.85m). Or design consists of a 1.3m long steel rotor with a weight of 440 N and two gyroscopic disks. The average rotor diameter (exclding the attached components) is 60 mm. The larger disc weighs 8.8 N, has a diameter of 41.3 mm and imparts a significant gyroscopic character to the rotor. There are a total of for radial AMBs two AMBs at the shaft ends to spport the shaft with a combined load capacity of 600N and two additional AMBs at the mid and qarter spans to allow for the application of simlated destabilizing flid or electromagnetic forces to the rotor. All for AMBs operate with a 0.38mm air gap and a typical bias crrent of.95 A. The analog switching power amplifiers sed to drive the AMBs operate from a 150V DC bs and have a maximm continos crrent rating of 10A. Differential pairs of eddy crrent displacement sensors are sed to determine the position of the rotor. The rotor can be driven p to 14,00 RPM sing an AC motor and belt-drive system. A decentralized PID control algorithm is exected on a cstom DSP platform at a 1 khz sample rate for rotor levitation. Fig. 1b shows a schematic of the complete test rig. (a) (b) Figre 1: (a) Photograph of the test rig; (b) Schematic of test rig showing the major components :- (1) Spport AMBs, () Exciter AMBs, (3) Rotor, (4) Gyroscopic discs, (5) Flexible copling, (6) Jack shaft, (7) Timing belt, (8) 5 hp AC motor, (9) Concrete base, (10) Steel base.

3 Plant Modeling and Rotor Levitation The modeling of high-speed rotor-amb systems for control has been presented in the literatre [5,10]. Typically a lateral rotordynamic analysis (ndamped mode shapes and critical speed map) is performed sing a finite-element rotor model as a starting point. Sbseqently, the AMB properties can be analyzed sing a magnetic relctance circit model to find a sitable bias crrent to perform bias flx linearization abot. The reslting negative position stiffness K x and crrent stiffness K i can be sed to approximate the AMB behavior when the rotor is within a neighborhood of the nominal air gap. Analytical models for the remaining electrical components and transdcers (i.e., power amplifiers, position sensors, signal conditioning and data acqisition components) are appended to this model. See [14] for a detailed procedre for modeling high-speed rotor-amb systems. Following or rotordynamic analysis, the 49-station finite-element rotor model was modally redced to retain two rigid body and three bending modes. Owing to the flexibility of the bearing pedestals, a fondation model with 6 modes was appended to the rotor model. The fondation model was identified experimentally and modal parameters were extracted sing a grey-box iterative prediction error estimate approach [10]. This rotor-fondation model now has 44 first-order states describing the x and y motion at the two radial bearings. A linearized AMB actator model with crrent gain, K i = N/A and negative position stiffness, K x = MN/m was then connected to the rotor-fondation model. The power amplifier in series with the AMB indctance was modeled with a 3 rd order low pass filter with approximately.0 khz small-signal bandwidth. The eddy crrent displacement sensors in series with an 8 th order inverse Chebychev anti-aliasing filter demonstrated a flat magnitde response p-to 3 khz, bt contribted a significant phase lag of 40 degrees at 1 khz. Lastly, a nd order Padé approximation for the 83.3 µs controller comptation delay (1 khz sampling rate) was appended to the model. The plant model now contains 100 states, and compared well with the experimentally measred system (see Fig. ). The initial levitation of the rotor-amb system at rest (0 RPM) is accomplished with a decentralized proportional-integral-derivative (PID) controller. The proportional gain (K p = 7.56), integration time (T i = 0.075) and derivative time (T d = 0.003) for each of the 4 control axes are obtained throgh experimental tning. The basic PID controller is agmented with phase lead, notch filters and high-freqency roll-off to enable sfficient damping of the rotor bending modes within the controller bandwidth. The controller is able to levitate and center the rotor within 0.15s. The plant modeling and PID controller design are discssed extensively in or recent work [15]. Figre : Bode plot from amplifier voltage inpt to sensor otpt showing dynamics of analytical rotor-amb system theoretical model verss an experimental transfer fnction measrement for 1 axis.

4 Stability Margin Measrement As mentioned earlier, ISO provides recommendations, based on experiment, for the evalation of the stability margin for AMB-spported machinery, and makes important steps to bridge the commnication gap between vendors, OEMs, and end-sers of the eqipment [6]. For newly commissioned machines, the standard sggests peak sensitivity less than 3 (in absolte terms or less than 9.5 db), for a machine to be designated Zone A. Higher vales of peak sensitivity vales progressively decrease the classical closed-loop gain and phase margins and increase the likelihood of isses with a machine. Zone B is normally acceptable for nrestricted long-term operation with absolte peak sensitivity from 3 4; Zone C is nsatisfactory for long-term continos operation with peak sensitivity from 4 5; and operation in Zone D is likely to case machine damage with a peak sensitivity greater than 5. Fig. 3 shows an otpt sensitivity fnction for -axes of the rotor-amb system measred at 0 RPM with an SR-785 dynamic system analyzer. The peak vale above 8 indicates Zone D stability margin and it is likely that operation at design speed wold case machine damage. This peak corresponds to excitation of the 1 st bending mode of the rotor and provides the engineer with an idea of where to place additional damping dring controller tning. Limitations on the application of the otpt sensitivity fnction as a measre of stability margin were first mentioned in a paper by Li et al. [7]. The athors showed that with respect to destabilizing mechanisms sch as cross-copled stiffness, low peak sensitivities do not necessarily garantee a good stability margin to this ncertainty, i.e., the measrements as proposed in ISO encompass a necessary bt not sfficient condition for stability margin for mltivariable systems. Therefore, to be more confident in stability margin assessment for certain machines, analysis of peak otpt sensitivity fnctions shold be followed by an evalation of the strctred singlar vales, which gives an additional measre of robstness to specific ncertainties. Figre 3: Experimental otpt sensitivity fnction at the nondriven end AMB while the rotor is levitated by a PID controller. A peak sensitivity vale greater than 5 indicates a very poor stability margin. Mixed-sensitivity and µ-optimal design The goal of mixed-sensitivity control design is to optimize trade-offs among several system sensitivity fnctions to achieve the desired stability and performance specifications. The synthesis of controllers which minimize the H norm of the transfer fnction from distrbance inpt to performance otpt provides a means to attain desired performance specifications. However, sch designs provide little garantee of robstness with respect to ncertainties in plant parameters. The synthesis of µ-optimal controllers takes direct accont of ser-defined ncertainties dring control design, and provides a garantee in the form of the strctred singlar vale for the amont of ncertainty the closed-loop system can

5 tolerate. For the sake of brevity, frther discssion mst be limited, however, the crios reader may find the following references insightfl [16,17]. Figre 4: The Gang of For, mixed-sensitivity framework block diagram. Inpt signals d and r are distrbance and reference inpts, respectively. Otpt signals z 1 and z are actator and position error performance signals, respectively. The se of freqency-dependent weighting fnctions to captre the performance specifications for rotor-amb systems is discssed extensively by several athors [9,10,11,1]. The for-block problem (see Fig. 4) ses the otpt sensitivity (S), process sensitivity (GS), control sensitivity (CS), and complementary otpt sensitivity (T) fnctions to form a performance constraint, W p SWr W CSW r W p GSW W TW d d <1, (1) where W p is the performance specification weight, W r is the reference inpt weight, W d is the distrbance inpt weight, and W is the actator dynamics weight. The selection of these freqency dependent weights is non-trivial and may be gided by insight into the system dynamics. A sb-optimal H or µ-optimal controller, K, which stabilizes G (stability constraint) and also satisfies Eq. 1, will garantee that 1/ W p W d bonds GS, 1/ W p W r bonds S, 1/ W W r bonds CS and 1/ W W d bonds T. For or control system the following weighting fnctions were adapted from [10], W s + ω = 0. 4 I p ( s), s + βω I M S ω s + ξω L + ω W ( s) =, M ω s + ξω + ω KS L L 1 Wd ( s) =, M GS 1 Wr =, () M S where M S = 3 enforces a bond on S, M GS = limits the peak vale of T, M CS = 10/M GS, ω B = 408rad/s, ω I = ω B /1500 is the integrator time constant, ω L = 10ω B defines the actator bandwidth limitations, and ω is the freqency of complex poles inserted to ensre the reciprocal of W is proper. The reslting µ-optimal Controller A (designed withot any CCS acting on the rotor) has a µ vale of and the peak otpt sensitivity over all for control axes is.9, indicative of an ISO Zone A stability margin. Compared with the PID controller in Fig. 3, the mixed-sensitivity optimization is able to improve damping of rotor and pedestal modes significantly. The only ncertainty considered in or plant model is the magnitde of the CCS force prodced at the rotor mid-span. For the sake of brevity, the linear parameterization of this force sed is 0 1 qxmid F xc = χ, (3) 1 0 q ymid where the scalar χ denotes the magnitde of the CCS force, and q xmid and q ymid are the

6 displacements at the rotor mid-span. We consider χ to be a real parametric ncertainty with vales of 0 to 40,000 lbf/in (7 N/µm) with a nominal vale of 0,000 lbf/in (3.5 N/µm). As we add 10,000 lbf/in and then 1,000 lbf/in of CCS to the rotor spported by Controller A, the peak sensitivity qickly rises throgh Zone B and Zone C at a freqency close to the conical rigid body mode (see Fig. 5a). As expected, the CCS generates a sb-synchronos excitation that eventally drives the system nstable. Another µ-optimal design Controller B was synthesized with a constant χ=0 000lbf/in, while Controller C was synthesized with χ as an ncertain parameter over the range described earlier. All controllers have the same weighting matrices as defined in Eq.. Fig. 5b shows the effect of varying CCS on the simlated peak otpt sensitivity of the three controllers. As Controller A is optimized for zero cross copled stiffness, it is able to garantee a Zone A stability margin p to a 6,000 lbf/in of CCS. Controller B is able to garantee Zone A stability margin from 13,500 6,000 lbf/in of CCS. At CCS vales far from their design points both controllers A and B, are completely nable to stabilize the system. Interestingly, while Controller C was able to stabilize to system over the entire range of CCS, it can only garantee a Zone B stability margin. The sb-optimal performance of Controller C may be improved by frther tning of the weighting fnctions to deal specifically with the low freqency sensitivity peak below 100 Hz cased by the CCS distrbance. By designing an online CCS estimator, it may also be possible to implement a gain schedling approach to ensre stability and performance over a wider range of CCS as stdied for changing rnning speed [10]. (a) (b) Fig. 5: (a) Peak sensitivity fnction for Controller A simlated with varying CCS, (b) Effect of CCS on peak otpt sensitivity fnctions of three µ-optimal controllers :- A designed for 0 lbf/in, B designed for lbf/in, C designed for the entire range lbf/in. Conclsions The se of sensitivity fnctions as a tool for the stability analysis and synthesis of control algorithms for rotor-amb systems with ncertainties and distrbances has been evalated sing a mixed-sensitivity framework. The contined development and characterization of an AMB test rig with a flexible rotor allows s to evalate the potential for model-based robst control design tools sch as µ-synthesis to stabilize ncertain systems with practical

7 limitations sch as actator slew rate. Perhaps the greatest challenge in harnessing the power of mixed-sensitivity and µ-synthesis is the selection of weighting fnctions. However, as these fnctions take into accont the properties and limitations of the control system their selection may be gided by engineering insight and ths tackled systematically. Frther work on this project will involve practical generation of CCS forces sing the mid-span AMB and the implementation of robst algorithms to stabilize the rotor-amb system. References [1] Alford, J., Protecting trbomachinery from self-excited rotor whirl, J. of Engineering for Power, Transactions of ASME, Vol. 87(4) (1965), p [] API 684, API Standard Paragraphs Rotordynamic Ttorial, nd edition, American Petrolem Institte, Washington D.C., U.S.A. (005). [3] Ulbrich, H., New test techniqes sing magnetic bearings, in Proc. 1 st Int. Symp. on Magnetic Bearings (ISMB-1), Zrich, Switzerland, (1988), p [4] Clod, H., Stability of Rotors Spported by Tilting Pad Jornal Bearings, PhD dissertation, University of Virginia (007). [5] Schweitzer, G., and Maslen, E.H., Magnetic Bearings, Theory, Design, and Application to Rotating Machinery, Springer-Verlag, Berlin, Germany (009). [6] ISO , Vibration of rotating machinery eqipped with active magnetic bearings Part 3: Evalation of stability margin, International Standards Organization, (006). [7] Li, G., Maslen, E., and Allaire, P., A note on ISO AMB stability margin, in Proc. 10 th Int. Symp. on Magnetic Bearings (ISMB-10), Martigny, Switzerland, (006), p [8] Swanson, E., Maslen, E., Li, G., and Clod, C., Rotordynamic design adit of AMB spported trbomachinery, Proc. 37 th Trbo. Symp., Hoston, Texas, Sept (008). [9] Schönhoff, U., Practical Robst Control of Mechatronic Systems with Strctral Flexibilities PhD dissertation, Technical University of Darmstadt (003). [10] Li, G., Robst Stabilization of Rotor-Active Magnetic Bearing Systems, PhD dissertation, University of Virginia (006). [11] Maslen, E.H., and Sawicki, J.T., µ-synthesis for magnetic bearings: Why se sch a complicated tool?, in Proc. IMECE007 ASME, Seattle, U.S.A., Nov (007). [1] Mystkowski, A., µ-synthesis control of flexible modes of AMB rotor, in Proc. SMASIS009 ASME, Oxnard, CA, U.S.A., Sept. 1-3 (009). [13] Mshi, S.E., An Active Magnetic Bearing Test Rig for Aerodynamic Cross-copling: Control Design and Implementation, MS Thesis, University of Virginia (008). [14] Li, G., Lin, Z., Allaire, P., and Lo, J., Modeling of a high speed rotor test rig with active magnetic bearings, J. Vibr. and Acostics ASME, Vol. 18(3) (006), p [15] Mshi, S., Lin, Z., and Allaire, P., Design, constrction and modeling of a flexible rotor active magnetic bearing test rig, in Proc. ASME Trbo Expo, Glasgow, Scotland, Jne (010). [16] Sidi, M., Design of Robst Control Systems: From Classical to Modern Practical Approaches, Krieger Pb. Co., U.S.A., (001). [17] Zho, K., Doyle, J., and Glover, K., Robst and Optimal Control, Prentice Hall, Upper Saddle River, NJ, U.S.A, (1996).