Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 46, 1, pp. 41-50, Warsaw 2008 EXPERIMENTAL IDENTIFICATION OF DYNAMIC PARAMETERS FOR ACTIVE MAGNETIC BEARINGS Dorota Kozanecka Zbigniew Kozanecki Tomasz Lech Technical University of Łódź, Insitute of Turbomachinery, Łódź, Poland e-mail: dkozan@p.lodz.pl; zkozan@p.lodz.pl; tlech@p.lodz.pl In the paper, an experimental identification procedure of dynamic pa- rameters for activemagnetic bearings is presented. In the proposedme- thod, analysis of variable components of displacements of the rotor for an excitation with the rotating vector force of unbalance, is conducted. Proportionality of the bearingmagnetic response force to the excitation force in the whole range of rotational frequency, is assumed. It allows for identification of equivalent dynamic coefficients of the bearing. The experimental investigations were carried out on test rigs with the shaft supported in activemagnetic bearings. Key words: magnetic bearing, identification, dynamic properties 1. Introduction In rotor dynamics, characteristics of bearings exert decisive influenceonmodes of vibrations and critical frequencies of the rotating shaft. In a systemof active support of the rotor in themagnetic field, there is a strict connection between dynamic parameters of the model and the shape of its frequency characteri- stics obtained for the fixed rotor. This feature is used in situations when it is necessary to test different control algorithms in order to optimise dynamic characteristics of the designed machine without the risk connected with its start-up and shut-down. Thus, it is very important from the viewpoint of the measurement procedure to elaborate amethodwhich allows for interpretation of the obtained characteristics and whose aim is to find dynamic parameters of the system (stiffness, damping) determining stable machine operation in 42 D. Kozanecka et al. the assumed range of rotational frequency. This is a complex problem that demands proper measurement and calculation techniques to be developed, and also access to defined levels of software of the bearing control system is needed. In the present paper, a concept of themethod and the investigation results obtained by the authors through the experimental procedure aimed at the de- termination of dynamic coefficients of the bearing are presented. Its important feature consists in the possibility of realization under nominal operating con- ditions of the bearing, and, therefore, reliable identification of actual values of these coefficients is possible. 2. Concept of the method In the measurement method, a dependence of the vector of the resultant ma- gnetic force acting on themachine shaft as a function of the journal position in themagnetic radial bearing and the currents that flow through thewindings of actuators (electromagnets) is employed. The magnetic bearing response vec- tor is a sum of forces generated by bearing electromagnets and alters in each control cycle (Kozanecki and Kozanecka 2003). The magnetic response components FXmag, FY mag for each control axis are related to themeasuredmean values of the current controlling the electro- magnets IXT , IXB, IYT , IYB in a given control period, values of themagnetic gaps sXT , sXB, sYT , sYB and to the values of the electromagnet constants KXT ,KXB,KYT ,KYB (top – index T , bottom– index B). Values of thema- gnetic gaps are found on the basis ofmeasurements of instantaneous positions of the journal with respect to the centre of the bush of the known clearan- ce. Themagnetic response components FXmag, FY mag for the axis X, Y are determined by the following relationships FXmag =KXT I2XT s2 XT −KXB I2XB s2 XB FY mag =KYT I2YT s2 YT −KYB I2YB s2 YB (2.1) Equation (2.1) holds on the assumption that the linear dependence of the magnetic flux on the induction is maintained. It means that the bearing ope- rates according to that part of the characteristics which is distant enough from the state of magnetic circuit saturation, when the induction does not exceed 50% of the saturation induction for the core material. The value of the con- stant K depends on electromagnet design parameters and can be calculated Experimental identification of dynamic parameters... 43 theoretically (Schweitzer et al., 1993). However, in the actual design of the journal bearing, the constants KXT , KXB, KYT , KYB can slightly differ for each pair of electromagnets of the bush. In order to increase the accuracy of the proposed measurement method, the constant values are verified experi- mentally for each electromagnet, and their real values are taken into account in the calculations (Kozanecki and Kozanecka, 2003). If the journal motion parameters are known and the magnetic response force is determined by an indirect method (Kozanecka et al., 2003a), it is possible to find the bearing dynamic parameters that relate the magnetic response force to the journal motion parameters. The idea of the method is based on the relationship between stiffness and damping coefficients and the linear response force for the bearing in which interactions between the axes do not occur. In the journal active magnetic bearing system, the interactions between the control axes X and Y can be neglected for small displacements of the journal (Kozanecka, 2001; Kozanecki and Kozanecka, 2003). In the case of the bearing under investigation, the control system performs two independently imposed control algorithms for both the axes X, Y , which also allows one to neglect the interactions between them. For one control axis, the relation between the stiffness coefficients KXX and the damping coefficients CXX and the linear magnetic response force of the bearing FXlin has the following form FXlin =KXXx+CXXVX (2.2) Changes in the nonlinearmagnetic response force FXmag measured for the given control axis andbeing the responseof the bearing system to the assumed synchronous excitation FZ are approximated with the linearised harmonic function FXlin. A difference between the nonlinear magnetic response force of the bearing along the given axis FXmag that is known from the model calculations and its linearised form FXlin determined on the basis of formula FXlin−FXmag =∆Fi (2.3) is sought with the least square method in such a way that ∑ ∆F2i =min. Thus, the equivalent values of thebearingdynamicparameters KXX,CXX for the axis X are obtained. An analogous situation refers to the coefficients KYY , CYY for the axis Y . The calculations are conducted for stable bearing operation, where the journal position oscillates around the assumed point of equilibrium (Fig.1). Figure 2 presents a comparison between the measured magnetic response force FXmag anda theoretical function,which is a sumof the forces of stiffness 44 D. Kozanecka et al. Fig. 1. Displacement versus time and the orbit and damping FXlin = KXXx+CXXVX. The curve FXlin has been plotted on the basis of the measured journal displacement X (Fig.2) and the journal velocity VX obtained through digital differentiation of the displacement and a selection of suitable values of the dynamic stiffness coefficients KXX and the damping coefficients CXX in such away as tomake the sumof squares of differences minimal for the selected part of the time history. Fig. 2. Measuredmagnetic response component along the control axis X−FXmag and its modelled time history FXlin with the identified dynamic coefficients KXX,CXX In the method of identification of the bearing dynamic coefficients, it is required that the theoretically calculated magnetic response force is the clo- sest approximation of its function obtained in themeasurements and that the share of synchronous components in the curves of displacement, current and magnetic response force is dominant (Kozanecka, 2005). Experimental identification of dynamic parameters... 45 3. Test rig Thetest rig of the longflexible shaft linehas amodule structureandallows one to investigate systems of a different number of supports and of various shaft lengths within the assumed range of frequency of rotations. It is driven by an electric motor connected to the shaft through an elastic membrane coupling with smooth rotation control and fed by a frequency converter. In Fig.3, the magnetic bearing is one of two supports of the shaft part whose length is approx.1000mm, the thin-wall pipe diameter is φ = 54mm and the wall thickness is equal to 2mm.There is a disk at the free end of the shaft, on which themasses of test unbalancing can bemounted. Fig. 3. Test rig In the second configuration, the test rig consists of the horizontal flexible power-transmission shaft supported on two rolling bearings mounted at both ends. An active magnetic bearing operates as an auxiliary bearing that mo- difies the dynamic properties of the shaft line. Between the magnetic bearing and the shaft right end, there is a rigid disk which allows one to mount the balancing weights for the real structure. The mass of the rotating system is equal to 4.85kg, the shaft line length equals 1923mm. The test rig allows one to investigate the effects of the magnetic bearing on dynamic properties (vibration level, displacement and the coefficient of vibration amplification of subsequent critical frequencies) and to control vibrations of the long flexible rotor (Kozanecka, 2005). 4. Results of the investigations Themodel test rig of the flexible power-transmission shaft was used to carry out an experimentwhose aimwas to verify the identification procedure of dy- namic coefficients of the active magnetic bearing. A kinematical exciter was fixed on the disk, on which the masses of test unbalancing can be mounted. After introducing a selected program for magnetic bearing control, harmonic vibrations of the shaft of frequencies (10,20,30, . . . ,80)Hz and the assigned 46 D. Kozanecka et al. amplitude were excited. For each frequency under analysis, the time histories of displacements and currents in the magnetic bearing, which were subject to respective calculation procedures, were recorded, and then the bearing dyna- mic parameters were estimated. To conduct themeasurement and calculation procedures, a measurement system with DBK 15 input systems made by IO- tech, operating with a PC and employing the Daq/112B type PCMCIA me- asurement card of the resolution equal to 12 bites and themaximum sampling frequency of 100kHz, was applied. The voltage time histories corresponding to displacements (positions) of the journal along both the control axes X, Y were recorded on-line on re- spective inputs of the measurement-control module. These were two voltage signals 0-24V fromBently-Nevada type 3300 eddy-current transducers of rela- tive vibrations. The voltage time histories corresponding to currents flowing in electromagnets were measured and recorded. These were four voltage signals 0-5V from current-voltage LEM type transducers (Kozanecka, 2005). TheDaqViewv.7.9.8 softwarewas used for recordingpurposes.Therewere 4000 measurements made, at the sampling frequency of 8kHz/channel. The resultswere stored inbinaryfiles of adataacquisition system, and thenconver- ted into text files. The programs for analysis of dynamics and identification procedures of the bearing dynamic parameters, according to themethodology proposed, were developed with theMS Excel spreadsheet. Exemplary timehistories of thequantitiesmeasuredare shown inFig.4and Fig.5, and of those calculated in Fig.6 and Fig.7 for the magnetic bearing of the selected configuration of the control program for the kinematic excitation of the frequency 40Hz and the assigned amplitude, whose value was such as to obtain the dominant share of synchronous components in the time histories under analysis and to obtain the linear range of magnetic response forces. Fig. 4. Displacements for both the control axes X, Y and the shaft motion trajectory The occasional disturbances which occur in the recorded time histories of displacements (Fig.4) are amplified by digital differentiation, and the effect Experimental identification of dynamic parameters... 47 Fig. 5. Time histories of the currents in electromagnet windings IXT , IXB, IYT , IYB – averaged Fig. 6. (a) Velocity components for both the control axes VX, VY – averaged; (b) variable gaps for individual electromagnets sXT , sXB, sYT , sYB of these disturbances is very distinct in the time history of the velocity com- ponent VY (Fig.6a). This does not affect the calculation accuracy, where the characteristics are approximated with the harmonic time history. Themeasurement and calculation cycleswere conducted for various excita- tion frequencies in the range (10-80)Hz, which allowed one to build functions representing values of the bearing dynamic coefficients, namely: the stiffness coefficients KXX,KYY and the damping coefficients CXX,CYY , as functions of the frequency at the given excitation frequency and the given configuration of the control program (Fig.8). 48 D. Kozanecka et al. Fig. 7. (a) Measuredmagnetic response component FXac and the force FXacT =FXlin modelled with the identified dynamic coefficients KXX,CXX; (b) components of the magnetic response related to the stiffness FX stiff and to the damping FXdamp Fig. 8. (a) Stiffness KXX,KYY versus frequency; (b) damping CXX,CYY versus frequency Itmeans that the start-upand shut-downcharacteristics of themodel shaft line under such magnetic bearing operating conditions should be overcritical characteristics. The experimentally determined dynamic coefficients of the be- aring showsomeanisotropy of theproperties for individual axes KXX ∼=KYY , CXX ∼=CYY . The analysis of this state showed that for isotropic properties of the energy transmission systems (symmetrical saturation-control current cha- racteristics), this scattering resulted from the scattering of constant values of electromagnets for the given axis. This conclusion was confirmed by calcula- Experimental identification of dynamic parameters... 49 tions conducted with the simulation model of the magnetic bearing, however its further verification is still needed (Kozanecka, 2005). 5. Conclusions The experiment was conducted for various configurations of the program con- trolling the auxiliary magnetic support used in the model shaft line system. The determined dynamic coefficients were employed in subsequent stages of the investigations in the modelling and numerical calculations of the start- up and shut-down characteristics of the flexible power-transmission shaft. The generated numerical characteristics were verified experimentally through com- parison to the start-up and shut-down curves recorded for the realmodel shaft line with supports of identified dynamic properties. The proposed methodology of measurement of the response and dynamic coefficients of themagnetic bearing is a very important tool in designingdyna- mics andvibration control ofmachine rotors inwhich activemagnetic bearings are applied. It allows one to find analogies to classical bearing systems and to employ professional calculation codes for evaluation of the effects of modifi- cation in the dynamic properties of shaft lines introduced through changes in the configuration of the program controlling its active magnetic supports. References 1. KozaneckaD., 2001,Dynamics of the flexible rotorwith an additional active magnetic bearing,Machine Dynamics Problems, 25, 2, 21-38 2. KozaneckaD., 2005,Final reporton theprojectno. 4T10B04423,Vibrations Control of the Shaft Line of the Turbomachine Supported in Active Magnetic Bearings, pp.153 [in Polish] 3. Kozanecka D., Kozanecki Z., Lech T., 2001, Modelling the dynamics of active magnetic bearing actuators,Proc. World Multiconference on Systemics, Cybernetics and Informatics, SCI 2001, USA, IX, Industrial – Parts I, 232-235 4. KozaneckaD.,KozaneckiZ., LechT., 2002,Theoretical andexperimental investigation of dynamics of the flexible rotor with active magnetic bearings, Advances in Vibration Engineering, India, 1, 4, 412-422 5. Kozanecka D., Kozanecki Z., Lech T., Kaczmarek A., 2003a, Identifi- cation of the external load of the rotaring shaft supported in active magnetic 50 D. Kozanecka et al. bearings,Proc. 7thConference onDynamical SystemsTheory andApplications, Łódź, Poland, II, 797-804 6. KozaneckaD.,KozaneckiZ., LechT., ŚwiderP., 2003b,Newconceptof the spin test systemwith activemagnetic bearings,Proc. of the 2nd Int. Symp. on Stability Control of Rotating Machinery, Bently Nevada Corporation, 199- 208 7. Kozanecki Z., KozaneckaD., 2003,Application of unconventional bearings inmodernturbomachinery,State of theArt onGasTurbineResearch inPoland, CAME-GT, 73-82 8. Schweitzer G., Traxler A., Bleuler H., 1993, Magnetlager, Springer- Verlag, Berlin [in German] Eksperymentalna identyfikacja parametrów dynamicznych aktywnego łożyska magnetycznego Streszczenie W artykule przedstawiono procedurę eksperymentalnej identyfikacji parametrów dynamicznych aktywnego łożyskamagnetycznego. Zaproponowanametoda wykorzy- stuje analizę zmian przemieszczenia wirnika dla wymuszenia wirującym wektorem niewyważenia. Założono proporcjonalność siły reakcji magnetycznej na wymuszenie w zadanymzakresie częstości. Pozwoliło to na identyfikację dynamicznychwspółczyn- ników łożyska. Badania eksperymentalne przeprowadzono na stanowisku badawczym układu wirującego podpartego w aktywnym łożyskumagnetycznym. Manuscript received February 21, 2007; accepted for June 27, 2007