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1 NONCOLOCATION EFFECTS ON THE RIGID BODY ROTORDYNAMICS OF ROTORS ON AMB Gancarlo Genta Department of Mechancs, Poltecnco d Torno, Torno, Italy, [email protected] Stefano Carabell Department of Automatc Control, Poltecnco d Torno, Torno, Italy, [email protected] ABSTRACT The eect of sensor-actuator non-colocaton on the behavour of machnes runnng on acve magnetc bearngs s studed under the assumptons that the rotor behaves as a rgd body, the controller s an deal decentralzed proportonal-dervatve one, the behavour of the bearngs can be lnearzed and the whole machne s axally symmetrcal. The possble presence of an nstablty range, whch n some cases can extend down to the zero-speed condton, s demonstrated. The eect of dampng on the nstablty range s studed, showng that t s stablzng and that, wth dampng hgh enough, t s possble to acheve stable runnng n the whole workng range. A smple centralzed controller whch cures the consequences of non-colocaton s shown to exst and ts gans are computed. The paper ncludes also an example related to an actual machne showng strong non-colocaton eects. INTRODUCTION Noncolocaton between sensors and actuators s a well known problem for exble strutures [, 3]. Wth magnetc bearng technology, t must be taken nto account n the desgn stage, as t can ntroduce nonneglble eects n the dynamc behavour of the machne, and even lead to nstabltes. There s no dculty n dong so when modellng the system usng the nte element method however t s usually not consdered when smpler models, as the well known four-degrees-of-freedom (twof workng usng complex coordnates), model are used. Such a smple approach s well suted for the very common case of machnes workng well below the exural crtcal speeds of the rotor and through the rgd-body crtcal speeds, whch n the case of AMB may have a farly low value. In such a speed range the rotor can be consdered as a rgd body. If the feedback loop s modeled wth and deal PD controller, the very smple nature of the model allows to perform a general rotordynamc study, yeldng nterestng results. In partcular t s possble to study the eect of the non-colocaton and to show that t ntroduces a type of behavour whch can be qute derent from that typcal of rotors runnng on conventonal bearngs. An nstablty range can be present, and n some extreme cases the system may even be unstable at standstll. The nstablty range can be shown to shrnk whth ncreasng dampng (.e. the dervatve gan of the control loop). As the analyss deals only wth the rgd-body dynamcs, t doesn't allow to predct possble spllover eects, n whch hgher modes may be excted by the control system, partcularly owng to non-colocaton eects. However, f the maxmum operatng speed s well below the rst crtcal speed lnked wth the deformaton of the rotor and the controller doesn't ntroduce a large phase loss n the vcnty of the hgher natural frequences of the system, the nternal dampng of the rotor can successfully deal wth spllover problems and the present analyss s well applcable, at least n producng a reduced order model for the desgn of the electromechancal parts and the control system. ANALISYS Equatons of moton Consder a rgd rotor runnng on n actve magnetc bearngs (AMB). Assume that z axs of the nertal refrence frame Gxyz centered n the poston of the center of mass of the rotor at rest G concdes wth the rotaton axs and let z and z be the z- coordnates of the sensor and the actuator of the -th bearng. The lateral behavour can be modeled usng the followng equaton of moton wrtten n complex coordnates where M Mq ;!G_q F c + F n +! F r () m G J t x + y q J p y ; x and F c, F n and F r are the control forces, the nonrotatng forces and the rotatng forces due to unbalance (see []). Assume that the controller s an deal decentralzed PD controller and that the law expressng force F (n complex notaton) exerted by the-th actuator as a functon of the dsplacements x + y and x + y at the -th sensor and actuator locatons and the velocty atthe-th sensor s:

2 F ;K (x + y ) ; K C (_x + _y )+K u (x + y ): () where and K and C are the gans of the control loop, whle K u s them open-loop destablzng stness of the bearng. By ntroducng the control force vector F c due to anumber n of actuators nto Eq. () and defynng the average dstance of the -th sensor-actuator par z (z + z) and the noncolocaton z d (z ; z ) of the same par, the equaton of moton of the system reduces to where Mq +(C ;!G) _q + Kq F n +! F r (3) k K k k k3 ;k4 + k4 k (K ; K u ) k4 K z d k (K z ; K u z) k3 (K [z ; z d ) ; K u z ] c C c c c3 ;c4 + c4 c C K c C K z c3 C K (z ; z d ) c4 C K z d : Owng to noncolocaton (z 6 z), matrces C and K are non symmetrcal and may be non-postve dened. The presence of the negatve termsdueto K u s usually not causng problems n colocated systems, owng to ther smallness, but n the present case they may contrbute to make the stness matrx non-postve dened. The skew-symmetrc part of matrx K s usually referred to as a crculatory matrx t contans only noncolocaton eects due to dstances z d. Study of the stablty Consder the homogeneous equaton assocated wth Eq. (). Assumng a soluton of the type q qe t, where vector q contans the complex coordnates x + y and y ; x, and solvng the related egenproblem, the followng nondmensonal characterstc equaton allowng to compute the whrlng frequences s obtaned 4 ; [! + ( + )] 3 + (4) ; + + ; ( ; ) ;! + +! +( + ) ; ( ; )( + ) + + ; + : where the nondmensonal complex whrl frequency and the nondmensonal spn speed!! have been dened wth reference to the natural frequency p km of a Jecott rotor wth the same mass and a stness equal to k. Eq. (4) depends only on eght nondmensonal parameters, namely -`elastc' parameters: + P m n (K z ; K u z P ) n J t (K ; K u ) P ; m n K z P d n J t (K ; K u ) r m J t (K z ; K u z ) (K ; K u ) K z d (K ; K u ) -`nertal' parameter: J p J t, -`dampng' parameters: r m J t C K C K C K C K : Note that: - s made of two parts, namely and. The rst one does not depend on the non-colocaton but only on the average postons z andsalways postve vanshes for colocated systems and s aways negatve. - does not depend on the non-colocaton as, can be ether postve or negatveandvanshes for symmetrcal systems (see below). - can be postve or negatve and vanshes for ether colocated or symmetrcal systems. - The sgn of and has no eect on the behavour of the system, as only the squares of these parameters are ncluded n the equatons. - s the usual parameter for gyroscopc eects ts value can span from (long rotors) to (dsc rotors) however a smaller varablty range s expected n actual applcatons. - concdes wth the dampng rato of the above mentoned Jecott rotor. If all the bearngs have the same dervatve ganc and the contrbutons due to the terms K u are small enough to be neglected, and the number of relevant nondmensonal parameters reduces to ve. As the equaton has complex coecents, the solutons are complex but not conjugate. Although lttle can be sad n general on the stablty of the system, Eq. (4) allows to assess numercally the stablty nany gven case. In the case of the undamped system, Eq. (4) reduces to 4 ;! 3 ;( + ) +! +; + (5)

3 whch depends on just four nondmensonal parameters Ṫhe latter equaton has real coecents: the solutons can be real numbers, n whch case the system s stable (n the sense that the ampltude of free whrlng nether decreases nor ncreases n tme), or complex conjugate numbers. In the latter case, at least one soluton wth negatve magnary part exsts and the system s unstable Symmetrcal system Consder a rotor on two equal bearngs wth ts center of mass at mdspan. Assume that also the sensors are smmetrcally located. The equatons of moton for the translatonal and rotatonal degrees of freedom uncouple (only four nondmensonal parameters are derent from zero, namely,, and ) and the characterstc Eq. equatons: (4) splts nto two ndependent ; + + ; +! + + (6) Cylndrcal whrlng s governed by the same equaton of the well known equaton of moton of the Jecott rotor. The equaton descrbng the concal whrlng ders from the usual equaton dealng wth co-located systems because the product zz can be negatve n the case the actuator on one sde s connected wth the sensor on the other one. In ths case s negatve. If s postve, the behavour of the system s equal to that of a co-located p system wth the actuator n the poston z zz. The case wth negatve has very lttle practcal nterest, as the system s unstable at standstll, behavng as a sprng, mass, damper system wth negatve stness and dampng coecent. However the gyroscopc moment can stablze the undamped system. The soluton of the second Eq. (6) s p! + (! +) +4 whch holds for both postve or negatve. If s negatve, t follows: (7) s pa <( )! + b + a (8) s pa + b ; a ( );jj (9) where a! ; 4 ; 4jj and b 4! jj. In the case of the undamped system, stablty occurs f p jj!> : () However, the presence of dampng make thesys- tem unstable at all speeds snce the magnary part of one of two values of s always negatve forany value of the spn speed! FIGURE : Nondmensonal campbell dagram the decay rate plot of a system wth :5, :5, :, :5 and :6. Non-symmetrcal system If the center of mass of the rotor s not at mdspan or f the symmetry assumed n the prevous secton s volated, the two equatons of moton do not uncouple and the modes do not reduce to concal and cylndrcal ones. Nevertheless often they are stll referred to as concal or cylndrcal, but only n a general way, as the rst one does not have tsvertex n the center of mass and the latter s not a true cylnder. The condton for stablty of the undamped system at standstll s ( ; ) +4 ; 4 > () whch sobvously vered for >, although beng less restrctve than that. The equatons become complcated enough to prevent from performng a closed form general study of the stablty, even n the undamped case. Although lttle can be sad n general on the stablty of the system, Eq. (4) allows to assess numercally the stablty nany gven case. Some typcal plots and conclusons drawn from numercal expermentaton on undamped systems wll be reported here. The nondmensonal campbell dagram and the decay rate plot of a system wth :5, :5, :, :5 and :6 are reported n Fg.. The system s stable for!,as(; ) +4 ; 4 :55 >. The curves ralated to cylndrcal and concal whrlng cross n the rst quadrant and, where they meet, a eld of nstablty starts. The unstable condtons persst up to a certan speed, whch sbeyond the crossng of the wth the lne!. The plot s repeated n Fg., wth the same values of the parameters, but wth :5 nstead of :6,.e., wth a dsc rotor nstead of a long rotor. The results are smlar to the one prevously seen, wth the derence that the curve related to the concal mode n forward whrlng (whose asymptote s

4 FIGURE : Nondmensonal campbell dagram the decay rate plot of a system wth :5, :5, :, :5 and : FIGURE 3: Nondmensonal campbell dagram the decay rate plot of a system wth, :5, :, :5 and :6. the straght lne wth equaton! J p J t )hasa greater slope. As a result the nstablty range moves toward lower speeds and les all n the subcrtcal range (on the left of the lne! ). The plot of Fg. 3 deals wth the same values of the parameters as n Fg. (long rotor), but for the values of whch snow greater than ( nstead of.5). The curves ralated to cylndrcal and concal whrlng now cross n the fourth quadrant and,consequently, the eld of nstablty occurs n backward whrlng condtons. The plot of Fg. 4 refers to the same case of Fg. 3, but for a dsc rotor ( :5 nstead of :6). As > the nstablty range les n the backward whrl zone of the plot, but t s dsplaced towards lower values of the speed. Note that n all cases studed above an nstablty range was present. Further numercal nvestgaton showed that ths s due to the fact that >. If, on the contrary <, no nstablty range was encountered, at least unless >. The conclusons drawn from the numercal experments run on undamped system are reported n the followng table crossng n < > < I quadrant no nst. unst. FWD modes > IV quadrant no nst. unst. BWD modes A further case, wth the same parameters of that studed n Fg., but wth :, s shown n Fg. 5. Note that now(;) +4 ;4 ;:475 < : the system s unstable even at standstll, for both forward and backward modes, to be stablzed at hgh speed by the gyroscopc eect. The eect of dampng s that of reducng the wdth of the nstablty range and, f the system s damped enough, no nstablty s encountered. GEOMETRIC RE-COLOCATION Consder a rotor runnng on two magnetc bearngs. If the rotor s rgd, the noncolacaton eect can be compensated for by usng a centralzed control system,.e. t s possble to desgn a centralzed control system whch causes the actuators to produce forces whch are proportonal to the dsplacements (or the velocty, for the dervatve branch of the control loop), at the actuator locatons nstead of that of the sensors. The complex dsplacements at the sensor and actuator locatons can be expressed as functons of the dsplacement and rotaton at the center of gravtyas where x + y x + y x + y x + y z T z T x + y y ; x T x + y y ; x T z z () (3) The proportonal part of the forces exerted by the actuators are proportonal to the dsplacements at the actuator locaton f Fx + F y K F x + F y K T T ; x + y x + y (4) The matrx of the gans of the control system requred perform the recolocaton s thus K K c K z ; z T T ; (5) K (z ; z ) K(z ; z ) K(z ; z ) K(z ; z ) The matrx of the dervatve gans can be obtaned n the same way, just substtutng C K for K.

5 FIGURE 4: Nondmensonal campbell dagram the decay rate plot of a system wth, :5, :, :5 and :5. EXAMPLE Consder a rotor wth the followng nertal data: m 9:7 kg J t :8 kg m J p :337 kg m. The center of mass of the rotor s at 34.5 mm from one end of the shaft whle the actuators and sensors are at 8.7 mm, 5.8 (actuators), 9. mm and 9.5 mm (sensors) respectvely. The gans of the sensor-actuator loop of the bearngs are K : 6 N/m and K :6 6 N/m, K u 3 N/m and K u 36 N/m. The nondmensonal parameters of the undamped system are: :6 ( :33, :4), :99, :43 and :4. The value of s 54:8 rad/s. The system s stable for!, as ( ; ) + 4 ;4 :378 >. The campbell dagram of the undamped system s shown n Fg. 6. As expected the branches of the meet n the rst quadrant ( < ), a eld of nstablty exsts ( > ) and s located manly n the supercrtcal eld ( <). The computaton of the was repeated wth derent non-colocatons and values of the dampng to obtan stablty maps wth the am of assessng stablty boundares. The results are reported n Fg. 7 n whch the spn speeds at whch the rotor becomes unstable and then stable agan are plotted as functons of the dstance d between the sensors and the actuators. The varous curves have been obtaned for dfferent values of the dampng rato. Note that the sensor-actuator dstance has been assumed to be the same for the two bearngs (whch s not the case n the actual system) and also the controllers have been assumed to supply the same dervatve acton (equal C ). Strctly speakng, the values of,, and are not exactly equal. If the sensor-actuator dstance s smaller than mm no nstablty occurs even f the system s undamped, whle larger sensor-actuator dstances lead to ncreasngly large nstablty ranges FIGURE 5: Nondmensonal campbell dagram the decay rate plot of a system wth :, :5, :, :5 and :6. As ( ; ) +4 ; 4 ;:475 < the system s unstable for!. By addng dampng the maxmum value of d for whch thesystem s stable ncreases and, f the unstable range s at any rate found, the threshold of nstablty ncreases wth the dampng. The value of the upper lmt of the nstablty range has a more complex behavour: the presence of dampng causes t to ncrease, but then t decreases wth further ncreases of dampng. As the average sensor-actuator dstance s of 35 mm, a dampng rato n excess of.75 s requred to guarantee stablty. A larger value of dampng,.e. C C ;3 s assumed, to account for the fact that the larger bearng whch has a larger noncolocaton (due to a greater bulk of the actuator). It leads to a stable system wth :78, :86, :6 and :79. The matrx of the gans of a centralzed control system able to recolocate the system, s :896 ;:696 :54 :446 6 N/m The of the undamped system s reported n Fg. 8: ts overall pattern s that of a conventonal rotor on soft bearngs and no noncolocaton eect s present CONCLUSIONS The sensor-actuator non-colocaton may have a detrmental eect on the behavour of machnes runnng on acve magnetc bearngs. Some bearng con- guratons, manly those based on optcal sensors, allow postonng the sensors and the actautors n the same locaton, thus avodng the problem from ts onset, but n the majorty of cases non-colocaton s the rule. The dstance between sensors and actuators depends on the actual layout of the machne, and n some cases cannot be reduced owng to the length of the pole peces of the actuators and, n a number

6 R(λ)[Hz] R(λ)[Hz] ω [rpm].5 3 x ω [rpm].5 3 x 4 FIGURE 6: of the system studed n the example (undamped system). Instablty range [rpm].8 x ζ ζ. ζ.5 ζ d [mm] FIGURE 7: Lower and upper lmts of the nstablty range as functons of the sensor-actuator dstance, for varous values of the dampng rato. of cases, the need of avodng nterferences on the sensors. As long as the rotor may be assumed as rgd, the sensor-actuator noncolocaton s commonly thought not to be a problem, n any case a problem to be consdered only for machnes desgned to work well above the rgd body crtcal speeds where exble modes come nto play. The eect of noncolocaton has been studed here under the assumptons that the rotor behaves as a rgd body, the controller s an deal decentralzed proportonal-dervatve one, the behavour of the bearngs can be lnearzed and the whole machne s axally symmetrcal. Under these condtons the eect of the non-colocaton s to ntroduce a skewsymmetrc part nto both the closed-loop stness and dampng matrces and even to make the overall matrces non-postve dened. The outcome s the possble presence of an nstablty range, whch FIGURE 8: of the same system of Fg. 7, but wth a a centralzed controller whch recolocates sensors and actuators (undamped system). n some cases may extend down to zero-speed. Wth some combnatons of the values of the parameters the mode whch can become unstable s a forward whrlng mode, n other cases a backward mode s unstablzed. The presence of dampng reduces the wdth of the nstablty range and, f the dampng s hgh enough, stable runnng can be acheved n the whole workng range. As only the rgd-body behavour has been consdered, t s possble to use a centralzed controller to cure the consequences of non-colocaton, obtanng the dynamc behavour typcal of colocated systems. Ths procedure has been here referred to as geometrc re-colocaton. The results here obtaned are lnked wth the rgd-body assumptons and hold only n the speed range extendng to speeds well below the rst crtcal speed lnked wth rotor deformatons. Many machnes runnng on magnetc bearngs however operate n these condtons, so they are applcable to many actual cases. An example related to a turbomolecular pump, on whch the eects of noncolocaton were rst observed, shows how the analytcal results apply to an actual machne. REFERENCES [] G. Genta, Vbraton of Structures and Machnes, 3rd ed., Sprnger, New York, 998. [] R. Cannon and D. Rosenthal, Experments n control of exble structures wth noncolocated sensors and actuators, AIAA Journal of Gudance, vol. 7, pp. 546{553, Sept.-Oct [3] V. Spector and H. Flashner, Modelng and desgn mplcatons of noncollocated control n exble systems, ASME Journal of Dynamc Systems, Measurement, and Control, vol., pp. 86{93, June 99.