FROM OIL TO MAGNETIC FIELDS: ACTIVE AND PASSIVE VIBRATION CONTROL

Transcription

1 ata mehania et automatia, vol. no. (8) FROM OIL TO MAGNETIC FIELDS: ACTIVE AND PASSIVE VIBRATION CONTROL Gianarlo Genta *, Andrea Festini *, Xavier Delepine * * Department of Mehanis, Politenio di Torino, Corso Dua degli Abruzzi 4, 119 Torino, Italy gianarlo.genta@polito.it, andrea.festini@polito.it, xavier.delepine@polito.it Abstrat: When a redution of vibration amplitude usually designers resort to dampers based either on dry frition, internal material damping or fluid visosity, eah one of these mehanisms having its peuliar field of appliation. To improve performane while at the same time reduing osts and environmental load, eletromagneti damping devies are finding new appliations, beyond the fields where they have a long history. Moreover, eletromagneti dampers an be easily ontrolled, obtain ative or even intelligent vibration ontrol. Three examples from the internal ombustion engines, automotive and gas turbine fields are disussed in some detail to show the potential advantages of this tehnology. Keywords: Vibration ontrol, eddy urrent dampers, ative damping. 1. INTRODUCTION Mehanial systs (atually not only mehanial) are prone to vibrate if they an store energy in two different forms, usually potential and kineti, in a way that energy an flow from one form to the other. Vibration is the proess in whih this energy exhange takes plae. Sine every time energy is transformed from one form to another some of it is dissipated (usually onverted into low tperature thermal energy, from whih it annot be transformed bak), any vibrating syst is bound to ome to rest eventually, unless it is onneted to some energy soure, providing to refurbish its energy level. This proess of energy dissipation is usually referred to as damping. The designer of any mahine has usually to deal with vibration. Although there are ases where vibration is a desired effet (vibrating sieves, vibration welding mahines, et.), usually the task of the designer is to minimize, or at least to ontrol, it. When it is not possible to at on what exites vibration or to insulate the relevant elent from it, the traditional approah to keep vibration under ontrol is to at on the elasti and inertial harateristis of the syst to modify the frequenies at whih free vibration takes plae (these modifiations may inlude the addition of a further mehanial syst operating as a vibration absorber) or to inrease the damping properties of the syst. Often both ations are required, like when using damped vibration absorbers. Several mehanisms an be used to dissipate energy during vibration. Those traditionally ployed are: Dry frition between two surfaes moving in ontat with eah other; Internal damping of some materials; Visous fores in a fluid. Dry frition is today relied upon only for small quantities of energy, and in very simple mahines. Its major disadvantage, introduing nonlinearities into the syst, is onsidered a serious drawbak, both for its performane and owing to diffiulties in modeling its behavior. Internal damping of most engineering materials is too low to be used to dissipate large quantities of energy, so that dampers of this kind have some restrited fields of appliation. Elastomeri materials may be tailored to have the required damping harateristis, but their low thermal ondutivity and poor high tperature harateristis limit their appliations. Nevertheless many small automotive diesel engines use torsional vibration dampers based on elastomeri elents. Visous dampers are widely used in many appliations, from torsional vibration dampers in reiproating engines to squeeze film dampers in turbines, from automotive shok absorbers to large dampers used in large buildings, just to name a few examples. In their basi form they are linear devises, sine the fore they supply is proportional to the veloity F = x, (1) to the point that visous damper has beome a synonymous of linear damper. It is however possible to obtain different law F(x ) by adequately designing the syst. Another way for dissipating energy in a vibrating syst is transforming the (mehanial) vibration energy into eletri energy and then dissipating it through Joule effet. This an be done both in a solid ondutor, by generating eddy urrents, or through purposely built eletri iruits in whih a urrent an flow. All these eletromagneti devies need a magneti field to operate: if this is produed by permanent magnets no external power is required, and a passive syst is obtained. If, on the ontrary, an eletromagnet is used, some external power is required and then the syst must be onsidered as ative (Some onfusion about this terminology is sometimes found. Here a passive syst is intended as a syst requiring no external energy to work, while 11

2 Gianarlo Genta, Andrea Festini, Xavier Delepine From oil to magneti fields: ative and passive vibration ontrol an ative syst is a syst that needs to reeive power. This has nothing to do with whether the syst is ontrolled or not: an eddy urrent damper in whih the magneti field is produed by an eletromagnet fed by a onstant urrent is ative but not ontrolled. The term siative is also used, often with the meaning of a ontrolled syst needing a limited amount of energy. What limited means is learly arbitrary). In priniple, the vibration energy extrated from the syst may be onverted into some useful form of energy to be used somewhere else: this require a transduer working on four quadrants instead of a simple energy dissipator (usually a resistor). Eletri and magneti iruits may be oupled to the mehanial syst through transduers of various kind, so that to modify the dynamis of the syst. Eletromagneti damped vibration absorbers may use eletri damped osillators instead of springs, masses and dampers. Ative devies, based on atuators exerting fores on the syst with the aim of reduing vibration, an nowadays be added to these more or less lassial ways of ontrolling vibration. Sine they supply energy to the syst, the atuators must be arefully ontrolled to avoid instabilities. While passive systs based on energy dissipators are intrinsially stable, the stability of ative systs must be heked arefully in eah ase.. OIL AND RUBBER VS. MAGNETIC FIELDS The trend towards eletromagneti dampers, be they ative or passive, ontrolled or not, is lear. It is due to both new needs and new opportunities. Visous and elastomeri dampers have basially 3 probls. The first is thermal stability. The visosity or oil and the internal damping of rubber are strongly dependent of tperature. Speifi probls are then avitation in oleodynami dampers (avitation is an important probl in automotive shok absorbers) and overeating with subsequent failure in elastomeri dampers. Elastomeri materials show also a large dependene of their damping harateristis on frequeny. The seond probl is linked with manufaturing and disposal. Environmental laws ause the ost of dealing with oils and above all to dispose of used oil to inrease steadily. Strit rules are also imposed on the manufature and disposal of rubber mahine omponents. These rules are bound to beome more strit in the future with further inreasing osts. A third point, linked with visous damping devies, is the diffiulty of ontrolling the harateristis of the fluid. Eletrorheologial and magnetorheologial fluids have been investigated showing attrative potentialities for tuning damping fores aording to the operating onditions. However they too have probls related to the ageing of the fluid and to the tuning required for the ompensation of the tperature and frequeny effets. Eletromagneti dampers improve muh this situation. Although the resistivity of ondutors (a property that determines the harateristis of eletromagneti dampers) is a funtion of the tperature, this dependene is muh weaker that that of the visosity of oils or the internal damping of rubbers. Their manufature and disposal is muh less subjet to restritions and they are easily ontrolled by eletroni, possibly digital, miroproessorbased, devies. Eletromagneti dampers were often disarded in the past for many appliations beause they were onsidered too heavy, bulky and ostly. Reently the situation has muh improved. New opportunities have opened with the introdution of high-performane rare-earths permanent magnets. They allow generating magneti fields muh more intense than those due to traditional magnets for a given quantity of magneti material. The mass and bulk of eletromagneti dampers is thus redued a great deal. After the expiry of the original patents, their ost started dereasing and today it is possible to build eletromagneti passive dampers that are ompetitive with traditional dampers. Even if slower, there has been also a progress in the field of soft magneti materials, whih helps in ontaining the size and weight of eletromagneti dampers. To desribe the potentialities of eletromagneti dampers, three appliations will be desribed in detail. They deal with vibration in widely different frequeny range, and are based on different layouts. Eletromagneti dampers an be built following two different shes: vibrational motion an ause eletri urrents to be generated, whih flowing either n a solid ondutor or a resistor of various type produes the required energy dissipation. Alternatively, eletromagneti fores an be produed by ontrolled atuators, either of the Lorentz or the Maxwell type. While in the first ase damping is automatially produed, in the seond ase the ontroller must ensure that the fores oppose vibrational motion, so that energy is dissipated and vibration is quenhed. 3. MOTIONAL EDDY CURRENT DAMPERS FOR TORSIONAL VIBRATION CONTROL The eletromehanial dynamis of a torsional eddy urrent damper of motional type an be studied using the model shown in Fig. 1. Suh a onfiguration is haraterized by a single magneti pole pair (Tonoli, 7; Graves et al., ). The rotor is made by two windings 1, 1 and, installed on orthogonal planes. It is rossed by the onstant magneti field (flux density Bs) generated by the stator. The analysis is performed under the following assumptions: The two rotor oils have the same eletri parameters and are shunted, The relutane of the magneti iruit is onstant. The analysis is therefore only appliable to motional eddy urrent devies and not to transformer ones (Graves et al., ; Kamerbeek, 1973), 1

3 The magneti flux generated by the stator is onstant as if it were produed by permanent magnets or by urrent driven eletromagnets, All quantities are assumed to be independent from the axial oordinate, Every eletri parameter is assumed to be lumped. pφ rs = and R r ata mehania et automatia, vol. no. (8) k pφ rs =. (6) L The model has been validated experimentally (Tonoli and Amati, 8) using a test rig based on a four pole pairs axial flux indution mahine (Fig., steady state tests). r F 1 θ 1' F θ 1 = θ F T ' 1 B s Fig. 1. Sketh of the eddy urrent damper F 1 Fig.. Test used for the identifiation of the indution mahine at steady state Angle θ(t) between the plane of winding and the diretion of the magneti field indiates the angular position of the rotor relative to the stator. When urrents i γ1 and i γ flow in the windings, they interat with the magneti field of the stator and generate a pair of Lorentz fores F 1, in Fig. 1. Eah fore is perpendiular to both the magneti field and the axis of the ondutors. Their magnitudes are: F = Nl i B, F = Nl i B, () 1 r r1 s r r s where N and l γ indiate the number of winding in eah oil and their axial length, respetively. The total torque T ating on the rotor is: [ sin ( ) os( ) ] T = φ θ i + θ i. (3) rs r1 r For onstant rotation speeds, the torque to speed harateristis is found: T =, with + ( p ) 1 / p φ R rs =, r R r = p, (4) Lr where,, p and p are respetively the damping at low frequenies, the number of magnet poles and the frequeny of the eletri pole of the syst, and φ γs is the equivalent rank radius. R γ and L γ are the resistane and indutane of the rotating oils. The mehanial impedane an also be obtained using the Laplae transform as: Z m ( s) ( ) ( s) 1 + s/ 1 + s( k / ) = T s θ = = p. (5) This impedane orresponds to a visous torsional damper and a torsional spring onneted in series, whose parameters are: In Fig., the magneti flux is generated by permanent magnets while energy is dissipated in a solid ondutive disk. The first array of 8 irular permanent magnets is bond on the iron disk (1) with alternate axial magnetization. The seond array is bond on the disk () with the same riterion. Three alibrated pins (3) are used to fae the two iron disk - permanent magnet assblies ensuring a 1 mm airgap between the ondutor and the magnet arrays. The latter are irumferentially oriented so that the magnets with opposite magnetization are faed to eah other. In the following suh an assbly is named "stator". The ondutor disk (4) is plaed in between the two arrays of magnets and is fixed to the shaft (5). It an rotate relative to the stator by means of two ball bearings installed in the hub. The magneti iruit has been designed using a simplified one-dimensional analysis with the aim of avoiding saturation in the iron parts. The main features of the indution mahine are summarized in Tab. 1. Tab. 1. Main features of the indution mahine used for the tests Number of pole pairs - 4 Diameter of the magnets mm 3 Thikness of the magnets mm 6 Magnets geometry - Cirular Magnets material - Nd-Fe- B(N45) Residual magnetization of the magnets T 1. Thikness of ondutor disk mm 7 Resistivity of ondutor (Cu) Ohm m Airgap mm 1 Experimental tests at onstant speed have been arried out to identify the slope of the torque to speed harateristi at zero or low speed and the pole frequeny p. While the former has been identified in quasi-stati tests, the latter has been identified as best fit of the experimental points reported in Fig. 3 (Tonoli and Amati, 8). The identified parameters are = 1.4 Nms/rad 13

4 Gianarlo Genta, Andrea Festini, Xavier Delepine From oil to magneti fields: ative and passive vibration ontrol and p = 51.1 Hz, from whih the indution mahine harateristis an be obtained. An interesting field of appliation of this tehnology is the vibration damping of the rankshafts in internal ombustion engines. In fat, tehnologial issues make ative solutions diffiult to be implented, whih naturally lead to passive solutions. Nowadays, the ommon solutions involve either elastomers or visous fluids. However, they undergo a large number of stress-strain yles, and onsequently their working life is redued to approximately half the working life of a motor. Instead, eddy urrent damping does not imply the deformation of the material of the atuator, Therefore, the working life is inreased, and the effiieny of the apparatus is improved (Fig. 4). 6 5 T = Ω Theoretial Run up test Constant speed test Quasi stati test 4 T [Nm] 3 T 1 + ( p Ω ) p = Ω Ω [rpm] Fig. 3. Experimental results of the indution mahine haraterization at steady state Fig. 4. Torsional damper: shati ross setion and omparison of the amplitude of three harmonis in the nondimensional torsional response of the engine using a visous and an eddy urrent damper 14

5 ata mehania et automatia, vol. no. (8) 4. ELECTROMAGNETIC AUTOMOTIVE SHOCK ABSORBERS Working priniple. Main tasks of automotive shok absorbers are the apability of reduing transmission of vibration and ontrolling the applied load. Ative solution an be used to ahieve these goals, but very good results may be obtained also from si-ative onfigurations, allowing to hange the value of damping, depending on the driving onditions. Conventional solutions are nowadays based on hydrauli dampers, whose harateristis may be variable as a result of modifiations in the hydrauli iruit or in the oil physial properties (magnetorheologi fluids). Solution based on linear eletri motors were proposed to solve the typial probls related to the use of fluids, but they lead to an inrease of size and mass (Karnopp, 1989). This is due to the fat that, while in rotary eletri motors all the magneti and ondutor material is always ative, in linear motors only a relatively small part of it is working at eah time. Moreover, rotating eletri motors are based on a more onsolidated design pratie and on a higher tehnology bakground. The eletromagneti tehnology based on brushless motors allows obtaining a tunable suspension damper with many advantages and limited disadvantages. The apability of being tunable, and regenerating energy (four-quadrant operation) with limited inrease of mass and size using an atuator that an beome fully ative are onsidered important features. The tunable eletromehanial damper desribed in the following setion is based on a DC eletri motor whose eletri terminals are shunted on a resistive load instead of being onneted to a onverter: the torque needed to rotate the shunted motor an be omputed from the harateristi equations of the eletri mahine. They link the bak eletromotive fore V f and the eletromehanial torque T to the rotating speed and the eletri urrent i V = K ; T = K i. (7) f e t K e and K t indiate the bak eletromotive fore onstant and the torque onstant, respetively. If the eletri terminals of the motor are shunted by a resistane R, the urrent i (eddy urrent) is indued by the bak eletromotive fore V f, so that (in ase of a onstant speed ) Vf = R i. (8) Substituting the Ohm equation (8), into the motor harateristi equations (7) allows to get rid of the bak eletromotive fore and of the eddy urrent. The bak eletromotive fore and the torque onstants are the same, both are indiated as K m. The eletromehanial torque T is then related to the angular speed as shown in Fig. 3a: T =, where K m =. (9) From the mehanial point of view the resistively shunted eletri motor behaves as a torsional visous damper (Graves et al., ; Karnoppm 1987, Amati et al., R 6). The torsional damping oeffiient an be tuned by ating on the shunt resistane. The lower is the resistane, the higher is the damping. The solution in Fig. 5 shows a linear eletromehanial damper inluding a rotary motor and a ball srew transmission. From the eletrial point of view, the motor of Fig. 5 is represented as a universal motor with two eletri terminals (suh as a brush motor). R inludes the resistane of the motor windings (R m ) and the external one (R ext ) that allows to tune the damping oeffiient (R = R m +R ext ). As the effet of the indutane of the motor oil L is unwanted, no external ontribution is usually added to it. From the mehanial point of view, the ontribution of the masses and inertias on the dynami performanes of the damper annot be negleted. Fig. 5. Sketh of the eletromehanial shok absorber The equivalent mehanial model of the devie shown in Fig. 5 is reported in Fig. 6. It takes into aount the ontribution of the eletri and mehanial part of the motor written at the level of the damper. A detailed desription of the model, that allows to take into aount its frequeny dependeny, is reported in Amati et al (6). To this end, the effet of the rotor inertia J, the motor indutane (L) and resistane (R) an be written as: m = τ J, = + τ, k = τ k, (1) eq eq s where =K m /R and k =K m /L, are the torsional damping and stiffness produed by the shunted motor. Fig. 6. Mehanial analogue of the eletri motor The dynami behavior of the damper an therefore be haraterized by the mehanial impedane given by the following equation: F = m v eq s + ζ s+ s+ p, (11) 15

6 Gianarlo Genta, Andrea Festini, Xavier Delepine From oil to magneti fields: ative and passive vibration ontrol where s is the Laplae variable. The pole frequeny p, the zero frequeny, and damping fator ζ are k = p, eq k =, meq m k eq ζ = (1) Fig. 7 holds in the ase where p is lower than ( ζ 1 ). Between the pole and the zeros the syst behaves as a spring of stiffness k as shown by the - db/de slope of the mehanial impedane. A detailed study of the frequeny behavior is reported in Amati et al (6). eq J By introduing Eq. (13) into Eq. (14), it follows that π ρ Γ =. (15) Coming bak to the layout of Fig. 5, the equivalent mehanial parameters of equation (1) are linked together similarly to torsional damping and rotor inertia. The ratio between the equivalent mass and damping due to the eletromehanial effets is onstant, its value being a funtion of the motor shape and tehnology meq J π ρ = = Γ. (16) eq s Fig. 7. Mehanial impedane of the damper 4.1. Eletromehanial design The rather typial design situation is strive to obtain the speified damping oeffiient while keeping the equivalent mass (Equation 1) as small as possible. Sine the equivalent mass is related to the moment of inertia of the rotor, the aim of this setion is to find a relationship between the eletromehanial damping oeffiient in equation 9 and the moment of inertia J of the rotor. The analysis is performed assuming a onventional brushless motor with permanent magnets on the rotor surfae running in a the toothed and slotted stator. The analysis performed in Amati et al (6) leads to the following expression of the damping oeffiient =Γ, (13) 4 r l where parameter Γ is a funtion of the motor shape and tehnology. From the mehanial point of view it is worth to notie that the torsional damping oeffiient is proportional to the moment of inertia of the rotor J. Under the assumption that the rotor is an homogeneous ylinder made of an material with density ρ this is π = ρ. (14) 4 J r l This is a very important result from the design point of view. One the equivalent damping is speified, the equivalent mass to be minimized is just a funtion of the motor shape and tehnology (ρ;γ ), and not of the transmission ratio τ. The only way to redue it is to improve onstant Γ. In addition to the rotor inertia, the overall mass of the eletri motor m is the other important mehanial feature of the damper. For what is onerned to the total mass, in Amati et al (6) with referene to the shok absorber of Fig. 5, the mass of the eletrial motor is expressed as funtion of the equivalent damping of Eq. (1) as m Γ ( ) m eq s 4 = r l. (17) Γ ( τ r) Equation (17) shows it is related to tehnology and (τ r) one the desired damping oeffiient is hosen. For a given equivalent damping ( eq ), motor shape and tehnology (this gives Γ m and Γ ) the larger the rotor radius r and the transmission ratio τ, the smaller the mass of the eletri motor. The apparently unreasonable lower mass orresponding to a larger radius is due to the smaller amount of ondutor neessary to obtain the desired damping eq. To onlude the present setion, equations (1), (13), (16), and (17) outline a design proedure of the eletromehanial damper. 4.. Appliation example To analyze the potentialities of eletromehanial dampers in automotive appliations, the previously outlined proedure has been applied to design the damper of a C-segment vehile front suspension. The appliation example was done on a M Pherson suspension using the same mehanial interfaes as the original syst, and using the inner spae of the oil spring with an appropriate allowane (devie data are reported 16

7 ata mehania et automatia, vol. no. (8) in Tab. ). With referene to the layout of Fig. 8 the eletromagneti damper is represented in shade. Fig. 8. Eletromehanial damper integrated in a M Pherson suspension and ross setion of the damper The eletri motor () (4) (5) is housed in upper part of the ylinder lose to the upper strut mount (1). This hoie allows to exploit the relatively large diameter inside the spring (8) while keeping a small diameter lose to the wheel. The srew (1) is rigidly onneted to the moving piston (6), bolted to the hub. The motor, the nut of the ball srew, and the guiding tubes (11) are onneted to the sprung mass. The rotor of the eletri motor and the nut of the ball srew are rotating on ball bearings (3a) (3b) and onneted together, this redues onsiderably the axial length of the devie as it exploits the room inside the rotor of the motor. The total mass of the eletri damper is shown in Fig. 8 is 5 kg. This mass has to be ompared to the 4.1 kg of a hydrauli solution with ontinuously variable damping. The proposed damper is still heavier than the hydrauli one but is muh lighter than the onfigurations found in the literature with omparable performanes. A passive eletromehanial damper based on a linear eletri motor was designed using the same speifiations (Karnopp, 1989) and lead to a devie of more than 15 kg of mass. The effets of the damper in the ar suspension have been investigated by integrating the model in a simple quarter ar model. Fig. 9 shows the transfer funtion between the displaent of the ontat point of the tire to the ground and the aeleration of the sprung mass. Tab.. Damper speifiations and parameters of the single orner suspension Max damp. Coeff. (rebound) >1 kns/m Max speed 1 m/s Max stroke (peak to peak) 15 Mm Spring stiffness K s 18 kn/m Tire radial stiffness K p 15 kn/m Tire radial damping C p 5 Ns/m Sprung mass M s 45 kg Unsprung mass M n 3 kg Axial length at midstroke 45 Mm Max fore 1.8 kn The undamped response urves is that of the open iruit damper (R ext = : no eletro-mehanial damping); the other ones to some values the external resistane R ext. The equivalent mass (m eq ;q. 1) due to the rotor inertia introdues a sort of inertial oupling between the sprung and unsprung masses. The effet is, at any rate, not large as donstrated by the fat that the two undamped natural frequenies of the suspension go from.95 Hz and Hz (no eletromehanial damper installed) to.94 Hz and 11.1 Hz (open iruited eletromehanial damper). 17

8 Gianarlo Genta, Andrea Festini, Xavier Delepine From oil to magneti fields: ative and passive vibration ontrol Fig. 9. Transfer funtion between displaent of the ground (input) and aeleration of the sprung mass (output) 5. ACTIVE AND PASSIVE DAMPERS FOR ROTATING MACHINES 5.1. Ative magneti damper (AMD) In the present setion the use of ative magneti bearings (AMB, Fig. 1a) for vibration damping (ative magneti damper referred to as AMD, Fig. 1b) (Amati et al., 6) is introdued. Those two onfigurations differ only by the funtion of the fore generated by the atuator. While for the AMB, the atuator is used for both suspension and damping, the AMD is used only to introdue damping into the syst and the of the suspension must be insured by mehanial means. The stati stability of the suspension is thus guaranteed if the mehanial stiffness is greater than the open-loop negative stiffness of the AMD. The AMD tehnology is partiularly well suited in the field of rotating mahines, where they an replae with advantages squeeze film dampers. A model of the atuator is needed for the design of the ontrol law of the AMD. To this aim, the usual expression of the fore generated by one eletromagnet is used: μ F = N A i( t) 4 ( ) q + q t, (18) where N is the number of turns of eah winding, A is the area of the magneti iruit at the air gap, μ is the magneti permeability of vauum, i is the urrent, q is the nominal air gap and q is the displaent. a) b) Fig. 1. Sketh of an ative magneti bearing (a) and of an ative magneti damper (b) operating also as an elasti support 18

9 ata mehania et automatia, vol. no. (8) 5.. Transformer eddy urrent dampers A transformer eddy urrent damper may use the same onfiguration as an AMD, although in this ase the oils are supplied with onstant voltage through whih the magneti field is generated. Damping is thus provided by eddy urrents instead by ontrol fores as in AMDs. Both AMD and transformer operation is possible in the ase shown in Fig. 1b. However, in the latter ase, the absene of a ontrol law an make the implentation easier. The basi priniple of transformer eddy urrent dampers is the following: the displaent with speed q of the anhor hanges the relutane of the magneti iruit ausing the flux linkage to hange in time, whih generates a bak eletromotive fore in the oils, and onsequently eddy urrents in the oils. The urrent in the oils has thus two ontributions: a fixed one due to the voltage applied, and a variable one indued by the bak eletromotive fore. The first ontribution generates a fore that inreases when the air-gap dereases, whih produes a negative stiffness. The damping fore is generated by the ontribution due to the anhor speed q, that ats against the motion of the moving elent. In these terms, this onfiguration an be alled si-ative, as power is initially required to produe the initial magneti field, but damping is introdued without a traditional feedbak law (i.e. using sensors). The transfer funtion between the speed q and the eletromagneti fore F shows a first order dynamis with pole frequeny RL due to the R-L nature of eah iruit F 1 K =, (19) q s 1 + s/ where ( ) K RL R R =, =, L RL L V / q RL μ N A =. q The mehanial impedane has the form of a band limited negative stiffness. The value of the negative stiffness is proportional to the eletrial power (V /R) dissipated at steady state by the eletromagnet. For given values of the number of turns (N), of the air gap area (A), and nominal air-gap (q ), the mehanial impedane and the pole frequeny are funtions of the voltage applied V and the resistane R. As mentioned earlier, a mehanial spring must be loated in parallel to the eletromagnets to ompensate their negative stiffness. For stati stability, the additional stiffness K m must be larger than the negative stiffness of the eletromagnets ( K m K ). For the analysis, the mehanial spring in parallel to the transformer damper an be onsidered as a part of the damper, and from now on the whole eletromagnet + mehanial spring will be referred to as transformer eletromehanial damper (TEMD). The mehanial impedane of the damper in parallel with the mehanial spring, i.e. of the TEMD, is studied: F 1 K Keq 1 + s / z = + Km = q s ( 1 + s/ ) s 1 + s/ RL RL where K = K + K ; eq m K eq =. z RL Km, () Apart from the pole at null frequeny, the impedane shows a zero-pole behavior. To ensure stability, the frequeny of the zero must to be smaller than the frequeny ( < z < RL ) of the pole. As shown in Fig. 11a, it is possible to identify three different frequeny ranges. Fig. 11. Transformer eddy urrent damper. A) Mehanial impedane of a transformer damper in parallel to a spring of stiffness K eq and b) mehanial equivalent 19

10 Gianarlo Genta, Andrea Festini, Xavier Delepine From oil to magneti fields: ative and passive vibration ontrol Equivalent stiffness range << z < RL : the syst behaves as a spring of stiffness K eq >. Damping range z << RL : the syst behaves as a visous damper with oeffiient C defined as: K m C =. (1) RL Mehanial stiffness range z < RL <<: the transformer damper ontribution vanishes and the raining stiffness ontribution is provided by the mehanial spring (K m ). It an be onluded that the assoiation of eletromagnets with a mehanial suspension opens interesting perspetives for the vibration ontrol of rotating mahines. As the AMD and TEMD onfigurations are idential, it is possible to swith from one strategy to the other while the mahine is running. 5. CONCLUSIONS More eletri or even All eletri are more than fashionable ath phrases. They synthesize a tendeny gaining momentum in many fields of tehnology towards a larger use of eletri mahines to ontrol motion and to distribute power, while trying to do without lubriating or damping fluids. Even in ases where it is impossible to avoid hydrauli transmission, the eletrohydrauli approah allows to mate the great ompatness of hydrauli mahinery and transmission lines with the ease of ontrol of eletri mahines. While the road to build an all eletri turbojet or power turbine (to quote two examples) is still long and full of obstales, the use of eletromagneti devies to ontrol vibration and to manage power is in many ases mature for appliations. Passive, siative and fully ative dampers find inreasing appliations not only in those high tehnology fields in whih they were first used, but also in low ost onsumer markets. The automotive shok absorber and the internal ombustion engine torsional dampers here shown are two examples of this trend. In eddy urrent dampers, usually passive or siative, the substitution of magneti fields instead of oil or elastomers is straightforward and the only diffiulties are osts and sometimes still mass and bulk. Fully ative devies on the ontrary, apart from being more omplex and usually ostly, require aurate studies sine their behavior is not intrinsially stable. As usual with ative systs, their ontrol laws must be designed with both ost and performane and stability in mind. Although being still far in the future, the vision of mahines with no part in ontat and hene requiring no lubriant and displaying no wear, and with vibration ontrol that does not involve energy dissipation, is a lear goal that an guide designers and sientists. REFERENCES 1. Tonoli, A. (7), Dynami harateristis of eddy urrent dampers and ouplers, Elsevier J. Sound Vib., 31, pp Graves, K. E., Tonih, D., and Iovenitti, P. G. (), Theoretial omparison of motional and transformer f devie damping effiieny, Journal of Sound and Vibration, vol. 33, no. 3, pp Kamerbeek, E. M. H. (1973), Eletri motors, Philips teh. Rev., vol. 33, pp Tonoli, A., and Amati, N. (8), Dynami modeling and experimental validation of eddy urrent dampers and ouplers, Journal of Vibration and Aoustis, vol Mahi, P., Silvagni, M., Amati, N., Carabelli, S. and Tonoli, A. (6), Transformer eddy urrent dampers for the vibration ontrol of rotating mahines, Proeedings of the 8th Biennial ASME Conferene on Engineering Syst Design and Analysis, Torino, Italy, pp Karnopp, D. (1989), Permanent magnets linear motors used as variable mehanial dampers for vehile suspensions. Vehile Syst Dynamis, 18, pp Karnopp, D. (1987). Fore generation in si-ative suspensions using modulated dissipative elents. Vehile Syst Dynamis, 16, pp Amati, N., Canova, A., Cavalli, F., Caviasso, G., Carabelli, S., Festini, A., and Tonoli, A. (6), Eletromagneti shok absorbers for automotive suspensions: eletromehanial design. ESDA-ASME Amati, N., Carabelli, S., Genta G., Mahi P., Niolotti F., Silvagni M., Tonoli, A., Vinsonti, M. (6), Vibration ontrol of rotors: trade off between ative, si-ative and passive solutions. The IX Finnish Mehanis Day. Lapperanta, Finland. June (pp. 1-15). ISBN/ISSN: /