DESIGN AND EXPERIMENT OF AN ELECTROMAGNETIC VIBRATIONAL INERTIAL ACTUATOR USING LINEARIZED MAGNETIC SPRING

Transcription

1 Rev. Rou. Sci. Tech. Électrotech. et Éerg. Vol. 63, 3, pp , Bucarest, 018 DESIGN AND EXPERIMENT OF AN ELECTROMAGNETIC VIBRATIONAL INERTIAL ACTUATOR USING LINEARIZED MAGNETIC SPRING RADU OLARU, MARIUS-MUGUREL MIHAI, BOGDAN GÎRTAN, CAMELIA PETRESCU, ALEXANDRU ARCIRE Key words: Electroagetic actuator, Magetic sprig, Vibratio iertial actuator. The paper is focused o the theoretical odelig, siulatio, optiizatio ad experietal ivestigatio of a vibratio electroagetic iertial actuator based o liearized agetic sprig. The actuator has a extreely siple desig ad cotais two differetially coupled agetic sprigs, two coils ad a agetic housig. A agetic shield ad two agetic rigs, fixed o the ovig agets, esure the force ad liearity perforace of the vibratio actuator. Theoretical odelig ad results fro siulatio ad optiizatio of a referece odel led to the costructio of a prototype which was studied experietally. Measured data show that the resoace frequecy of the prototype is 4 Hz. This frequecy ca be adjusted i the doai Hz by extedig the gap betwee the statioary agets ad the obile agetic assebly. The device, outed o a echaical structure, could geerate vibratio forces higher tha 4 N i the frequecy rage Hz. 1. INTRODUCTION The issue of vibratory otio ad electroagetic actuators for vibratio is very broad ad covers applicatios fro sall to great powers ad low to high frequecies. Applicatios ca be foud i: cellular phoes [1], eergy harvesters [], vibratio suppressio [3] ad shakers [4]. Most vibratio actuators cosist of a ass suspeded o a fixed base through a elastic eleet which is put i vibratio by a iteral force. This force ca be geerated usig differet physical priciples. Geerally, the electroagetic priciple is preferred because electroagetic devices have a log lifespa copared to piezoelectric ad electrostatic devices that suffer fro degradatio ad leakage effect, respectively. Vibratio electroagetic actuators ca be with ovig iro (electroaget, ovig coil (voice coil actuator ad ovig aget(. Movig coil devices are ow the ost used i vibratio systes. However, the research o vibratio actuators ad geerators with ovig agets is itese ad growig [5], ay copaies coercializig such devices. A copariso betwee ovig coil ad ovig aget actuators shows that i soe applicatios usig the ovig aget actuator ca be the optial solutio [6]. Moreover, researches cocerig the replaceet of the classical echaical suspesio syste, with agetic sprigs, are uderway, eliiatig the elastic ovig eleet subjected to echaical wear. Such applicatios are foud i otors drivig copressors [7], vibratio geerators or shakers [8]. Covetioal echaic sprigs ipose soe liitatios o the syste regardig aily copactess, frictio, aterial fatigue ad failure. Magetic sprigs, based o repulsive or attractive peraet agets, show the potetial to overcoe these issues especially for ediu strokes ad frequet operatio. A iertial vibratioal actuator is a reactio type actuator geeratig dyaic forces for excitatio of echaical structures. Ulike the direct-actio actuators, where the iertial ass is coected to the structure by eas of a shaft that eeds a support surface, usually actig i oe directio, a iertial vibrator ca be outed i ay positio, directly to the structure, without exteral support, the reactio forces supported by the vibrator housig beig trasitted to the echaical structure. Iertial electroagetic actuators are widely used i active vibratio cotrol of echaical structures, where the largest output force of the covetioal electroagetic actuator is restricted to a very arrow badwidth aroud the atural frequecy [9]. A variety of desig solutios have recetly bee proposed, all of the usig echaical copoets i order to cotrol the stiffess [10]. I this paper a electroagetic vibratioal iertial actuator (EVIA) based o electrically activated liearized agetic sprig, which esures both the elastic suspesio of the obile agetic part ad geerates the vibratio forces, is studied ad developed. The ageto-elastic actuator offers a siple procedure, without usig echaical eleets, for adjustig the atural frequecy by chagig the stiffess of the agetic sprig.. MATHEMATICAL MODELING OF EVIA WITH LINEARIZED MAGNETIC SPRING To predict the perforace of the iertial vibratio actuator usig a liearized agetic sprig, the goverig equatios of otio are derived. I the electroechaical odel illustrated i Fig. 1 the iertial ass,, of the ovig aget is coected to the support structure by a agetic sprig with the stiffess coefficiet k, ad a daper havig the dapig coefficiet c. The actuator is coposed of two differetially coupled agetic sprigs so as to obtai a liear characteristic that reders a force proportioal to the displaceet, f s kz, as will be deostrated i &3.1. A drivig force that follows fro the Laplace force f dr ( i(, (1) where α is the actuator costat, is geerated whe a siusoidal curret i( I si t passes through the actuator s coil ad produces the vibratory displaceet, z(. The goverig equatio of otio ca be obtaied fro the balace of forces as: Techical Uiversity of Iasi, Faculty of Electrical Egieerig, Bd. Diitrie Magero o.1-3, Iasi, capet@ee.tuiasi.ro

2 54 Desig of a electroagetic vibratioal iertial actuator G F F( ( i( s s s. (8) I the tie doai, the respose of the syste due to a siusoidal curret is z( Z( ) si t, (9) Fig. 1 Electroechaical odel of iertial vibratio actuator. The actuator costat α which appears i (1) is give by the followig relatio α Bl c coef, (3) where l c is the coil legth, B the average radial agetic flux desity i the space occupied by the coil widigs ad coef a coefficiet depedig o the ier ad outer radii of the coil. The curret i the coil decreases with frequecy due to the icrease i coil reactace (whe usig a voltage source) ad due to the electrootive force geerated by the otio of the cetral aget iside the coil, that follows fro Faraday s law, e αvcoef, (4) so that, i order to keep the geeral for of (), a curret excitatio source is used i practice. For a accurate aalysis of the vibratio actuator i the frequecy doai, two cases ust be cosidered: voltage excitatio source ad curret excitatio source, respectively, i order to coply with the right-had side ter of ()..1. EVIA WITH CURRENT EXCITATION SOURCE The excitatio source supplies a siusoidal curret of costat aplitude ad frequecy. I this case eq. () becoes: z c kz i(, (5) where c is the echaical dapig coefficiet. Usig the Laplace trasfor, the vibratio displaceet has the expressio where z cz kz i(. () i( i( 1 Z(, (6) s c k s s k / is the atural agular frequecy ad c / k is the echaical dapig ratio, k beig the agetic sprig coefficiet. The acceleratio force, F( s s z( is equal ad opposite to that applied to the ovig ass: s F( i(. (7) s s The trasfer fuctio betwee the total force F ad the curret i applied to the coil will be where Z(ω) is the oscillatio aplitude of the iertial ass ad ca be expressed as I Z ( ) k (10) ad φ is the phase of the ass oscillatio relative to the curret. The fuctio described by (10) has a axiu for the resoace agular frequecy ω r r 1 (11) ad the aplitude of the ass displaceet or the resoace aplitude is give by the expressio Zr I 1, k 1 which is slightly higher tha for the atural pulsatio, I Z k (1). (13) The ass acceleratio, a(= z ad the acceleratio force ( z, are expressed i the frequecy doai as: ad f a I A ( ) k F ( ) I a (14) (15) respectively. O the other had, the acceleratio force produces a reactio force fro the vibrator casig, f ( f (... EVIA WITH VOLTAGE EXCITATION SOURCE I the case of a applied voltage source, v(= Vsiωt, where V is the axiu value=costat, the feedback effect of the iduced voltage ca be odeled usig the electrical circuit i Fig., so that the voltage equatio ca be writte as: di z Lc i v(, (16) dt where L c ad R c are the iductace ad the resistace of the coil, respectively. r a

3 3 Radu Olaru et al. 55 Fig. Electrical circuit with a ideal voltage source. Assuig that the coil iductace is very sall ad ca be eglected, L 0, the curret has the expressio i ( v e) / R c or c ad equatio () becoes: v( z ( c ) z kz (17) z ct z kz ic (, (18) where i c ( v( /, is the curret copoet give by the voltage source, beig equal to the curret supplied i the absece of oveet (fixed age or at very low frequecies whe the iduced voltage (dyaic copoe ca be eglected. The costat ct c ce is the dapig coefficiet that describes the total dapig of the syste as a cobied echaical ad electric dapig. The echaical dapig coefficiet, c, approxiates forces of viscous dapig ad frictio, while the electric dapig coefficiet approxiates the feedback electric force due to the electrootive force geerated by the otio of the cetral aget iside the coil, beig expressed as ce. (19) I (17) ad (18) the force that acts o the vibrator has a costat value regardless of the e..f. value. I statioary state, Fdr. s Ic, where I c V / is the direct curret equal to the ratio betwee the peak value of the siusoidal voltage ad the coil resistace. I the frequecy doai, equatios that express the ai quatities ad paraeters of EVIA with costat voltage excitatio source have the sae for as i the case of EVIA excited by a curret geerator, eqs. (10) (15), the oly differeces beig that I is replaced by I c ad the echaical dapig ratio is replaced by the total dapig ratio, t = e. I the case whe the coil reactace X c Lc caot be eglected, the curret i the circuit preseted i Fig. is: V E I. R c L c I these coditios (17) becoes: (0) v( t ) z c z kz Lc, (1) Lc which correspods to a oliear dyaical syste. Equatio (1) shows that the electrical dapig ad the active force o the vibrator decrease with the icrease of coil ipedace for higher frequecies. Relatio (1) also idicates that the coditio Lc ust be iposed so that the syste ca be described with sufficiet accuracy by (17). 3. NUMERICAL ANALYSIS OF THE EVIA MODEL 3.1. LINEARIZED MAGNETIC SPRINGS Several solutios for obtaiig liearized agetic sprigs are described i this sectio. Figure 3 shows that the curve force versus displaceet for a siple agetic sprig coposed of two rig agets (also for disc aget presets a high oliearity. A iportat liearizatio of the agetic sprig is obtaied by usig two siple sprigs coupled differetially, coposed of two or three idetical (or o-idetical) agets. Cosiderig the differetial sprig with idetical agets as a referece (Fig.4), the liear doai of the differetial sprig ca be icreased by applyig two rigs ade of soft agetic aterial o the faces of the obile aget. I this case, the force ad rigidity of the agetic sprig decrease as the thickess of the agetic sprig icreases. The experietal tests coducted i the laboratory showed that usig agets of differet sizes (Fig. 5.a ad Fig. 5.b), such as the case whe the obile aget height is to 5 ties loger tha i the iitial case (Fig. 4) leads to a iproveet of the F(z) liearity ad also icreases the sprig rigidity, with o sigificat decrease of the axiu force F, (which is obtaied whe the obile aget is i cotact with the fixed oe. The drawback of the cofiguratios i Fig. 5 is that the total ass of the device is icreased. Other siulatios (which are ot preseted i this paper) show that the slope of the characteristic F(z) ad the stiffess coefficiet icrease for a decreasig distace z 0 betwee the obile ad the fixed agets. 3.. SIMULATION AND OPTIMIZATION OF THE REFERENCE MODEL A aalysis ad optiizatio based o uerical siulatio for the proposed actuator odel is preseted i this sectio. The rig agets, both fixed ad obile, type R N fro Superagete, for a ageto-elastic syste, ore precisely a differetial agetic sprig. The ai objective of the optiizatio is axiizig the electroagetic force F dr i i the coditio of aitaiig a height/diaeter ratio of the device approxiatively equal to 1, as is custoary for ay covetioal vibratio geerators coercially available. I order to icrease the actuator costat α ad the drivig force a icrease of the ea radial agetic flux desity B ust be achieved, sice the other two costats i (3) are liited by the coil diaeter ad sectio. I order to icrease the agetic flux desity, the structure of the vibratio actuator preseted i Fig. 6, which icludes two agetic rigs attached to two ovig agets, was adopted. The device is shielded by a cylidrical ferroagetic tube ad two lids. This structure esures a liear characteristic force versus displaceet,

4 56 Desig of a electroagetic vibratioal iertial actuator 4 Fig. 3 Magetic force versus displaceet for a siple agetic sprig. Fig. 4 Differetial agetic sprig: two agetic sprigs coupled differetially; agetic force vs. displaceet with ad without agetic rigs. Magetic discs sigificatly chage the aspect of the agetic sprig characteristic (copared to the o disc cofiguratio) rederig a liear F(z) depedece for displaceets of the obile part up to as close as 1 fro the fixed agets (Fig. 7). The shield, the lids ad the ferroagetic rigs icrease the electroagetic force of the vibratio actuator as show i Fig. 8. Accordig to Fig. 8 ad the data i Table 1, a force F = 16. N is obtaied i the o shield, o iro rigs cofiguratio for a d.c. curret I = A. If iro rigs are added the F = N, i.e. a icrease of 8. % is obtaied. If the exteral agetic cylider is added, the the force icreases to N (a icrease of 11. % copared to the iitial case), ad, fially, if the top ad botto lids are added the force is F = 3.4 N, i.e. a total icrease of 44.6 % copared to the first case. These figures show the ajor effect of usig the ferroagetic lids, with the observatio that this effect is axiized whe the lids are as close as possible to the coils. Table 1 Drivig forces i several cases μ r tube μ r caps μ r rigs. I (A) F (N) (a) Fig. 6 Referece odel for EVIA with agetic shield used i siulatios. (b) Fig. 5 Differetial agetic sprigs with differet ratios - radius of obile aget / radius of fixed agets: a) >1; b) <1. F(z), as deostrated i &3.1. The two coils are coected i series ad create opposite agetic fields. Preliiary siulatios for z 0 = 6 7 (distace betwee the fixed aget ad the obile oe i cetral positio) showed that: i the absece of the agetic housig ad ferroagetic rigs, the static force i iddle positio has the largest value for a coil sectio S w = 360 whe the widow width / height ratio is betwee 0.7 ad 0.9, ad the distace betwee the two coils is betwee 3 4 ; the curves for the passive differetial agetic sprig are ot odified whe the agetic shield is itroduced. Fig. 7 Magetic sprig curves: with agetic rigs (cotiuous lie) ad without agetic rigs (dotted lie). Fig. 8 Drivig force versus curret with ovig assebly blocked i cetral positio.

5 5 Radu Olaru et al EXPERIMENTAL INVESTIGATION Based o the results obtaied usig the atheatical odel ad the uerical siulatio, a EVIA prototype was desiged ad realized practically. The desig uses a ovig agetic assebly that ca glide i both directios o a shaft with two bearigs, Fig. 9. The four agets are the sae as those described i &3., each coil has N = 800 copper wires, 0.55 i diaeter. Each lid is i direct cotact with a fixed aget. The gap betwee the obile agetic assebly ad the fixed aget is z 0 = 6.5. The depedece of the displaceet versus frequecy for the ovig assebly is illustrated i Fig. 10. The axiu value of the displaceet was z r = 5.7, obtaied for a resoace frequecy of 4 Hz ad a excitatio siusoidal curret of 0.1 A i aplitude. The acceleratio whe the obile part is blocked i cetral positio, for a siusoidal voltage of 30 V i aplitude ad for a frequecy rage f = Hz, is illustrated i Fig. 11. The decrease i the electroagetic force is caused by the icrease i the coil ipedace ad the icrease of the iduced voltage (dyaic copoe. The experietal setup used to dyaically test EVIA whe outed i the ceter of a etallic diaphrag is show i Fig. 1. The vibratio paraeters (displaceet, velocity ad acceleratio) of the diaphrag activated with EVIA are plotted versus frequecy i Fig. 13, for a excitatio siusoidal curret of 0. A i aplitude. The plots correspod to a syste with two degrees of freedo. The two peak values obtaied for the frequecies f 1 = 14 Hz ad f = 4 Hz correspod to the atural frequecies of the EVIA diaphrag. Figure 14 shows the depedece o frequecy of the force geerated by EVIA. Usig the siusoidal source available i the laboratory ±30 V (peak), the vibrator geerates acceleratio forces of over 4 N i the frequecy rage Hz, ad at least 3 N i the frequecy rage [00, 1000] Hz. The approach used to adjust the resoace frequecy by chagig the stiffess due the chage i the gap z 0 betwee the ovig assebly ad the fixed agets was experietally tested ad the results are show i Fig. 15 ad i Table. Results show that i the cosidered rage for z 0, , the resoace frequecy decreases Fig. 10 Displaceet versus frequecy of the obile assebly for I peak = 0.1 A. Fig. 11 Blocked acceleratio force versus frequecy. Fig. 1 Experietal setup for dyaic testig EVIA. (a) approxiately liearly with the icrease of the gap z 0. (b) Fig. 9 Desig of EVIA. (c) Fig. 13 Frequecy respose curves for displaceet (a), velocity (b) ad acceleratio (c) of the etallic diaphrag.

6 58 Desig of a electroagetic vibratioal iertial actuator 6 Fig. 14 Force geerated by EVIA outed o diaphrag. Fig. 15 Chagig resoace frequecy of EVIA with the gap legth. Table Resoace frequecy for differet gap legths z 0 () Resoace frequecy (Hz) CONCLUSIONS I this paper a electroagetic vibratio iertial actuator (EVIA) based o electrically activated liearized agetic sprig was developed ad ivestigated theoretically ad experietally. The key eleet that esures the liear behavior of the vibratig syste is the agetic sprig which is usually oliear. Usig uerical siulatios three cofiguratios of the liear agetic sprig were foud: the first oe usig two siple agetic sprigs coupled differetially (differetial agetic sprig), the secod oe usig a ovig aget with a legth ad ass larger tha those of the statioary agets (i this experietal tests the fixed ad obile agets were idetical i pairs of two), ad the third oe applyig two ferroagetic rigs o the faces of the obile aget, a fact that also icreases the actuator drivig force. Thus, the liear characteristic F(z) of the obile assebly exteded to alost 90 % of the axiu distace z 0 (Fig. 7). The atheatical odelig ad aalysis i the frequecy doai of the vibratig syste was doe for two excitig odes: siusoidal curret ad siusoidal voltage. The atheatical odel with curret excitatio source shows that the theoretical ad practical advatage of this vibratig syste is that it has a liear behavior, regardless of the voltage iduced by the ovig aget i the coils, while a vibratio syste with a voltage excitatio source behaves liearly oly if the coils reactace is uch saller tha their resistace, Lc i the cosidered frequecy rage. The atheatical odel with voltage excitatio source showcases the total dapig coefficiet, ct c ce ad the expressio of the electrical dapig coefficiet, c e / R c. It is to be oted that the aalysis based o a atheatical odel described i this paper is also applicable to other electroagetic actuators with obile aget ad statioary coil, or vice versa, statioary aget ad ovig coil, such as voice actuators ad vibratio eergy harvesters based o agetic levitatio. The developed atheatical odel, the uerical siulatios ad the optiizatio of a referece odel led to the costructio of a laboratory prototype, ivestigated experietally. The theoretical aalysis showed the ajor effect of usig a agetic housig (shield) ad agetic rigs resultig i the icrease of the actuator drivig force by 44 %. The resoace frequecy of the EVIA prototype tested experietally as a vibrator was 4 Hz, ad the drivig force i cetral blocked positio was over 4 N for frequecies betwee 70 Hz ad 00 Hz ad for a peak siusoidal excitatio voltage of 30 V ( V peak ). Vibratio forces geerated by the EVIA actuator outed o a etallic diaphrag were over 4 N i the frequecy rage Hz. Obviously, usig a higher siusoidal voltage source leads to higher proportioal forces. The electroagetic actuator ca adjust the atural frequecy by chagig the gap betwee the statioary agets ad the obile assebly at equilibriu, resultig i a chage of the agetic sprig stiffess. The easured resoace frequecies were 4, 40.3, 38., 35.8, 34.3 ad 3.3 Hz, obtaied by extedig the gap z 0 betwee 6.5 to 11.5, i steps of oe illieter. The proposed odel for the electroagetic vibratio iertial actuator has a siple ad cheap costructio, is easily cotrollable ad ca fid ay applicatios i active vibratio cotrol. Received o May 8, 018 REFERENCES 1. K. B. Lee, J. H. Ki, H. J. Ki, Horizotal liear vibratig actuator to reduce sart phoe thickess, Joural of Vibroegieerig, 15, 4, pp (013).. R. Olaru, R. Gherca, Caelia Petrescu, Aalysis ad desig of a vibratio eergy harvester usig peraet agets, Rev. Rou. Sci. Tech. Électrotech. et Éerg., 59,, pp (014). 3. Jiao X., Zhag Y., Chai S., The Desig of a Electroagetic Dapig Vibratio Absorber, Applied Mechaics ad Materials, 68 70, pp (01). 4. H. Je Ki, H Ji Ki, S.H. Jeo, B.W. Ha, Desig ad Experiet of a Electroagetic Vibratio Exciter for the Rappig of a Electrostatic Precipitator, Joural of Magetics, 17, 1, pp (01). 5. C. Paulitsch, P. Gardoio, S. J. Elliott, Iteral Velocity Feedback For Stabilizatio of Iertial Actuators for Active Vibratio Cotrol, H. Ulbrich ad W. Guther (eds.), IUTAM Syposiu o Vibratio Cotrol of Noliear Mechaiss ad Structures, pp , G. Loussert G, A efficiet ad optial ovig aget actuator for active vibratio cotrol, 15 th Iteratioal Coferece o New Actuators (ACTUATORS 016), Bree, Geray. 7. I. Abdalla, T. Ibrahi, N. Nor, Desig ad odelig of a sigle-phase liear peraet aget otor for household refrigerator applicatios, Busiess Egieerig ad Idustrial Applicatios Colloquiu (BElAC), 013 IEEE, pp R. Olaru, A. Arcire, Caelia Petrescu, Marius Mugurel Mihai, Bogda Gîrta, A ovel vibratio actuator based o active agetic sprig, Sesors ad Actuators A Physical, 64, 1, pp (017). 9. X. Liu, C. Ha, Ye Wag, Desig of atural frequecy adjustable electroagetic actuator ad active vibratio cotrol test, Low Frequecy Noise, Vibratio ad Active Cotrol, 35, 3, pp (016). 10. A. Jafari, N. Tsagarakis, D. Caldwell, A ew actuator with adjustable stiffess based o a variable ratio lever echais, IEEE/ASME Tras. Mechatro, 19, 1, pp (014).