Magnetic levitation technique for active vibration control

Transcription

1 Magnetic levitation technique for active vibration control 41 X 3 Magnetic levitation technique for active vibration control Md. Emdadul Hoque and Takehi Miuno Saitama Univerity Jaan 1. Introduction Thi chater reent an alication of ero-ower controlled magnetic levitation for active vibration control. Vibration iolation are trongly required in the field of high-reolution meaurement and micromanufacturing, for intance, in the ubmicron emiconductor chi manufacturing, canning robe microcoy, holograhic interferometry, cofocal otical imaging, etc. to obtain recie and reeatable reult. The growing demand for tighter roduction tolerance and higher reolution lead to the tringent requirement in thee reearch and indutry environment. The microvibration reulted from the tableto and/or the ground vibration hould be carefully eliminated from uch ohiticated ytem. The vibration control reearch ha been advanced with aive and active technique. Conventional aive technique ue ring and damer a iolator. They are widely ued to uort the invetigated art to rotect it from the evere ground vibration or from direct diturbance on the table by uing oft and tiff uenion, reectively Hari & Pierol, ; Rivin, 3. Soft uenion can be ued becaue they rovide low reonance frequency of the iolation ytem and thu reduce the frequency band of vibration amlification. However, it lead to otential roblem with tatic tability due to direct diturbance on the table, which can be olved by uing tiff uenion. On the other hand, aive ytem offer good high frequency vibration iolation with low iolator daming at the cot of vibration amlification at the fundamental reonance frequency. It can be olved by uing high value of iolator daming. Therefore, the erformance of aive iolator are limited, becaue variou trade-off are neceary when excitation with a wide frequency range are involved. Active control technique can be introduced to reolve thee drawback. Active control ytem ha enhanced erformance becaue it can adat to changing environment Fuller et al., 1997; Preumont, ; Karno, Although conventional active control ytem achieve high erformance, it require large amount of energy ource to drive the actuator to roduce active daming force Benai et al., 4a & 4b; Yohioka et al., 1; Preumont et al., ; Daley et al., 6; Zhu et al., 6; Sato & Trumer,. Aart from thi, mot of the reearche ue high-erformance enor, uch a ervo-tye accelerometer for detecting vibration ignal, which are rather exenive. Thee are the difficultie to exand the alication field of active control technique.

2 4 Magnetic Bearing, Theory and Alication The develoment and maintenance cot of vibration iolation ytem hould be lowered in order to exand the alication field of active control. Conidering the oint of view, a vibration iolation ytem have been develoed uing an actively ero-ower controlled magnetic levitation ytem Hoque et al., 6; Miuno et al., 7a; Hoque et al., 1a. In the rooed ytem, eddy-current relative dilacement enor were ued for dilacement feedback. Moreover, the control current converge to ero for the ero-ower control ytem. Therefore, the develoed ytem become rather inexenive than the conventional active ytem. An active ero-ower controlled magnetic uenion i ued in thi chater to realie negative tiffne by uing a hybrid magnet conit of electromagnet and ermanent magnet. Moreover, it can be noted that realiing negative tiffne can alo be generalied by uing linear actuator voice coil motor intead of hybrid magnet Miuno et al., 7b. Thi control achieve the teady tate in which the attractive force roduced by the ermanent magnet balance the weight of the uended object, and the control current converge to ero. However, the conventional ero-ower controller generate contant negative tiffne, which deend on the caacity of the ermanent magnet. Thi i one of the bottleneck in the field of alication of ero-ower control where the adjutment of tiffne i neceary. Therefore, thi chater will invetigate on an imroved ero-ower controller that ha caability to adjut negative tiffne. Aart from thi, ero-ower control ha inherently nonlinear characteritic. However, comenation to ero-ower control can olve uch roblem Hoque et al., 1b. Since there i no teady energy conumtion for achieving table levitation, it ha been alied to ace vehicle Sabni et al., 1975, to the magnetically levitated carrier ytem in clean room Morihita et al., 1989 and to the vibration iolator Miuno et al., 7a. Six-axi vibration iolation ytem can be develoed a well uing thi technique Hoque et al., 1a. In thi chater, an active vibration iolation ytem i develoed uing ero-ower controlled magnetic levitation technology. The iolation ytem i fabricated by connecting a mechanical ring in erie with a uenion of negative tiffne ee Section 4 for detail. Middle table are introduced in between the bae and the iolation table. In thi context, the nomenclature on the vibration diturbance, comliance and tranmiibility are dicued for better undertanding. The underlying concet on vibration iolation uing magnetic levitation technique, realiation of ero-ower, tiffne adjutment, nonlinear comenation of the maglev ytem are reented in detail. Some exerimental reult are reented for tyical vibration iolation ytem to demontrate that the maglev technique can be imlemented to develo vibration iolation ytem.. Vibration Sureion Terminology.1 Vibration Diturbance The vibration diturbance ource are categoried into two grou. One i direct diturbance or tableto vibration and another i ground or floor vibration. Direct diturbance i defined by the vibration that alie to the tableto and generate deflection or deformation of the ytem. Ground vibration i defined by the detrimental vibration that tranmit from floor to the ytem through the uenion. It i worth noting that ero or low comliance for tableto vibration and low tranmiibility le than unity are ideal for deigning a vibration iolation ytem.

3 Magnetic levitation technique for active vibration control 43 Almot in every environment, from laboratory to indutry, vibrational diturbance ource are common. In modern reearch or alication arena, it i certainly neceary to conduct exeriment or make meaurement in a vibration-free environment. Think about a indutry or laboratory where a number of energy ource exit imultaneouly. Conider the ilicon wafer hotolithograhy ytem, a rincial equiment in the emiconductor manufacturing roce. It ha a tage which move in te and caue diturbance on the table. It uort electric motor, that generate eriodic diturbance. The floor alo hold ome rotating machine. Moreover, earthquake, movement of emloyee with trolley tranmit eimic diturbance to the tage. Aume a laboratory meaurement table in another cae. The table uort ome machine tool, and change in load on the table i a common henomena. In addition, air comreor, vacuum um, ocillocoe and dynamic ignal analyer with cooling fan ret on the floor. Some more otential energy ouce are elevator mechanim, air conditioning, rail and road tranort, heat um that contribute to the vibrational background noie and that are couled to the foundation and floor of the urrounding building. All the above ource of vibration affect the ytem either directly on the table or tranmit from the floor.. Comliance Comliance i defined a the ratio of the linear or angular dilacement to the magnitude of the alied tatic or contant force. Moreover, in cae of a varying dynamic force or vibration, it can be defined a the ratio of the excited vibrational amlitude in any form of angular or tranlational dilacement to the magnitude of the forcing vibration. It i the mot extenively ued tranfer function for the vibrational reone of an iolation table. Any deflection of the iolation table i demontrated by the change in relative oition of the comonent mounted on the table urface. Hence, if the iolation ytem ha virtually ero or lower comliance infinite tiffne value, by definition, it i a better-quality table becaue the deflection of the urface on which fabricated art are mounted i reduced. Comliance i meaured in unit of dilacement er unit force, i.e., meter/newton m/n and ued to meaure deflection at different frequencie. The deformation of a body or tructure in reone to external ayload or force i a common roblem in engineering field. Thee external diturbance force may be tatic or dynamic. The develoment of an iolation table i a good examle of thi roblem where uch tatic and dynamic force may exit. A tatic laod, uch a that caued by a large, concentrated ma loaded or unloaded on the table, can caue the table to deform. A dynamic force, uch a the eriodic diturbance of a rotating motor laced on to of the table, or vibration induced from the building into the iolation table through it mounting oint, can caue the table to ocillate and deform. Aume the imlet model of conventional ma-ring-damer ytem a hown in Fig. 1a, to undertand comliance with only one degree-of-freedom ytem. Conider that a ingle frequency inuoidal vibration alied to the ytem. From Newton law, the general equation of motion i given by m x cx kx F int, 1 where m : the ma of the iolated object, x : the dilacement of the ma, c : the daming, k : the tiffne, F : the maximum amlitude of the diturbance, ω : the rotational frequency of diturbance, and t : the time.

4 44 Magnetic Bearing, Theory and Alication F in t x X in t X in t n c k m km X in t F in t a b Fig. 1. Conventional ma-ring-damer vibration iolator under a direct diturbance b ground vibration. The general exreion for comliance of a ytem reented in Eq. 1 i given by Comliance x F 1. k m c The comliance in Eq. can be rereented a x 1/ k Comliance F 1 / n 4 / n, 3 where n : the natural frequency of the ytem and : the daming ratio..3 Tranmiibility Tranmiibility i defined a the ratio of the dynamic outut to the dynamic inut, or in other word, the ratio of the amlitude of the tranmitted vibration or tranmitted force to that of the forcing vibration or exciting force. Vibration iolation or elimination of a ytem i a two-art roblem. A dicued in Section.1, the tableto of an iolation ytem i deigned to have ero or minimal reone to a diturbing force or vibration. Thi i itelf not ufficient to enure a vibration free working urface. Tyically, the entire table ytem i ubjected continually to vibrational imule from the laboratory floor. Thee vibration may be caued by large machinery within the building a dicued in Section.1 or even by wind or traffic-excited building reonance or earthquake.

5 Magnetic levitation technique for active vibration control 45 The model hown in Fig. 1a i modified by alying ground vibration, a hown in Fig. 1b. The abolute tranmiibility, T of the ytem, in term of vibrational dilacement, i given by 1 4 / n. 4a 1 / n 4 / n X X Similarly, the tranmiibility can alo be defined in term of force. It can be defined a the ratio of the amlitude of force tranmitted F to the amlitude of exciting force F. Mathematically, the tranmiibility in term of force i given by F F n n n 1 4 / 1 / 4 /. 4b 3. Zero-Power Controlled Magnetic Levitation 3.1 Magnetic Suenion Sytem Since lat few decade, an active magnetic levitation ha been a viable choice for many indutrial machine and device a a non-contact, lubrication-free uort Schweiter et al., 1994; Kim & Lee, 6; Schweiter & Malen, 9. It ha become an eential machine element from high-eed rotating machine to the develoment of reciion vibration iolation ytem. Magnetic uenion can be achieved by uing electromagnet and/or ermanent magnet. Electromagnet or ermanent magnet in the magnetic uenion ytem caue flux to circulate in a magnetic circuit, and magnetic field can be generated by moving charge or current. The attractive force of an electromagnet, F can be exreed aroximately a Schweiter et al., 1994 I K F, 5 where K : attractive force coefficient for electromagnet, I : coil current, : mean ga between electromagnet and the uended object. Each variable i given by the um of a fixed comonent, which determine it oerating oint and a variable comonent, uch a I I i, 6 D x, 7 where I : bia current, i : coil current in the electromagnet, D : nominal ga, x : dilacement of the uended object from the equilibrium oition.

6 46 Magnetic Bearing, Theory and Alication 3. Magnetic Suenion Sytem with Hybrid Magnet In order to reduce ower conumtion and continuou ower uly, ermanent magnet are emloyed in the uenion ytem to avoid roviding bia current. The uenion ytem by uing hybrid magnet, which conit of electromagnet and ermanent magnet i hown in Fig.. The ermanent magnet i ued for the uroe of roviding bia flux Miuno & Takemori,. Thi control realie the teady tate in which the electromagnet coil current converge to ero and the attractive force roduced by the ermanent magnet balance the weight of the uended object. i Electromagnet Permanent magnet N S S N m N S x Fig.. Model of a ero-ower controlled magnetic levitation It i aumed that the ermanent magnet i modeled a a contant-current bia current and a contant-ga electromagnet in the magnetic circuit for imlification in the following analyi. Attractive force of the electromagnet, F can be written a F I i K, 8 D x where bia current, I i modified to equivalent current in the teady tate condition rovided by the ermanent magnet and nominal ga, D i modified to the nominal air ga in the teady tate condition including the height of the ermanent magnet. Equation 8 can be tranformed a Uing Taylor rincile, Eq. 9 can be exanded a f d I x i F K D D I 3 I x x x i i F K D D D D I I

7 Magnetic levitation technique for active vibration control 47 For ero-ower control ytem, control current i very mall, eecially, in the hae aroache to teady-tate condition and therefore, the higher-order term are not conidered. Equation 1 can then be written a 3 F Fe kii k x x 3x..., 11 where I F e K, 1 D I k i K, 13 D I k K, 14 3 D 3, 15 D D For ero-ower control ytem, the control current of the electromagnet i converged to ero to atify the following equilibrium condition F e mg, 17 and the equation of motion of the uenion ytem can be written a From Eq. 11, 17 and 18, m x F mg m x kii k x x 3x Thi i the fundamental equation for decribing the motion of the uended object. 3.3 Deign of Zero-Power Controller Negative tiffne i generated by actively controlled ero-ower magnetic uenion. The baic model, controller and the characteritic of the ero-ower control ytem i decribed below Model A baic ero-ower controller i deigned for imlicity baed on linearied equation of motion. It i aumed that the dilacement of the uended ma i very mall and the

8 48 Magnetic Bearing, Theory and Alication nonlinear term are neglected. Hence the linearied motion equation from Eq. 19 can be written a m x kii k x. The uended object with ma of m i aumed to move only in the vertical tranlational direction a hown by Fig.. The equation of motion i given by mx k x kii fd, 1 where x : dilacement of the uended object, k : ga-force coefficient of the hybrid magnet, k i : current-force coefficient of the hybrid magnet, i : control current, f d : diturbance acting on the uended object. The coefficient k and ki are oitive. When each Lalace-tranform variable i denoted by it caital, and the initial value are aumed to be ero for imlicity, the tranfer function rereentation of the dynamic decribed by Eq. 1 become 1 X b I dw, a where a k m, b k /, and d 1/. / i m m 3.3. Suenion with Negative Stiffne Zero-ower can be achieved either by feeding back the velocity of the uended object or by introducing a minor feedback of the integral of current in the PD roortionalderivative control ytem Miuno & Takemori,. Since PD control i a fundamental control law in magnetic uenion, ero-ower control i realied from PD control in thi work uing the econd aroach. In the current controlled magnetic uenion ytem, PD control can be rereented a where d : roortional feedback gain, I d v X, 3 v : derivative feedback gain. Figure 3 how the block diagram of a current-controlled ero-ower controller where a minor integral feedback of current i added to the roortional feedback of dilacement. 1 i k i w 1 m k x d v Fig. 3. Tranfer function rereentation of the ero-ower controller of the magnetic levitation ytem

9 Magnetic levitation technique for active vibration control 49 The control current of ero-ower controller i given by X I v d, 4 where : integral feedback in the minor current loo. From Eq. to 4, it can be written a, 3 v d v a a b b b d W X 5. 3 v d v v d v a a b b b d W I 6 To etimate the tiffne for direct diturbance, the direct diturbance, W on the iolation table i conidered to be tewie, that i, F W F : contant. 7 The teady dilacement of the uenion, from Eq. 5 and 7, i given by. lim lim t k F F a d X t x 8 The negative ign in the right-hand ide illutrate that the new equilibrium oition i in the direction ooite to the alied force. It mean that the ytem realie negative tiffne. Aume that tiffne of any uenion i denoted by k. The tiffne of the eroower controlled magnetic uenion i, therefore, negative and given by k k Realiation of Zero-Power From Eq. 6 and 7. lim lim I t i t 3 It indicate that control current, all the time, converge to ero in the ero-ower control ytem for any load.

10 5 Magnetic Bearing, Theory and Alication 3.4 Stiffne Adjutment The tiffne realied by ero-ower control i contant, a hown in Eq. 9. However, it i neceary to adjut the tiffne of the magnetic levitation ytem in many alication, uch a vibration iolation ytem. There are two aroache to adjut tiffne of the eroower control ytem. The firt one i by adding a minor dilacement feedback to the eroower control current, and the other one i by adding a roortional feedback in the minor current feedback loo Ihino et al., 9. In thi reearch, tiffne adjutment caability of ero-ower control i realied by the firt aroach. Figure 4 how the block diagram of the modified ero-ower controller that i caable to adjut tiffne. The control current of the modified ero-ower controller i given by where I d v X, 31 : roortional dilacement feedback gain acro the ero-ower controller. 1 i k i w 1 m k x The tranfer-function rereentation of the dynamic hown in Fig. 4 i given by X d. 3 W 3 b v b d b a a b From Eq. 7 and 3, the teady dilacement become d F lim x t lim X F 33 a b k k t Therefore, the tiffne of the modified ytem become k k ki. 34 It indicate that the tiffne can be increaed or decreaed by changing the feedback gain. d v Fig. 4. Block diagram of the modified ero-ower controller that can adjut tiffne i

11 Magnetic levitation technique for active vibration control Nonlinear Comenation of Zero-Power Controller x Zero-ower controller + _ i k 1 d x. k i D Nonlinear comenator Fig. 5. Block diagram of the nonlinear comenator of the ero-ower controlled magnetic levitation It i hown that the ero-ower control can generate negative tiffne. The control current of the ero-ower controlled magnetic uenion ytem i converged to ero for any added ma. To counterbalance the added force due to the ma, the table oition of the uended object i changed. Due to the air ga change between ermanent magnet and the object, the magnetic force i alo changed, and hence, the negative tiffne generated by thi ytem varie a well according to the ga ee Eq. 14. To comenate the nonlinearity of the baic ero-ower control ytem, the firt nonlinear term of Eq. 19 i conidered and added to the baic ytem. From Eq. 19, the control current can be exreed a k 1 i i d x, 35 ZP where d : the nonlinear control gain and, i. k D i : the current in the ero-ower controller, k, k i and D are contant for the ytem. The quare of the dilacement x i fed back to the normal ero-ower controller. The block diagram of the nonlinear controller arrangement i hown in Fig. 5. The air ga between the ermanent magnet and the uended object can be changed in order to chooe a uitable oerating oint. It i worth noting that the nonlinear comenator and the tiffne adjutment controller can be ued imultaneouly without intability. Moreover, erformance of the nonlinear comenation could be imroved furthermore if the econd and third nonlinear term and o on are conidered together. 4. Vibration Sureion Uing Zero-Power Controlled Magnetic Levitation 4.1 Theory of Vibration Control Table k c 3 k 3 k 1 c 1 Bae Fig. 6. A model of vibration iolator that can ure both tableto and ground vibration

12 5 Magnetic Bearing, Theory and Alication The vibration iolation ytem i develoed uing magnetic levitation technique in uch a way that it can behave a a uenion of virtually ero comliance or infinite tiffne for direct diturbing force and a uenion with low tiffne for floor vibration. Infinite tiffne can be realied by connecting a mechanical ring in erie with a magnetic ring that ha negative tiffne Miuno, 1; Miuno et al., 7a & Hoque et al., 6. When two ring with ring contant of k 1 and k are connected in erie, the total tiffne k c i given by k k 1 k c. 36 k1 k The above baic ytem ha been modified by introducing a econdary uenion to avoid ome limitation for ytem deign and uorting heavy ayload Miuno, et al., 7a & Hoque, et al., 1a. The concet i demontrated in Fig. 6. A aive uenion k 3, c 3 i added in arallel with the erial connection of oitive and negative ring. The total tiffne k ~ c i given by ~ k1k k c k3. 37 k1 k However, if one of the ring ha negative tiffne that atifie k1 k, 38 the reultant tiffne become infinite for both the cae in Eq. 36 and 37 for any finite value of k 3, that i ~ k. 39 c Equation 39 how that the ytem may have infinite tiffne againt direct diturbance to the ytem. Therefore, the ytem in Fig. 6 how virtually ero comliance when Eq. 38 i atified. On the other hand, if low tiffne of mechanical ring for ytem k 1, k 3 are ued, it can maintain good ground vibration iolation erformance a well. 4. Tyical Alication of Vibration Sureion In thi ection, tyical vibration iolation ytem uing ero-ower controlled magnetic levitation are reented, which were develoed baed on the rincile dicued in Eq. 37. The iolation ytem conit mainly of two uenion with three latform- bae, middle table and iolation table. The lower uenion between bae and middle table i of oitive tiffne and the uer uenion between middle table and bae i of negative tiffne realied by ero-ower control. A aive uenion directly between bae and iolation table act a weight uort mechanim. A tyical ingle-axi and a tyical ix-axi vibration iolation aaratue are demontrated in Fig. 7. The ingle-axi aaratu Fig. 7a conited of a circular bae, a circular middle table and a circular iolation table. The height, diameter and weight of the ytem were 3mm, mm and kg, reectively. The oitive tiffne in the lower art wa realied by three mechanical ring and an electromagnet. To reduce coil current in the electromagnet, four ermanent magnet 15mm mm were ued. The ermanent magnet are made of Neodymium-Iron-Boron NdFeB. The tiffne of each coil ring wa 3.9

13 Magnetic levitation technique for active vibration control 53 N/mm. The electromagnet coil had 18-turn and 1.3Ω reitance. The wire diameter of the coil wa.6 mm. The relative dilacement of the bae to middle table wa meaured by an eddy-current dilacement enor, rovided by Swi-made Baumer electric. The negative tiffne uenion in the uer art wa achieved by a hybrid magnet conited of an electromagnet that wa fixed to the middle table, and ix ermanent magnet attached to the electromagnet target on the iolation table. Another dilacement enor wa ued to meaure the relative dilacement between middle table to iolation table. The iolation table wa alo uorted by three coil ring a weight uort mechanim, and the tiffne of the each ring wa.35 N/mm. Vibration iolation table Leaf ring Hybrid magnet for negative tiffne Coil ring for weight uort mechanim Middle table Hybrid magnet for oitive tiffne Coil ring for oitive tiffne Bae a Iolation table Hybrid magnet Middle table Coil ring Bae b Fig. 7. Tyical alication of ero-ower controlled magnetic levitation for active vibration control a ingle-degree-of-freedom ytem b ix-degree-of-freedom ytem

14 54 Magnetic Bearing, Theory and Alication The ix-axi vibration iolation ytem with magnetic levitation technology i hown in Fig. 7b Hoque, et al., 1a. It conited of a rectangular iolation table, a middle table and bae. A oitive tiffne uenion realied by electromagnet and normal ring wa ued between the bae and the middle table. On the other hand, a negative tiffne uenion generated by hybrid magnet wa ued between the middle table and the iolation table. The height, length, width and ma of the aaratu were 3 mm, 74 mm, 59 mm and 4 kg, reectively. The iolation and middle table weighed 88 kg and 158 kg, reectively. The iolation table had ix-degree-of-freedom motion in the x, y,, roll, itch and yaw direction. The bae wa equied with four air of coil ring and electromagnet to uort the middle table in the vertical direction and ix air of coil ring and electromagnet two air in the x-direction and four air in the y-direction in the horiontal direction. The middle table wa equied with four et of hybrid magnet to levitate and control the motion of the iolation table in the vertical direction and ix et of hybrid magnet two et in the x-direction and four et in the y-direction to control the motion of the table in the horiontal direction. The iolation table wa alo uorted by four coil ring in the vertical direction and ix coil ring two in the x-direction and four in the y-direction in the horiontal direction a weight uort mechanim. Each et of hybrid magnet for eroower uenion conited of five quare-haed ermanent magnet mm mm mm and five 585-turn electromagnet. The ring contant of each normal ring wa 1.1 N/mm and that of weight uort ring wa 5.5 N/mm. There wa flexibility to change the oition of the weight uort ring both in the vertical and horiontal direction to make it comatible for deigning table magnetic uenion ytem uing ero-ower control. The relative dilacement of the iolation table to the middle table and thoe of the middle table to the bae were detected by eight eddy-current dilacement enor attached to the corner of the iolation table and the bae. A DSP-baed digital controller DS113 wa ued for the imlementation of the deigned control algorithm by imulink in Matlab. The amling rate wa 1 kh. 4.3 Exerimental Demontration Several exeriment have been conducted to verify the aforeaid theoretical analyi. The nonlinear comenation of ero-ower controlled magnetic levitation, tiffne adjutment of the levitation ytem are confirmed initially. Then the characteritic of the develoed iolation ytem are meaured in term of comliance and tranmiibility Nonlinear Comenation of Magnetic Levitation Sytem Firt of all, ero-ower control wa realied between the iolation table and the middle table for table levitation. Static characteritic of the ero-ower controlled magnetic levitation wa meaured a hown in Fig. 8 when the ayload were increaed to roduce tatic direct diturbance on the table in the vertical direction. In thi cae, the middle table wa fixed and the table wa levitated by ero-ower control. The reult reent the load-tiffne characteritic of the ero-ower control ytem. The figure without nonlinear comenation indicate that there wa a wide variation of tiffne when the downward load force changed. For the uniform load increment, the change of ga wa not equal due to the nonlinear magnetic force. Therefore, the negative tiffne generated from ero-ower control wa nonlinear which may everely affect the vibration iolation ytem.

15 Magnetic levitation technique for active vibration control 55 To overcome the above roblem, the nonlinear comenator wa introduced in arallel with the ero-ower control ytem. The nonlinear control gain d wa choen by trial and error method. The ga D between the table and the electromagnet wa 5.1 mm after table levitation by ero-ower control. The value of k and k i were determined from the ytem characteritic. The load-tiffne characteritic uing nonlinear comenation i alo hown in the figure. It i obviou from the figure that the linearity error wa reduced when control gain d wa increaed. For d 55, the linearity error wa very low and the tiffne generated from the ytem wa aroximately contant. Thi reult how the otential to imrove the tatic reone erformance of the iolation table to direct diturbance. Fig. 8. Nonlinear comenation of the conventional ero-ower controlled magnetic levitation ytem Fig. 9. Load-dilacement characteritic of the modified ero-ower controlled magnetic levitation ytem 4.3. Stiffne Adjutment of Zero-Power Controlled Magnetic Levitation The exeriment have been carried out to meaure the erformance of the modified eroower controller. Figure 9 how the load-dilacement characteritic of the ytem with the imroved ero-ower controller Fig. 4. When the roortional feedback gain,, it can be conidered a a conventional ero-ower controller Fig. 3. The reult how that when the ayload were ut on the uended object, the table moved in the direction ooite to the alied load, and the ga wa widened. It indicate that the ero-ower control realied negative dilacement, and hence it tiffne i negative, a decribed by Eq. 8 and 9. The conventional ero-ower controller realied fixed negative tiffne of magnitude -9. N/mm. When the roortional feedback gain, wa changed, the tiffne alo gradually increaed. When 4 A/m, negative tiffne wa increaed to -1.5 N/mm. It confirm that roortional feedback gain, the ero-ower controller, a exlained in Eq. 34. can change the tiffne of Exerimental Reult with Vibration Iolation Sytem Further exeriment were conducted with the linearied ero-ower controller with the vibration iolation ytem, a hown in Fig. 1. In thi cae, the oitive and negative tiffne ring were, then, adjuted to atify Eq. 38. The tiffne could either be adjuted in the oitive or negative tiffne art. In the former, PD control could be ued in the electromagnet that were emloyed in arallel with the coil ring. The latter technique wa reented in Section For better erformance, the latter wa adoted in thi work.

16 56 Magnetic Bearing, Theory and Alication Stiffne dominated region Fig. 1. Static characteritic of the iolation Fig. 11. Dynamic characteritic of the iolation table with and without nonlinear control table in the vertical direction Figure 1 demontrate the erformance imrovement of the controller for tatic reone to direct diturbance. The dilacement of the iolation table and middle table were lotted againt diturbing force roduced by ayload in the vertical direction. It i clear that erocomliance to direct diturbance wa realied u to 1 N ayload with nonlinear controller d =55. The tiffne of the iolation ytem wa increaed to 96 N/mm which wa aroximately.8 time more than that of without nonlinear control. The figure illutrate ignificant imrovement in rejecting on-board-generated diturbance. The dynamic erformance of the iolation table wa meaured in the vertical direction a hown in Fig. 11. In thi cae, the iolation table wa excited to roduce inuoidal diturbance force by two voice coil motor which were attached to the bae and can generate force in the Z-direction. The dilacement of the table wa meaured by ga enor and the data wa catured by a dynamic ignal analyer. It i found from the figure that high tiffne, that mean virtually ero-comliance, wa realied at low frequency region -66 db[mm/n] at.15 H. It alo demontrate that direct diturbance rejection erformance wa not worened even nonlinear ero-ower control wa introduced. Finally a comarative tudy of the diturbance ureion erformance wa conducted with ero-comliance control and conventional aive uenion technique a hown in the figure. The exeriment wa carried out with ame lower uenion for ground vibration iolation. Firt, the iolation table wa uended by oitive uenion conventional ring-damer and frequency reone to direct diturbance wa meaured. The tiffne dominated region i marked in the figure, and it i een from the figure that the dilacement of the iolation table wa almot ame below 1 H aroximately -46 db. However, when the iolation table wa uended by ero-comliance control atifying Eq. 38 and 39, dilacement of the table wa abrutly reduced at the low frequency region below 1 H -66 db at.15 H. It i confirmed from the figure that the develoed ero-comliance ytem had better direct diturbance rejection erformance over the conventional aive uenion even both the ytem ued imilar vibration iolation erformance.

17 Magnetic levitation technique for active vibration control 57 Fig. 1. Dynamic characteritic of the iolation table in the vertical direction. The characteritic of the iolation table were further invetigated by meauring the reone of the table to direct diturbance in the horiontal direction a hown in Fig. 1. In thi cae, four voice coil motor were ued to excite the iolation table along the horiontal direction. The reult how the dynamic reone of the iolation table when the table wa excited along yaw mode. The reone of the table to direct dynamic diturbance wa catured by dynamic ignal analyer. The reult jutify that the dilacement of the table to direct diturbance in the horiontal rotational motion were alo low at the low frequency region. The reult confirmed that the iolation table wa realied high tiffne againt diturbing force in the motion aociated with horiontal direction. Overhoot After te load Original oition Tranient eriod Original oition teady-tate Fig. 13. Ste reone of the iolation table with magnetic levitation technology The te reone of the iolation table i hown in Fig. 13. In thi exeriment, a tewie diturbance wa generated by uddenly removing a certain amount of load from the table and the reone wa meaured. The reult howed that the table moved uward in the direction of load removal and returned to the original oition teady-tate after certain eriod. However, there wa a revere action in cae of te wie diturbance. Therefore, a

18 58 Magnetic Bearing, Theory and Alication eak wa aeared due to the reone of the te load. Thi unleaant reone might hamer the objective function of many advanced ytem. It can be noted that a feedforward controller can be added in combination with ero-ower control to overcome thi roblem. Fig. 14. Tranmiibility characteritic of the iolation table. Figure 14 how the abolute tranmiibility of the iolation table from the bae of the develoed ytem. In thi cae, the bae of the ytem wa inuoidally excited in the vertical direction by a high-owered neumatic actuator attached to the bae, and the dilacement tranfer function tranmiibility of the iolation table wa meaured from the bae. The bae dilacement in the vertical direction wa conidered a inut, and the outut ignal wa the dilacement of the iolation table. The daming coefficient c between the bae and the middle table layed imortant role to ure the reonance eak. The figure how that the reonant eak wa almot ureed when c wa choen a.9. It i clear from the figure that the develoed ytem can effectively iolate the floor vibration that tranmitted through the uenion, uch a active-aive oitive uenion and active ero-ower controlled magnetic levitation. 5. Concluion A ero-ower controlled magnetic levitation ytem ha been reented in thi chater. The unique characteritic of the ero-ower control ytem i that it can generate negative tiffne with ero control current in the teady-tate which i realied in thi chater. The detail characteritic of the levitation ytem are invetigated. Moreover, two major contribution, the tiffne adjutment and nonlinear comenation of the uenion ytem have been introduced elaborately. Often, there i a challenge for the vibration iolator deigner to tackle both direct diturbance and ground vibration imultaneouly with minimum ytem develoment and maintenance cot. Taking account of the oint of view, tyical alication of active vibration iolation uing ero-ower controlled magnetic levitation ha been reented. The vibration iolation ytem i caable to ure the effect of tableto vibration a well a to iolate ground vibration. Some exerimental demontration are reented that verifie the feaibility of it alication in many indutrie and ace related intrument. Moreover, it can be noted that a feedforward controller in combination with the ero-ower controller can be ued to imrove the erformance of the iolator to ure direct diturbance.

19 Magnetic levitation technique for active vibration control Acknowledgment The author gratefully acknowledge the financial uort made available from the Jaan Society for the Promotion of Science a a Grant-in-Aid for cientific reearch Grant no..838 for the foreign reearcher and the Minitry of Education, Culture, Sort, Science and Technology of Jaan, a a Grant-in-Aid for Scientific Reearch B. 7. Reference Benai, L. ; Elliot, S. J. & Gardonio, P. 4a. Active vibration iolation uing an inertial actuator with local force feedback control, Journal of Sound and Vibration, Vol. 76, No. 3, Benai, L. & Elliot, S. J. 4b. Active vibration iolation uing an inertial actuator with local dilacement feedback control, Journal of Sound and Vibration, Vol. 78, No. 4-5, Daley, S. ; Hatonen, J. & Owen, D. H. 6. Active vibration iolation in a mart ring mount uing a reetitive control aroach, Control Engineering Practice, Vol. 14, Fuller, C. R. ; Elliott, S. J. & Nelon, P. A Active Control of Vibration, Academic Pre, ISBN , New York, USA Harri, C. M. & Pierol, A. G.. Shock and Vibration Handbook, McGraw Hill, Fifth Ed., ISBN , New York, USA Hoque, M. E. ; Takaaki, M. ; Ihino, Y. & Miuno, T. 6. Develoment of a three-axi active vibration iolator uing ero-ower control, IEEE/ASME Tranaction on Mechatronic, Vol. 11, No. 4, Hoque, M. E. ; Miuno, T. ; Ihino, Y. & Takaaki, M. 1a, A ix-axi hybrid vibration iolation ytem uing active ero-ower control uorted by aive uort mechanim, Journal of Sound and Vibration, Vol. 39, No. 17, Hoque, M. E. ; Miuno, T. ; Kihita, D. ; Takaaki, M. & Ihino, Y. 1b. Develoment of an Active Vibration Iolation Sytem Uing Linearied Zero-Power Control with Weight Suort Sring, ASME Journal of Vibration and Acoutic, Vol. 13, No. 4, /9 Ihino, Y. ; Miuno, T. & Takaaki, M. 9. Stiffne Control of Magnetic Suenion by Local Feedback, Proceeding of the Euroean Control Conference 9, , Budaet, Hungary, 3-6 Augut, 9 Karno, D Active and emi-active vibration iolation, ASME Journal of Mechanical Deign, Vol. 117, Kim, H. Y. & Lee, C. W. 6. Deign and control of Active Magnetic Bearing Sytem With Lorent Force-Tye Axial Actuator, Mechatronic, vol. 16,. 13 Miuno, T. 1. Prooal of a Vibration Iolation Sytem Uing Zero-Power Magnetic Suenion, Proceeding of the Aia Pacific Vibration Conference 1, , Hanghau, China Miuno, T. & Takemori, Y.. A tranfer-function aroach to the analyi and deign of ero-ower controller for magnetic uenion ytem, Electrical Engineering in Jaan, Vol. 141, No., Miuno, T. ; Takaaki, M. ; Kihita, D. & Hirakawa, K. 7a. Vibration iolation ytem combining ero-ower magnetic uenion with ring, Control Engineering Practice, Vol. 15, No., Miuno, T. ; Furuhima, T. ; Ihino, Y. & Takaaki, M. 7b. General Form of Controller Realiing Negative Stiffne, Proceeding of the SICE Annual Conference 7, , Kagawa Univerity, Jaan, 17- Setember, 7

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