Effect of Optimal Placement of Permanent Magnets on the Electromagnetic Force in the Horizontal Direction

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1 actuators Article Effect Optimal Placement Permanent Magnets on Electromagnetic Force in Horiontal Direction Yasuaki Ito 1, Yoshiho Oda 1, Takayoshi Narita, * ID and Hideaki Kato 1 Course Mechanical Engineering, Tokai University, Kitakaname 4-4-1, Hiratsuka-shi, Kanagawa 59-19, Japan; 8bemm8@mail.u-tokai.ac.jp (Y.I.); 7bemm5@mail.u-tokai.ac.jp (Y.O.) Department Prime Mover Engineering, Tokai University, Kitakaname 4-4-1, Hiratsuka-shi, Kanagawa 59-19, Japan; hkato@tokai-u.jp * Correspondence: narita@tsc.u-tokai.ac.jp; Tel.: Received: 15 June 18; Accepted: 4 August 18; Published: 9 August 18 Abstract: The surface quality plates is deteriorated as y contact rollers while being conveyed during manufacturing processes. To solve this problem, we previously proposed a hybrid electromagnetic levitation system comprising electro, permanent, and a horiontal positioning control system for plates. Moreover, to increase stability, we proposed integrating se levitation systems. In this study, we aim to determine placement permanent in levitation system to suppress deflection a levitated plate for cases where magnetic field in horiontal direction changes. Using a genetic algorithm, gap, number, and placement permanent in system are obtained. Keywords: magnetic levitation; genetic algorithm; permanent magnet 1. Introduction Permanent can generate constant attractive force and are actively used in various industrial fields. Several studies on electromagnetic levitation technology for ferromagnetic objects have been performed focusing on this feature [1]. In current plate production line, transportation is achieved by contact conveyance by rollers. However, surface plate is damaged by this method, and surface quality plate deteriorates. With non-contact magnetic levitation transport using electromagnet proposed by us, se problems can be prevented. These studies consider using attractive force generated by permanent as a constant suspension force for levitation [,3]. Although this technology is epected to be applied in production high-surface-quality plates, thickness plate must be decreased to reduce its weight. A plate with reduced fleural rigidity owing to its thinness is difficult to levitate in a conventional magnetic levitation system because eciting elastic vibration. To solve this problem, we proposed a hybrid electromagnetic levitation system for thin plates where permanent are installed around electro for levitation [4]. This system can suppress elastic vibration levitated plates by generating an attractive force on entire surface plate using electro and permanent [5]. However, number arrangement patterns permanent is very enormous, and it is difficult to eperimentally obtain a more effective arrangement. Therefore, we focused on genetic algorithm (GA) which is one optimiation algorithms effective for nonlinear objective function [6], which is used to obtain optimum shape electrical motor in several studies [7,8]. Furrmore, we confirmed GA we used to obtain placement permanent, considering interactions with each or, improved levitation stability system [9,1]. In addition, we proposed adding anor Actuators 18, 7, 54; doi:1.339/act7354

2 system generates tension on edge plate due to electromagnetic field generated by horiontal electro. This tension furr suppresses vibration plate and improves levitation. In this study, we propose using a hybrid electromagnetic levitation system by applying horiontal positioning control and determine placement permanent Actuators using a 18, GA for 7, 54a case where a horiontal electromagnetic field is acting on plate. Furrmore, 1 we performed a levitation eperiment and considered levitation stability placement. electromagnet in horiontal direction and positioning controls to electromagnetic levitation. Outline Electromagnetic Levitation System Integrating Permanent Magnets and system using vertical electro for levitation [7]. This system generates tension on edge Horiontal Positioning Control System plate due to electromagnetic field generated by horiontal electro. This tension furr The suppresses hybrid electromagnetic vibration levitation plate system and integrating improvespositioning levitation. Incontrol this study, in we horiontal propose using direction a hybrid is shown electromagnetic in 1. The levitation object system electromagnetic by applying levitation horiontal is a positioning rectangular, control inc-coated and determine plate (SS4) with placement length 8 mm, permanent width 6 mm, using and thickness a GA for a case.4 where mm. To aaccomplish horiontal electromagnetic noncontact support field isthis acting plate, on as if it were plate. hoisted Furrmore, by strings, we we performed use five apairs levitation electro eperiment and (No. considered 1 5 in levitation 1). The displacement stability placement. plate is measured using five eddy-current gap sensors. Here, electric circuits paired electro are connected in series, whereas an eddycurrent Outline gap sensor Electromagnetic is positioned between Levitation two System Integrating each Permanent pair. The detected Magnets displacement and is. Horiontal Positioning Control System converted to velocity using digital differentiation. A regulated voltage from digital-to-analog converter The hybrid is supplied electromagnetic to a current-supply levitationamplifier system integrating to control positioning attractive control force in five horiontal pairs direction electro is shown to inensure 1. The object plate electromagnetic is levitated levitation by 5 mm is below a rectangular, surface inc-coated electro. plate (SS4) In with this length model, independent 8 mm, width control 6 is used, mm, and wherein thickness information.4 mm. on detected To accomplish values noncontact displacement, support velocity, thisand plate, coil ascurrent if it were hoisted electromagnet by strings, under we usestudy five pairs are fed back electro only to (No. same 1 5 electromagnet. in 1). The displacement plate is measured using five eddy-current gap sensors. The horiontal Here, displacement electric circuits plate pairedis electro measured using arefour connected laser beam in series, displacement whereas an sensors. eddy-current The four gapvelocities sensor is positioned plate between are detected two by differentiating each pair. signals The from detected displacement is sensors converted using toa velocity computer. using The digital current differentiation. flowing Aelectromagnet regulated voltage is obtained from by digital-to-analog measuring voltage converter is supplied resistor toconnected a current-supply in series amplifier to electric to control circuit. attractive The permanent force five pairs are installed electro around electromagnet to ensure unit for levitation plate is levitated as shown by in 5 mm below. The sie surface permanent electro. for In levitation this model, assistance independent is 3 mm control 3 mm is used, 15 mm wherein and information material is on ferrite. detected The values surface magnetic displacement, flu density velocity, is.1 and T. coil Permanent current electromagnet are placed under such study are deflection fed back only to floating same electromagnet. sheet is suppressed while using above system. AMP5 AMP4 AMP3 AMP AMP1 Levitation controller D/A converter DSP (ADSP674-)15MH A/D converter i 1 i i 3 i 4 i Permanent magnet (3 mm 3 mm 15 mm) y Laser type sensor Eddy current type sensor Electromagnet for levitation control Steel plate (8 mm 6 mm.4 mm) Electromagnet for horiontal positioning control i 1 i i 3 i 4 A/D converter DSP (ADSP34-A)5MH D/A converter Horiontal positioning controller Hybrid electromagnetic levitation system integrating positioning control in horiontal direction. AMP4 AMP3 AMP AMP1 The horiontal displacement plate is measured using four laser beam displacement sensors. The four velocities plate are detected by differentiating signals from displacement sensors using a computer. The current flowing electromagnet is obtained by measuring voltage resistor connected in series to electric circuit. The permanent are installed around

3 Actuators 18, 7, electromagnet unit for levitation as shown in. The sie permanent magnet for levitation assistance is 3 mm 3 mm 15 mm and material is ferrite. The surface magnetic flu density is.1 T. Permanent are placed such deflection floating sheet is suppressed while using above system. Actuators 18, 7, FOR PEER REVIEW 3 1. Photograph placement permanent around levitation electro.. Photograph placement permanent around levitation electro. 3. Control Model 3. Control Model 3.1. Electromagnetic Levitation Control System 3.1. Electromagnetic Levitation Control System mamatical control model is formulated to construct hybrid electromagnetic levitation A mamatical control model is formulated to construct hybrid electromagnetic levitation control system. In this study, each electromagnet unit is controlled by considering displacement control system. In this study, each electromagnet unit is controlled by considering displacement from gap sensor installed within and velocity and current electromagnet unit. Therefore, from gap sensor installed within and velocity and current electromagnet unit. Therefore, control model for levitation is established in each unit. shows model levitation control model for levitation is established in each unit. 3 shows model levitation control for one electromagnet unit using lumped parameter system. From previous study, it is control for one electromagnet unit using lumped parameter system. From previous study, it is confirmed by placing permanent, deflection plate is suppressed, and confirmed by placing permanent, deflection plate is suppressed, and levitation stability improves [9 11]. From this result, plate was assumed to be rigid body. levitation stability improves [9 11]. From this result, plate was assumed to be a rigid body. The plate is virtually divided into five hypotical masses. In an equilibrium levitation state, a The plate is virtually divided into five hypotical masses. In an equilibrium levitation state, constant attractive force equal to weight virtually divided plate needs to be generated a constant attractive force equal to weight virtually divided plate needs to be generated by electromagnet, and change attractive force from equilibrium state causes by electromagnet, and change attractive force from equilibrium state causes motion virtually divided plate. The vertical motion plate is epressed as follows. motion virtually divided plate. The vertical motion plate is epressed as follows. m = f.. (1) m = f (1) 4F f i + = 4F + 4F i Z I () I We defined on assumption inductance one coil is epressed by sum We defined on assumption inductance one coil is epressed by sum effective inductance inversely proportional to gap and leakage inductance [1]. Since two effective inductance inversely proportional to gap and leakage inductance [1]. Since two electro are not magnetically connected, mutual inductance is not considered. electro are not magnetically connected, mutual inductance is not considered. d d dt i = L e f f i dt L I. eff I = R i L Z + 1 v (3) L L (3) L Z L L L = L e f f + L Z lea (4) L eff L = + Llea (4) Z Using state vector, Equations (1) (4) are written as following state equations. = A + B v [ ] = i Τ (5)

4 Actuators 18, 7, FOR PEER REVIEW 4 1 Actuators 18, 7, Using state vector, Equations (1) (4) 1 are written as following state equations. F. F A = = A + B v, m Z [ ]. T m I = i Leff I R L1 Z L F A =, F m Z m I L Τ, e f f L I 1 R Z B = L [ L] T 1 B = L Furrmore, v is calculated by state feedback control Equation (6). Furrmore, v is calculated by state feedback control Equation (6). v = k [ ] v = k k = p d p v p i k = [ p p p ] d v i The value each parameter and feedback gain levitation control system are shown in The value each parameter and feedback gain levitation control system are shown in Tables 1 and. Tables and. Table The value each parameter. Parameters Values Parameters Values m.864 kg m.864 kg E 6 GPa E 6 GPa ν ν.3.3 Z Z m 3 m R n R n Ω Ω L L eff eff.55 1 Hm 4 Hm L lea.9 H L lea.9 H T s.1 s.1 s T s Table. The feedback gain levitation control system. Table. The feedback gain levitation control system. Parameters Values Values (No. (No in in 1) 1) Values (No. (No. 5 in 5 in 1) pd p d pv p v pi p i (5) (6) (6) Eddy current type gap sensor Permanet magnet I +i Electromagnet f PM f m f f PM Steel plate (rigid body) 3. Modeling levitation control for one electromagnet unit by lumped parameter system. 3. Modeling levitation control for one electromagnet unit by lumped parameter system.

5 The horiontal motion plate was modeled to have a single degree freedom, as shown in 4. Therefore, same attractive forces were generated from two electro placed at one side plate. The equation for small horiontal motion around equilibrium state plate subjected to same static magnetic forces from electro at two Actuators edges is 18, epressed 7, 54 as follows Horiontal Positioning Control System 4 F The horiontal motion plate was modeled 4 F to f = + have i a single degree freedom, as shown (8) X in 4. Therefore, same attractive forces were I generated from two electro placed at one side plate. The equation for small horiontal motion around equilibrium state d L eff I plate subjected to same static magnetic R forces 1 i = from i + velectro at two edges is dt L X L L (9) epressed as follows. m.ẋ = f 1 f = f (7) f = 4F L eff L = + + 4F L lea (1) X i (8) X I Using state vector, Equations (7) (1) d dt i = L can be written e f f I. R as i + 1 following state equations. v (9) L = A + BL v L X L = L e f f = + Li X lea (1) Using state vector, Equations (7) (1) can be written as following state equations. 1. = A + B v 4F[ ]. T 4F A = = i (11) mx mi L1 4F eff I A = R 4F m X (11) m I L L e f f X L L I X [ 1 1 B = B = L L Here, Here, control control signal signal is is epressed epressed as as follows. follows. m f f = (7) = 1 [ ] T f R L T ] T v = k (1) v = k (1) I + i m f f 1 Steel plate (rigid body) I - i Electromagnet for horiontal positioning control Laser type sensor Theoretical Theoretical model model horiontal horiontal positioning positioning control control plate. plate. 4. Determination Optimal Placement In proposed system, by applying attractive force permanent magnet to area where attractive force electromagnet is lacking, deflection levitating thin plate is suppressed and levitation stability system is improved, which is aim this

6 4. Determination Optimal Placement In proposed system, by applying attractive force permanent magnet to area where attractive force electromagnet is lacking, deflection levitating thin Actuators 18, 7, plate is suppressed and levitation stability system is improved, which is aim this study. However, eperimentally searching for placement permanent is practically study. However, impossible eperimentally because searching large number for combinations placement search permanent patterns. Searching is is practically performed impossible in range because where largeplate number is divided combinations into ¼, and position search patterns. PM Searching can be arranged is performed at intervals range 1 mm, where considering plate sie is divided PM and into 1 4, and range position EM. Furrmore, PM can maimum be arranged number at intervals PMs is 1set mm, to 15, considering and search sie is performed PM and within range range EM. Furrmore, 4 mm to 8 mm maimum for Gap number [9,1]. In addition, PMs is set to 15, and placement search is sought is performed by considering within interaction range 4PM mmby to past 8 mm research for Gap [11]. [9,1]. The Inmagnetic addition, field applied placement by ishoriontal sought by electromagnet considering interaction was analyed PMby JMAG, past research as shown [11]. in The magnetic 5. field applied by horiontal electromagnet was analyed by JMAG, as shown in 5. Vertical force (N) (mm) 5 1 y (mm) 15 Horiontal tension (N) y (mm) (a) Steady current I =.3 A Vertical force (N) (mm) 5 1 y (mm) 15 Horiontal tension (N) y (mm) (b) Steady current I =.4 A 5. Analysis result magnetic field (horiontal electromagnet). 5. Analysis result magnetic field (horiontal electromagnet). We calculated deflection thin plate by solving Equation (13), which epress bending We calculated plate when deflection lateral load thinand plate force byin solving central Equation plane (13), coeist which [13], epress using finite bending differential plate method. whenmoreover, lateralwe load used andequation force(13), considering central plane permanent coeist [13], usingand horiontal finite differential electro, method. to obtain used placement considering permanent permanent by GA during levitation horiontal when electro, attractive torce obtain permanent placement is applied. permanent Owing to large by GAnumber during levitation combinations when attractive search patterns, permanent evaluation function applied. J is defined Owingas toequation large (14) to evaluate combinations average deflection search patterns, plate evaluation and local function deflection. J is defined as Equation (14) to evaluate average deflection plate and local deflection. 4 D = f + f + f PM ρhg (13) D 4 = f + f + f PM ρhg (13) J = J J w J + J D JD w J D N D J = w + wd J i i=1 J = J D (14) N J D = ma N w = w D =.5 i i= The evaluation value J becomes 1 when J = permanent 1 are not installed. It is shown (14) N shape thin plate is improved when evaluation value J is small. The shape thin plate reduces evaluationjvalue D = J is determined from deflection plate. ma 6a,b show relation between gap, defined as distance from permanent to plate, and evaluation value wj in = w search D =.5 result. The left sides show obtained placement permanent. 7a,b show results when steady currents horiontal The evaluation electromagnet value J becomes I are.3 1 when A and.4permanent A, respectively. Theare results not installed. this analysis It shown shape number thin permanent plate is improved at when.4 A is larger evaluation than value at J.3 is A. small. It isthe reasonable shape to assume gap at.4 A is larger than at.3 A because increase attractive force from horiontal electro. Therefore, as attractive force each permanent magnet decreases, placement with gap requires more permanent.

7 Evaluation value J Evaluation value J Evaluation value J Evaluation value J horiontal horiontalelectromagnet electromagnetiiare are.3.3aaand and.4.4a, A,respectively. respectively.the Theresults resultsthis thisanalysis analysisshow show number permanent at.4 A is larger than at.3 A. It is reasonable number permanent at.4 A is larger than at.3 A. It is reasonabletoto assume assume gap gapatat.4.4aaisislarger largerthan than atat.3.3aabecause because increase increase attractive attractive force forcefrom from horiontal horiontalelectro. electro.therefore, Therefore,asas attractive attractiveforce forceeach eachpermanent permanentmagnet magnet decreases, placement with gap requires more permanent. Actuators 18, 7, 54placement with gap requires more permanent. 7 1 decreases, Gap(mm) Gap(mm) Gap(mm) Gap(mm) (a) (a)steady Steadycurrent currentii==.3.3aa (b) (b)steady Steadycurrent currentii==.4.4aa 6. Evaluation value vs. gap in consideration changed horiontal steady current. 6.6.Evaluation Evaluationvalue valuevs. vs.gap gapin inconsideration considerationchanged changedhoriontal horiontalsteady steadycurrent. current. ] Attractive Attractive force fo rce ffpm pm [N/m (N/m ) Attractive Attractive force fo rce ffpm pm [N/m (N/m)] [mm] yy(mm) [mm] yy(mm) [mm] (mm) [mm] (mm) Attractive fo force ]) Attractive rcefpm f pm[n/m (N/m Attractive fo force Attractive rcefpm f pm[n/m (N/m]) (a) (a)steady Steadycurrent currentii= =.3.3AA yy [mm] (mm) yy [mm] (mm) [mm] (mm) [mm] (mm) (b).4aa (b)steady Steadycurrent currentii==.4 7. Optimal placement permanent considering horiontal attractive force and 7. Optimal placement permanent considering horiontal attractive force and 7. Optimal placementforce permanent considering horiontal attractive force and distribution attractive acting on plate. distribution attractive force acting on plate. distribution attractive force acting on plate. 5.5.Magnetic MagneticLevitation LevitationEperiment EperimentUsing UsingOptimal OptimalPlacement Placement 5. Magnetic Levitation Eperiment Using Optimal Placement To evaluate placement obtained by a To evaluate placement obtained by aga, GA,levitation levitationeperiments eperimentswere wereconducted conductedfor for To evaluate placement obtained by a GA, levitation eperiments were conducted cases caseswhere wherepermanent permanent were wereinstalled installedand andthose thosewherein whereinpermanent permanent were werenot not for cases in wherelevitation permanent were installed and those whereinbetween permanent surface were not installed installed in levitationsystem. system.aadistance distance55mm mmwas wasmaintained maintained between surface installed in levitation system. A distance 5 mm was maintained between surface plate plateand and levitation levitationelectro electroand andbetween between edge edge plate plateand and horiontal horiontal plate and levitation electro and between results, edge steady plate andi horiontal positioning control electro. Based on analytical current was positioning control electro. Based on analytical results, steady current I wasset settoto.3.3 positioning control electro. Based onanalytical results, steady current I was setby to.3 and.4 A. 8 shows time histories measured displacement plate and.4 A. 8 shows time histories measured displacement plate by and.4 A. 8 shows intime histories measured displacement plate by eddy-current eddy-currentsensor sensorinstalled installed in central centralelectromagnet electromagnetunit unitand and eperimental eperimentalresults resultswith withand and eddy-current sensor installed in central electromagnet unit and eperimental results with and without permanent. 8a,b show results when steady current I is.3 A and 8c,d show results when I is.4 A. For each steady current, when permanent are installed in placement, vibration levitated plate can be suppressed. 9 shows comparison displacement standard deviations for each steady current. In this system, plate deflects where electromagnetic force does not reach plate. It is considered this bending portion is cause vibration plate. Furrmore, we consider bending was suppressed by attaching PM and vibration was suppressed. These results show validity searching for placement permanent by a GA and effectiveness

8 installed 8c,d in show results placement, when I is vibration.4 A. For each levitated steady current, plate when can permanent be suppressed. are installed 9 shows in comparison placement, displacement vibration standard levitated deviations for plate each can steady be suppressed. current. In this 9 system, shows comparison plate deflects displacement where electromagnetic standard deviations force does for not each reach steady current. plate. In It this is system, considered this plate bending deflects portion where is electromagnetic cause vibration force does not reach plate. Furrmore, plate. It we is considered consider Actuators 18, 7, bending this bending 54 was suppressed portion is by attaching cause PM vibration and vibration was suppressed. plate. Furrmore, These results we 8 1 consider show validity bending searching was suppressed for by attaching placement PM and vibration permanent was suppressed. by a These GA and results show effectiveness validity permanent searching for for vibration placement suppression permanent when magnetic field by a generated GA and by effectiveness permanent horiontal electro permanent for vibration is changed. suppression for vibration when suppression magnetic when field magnetic generatedfield by generated horiontal by electro horiontal electro is changed. is changed Displacement (mm) (mm) Displacement (mm) (mm) Time (s) 1 15 Time (s) (a) Without permanent (Steady (a) Without current permanent I =.3 A) (Steady current I =.3 A) Time (s) 1 15 Time (s) (b) With permanent (Steady current (b) With permanent I =.3 A) (Steady current I =.3 A).5.5 Displacement (mm) (mm) Displacement (mm) (mm) Time (s) 1 15 Time (s) Time (s) 1 15 Time (s) (c) Without permanent (Steady (d) With permanent (Steady current (c) Without current permanent I =.4 A) (Steady (d) With permanent I =.4 A) (Steady current current I =.4 A) I =.4 A) 8. Time history displacement at each steady current horiontal electromagnet Time Time history history displacement displacement at at each each steady steady current current horiontal horiontal electromagnet. electromagnet Without PM (.3 A) With PM (.3 A) Without PM (.4 A) With PM (.4 A) Without PM (.3 A) With PM (.3 A) Without PM (.4 A) With PM (.4 A) 9. Displacement standard deviation levitation direction each horiontal steady current Displacement standard deviation levitation direction each each horiontal steady steady current. current. Standard deviation (mm) (mm) 6. Conclusions We determined number, gap, and placement permanent in levitation system to suppress deflection a levitated plate when magnetic field acting on it is generated from horiontal direction. The levitation eperiments were performed by applying obtained placements permanent. The results show placement permanent can improve levitation stability system even if horiontal magnetic

9 Actuators 18, 7, field changes. This supports possibility more stable transport methods for levitated plates using placements permanent. Author Contributions: Y.I., Y.O., T.N. and H.K. wrote paper; Y.I. and Y.O. performed eperiments; Y.I. and Y.O. contributed analysis data; and T.N. and H.K. contributed to design controller and electric circuit. T.N. and H.K. contributed to analysis placements permanent. Funding: This research received no eternal funding. Acknowledgments: We would like to show our greatest appreciation to Toshiki Suuki and Masahiro Kida. Conflicts Interest: The authors declare no conflict interest. The founding sponsors had no role in design study; collection, analyses, or interpretation data; writing manuscript; and decision to publish results. Nomenclature The definitions symbols used in this study are as follows: E Young s modulus thin plate (N/m ) f vertical static magnetic force applied to plate, which is generated by permanent (N/m ) f dynamic magnetic force (N) F magnetic force coupled in equilibrium state (N) F magnetic force coupled in equilibrium state (N) g acceleration due to gravity (m/s ) h plate thickness (m) i dynamic current coupled electro (A) i dynamic current coupled (A) I current coupled in equilibrium state (A) I current coupled electro in equilibrium state (A) J D evaluation function maimum deflection (m) J evaluation function average absolute deflection (m) L lea leakage inductance one magnet coil (H) L inductance one magnet coil in equilibrium state (H) L lea leakage inductance one magnet coil (H) L eff /X effective inductance one magnet coil (H) L inductance one electromagnet coil in equilibrium state (H) m virtually divided plate (kg) N total number analysis points R resistance coupled magnet coils (Ω) R resistance coupled magnet coils (Ω) T s sampling time (s) ν Poisson ratio v dynamic voltage coupled (V) v dynamic voltage coupled (V) coordinates in width direction (m) X gap between plate and electromagnet in equilibrium state (m) y coordinates in longitudinal direction (m) vertical displacement from equilibrium state (m) i displacement at each analysis point on thin plate (m) ma maimum deflection thin plate (m) Z gap between plate and electromagnet in equilibrium state (m) ρ plate density (kg/m 3 ) References 1. Onuki, T.; Toda, Y. Optimal design hybrid magnet in maglev system with both permanent and electro. IEEE Trans. Magn. 1993, 9, [CrossRef]

10 Actuators 18, 7, Oka, K.; Higuchi, T. 3 Degree freedom maglev system with actuators and permanent. IEEJ Trans. Ind. Appl. 1995, 115, [CrossRef] 3. Morishita, M.; Ito, H. The self-gap-detecting ero power controlled electromagnetic suspension system. IEEJ Trans. Ind. Appl. 6, 16, [CrossRef] 4. Kamijo, Y.; Ito, H.; Maruyama, Y. Structure and characteristic magnetic bearing with ero power controlled electromagnetic suspension. Trans. Jpn. Soc. Mech. Eng. Ser. C 13, 79, [CrossRef] 5. Oshinoya, Y.; Ishibashi, K. Development electromagnetic levitation control device for a rectangular sheet. Trans. Jpn. Soc. Mech. Eng. Ser. C 1, 661, [CrossRef] 6. Goldberg, D.E. Genetic Algorithms in Search, Optimiation, and Machine Learning; Addison-Wesley Pressional: Boston, MA, USA, 1989; ISBN Mallik, S.; Mallik, K.; Barman, A.; Maiti, D.; Biswas, S.K.; Deb, N.K.; Basu, S. Efficiency and cost optimied design an induction motor using genetic algorithm. IEEE Trans. Ind. Electron. 17, 64, [CrossRef] 8. Okamoto, Y.; Tominaga, Y.; Wakao, S.; Sato, S. Topology optimiation rotor core combined with identification current phase angle in IPM motor using multistep genetic algorithm. IEEE Trans. Magn. 14, 5, [CrossRef] 9. Narita, T.; Hasegawa, S.; Oshinoya, Y. Optimal placement permanent in a hybrid magnetic levitation system for thin plate (eperimental consideration on levitation probability). Proc. Sch. Eng. Tokai Univ. 13, 38, Narita, T.; Hasegawa, S.; Oshinoya, Y. Hybrid electromagnetic levitation system for thin plates using permanent. J. Magn. Soc. Jpn. 13, 37, [CrossRef] 11. Ishii, H.; Narita, T.; Kato, H. Hybrid magnetic levitation system for thin plate by electro and permanent (basic study on placement search considering interaction magnetic field). J. Jpn. Soc. Appl. Electromagn. Mech. 16, 4, [CrossRef] 1. Chiba, A.; Fukao, T.; Ichikawa, O.; Oshima, M.; Takemoto, M.; Dorrell, D.G. Magnetic Bearings and Bearingless Drives; Elsevier: Boston, MA, USA, 5; p.. ISBN Timoshenko, S.P.; Woinowsky-Krieger, S. Theory Plate and Shells; McGraw-Hill Publishing Company: New York, NY, USA, by authors. Licensee MDPI, Basel, Switerland. This article is an open access article distributed under terms and conditions Creative Commons Attribution (CC BY) license (