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

actuators

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

Yasuaki Ito 1, Yoshiho Oda 1, Takayoshi Narita 2,* ID and Hideaki Kato 2 1 Course of Mechanical Engineering, Tokai University, Kitakaname 4-4-1, Hiratsuka-shi, Kanagawa 259-1292, Japan; [email protected] (Y.I.); [email protected] (Y.O.) 2 Department of Prime Mover Engineering, Tokai University, Kitakaname 4-4-1, Hiratsuka-shi, Kanagawa 259-1292, Japan; [email protected] * Correspondence: [email protected]; Tel.: +81-463-58-1211

Received: 15 June 2018; Accepted: 24 August 2018; Published: 29 August 2018

Abstract: The surface quality of steel plates is deteriorated as they contact rollers while being conveyed during manufacturing processes. To solve this problem, we previously proposed a hybrid electromagnetic levitation system comprising electromagnets, permanent magnets, and a horizontal positioning control system for steel plates. Moreover, to increase stability, we proposed integrating these levitation systems. In this study, we aim to determine the optimal placement of permanent magnets in the levitation system to suppress the deﬂection of a levitated steel plate for cases where the magnetic ﬁeld in the horizontal direction changes. Using a genetic algorithm, the optimal gap, number, and placement of permanent magnets in the system are obtained.

Keywords: magnetic levitation; genetic algorithm; permanent magnet

1. Introduction Permanent magnets 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 the current steel plate production line, transportation is achieved by contact conveyance by rollers. However, the surface of the steel plate is damaged by this method, and the surface quality of the steel plate deteriorates. With non-contact magnetic levitation transport using the electromagnet proposed by us, these problems can be prevented. These studies consider using the attractive force generated by permanent magnets as a constant suspension force for levitation [2,3]. Although this technology is expected to be applied in the production of high-surface-quality steel plates, the thickness of the steel plate must be decreased to reduce its weight. A steel plate with reduced flexural rigidity owing to its thinness is difficult to levitate in a conventional magnetic levitation system because of exciting elastic vibration. To solve this problem, we proposed a hybrid electromagnetic levitation system for thin steel plates where permanent magnets are installed around electromagnets for levitation [4]. This system can suppress the elastic vibration of levitated steel plates by generating an attractive force on the entire surface of the steel plate using electromagnets and permanent magnets [5]. However, the number of arrangement patterns of the permanent magnets is very enormous, and it is difficult to experimentally obtain a more effective arrangement. Therefore, we focused on genetic algorithm (GA) which is one of optimization algorithms effective for nonlinear objective function [6], which is used to obtain optimum shape of electrical motor in several studies [7,8]. Furthermore, we confirmed that the GA we used to obtain the optimal placement of permanent magnets, considering the interactions of the magnets with each other, improved the levitation stability of the system [9,10]. In addition, we proposed adding another

Actuators 2018, 7, 54; doi:10.3390/act7030054 www.mdpi.com/journal/actuators Actuators 2018, 7, x FOR PEER REVIEW 2 of 10

controls to the electromagnetic levitation system using vertical electromagnets for levitation [7]. This system generates tension on the edge of the steel plate due to the electromagnetic field generated by the horizontal electromagnets. This tension further suppresses the vibration of the steel plate and improves levitation. In this study, we propose using a hybrid electromagnetic levitation system by applying horizontal positioning control and determine the optimal placement of permanent magnets Actuatorsusing a 2018GA, for7, 54 a case where a horizontal electromagnetic field is acting on the steel plate. Furthermore,2 of 10 we performed a levitation experiment and considered the levitation stability of the optimal placement. electromagnet in the horizontal direction and positioning controls to the electromagnetic levitation 2. Outline of the Electromagnetic Levitation System Integrating Permanent Magnets and the system using vertical electromagnets for levitation [7]. This system generates tension on the edge of the Horizontal Positioning Control System steel plate due to the electromagnetic field generated by the horizontal electromagnets. This tension furtherThe suppresses hybrid electromagnetic the vibration of levitation the steel platesystem and integrating improves positioning levitation. Incontrol this study, in the we horizontal propose usingdirection a hybrid is shown electromagnetic in Figure 1. The levitation object of system electromagnetic by applying levitation horizontal is a positioningrectangular, controlzinc-coated and determinesteel plate (SS400) the optimal with placementlength of 800 of mm, permanent width of magnets 600 mm, using and thickness a GA for of a case0.24 wheremm. To a accomplish horizontal electromagneticnoncontact support field of is this acting plate, on theas if steel it were plate. hois Furthermore,ted by strings, we we performed use five apairs levitation of electromagnets experiment and(No. considered 1–5 in Figure the levitation1). The displacement stability of theof the optimal steel placement.plate is measured using five eddy-current gap sensors. Here, the electric circuits of paired electromagnets are connected in series, whereas an eddy- 2.current Outline gap of sensor the Electromagnetic is positioned between Levitation the two System magnets Integrating of each Permanentpair. The detected Magnets displacement and the is Horizontal Positioning Control System converted to velocity using digital differentiation. A regulated voltage from the digital-to-analog converterThe hybrid is supplied electromagnetic to a current-su levitationpply amplifier system integratingto control the positioning attractive control force of in the the five horizontal pairs of directionelectromagnets is shown to inensure Figure that1. The the object steel of plate electromagnetic is levitated levitationby 5 mm is abelow rectangular, the surface zinc-coated of the steelelectromagnets. plate (SS400) In with this lengthmodel, of independent 800 mm, width control of 600 is used, mm, andwherein thickness information of 0.24 mm. on detected To accomplish values noncontactof displacement, support velocity, of this and plate, coil as current if it were of hoistedthe electromagnet by strings, under we use study five pairsare fed of back electromagnets only to the (No.same 1–5 electromagnet. in Figure1). The displacement of the steel plate is measured using five eddy-current gap sensors.The horizontal Here, thedisplacement electric circuits of the of plate paired is electromagnetsmeasured using are four connected laser beam in series, displacement whereas ansensors. eddy-current The four gap velocities sensor isof positioned the plate betweenare detected the twoby magnetsdifferentiating of each the pair. signals The from detected the displacementdisplacement issensors converted using to a velocity computer. using The digital current differentiation. flowing the Aelectromagnet regulated voltage is obtained from theby digital-to-analogmeasuring the voltage converter of isthe supplied resistor to connected a current-supply in series amplifier to the electric to control circuit. the attractive The permanent force of themagnets five pairs are installed of electromagnets around electromagnet to ensure that unit the for steel levitation plate is levitatedas shown by in 5Figure mm below 2. The the size surface of the ofpermanent the electromagnets. magnet for Inlevitation this model, assistance independent is 30 mm control × 30 mm is used, × 15 mm wherein and the information material is on ferrite. detected The valuessurface of magnetic displacement, flux density velocity, is 0.12 and T. coil Permanent current of magnets the electromagnet are placed undersuch that study the are deflection fed back of only the tofloating the same steel electromagnet. sheet is suppressed while using the above system.

Levitation controller AMP5 AMP4 D/A DSP AMP3 AMP2 converter (ADSP674-00)150MHz AMP1 A/D converter

iz1 iz2 iz3 iz4 iz5 z1 z2 z3 z4 z5

y Laser type sensor Eddy current type sensor Electromagnet for levitation control Permanent magnet z (30 mm×30 mm×15 mm) Steel plate (800 mm×600 mm×0.24 mm) x Electromagnet for horizontal positioning control

x1 x2 x3 x4 ix1 ix2 ix3 ix4 A/D converter AMP4 DSP D/A AMP3 (ADSP324-00A)50MHz converter AMP2 AMP1 Horizontal positioning controller

FigureFigure 1.1. HybridHybrid electromagnetic electromagnetic levita levitationtion system system integrating integrating positioning control in thethe horizontalhorizontal direction.direction.

The horizontal displacement of the plate is measured using four laser beam displacement sensors. The four velocities of the plate are detected by differentiating the signals from the displacement sensors using a computer. The current flowing the electromagnet is obtained by measuring the voltage of the resistor connected in series to the electric circuit. The permanent magnets are installed around Actuators 2018, 7, 54 3 of 10 electromagnet unit for levitation as shown in Figure2. The size of the permanent magnet for levitation assistance is 30 mm × 30 mm × 15 mm and the material is ferrite. The surface magnetic flux density is 0.12 T. Permanent magnets are placed such that the deflection of the floating steel sheet is suppressed while using the above system. Actuators 2018, 7, x FOR PEER REVIEW 3 of 10

Figure 2. Photograph of the placement of permanent magnets around the levitation electromagnets. Figure 2. Photograph of the placement of permanent magnets around the levitation electromagnets. 3. Control Model 3. Control Model 3.1. Electromagnetic Levitation Control System 3.1. Electromagnetic Levitation Control System A mathematical control model is formulated to construct the hybrid electromagnetic levitation A mathematical control model is formulated to construct the hybrid electromagnetic levitation control system. In this study, each electromagnet unit is controlled by considering the displacement control system. In this study, each electromagnet unit is controlled by considering the displacement from the gap sensor installed within and the velocity and current of the electromagnet unit. Therefore, from the gap sensor installed within and the velocity and current of the electromagnet unit. Therefore, the control model for levitation is established in each unit. Figure 3 shows the model of levitation the control model for levitation is established in each unit. Figure3 shows the model of levitation control for one electromagnet unit using the lumped parameter system. From previous study, it is control for one electromagnet unit using the lumped parameter system. From previous study, it is confirmed that by placing permanent magnets, the deflection of the steel plate is suppressed, and the confirmed that by placing permanent magnets, the deflection of the steel plate is suppressed, and the levitation stability improves [9–11]. From this result, the steel plate was assumed to be a rigid body. levitation stability improves [9–11]. From this result, the steel plate was assumed to be a rigid body. The steel plate is virtually divided into five hypothetical masses. In an equilibrium levitation state, a The steel plate is virtually divided into five hypothetical masses. In an equilibrium levitation state, constant attractive force equal to the weight of the virtually divided steel plate needs to be generated a constant attractive force equal to the weight of the virtually divided steel plate needs to be generated by the electromagnet, and the change of the attractive force from the equilibrium state causes the by the electromagnet, and the change of the attractive force from the equilibrium state causes the motion of the virtually divided steel plate. The vertical motion of the steel plate is expressed as follows. motion of the virtually divided steel plate. The vertical motion of the steel plate is expressed as follows. m z = 2 f ..z z (1) mzz = 2 fz (1) 44FF =+4Fz zz4Fz fzifzzz= z + iz (2) ZZI00 Iz z We defined on the assumption that the inductance of one coil is expressed by the sum of the We defined on the assumption that the inductance of one coil is expressed by the sum of the effective inductance inversely proportional to the gap and the leakage inductance [12]. Since the two effective inductance inversely proportional to the gap and the leakage inductance [12]. Since the two electromagnets are not magnetically connected, mutual inductance is not considered. electromagnets are not magnetically connected, mutual inductance is not considered.

d Le f f Iz . Rz 1 i d= − Leff· I z − R i + 1 v (3) z = − ⋅ 2z  − z z + z dt iz Lz Z0 2 z 2Lz iz 2Lzvz (3) dt Lz Z0 2Lz 2Lz Le f f Lz = + L (4) ZL lea = 0eff + Lz Llea (4) Z 0

Using the state vector, Equations (1)–(4) are written as the following state equations. = + z A z z B z v z (5) = []Τ z zzi z

Actuators 2018, 7, x FOR PEER REVIEW 4 of 10 Actuators 2018, 7, 54 4 of 10   0 1 0 Using the state vector, Equations (1)–(4) are written as the following state equations.    2F. z 2Fz  A = z = Azz + B0zvz , z m Z m I   z 0 h . iT z z  z = z Lz iz I R  0 − eff ⋅ z − z   2    0L 1z Z 0 0 2Lz  (5)  2Fz, 0 2Fz  Az =  mz Z0 mΤz Iz , L  e f f Iz 1 Rz  0 = − · 2 − Bz 00Lz Z 2L z 20 L h izT B = 0 0 1 z 2Lz Furthermore, vz is calculated by the state feedback control of Equation (6). Furthermore, vz is calculated by the state feedback=− control of Equation (6). vzzkz h i (6) vz = −k=zzk[ z = p pvz] piz (6) kzdzvzizpppdz The value of each parameter and the feedback gain of the levitation control system are shown in The value of each parameter and the feedback gain of the levitation control system are shown in Tables 1 and 2. Tables1 and2.

Table 1.1. TheThe valuevalue ofof eacheach parameter.parameter. Parameters Values Parameters Values m 0.864 kg E m 0.864206 kg GPa E 206 GPa ν ν 0.3 0.3 −3 −3 Z0 Z0 5 × 105 × 10m m Rn Rn 21.021.0Ω Ω L × −4 Leff eff 2.55 2.5510 × 10Hm−4 Hm Llea 0.090 H Llea 0.090 H Ts 0.001 s Ts 0.001 s

Table 2. The feedback gain of the levitation control system. Table 2. The feedback gain of the levitation control system.

ParametersParameters Values Values (No. (No. 1–4 1–4in Figure in Figure 1)1 ) Values Values (No. (No. 5 5in in Figure Figure 11)) 4 4 4 4 ppdzdz 1.411.41 × 10× 10 1.521.52 × ×1010 2 2 ppvzvz 1.371.37 × 10×2 10 1.551.55 × ×10102 2 2 ppiziz 0.4440.444 × 10×2 10 0.4430.443 × ×10102

Permanet magnet Eddy current type gap sensor I z+i z Electromagnet

z f PM ffzzfPM mz Steel plate (rigid body) Figure 3. Modeling of levitation control for one electromagnet unit by lumped parameter system. Figure 3. Modeling of levitation control for one electromagnet unit by lumped parameter system.

Actuators 2018, 7, x FOR PEER REVIEW 5 of 10

3.2. Horizontal Positioning Control System The horizontal motion of the steel plate was modeled to have a single degree of freedom, as shown in Figure 4. Therefore, the same attractive forces were generated from the two electromagnets placed at one side of the steel plate. The equation for small horizontal motion around the equilibrium state of the steel plate subjected to the same static magnetic forces from the electromagnets at two Actuatorsedges is2018 expressed, 7, 54 as follows. 5 of 10 = − = mx f1 f 2 f x (7) 3.2. Horizontal Positioning Control System 4F 4F The horizontal motion of the steel platef was= modeledx x + tox havei a single degree of freedom, as shown x X I x (8) in Figure4. Therefore, the same attractive forces were0 generatedx from the two electromagnets placed at one side of the steel plate. The equation for small horizontal motion around the equilibrium state of d L xeff I R 1 the steel plate subjected to the samei static= − magnetic⋅ x x forces− x fromi + thev electromagnets at two edges is dt x L 2 2 L x 2 L x (9) expressed as follows. x X 0 x x .. mx = f1 − f2 = fx (7) L = xeff + L x L xlea (10) 4Fx X 4Fx fx = x 0+ ix (8) X0 Ix Using the state vector, Equations (7)–(10) can be written as the following state equations. d Lxe f f I . R 1 i = − · x x − x i + v (9) x  = 2 + x x dt Lx x XA0x x B2xLxv x 2Lx

L T x = [xex f f xi ] Lx = + Lxleax (10) X0 Using the state vector, Equations (7)–(10) can be written as the following state equations. 010 . x = Ax x + Bxvx 44FF A = xxh . 0 iT (11) x x = x x ix mXxxx0 mI   0 1 0 Lxeff I xxR 4Fx −⋅4F −x (11)  0 0 2  Ax =  mx X0 mx Ix  L LxxXL0 2  0 − xe f f · Ix − Rx Lx X2 2Lx 0 T  1 T h= 1 i B B=x 00 00 x  2Lx   2Lx 

Here, thethe controlcontrol signalsignal isis expressedexpressed asas follows.follows.

v=−x = −kxx (12) vxxkx (12)

Ix + ix

x m x

f2 Steel plate (rigid body)

Electromagnet for horizontal

I x - i x positioning control

Laser type sensor

Figure 4. Theoretical model of the horizontal positioningpositioning control ofof thethe steelsteel plate.plate.

4. Determination of Optimal Placement In the proposed system, by applying the attractive force of the permanent magnet to the area where the attractive force of the electromagnet is lacking, the deflection of the levitating thin steel Actuators 2018, 7, 54 6 of 10 plate is suppressed and the levitation stability of the system is improved, which is the aim of this study. However, experimentally searching for the optimal placement of the permanent magnets is practicallystudy. However, impossible experimentally because of th searchinge large number for the of optimal combinations placement of the of search the permanent patterns. magnetsSearching is ispractically performed impossible in the range because where of the the steel large plate number is di ofvided combinations into ¼, and of the the position search patterns. of the PM Searching can be arranged at intervals of 10 mm, considering the size of PM and the1 range of EM. Furthermore, the is performed in the range where the steel plate is divided into 4 , and the position of the PM can maximumbe arranged number at intervals of PMs of is 10 set mm, to 15, considering and the search the size is ofperformed PM and thewithin range the of range EM. Furthermore,of 40 mm to 80the mm maximum for Gap number[9,10]. In of addition, PMs is set optimal to 15, andplacement the search is sought is performed by considering within theinteraction range of of 40 PM mm by to past80 mm research for Gap [11]. [9,10 The]. In magnetic addition, field optimal applied placement by the is horizontal sought by electromagnet considering interaction was analyzed of PM by by JMAG,past research as shown [11]. in The Figure magnetic 5. ﬁeld applied by the horizontal electromagnet was analyzed by JMAG, as shown in Figure5.

0.08 0.08

-3 0.07 -3 0.07 x 10 x 10 8 8 0.06 0.06 6 6 0.05 0.05 4 4 0.04 0.04 2 2 0.03 0.03 Vertical force (N) Vertical force (N) 0 0 Horizontal (N) tension 0.02 Horizontal (N) tension 0.02 -2 -2 50 0.01 50 0.01 40 150 40 150 30 0 30 100 100 0 20 20 10 50 10 50 -0.01 -0.01 x (mm) 0 0 y (mm) 0 50 100 150 x (mm) 0 0 y (mm) 0 50 100 150 y (mm) y (mm) (a) Steady current Ix = 0.3 A (b) Steady current Ix = 0.4 A

Figure 5. Analysis result of magnetic field (horizontal electromagnet). Figure 5. Analysis result of magnetic field (horizontal electromagnet). We calculated the deflection of the thin steel plate by solving Equation (13), which express the bendingWe of calculated the plate the when deflection the lateral of the load thin and steel the plate force by in solving the central Equation plane (13),coexist which [13] express, using the the finitebending differential of the plate method. when Moreover, the lateral we load used and Eq theuation force (13), in the considering central plane permanent coexist [13magnets], using and the horizontalfinite differential electromagnets, method. to Moreover, obtain the we optimal used Equationplacement (13), of the considering permanent permanent magnets by magnets GA during and levitationhorizontal when electromagnets, the attractive to force obtain of the the optimal perman placementent magnets of is the applied. permanent Owing magnets to the large by GA number during oflevitation combinations when theof attractivethe search force patterns, of the the permanent evaluation magnets function is applied. J is defined Owing as to Equation the large number(14) to evaluateof combinations average ofdeflection the search ofpatterns, the steel theplate evaluation and local function deflection.J is defined as Equation (14) to evaluate average deflection of the steel plate and local deflection. ∂ 2 ∇ 4 = + + − ρ D z f z f x 2 z f PM hg 4 ∂ ∂ 2 (13) D∇ z = fz + fx zx+ f − ρhg (13) ∂2x PM

Jz JD J = wz + wD JzJ0 JD0 J J = z wN + D w ∑z|zi| D J z0 i=1 J D0 Jz = N (14) JD = |zmaxN | = = wz wD 0.5zi = i=1 (14) J J z The evaluation value becomes 1 when the permanentN magnets are not installed. It is shown that the shape of the thin steel plate is improved when the evaluation value J is small. The shape of the thin steel plate that reduces the evaluation value= J is determined from the deﬂection of the plate. J D z max Figure6a,b show the relation between the gap, deﬁned as the distance from the permanent magnets to the steel plate, and the evaluation value J in= the search= result. The left sides show the obtained wz wD 0.5 optimal placement of permanent magnets. Figure7a,b show the results when the steady currents of the horizontalThe evaluation electromagnet value J becomesIx are 0.3 1 when A and the 0.4 permanent A, respectively. magnets The are results not installed. of this analysis It is shown show thatthat the the shape optimal of numberthe thin ofsteel permanent plate is improved magnets at when 0.4 A the is larger evaluation than thatvalue at J 0.3 is A.small. It is The reasonable shape of to assume that the optimal gap at 0.4 A is larger than that at 0.3 A because of the increase of the attractive force from the horizontal electromagnets. Therefore, as the attractive force of each permanent magnet decreases, the placement with the optimal gap requires more permanent magnets. Actuators 2018, 7, x FOR PEER REVIEW 7 of 10 Actuators 2018, 7, x FOR PEER REVIEW 7 of 10 thethe thin thin steel steel plate plate that that re reducesduces the the evaluation evaluation value value J Jis is determined determined from from the the deflection deflection of of the the plate. plate. FigureFigure 6a,b 6a,b show show the the relation relation between between the the gap, gap, de definedfined as as the the distance distance from from the the permanent permanent magnets magnets toto the the steel steel plate, plate, an andd the the evaluation evaluation value value J Jin in the the search search result. result. The The left left sides sides show show the the obtained obtained optimaloptimal placement placement of of permanent permanent magnets. magnets. Figure Figure 7a,b 7a,b show show the the results results when when the the steady steady currents currents of of the horizontal electromagnet Ix are 0.3 A and 0.4 A, respectively. The results of this analysis show that the horizontal electromagnet Ix are 0.3 A and 0.4 A, respectively. The results of this analysis show that thethe optimal optimal number number of of permanent permanent magnets magnets at at 0.4 0.4 A A is is larger larger than than that that at at 0.3 0.3 A. A. It It is is reasonable reasonable to to assumeassume that that the the optimal optimal gap gap at at 0.4 0.4 A A is is larger larger than than that that at at 0.3 0.3 A A because because of of the the increase increase of of the the attractive attractive forceforce from from the the horizontal horizontal electromagnets. electromagnets. Therefore, Therefore, as as the the attractive attractive force force of of each each permanent permanent magnet magnet decreases, the placement with the optimal gap requires more permanent magnets. Actuatorsdecreases,2018 ,the7, 54 placement with the optimal gap requires more permanent magnets. 7 of 10

0.50 0.50 0.50 0.50

J 0.45 J 0.45

J 0.45 J 0.45 0.40 0.40 0.40 0.40 0.35 0.35 0.35 0.35 0.30 0.30 0.30 0.30 0.25 0.25 0.25 0.25 0.20 0.20 0.20 0.20 0.15 0.15 0.15 0.15 0.10 0.10 0.10 0.10

Evaluation value Evaluation 0.05 value Evaluation 0.05

Evaluation value Evaluation 0.05 value Evaluation 0.05 0.00 0.00 0.0035 40 45 50 55 60 65 70 75 80 85 0.0035 40 45 50 55 60 65 70 75 80 85 35 40 45 50 55 60 65 70 75 80 85 35 40 45 50 55 60 65 70 75 80 85 Gap(mm) Gap(mm) Gap(mm) Gap(mm)

(a) Steady current Ix = 0.3 A (b) Steady current Ix = 0.4 A (a) Steady current Ix = 0.3 A (b) Steady current Ix = 0.4 A Figure 6. Evaluation value vs. gap in consideration of changed horizontal steady current. FigureFigure 6. 6.Evaluation Evaluation value value vs. vs. gap gap in in consideration consideration of of changed changed horizontal horizontal steady steady current. current. ]

) 200 2 2 ]

) 200 2 2

(N/m 150 [N/m pm (N/m

pm 150 [N/m pm 100pm 100 50 50 Attractiv e fo rcef

Attractive force f 0

800Attractiv e fo rcef

Attractive force f 0 800 600 600 500 600 400 400 600 300 500 400 200 200 400 100 300 200 0 0 200 y [mm] 100 y (mm) 0 0 x x[mm](mm) y [mm] y (mm) x x[mm](mm)

(a) Steady current Ix = 0.3 A (a) Steady current Ix = 0.3 A ) 2 ] 200 2 ) 2 ] 200 2 (N/m

[N/m 150 pm (N/m pm

[N/m 150 pm

100pm 100 50 50

Attractive force f force Attractive 0 Attractiv e 800 fo rce f

Attractive force f force Attractive 0

Attractiv e fo rce f 600 600 800 500 600 400 400 600 300 500 400 200 200 400 100 300 200 0 0 200 y [mm] 100 y (mm) 0 0 x [mm]x (mm) y [mm] y (mm) x [mm]x (mm)

(b) Steady current Ix = 0.4 A (b) Steady current Ix = 0.4 A Figure 7. Optimal placement of permanent magnets considering the horizontal attractive force and Figure 7. Optimal placement of permanent magnets considering the horizontal attractive force and distributionFigure 7. Optimal of the attractive placement force of permanent acting on the magnets steel plate. considering the horizontal attractive force and distribution of the attractive force acting on the steel plate. distribution of the attractive force acting on the steel plate.

5.5. Magnetic Magnetic Levitation Levitation Experiment Experiment Using Using Optimal Optimal Placement Placement 5. Magnetic Levitation Experiment Using Optimal Placement ToTo evaluate evaluate the the optimal optimal placement placement obtained obtained by by a a GA, GA, levitation levitation experiments experiments were were conducted conducted for for To evaluate the optimal placement obtained by a GA, levitation experiments were conducted casescases where where permanent permanent magnets magnets were were installed installed and and those those wherein wherein permanent permanent magnets magnets were were not not for cases where permanent magnets were installed and those wherein permanent magnets were not installedinstalled in in the the levitation levitation system. system. A A distance distance of of 5 5 mm mm was was maintained maintained between between the the surface surface of of the the steel steel installed in the levitation system. A distance of 5 mm was maintained between the surface of the steel plateplate and and the the levitation levitation electromagnets electromagnets and and between between the the edge edge of of the the steel steel plate plate and and the the horizontal horizontal positioningplate and the control levitation electromagnets. electromagnets Based and on between analytical the results, edge of the the steady steel platecurrent and Ix thewas horizontal set to 0.3 positioning control electromagnets. Based on analytical results, the steady current Ix was set to 0.3 positioning control electromagnets. Based on analytical results, the steady current I was set to 0.3 andand 0.4 0.4 A. A. Figure Figure 8 8 shows shows the the time time histories histories of of th thee measured measured displacement displacement of of the the steel steelx plate plate by by the the and 0.4 A. Figure8 shows the time histories of the measured displacement of the steel plate by the eddy-currenteddy-current sensor sensor installed installed in in the the central central electr electromagnetomagnet unit unit and and the the experimental experimental results results with with and and eddy-current sensor installed in the central electromagnet unit and the experimental results with and without permanent magnets. Figure8a,b show the results when the steady current Ix is 0.3 A and Figure8c,d show the results when Ix is 0.4 A. For each steady current, when permanent magnets are installed in the optimal placement, the vibration of the levitated steel plate can be suppressed. Figure9 shows the comparison of the displacement standard deviations for each steady current. In this system, the steel plate deﬂects where electromagnetic force does not reach the steel plate. It is considered that this bending portion is the cause of the vibration of the steel plate. Furthermore, we consider that bending was suppressed by attaching PM and vibration was suppressed. These results show the validity of searching for the optimal placement of permanent magnets by a GA and the effectiveness Actuators 2018, 7, x FOR PEER REVIEW 8 of 10 Actuators 2018, 7, x FOR PEER REVIEW 8 of 10

without permanent magnets. Figure 8a,b show the results when the steady current Ix is 0.3 A and without permanent magnets. Figure 8a,b show the results when the steady current Ix is 0.3 A and Figure 8c,d show the results when Ix is 0.4 A. For each steady current, when permanent magnets are Figureinstalled 8c,d in theshow optimal the results placement, when Ithex is vibration0.4 A. For of each the levitatedsteady current, steel plate when can permanent be suppressed. magnets Figure are installed9 shows inthe the comparison optimal placement, of the displacement the vibration standa of therd levitated deviations steel for plate each can steady be suppressed. current. InFigure this 9system, shows thethe steelcomparison plate deflects of the displacementwhere electromagne standardtic deviationsforce does fornot each reach steady the steelcurrent. plate. In Itthis is system,considered the that steel this plate bending deflects portion where is theelectromagne cause of thetic vibration force does of thenot steel reach plate. the Furthermore,steel plate. It we is considered that this bending portion is the cause of the vibration of the steel plate. Furthermore, we considerActuators 2018 that, 7 ,bending 54 was suppressed by attaching PM and vibration was suppressed. These results8 of 10 considershow the that validity bending of searching was suppressed for the byoptimal attaching placement PM and of vibration permanent was magnets suppressed. by a These GA and results the showeffectiveness the validity of permanent of searching magnets for the for optimalvibration placement suppression of permanentwhen the magnetic magnets field by a generated GA and the by effectivenesstheof permanent horizontal of magnetselectromagnets permanent for vibration magnets is changed. suppressionfor vibration when suppression the magnetic when ﬁeldthe magnetic generated field by thegenerated horizontal by theelectromagnets horizontal electromagnets is changed. is changed.

0.5 0.5 0.5 0.5

0 0 0 0 Displace me nt (mm) Displace me nt (mm) Displace me nt (mm) Displace me nt (mm)

-0.5 -0.5 0 5 10 15 0 5 10 15 -0.5 -0.5 0 5 Time (s) 10 15 0 5 Time (s) 10 15 Time (s) Time (s) (a) Without permanent magnets (Steady (b) With permanent magnets (Steady current (a) Without permanent magnets (Steady (b) With permanent magnets (Steady current current Ix = 0.3 A) Ix = 0.3 A) 0.5 current Ix = 0.3 A) 0.5 Ix = 0.3 A) 0.5 0.5

0 0 0 0 Displace me nt (mm) Displace me nt (mm) Displace me nt (mm) Displace me nt (mm)

-0.5 -0.5 0 5 10 15 0 5 10 15 -0.5 -0.5 0 5 Time (s) 10 15 0 5 Time (s) 10 15 Time (s) Time (s) (c) Without permanent magnets (Steady (d) With permanent magnets (Steady current (c) Without permanent magnets (Steady (d) With permanent magnets (Steady current current Ix = 0.4 A) Ix = 0.4 A) current Ix = 0.4 A) Ix = 0.4 A) Figure 8. Time history of displacement at each steady current of the horizontal electromagnet. FigureFigure 8. 8. TimeTime history history of of displacement displacement at at each each stea steadydy current current of the horizontal electromagnet. 0.16 0.16 0.14 0.14 0.12 0.12 0.1 0.1 0.08 0.08 0.06 0.06 0.04 0.04

Standarddeviation (mm) 0.02

Standarddeviation (mm) 0.02 0 0 Without PM (0.3 A) With PM (0.3 A) Without PM (0.4 A) With PM (0.4 A) Without PM (0.3 A) With PM (0.3 A) Without PM (0.4 A) With PM (0.4 A)

Figure 9. Displacement standard deviation of the levitation direction of each horizontal steady current. FigureFigure 9. 9. DisplacementDisplacement standard standard deviation deviation of of the the levitation levitation direction direction of ofeach each horizontal horizontal steady steady current. current.

6. Conclusions

We determined the optimal number, gap, and placement of permanent magnets in the levitation system to suppress the deﬂection of a levitated steel plate when the magnetic ﬁeld acting on it is generated from the horizontal direction. The levitation experiments were performed by applying the obtained optimal placements of permanent magnets. The results show that the optimal placement of permanent magnets can improve the levitation stability of the system even if the horizontal magnetic Actuators 2018, 7, 54 9 of 10

ﬁeld changes. This supports the possibility of more stable transport methods for levitated steel plates using optimal placements of permanent magnets.

Author Contributions: Y.I., Y.O., T.N. and H.K. wrote the paper; Y.I. and Y.O. performed the experiments; Y.I. and Y.O. contributed the analysis of the data; and T.N. and H.K. contributed to the design of the controller and electric circuit. T.N. and H.K. contributed to analysis optimal placements of permanent magnets. Funding: This research received no external funding. Acknowledgments: We would like to show our greatest appreciation to Toshiki Suzuki and Masahiro Kida. Conﬂicts of Interest: The authors declare no conﬂict of interest. The founding sponsors had no role in the design of the study; collection, analyses, or interpretation of data; writing the manuscript; and decision to publish the results.

Nomenclature The deﬁnitions of symbols used in this study are as follows:

E Young’s modulus of the thin steel plate (N/m2) f vertical static magnetic force applied to the plate, which is generated by the permanent magnets (N/m2) fz dynamic magnetic force (N) Fx magnetic force of the coupled magnets in the equilibrium state (N) Fz magnetic force of the coupled magnets in the equilibrium state (N) g acceleration due to gravity (m/s2) h plate thickness (m) iz dynamic current of the coupled electromagnets (A) ix dynamic current of the coupled magnets (A) Ix current of the coupled magnets in the equilibrium state (A) Iz current of the coupled electromagnets in the equilibrium state (A) JD evaluation function of the maximum deflection (m) Jz evaluation function of the average absolute deflection (m) Llea leakage inductance of one magnet coil (H) Lx inductance of one magnet coil in the equilibrium state (H) Lxlea leakage inductance of one magnet coil (H) Lxeff/X0 effective inductance of one magnet coil (H) Lz inductance of one electromagnet coil in the equilibrium state (H) mz virtually divided steel plate (kg) N the total number of analysis points Rx resistance of the coupled magnet coils (Ω) Rz resistance of the coupled magnet coils (Ω) Ts sampling time (s) ν Poisson ratio vx dynamic voltage of the coupled magnets (V) vz dynamic voltage of the coupled magnets (V) x coordinates in the width direction (m) X0 gap between the steel plate and electromagnet in the equilibrium state (m) y coordinates in the longitudinal direction (m) z vertical displacement from the equilibrium state (m) zi displacement at each analysis point on the thin steel plate (m) zmax maximum deflection of the thin steel plate (m) Z0 gap between the steel plate and electromagnet in the equilibrium state (m) ρ plate density (kg/m3)

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