actuators Article
Noncontact Guide System for Traveling Continuous Steel Plate Using Electromagnet and Supplement of Permanent Magnets for Suppressing Vibration Takahiko Bessho 1 , Sora Ishihara 2 , Yasuhiro Narawa 2 , Ryo Yamaguti 2 , Takayoshi Narita 2, * and Hideaki Kato 2 1 2
*
ID
Course of Mechanical Engineering, Tokai University, Kitakaname 4-4-1, Hiratsuka-shi, Kanagawa 259-1292, Japan;
[email protected] Department of Prime Mover Engineering, Tokai University, Kitakaname 4-4-1, Hiratsuka-shi, Kanagawa 259-1292, Japan;
[email protected] (S.I.);
[email protected] (Y.N.);
[email protected] (R.Y.);
[email protected] (H.K.) Correspondence:
[email protected]; Tel.: +81-463-58-1211
Received: 15 June 2018; Accepted: 9 August 2018; Published: 13 August 2018
Abstract: In a plating process, the steel plate is conveyed 20–50 m in the vertical direction for drying, during which it is negligibly supported by rollers and other mechanisms. This produces plating without uniformity owing to the generation of vibration and other factors, which prevent the increase in productivity. We have developed a noncontact guide system for a high-speed traveling elastic steel plate in which electromagnetic forces are applied by actuators at the edges of the plate to control the plate’s position. In this study, we investigated the vibration phenomenon when changing the steady current value of the electromagnet used for controlling the position. In addition, we conducted mode analysis of the steel plate to enable stable control even at low steady current values and verified whether stable guide can be provided by using it together with a permanent magnet. As a result, by arranging the permanent magnets, stable guidance was possible even at a low steady current value. In addition, it became clear that vibration damping performance is also improved. Keywords: electromagnetic guideway; continuous steel plate; permanent magnet; finite element method
1. Introduction Primary products, which are the basic materials for various industrial products such as paper, film, thin steel plates, etc., are several kilometers in length. During the manufacturing process, the primary products are carried in a long continuum of states by a number of rollers. This long manufacturing process consists of many linear parts and the part where traveling direction is changed. For improving the quality of the final product and reducing cost, it is necessary to improve the quality of the primary product. A guidance technique for preventing wrinkles and meandering is proposed. Magnetic guideways that are capable of guiding a target in a noncontact manner, i.e., by utilizing electromagnetic force, have been actively considered for the manufacturing process of a thin steel plate that particularly considers the deterioration of surface quality caused by contact with a roller. In some of the existing techniques, electromagnets are installed on the upper and lower surfaces of a steel plate to support continuous steel plates in a noncontact manner by utilizing an attractive force [1–5]; however, to the best of our knowledge, suppression of out-of-plane vibrations have not been reported. Furthermore, these studies on this subject considered only linearly transported continuous steel plates. The problem of meandering persists due to lack of control in the perpendicular direction compared to that in the running direction of the plane of the steel plate. All studies on this subject considered Actuators 2018, 7, 47; doi:10.3390/act7030047
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only linearly transported continuous steel plates. Therefore, it is necessary to consider the noncontact guidance in the part where traveling direction is changed for realization of the noncontact guide system in actual process. Changing the traveling direction leads to a complex behavior of the continuous steel plate because of several factors such as the inertial force, centrifugal force, Coriolis force, etc. Our current study establishes that a noncontact guideway can be provided on a direction-changing traveling continuous steel plate, and the vibrations outside the plane can be suppressed by applying tension on the steel plate [6]. Although the loop shape of the direction-changing part becomes asymmetric while running, it has been experimentally clarified that the vibrational damping effects can be improved by dynamically relocating the electromagnet according to the shape [7]. In addition, our research group established a method for numerically computing the asymmetric shape of a traveling steel plate using multibody dynamics (MBD) and finite element method (FEM) [8]. Furthermore, structural analysis by FEM reveals that out-of-plane vibrations in the loop-shape part can be suppressed by placing an electromagnet in the area under high stress on the traveling steel plate [9]. In recent years, research on zero power control using permanent magnets has been actively investigated. Thus, we conducted an experiment using permanent magnets in noncontact guideway system. As a result, it was confirmed that vibration can be suppressed by installing a permanent magnet on the continuous steel plate [10]. In this paper, we considered the vibration phenomena that appear when changing the steady current value of the electromagnet used for guiding the steel plate. At that time, to realize stable control with a small control current, we considered whether it can be stabilized by using a permanent magnet in combination. 2. Noncontact Guide System We supported the continuous steel plate production line with a deflector and a sink roll. As shown in Figure 1, to form a noncontact guideway using an electromagnet for the roll part, we fabricated a device simulating a part through which the continuous steel plate passes through the plating tank. A continuous steel plate was welded to a 6894 mm long, 150 mm wide, and 0.3 mm thick belt made of stainless steel material (SUS632). The width of the steel plate in the current experiments was chosen to reduce the vertical deformation and the elastic vibration of the steel plate plane and to consider the impact of the edge control system on the loop-shaped part. The belt was hung from a pulley with a diameter of 700 mm and a width of 154 mm. A DC servomotor, enabling endless movement of the continuous steel plate, drove the pulley. In addition, electromagnets were installed in two locations when the experiments were conducted. Figure 2 shows schematic illustration of electromagnet. The number of turns of the electromagnet coil is 1005 times, and the material of the E type core is ferrite. The dimensions of the electromagnet are as shown in Figure 2. A noncontact edge control system, which was constructed to perform the active control with the aid of the electromagnet with respect to the edge direction of the running steel plate, was experimentally established. To compensate for the attractive force shortage, the electromagnets were connected in two series connections and arranged to face the adjacent opposite edges of the steel plate. As shown in Figure 3, the permanent magnets were also for the magnetic poles to oppose to each other placed near both the edges of the steel plate.
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Laser sensor ( Laser z-direction) sensor x2 ( z-direction) Electromagnet x2 No. 2 Laser sensor ( z-direction) y2 Electromagnet No. 2 x2 yElectromagnet 2 z2 No. 2 y2 z x2 Laser sensor 2 x2 ( Laser y-direction) sensorz2 ( y-direction) x2 Laser sensor
No. 1 No. 1 No. 1 z1 z1 zx11 x1 x1
( y-direction)
A' Front view A' A' Front view
Side view ( A-A' ) Side view ( A-A' )
Front view
Side view ( A-A' )
Figure 1. Non-contact guide system. Figure Non-contact guide system. Figure guide system. Figure1.1. 1.Non-contact Non-contact guide system.
Coil (1005 turns)
15
88 8
8 88
6060 60
1515
Coil (1005 turns) Coil (1005 turns)
36 3636
15 8 14 16 15158 81414 1616
Ferrite core Ferrite core
Ferrite core
Figure Schematic illustration illustration ofofelectromagnet. Figure 2.2.Schematic Schematic electromagnet. Figure 2. illustration of electromagnet. Figure 2. Schematic illustration of electromagnet.
Steel plate
Attractive force
S S
force S Attractive N Attractive force
Steel plate Steel plate
N N
Attractive force
N
Attractive force Attractive force
S
N N
Permanent magnet
S S
Permanent magnet
Permanent magnet Permanent magnet
Permanent magnet Permanent magnet
(a)
(a) (a) Figure 3. Cont.
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(b)edges of the steel plate. (a) Schematic drawing; (b) Figure 3. Permanent magnets placed near both the Figure 3. Permanent magnets placed near both the edges of the steel plate. (a) Schematic drawing; Photograph of arrangement. Figure 3. Permanent magnets placed near both the edges of the steel plate. (a) Schematic drawing; (b) (b) Photograph of arrangement. Photograph of arrangement. 3. Noncontact Edge Control System
3. Noncontact Edge 3. Noncontact EdgeControl ControlSystem System 3.1. Edge Control System
3.1. Edge A Control System is caused in the y direction of the continuous steel plate by the active control displacement 3.1. Edge Control System from electromagnets. The control system is shown in Figure 4. In thissteel experiment, pair of serially AA displacement the continuous platebybyathe theactive active control displacementisiscaused causedin inthe the yy direction direction of of the continuous steel plate control connected electromagnets is placed facing each other near the two edges of the continuous steel plate from electromagnets. The control system is shown in Figure 4. In this experiment, a pair of serially from electromagnets. control systemforce. is shown in Figure 4. In this experiment, a pair of serially to supplement for theThe lack of attractive For both the electromagnets, the distance between the connected electromagnets is placedfacing facingeach each other near the two edges of the continuous steel connected electromagnets is placed other near the two edges of the continuous steel plate electromagnet surface and the edge of the continuous steel plate is maintained at 5 mm. The plate to supplement for the lack of attractive force. For both the electromagnets, the distance to information supplement on for the lack of attractive force. For both electromagnets, between the displacement, traveling speed, and the current are locally the feddistance back only to the between the electromagnet surface and the edge of the continuous steel plate is maintained electromagnet surface and the edge of continuous steel plate is maintained at 5 mm. The electromagnets. In the z direction, the steel plate is guided from the initial position in the uncontrolled at 5 mm. information displacement, traveling and are fed back only information on displacement, traveling andspeed, current arecurrent locally fedlocally back to the stateThe to the center of theon electromagnet. Thisspeed, phenomenon occurs due to the magnetic fieldonly generated In the z direction, the steel plate is guided from the initial position in the uncontrolled to electromagnets. the electromagnets. In the z direction, the steel plate is guided from the initial position in for positioning control in the y direction. This phenomenon causes as an attractive force on the steel the state to the ofitthe the electromagnet. This phenomenon due to theoccurs magnetic generated uncontrolled state center the electromagnet. Thisoccurs phenomenon duefield to the magnetic plate thatcenter drivesto toward theofelectromagnet center. Thus, it is experimentally confirmed that the forrange positioning the control y direction. This phenomenon as an attractive on the steel in which this in attractive force the z direction of the steel plate is the outermost field generated forcontrol positioning in acts the yin direction. This causes phenomenon causes asforce an attractive force that plate drivesthat toward the electromagnet center. Thus, it is Thus, experimentally confirmed confirmed that the circumference ofitthe electromagnet coil. onplate the steel drives it toward the electromagnet center. it is experimentally range in which this attractive force force acts in direction of the steel plate is is thethe outermost that the range in which this attractive actsthe in zthe z direction of the steel plate outermost circumference of the electromagnet coil. DSP(ADSP324-00A) AMP4 circumference of the electromagnet coil. D/A AMP3 AMP2 AMP1 AMP4 AMP3 AMP2 AMP1
converter
D/A converter
50MHz y2 y1 Digital DSP(ADSP324-00A) differentiator 50MHz y2 y1 A/D converter Digital differentiator y2 y1
A/D converter y2
y1
Traveling direction
Laser sensor No.2 Electromagnet No.2
Traveling direction
Laser sensor No.1 Electromagnet No.1
Figure 4. Control system of noncontact guide.
Laser sensor No.2 Electromagnet No.2
Laser sensor No.1 Electromagnet No.1
Figure4.4.Control Control system system of Figure of noncontact noncontactguide. guide.
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3.2. Modelling System 3.2. Modelling System The parts where the direction of travel is changed (indicated by curves) and the linear-traveling Theare parts whereinto thesections, directionas ofshown travel is by curves) and the parts divided inchanged Figure 5.(indicated Each section is modeled as alinear-traveling target of control. parts divided into sections, shown inwhere Figurethe 5. Each section is modeled a target ofby control. For are a model with one degree as of freedom weight of each section is as represented mn and For a model onegenerated degree of freedom where the weight of each section by m and the is the controlwith force by the electromagnet is represented by isfn,represented the equation ofn motion control force generated by the electromagnet is represented by fn , the equation of motion is expressed as expressed as
y=n 2=f n2 f n mm n ynn ..
(1) (1)
Electromagnet
Laser sensor I0 + vn
F0 - fn
F0 + fn
mn
yn
lxn
120
I0 - vn
xn Steel belt Figure 5. Model of of noncontact guide. Figure 5. Model noncontact guide.
The inductance electromagnetic coil expressed a sum component inversely The inductance ofof thethe electromagnetic coil is is expressed asas a sum of of thethe component inversely proportional to the gap between the steel plate and magnet and the component of leakage inductance. proportional to the gap between the steel plate and magnet and the component of leakage inductance. deviation around static equilibrium state very small, characteristic equations If If deviation around thethe static equilibrium state is is very small, thethe characteristic equations of of thethe electromagnet are linearized as electromagnet are linearized as 2F0 2F0 f n = 2F y0n + (2) 2vFn f =Γ0 y I+0 R 0 v (2) n
n
Γ0
where
n
I0 R
vn = Rin
where
(3)
F0 : magnetic force in the equilibrium state (0.83 N), Γ0 : gap between steel plate and electromagnet v = Rin state (0.5 A), R: resistance of the coupled(3) in the equilibrium state (59 mm), I0 : current in the nequilibrium magnetFcoils (10 Ω), yn : horizontal displacement [m], vn : dynamic voltage of the coupled magnets [V], 0: magnetic force in the equilibrium state (0.83 N), Γ0: gap between steel plate and electromagnet in :in dynamic current ofstate the coupled magnets [A]. in the equilibrium state (0.5 A), R: resistance of the the equilibrium (59 mm), I0: current The state equation of the system is introduced from the from motion, coupled magnet coils (10 Ω), yn: horizontal displacement [m], the vn: equation dynamic of voltage of that the is, coupled . magnets [V], in: dynamic current of the coupled magnets [A]. yn = Ay yn + By v n (4) The state equation of the system is introduced from the from the equation of motion, that is,
where
y n =h Ay yn. +iB T y vn
yn =
where
yn
"
0
y n = [4FΓy00n
Ay =
1 0
(4) (5)
,
yn
#
y n ] ,
T
,
(6) (5)
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I0 R
4.4.Vibration Change VibrationPhenomenon PhenomenonAccompanying AccompanyingSteady SteadyCurrent CurrentValue Value Change First, First,we wecompared comparedthe thevibrations vibrationsofofthe thecontinuous continuoussteel steelplate platecaused causedby bythe thedifferences differencesininthe the steady current value. From the basic experiments, it has been confirmed that the steel sheet can be steady current value. From the basic experiments, it has been confirmed that the steel sheet can be controlled controlledstably stablyfor for3030min minorormore morewhen whenthe thesteady steadycurrent currentisis0.70.7AAororless. less.InInaddition, addition,ititwas was confirmed that steel plates with a traveling velocity of 1000 m/min could be controlled when the confirmed that steel plates with a traveling velocity of 1000 m/min could be controlled when the steady steadycurrent currentisis0.3 0.3AAorormore. more.InInthe theexperiment, experiment,two twosteady steadycurrent currentvalues valuesofof0.3 0.3AAand and0.7 0.7AAwere were used. used.From Fromthe theprevious previousstudy, study,the thestatic staticattractive attractiveforce forcewhich whichthe theelectromagnet electromagnetgenerates generateson onthe the steel plate is 0.1 N when steady current is 0.3 A, and 0.6 N when steady current is 0.7 A [11]. We have steel plate is 0.1 N when steady current is 0.3 A, and 0.6 N when steady current is 0.7 A [11]. We have confirmed confirmedthat thatthe theRMS RMSinincontrol controldirection direction(y-axis) (y-axis)isissame sameinineach eachsteady steadycurrent. current.When Whenrunning running the was onon the theexperiments, experiments,the theelectromagnet electromagnet wasinstalled installed theloop loopshaped shapedpart partofofthe thesteel steelplate. plate.As Asshown shown ininFigure 6, the laser sensor was installed radially from the center assuming that a pulley is installed Figure 6, the laser sensor was installed radially from the center assuming that a pulley is installed ◦ . In this section, the sensor was under measured every underthe theloop loopofofthe thesteel steelplate, plate,and andit itcan canbebe measured every1515°. In this section, the sensor was ◦ ◦ ◦ ◦ ◦ ◦ installed at 30 , 45 , 75 , 90 , 135 , and 150 on the lower part of the steel installed at 30°, 45°, 75°, 90°, 135°, and 150° on the lower part of the steelplate, plate,and andmeasured measuredthe the displacement displacementininthe theout-of-plane out-of-planedirection directionfor for3030s.s.The Thevibration vibrationofofthe thesteel steelplate plateincludes includesvarious various frequency measured displacement displacementof ofthe thesteel steelplate plateto frequencyvibration. vibration. Therefore, Therefore, we we obtained RMS from measured toevaluate evaluatethe thevibration vibration of the steel plate. During the running experiment, the running speed of the steel plate. During the running experiment, the running speed waswas 1000 1000 m/min, the data were measured times, then average was evaluated. Figure 7 shows m/min, the data were measured ten ten times, andand then thethe average was evaluated. Figure 7 shows the the amplitude measurement point at each steady current value. a result, it was confirmed amplitude of of thethe measurement point at each steady current value. As As a result, it was confirmed that that A was to suppress vibration compared to 0.3 0.7 0.7 A was ableable to suppress vibration compared to 0.3 A. A. Therefore, Therefore,it itwas wasconfirmed confirmedthat thatstable stableguidance guidancecan canbebeperformed performedononthe thesteel steelplate plateconveyance conveyance with a steady current value of 0.7 A. When the steady current value is strong, stable guidance with a steady current value of 0.7 the steady current value is strong, stable guidance of of the the steel sheet performed. However, heat was generated in the electromagnet under control steel sheet cancan bebe performed. However, heat was generated in the electromagnet under control due due to increase an increase in the current value. Furthermore, problems an increase inresistance the resistance to an in the current value. Furthermore, problems suchsuch as anasincrease in the value value due to the heat generation of the electromagnet and a decrease in the overall cost performance due to the heat generation of the electromagnet and a decrease in the overall cost performance were were noted. Therefore, guidance control a state wherethe thesteady steadycurrent current value value is low noted. Therefore, guidance control in in a state where low isisdesired. desired. Moreover, a stable guidance is desired with small amplitude. Therefore, a permanent magnet Moreover, a stable guidance is desired with vibration small vibration amplitude. Therefore, a permanent was used in combination with a steady current value of 0.3 A. The of performance this combination magnet was used in combination with a steady current value 0.3 A. The of performance of this was then compared withcompared that usingwith a steady current value of 0.7 A,value and we the combination was then that using a steady current of investigated 0.7 A, and wewhether investigated performance increased or remained equal. whether the performance increased or remained equal.
Figure Figure6.6.Sensing Sensingposition positionononthe thelower lowerpart partofofthe thesteel steelplate. plate.
RMS RMSofofdisplacement displacementz (mm) z (mm)
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30 30
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75 90 75θ (°) 90 θ (°)
135 135
150 150
Figure 7. The RMS of displacement in the z direction of the steel plate at each steady current value. Figure Figure 7. 7. The The RMS RMS of of displacement displacement in in the the z direction of the steel plate at each steady current value.
5. Consideration on Arrangement of Permanent Magnets 5. 5. Consideration Consideration on on Arrangement Arrangement of of Permanent Permanent Magnets Magnets 5.1. Permanent Magnet Analysis 5.1. Permanent Magnet Analysis Until now, we determined the arrangement of electromagnets from the analysis of the stationary Until now, we determined the arrangement of electromagnets from the analysis of the stationary stationary shape of the steel plate and we damped the vibration of the steel plate. We investigated the calculation shape of the steel steel plate plate and and we we damped damped the the vibration vibration of of the the steel steel plate. plate. We We investigated investigated the the calculation calculation of vibration mode shape of steel plate from the dynamic analysis and placement of permanent of vibration vibrationmode modeshape shape steel plate the dynamic analysis and placement of permanent of of steel plate fromfrom the dynamic analysis and placement of permanent magnets magnets at the position which becomes the antinode of the mode shape of the characteristic frequency magnets at the position which becomes the antinode of theshape mode of shape of the characteristic frequency at the position which becomes the antinode of the mode the characteristic frequency band. band. First, electromagnetic analysis software JMAG was used to consider permanent magnets. We band. First, electromagnetic software JMAG was used to consider permanent We First, electromagnetic analysisanalysis software JMAG was used to consider permanent magnets. magnets. We analyzed analyzed the static characteristics of the permanent magnet acting on the steel plate. The material was analyzed static characteristics of the permanent acting on the steelThe plate. The material was the static the characteristics of the permanent magnet magnet acting on the steel plate. material was set as set as follows. The steel plate was SS400, the coil was copper, the core was ferrite with TDK PC 47 set as follows. Theplate steelwas plate was the SS400, was the copper, the core was ferrite PC 47 follows. The steel SS400, coil the wascoil copper, core was ferrite with TDKwith PC 47TDK (remanent (remanent flux density Br = 180 mT, coercive Hc = 13 A/m), the permanent magnet was TDK FB5B (remanent density Brcoercive = 180 mT, Hc =the 13 permanent A/m), the permanent magnet was TDK FB5B flux densityflux Br = 180 mT, Hccoercive = 13 A/m), magnet was TDK FB5B (remanent (remanent flux density Br = 420 mT, coercive Hc = 253 kA/m). The size of the permanent magnet is (remanent flux Br =coercive 420 mT,Hc coercive Hc = 253The kA/m). sizepermanent of the permanent is flux density Br density = 420 mT, = 253 kA/m). sizeThe of the magnet magnet is 30 mm 30 mm in length, 30 mm in width and 15 mm in thickness. From the analysis result, Figure 8 shows 30 mm in length, in width in thickness. the analysis result, Figure 8 shows in length, 30 mm30 in mm width and 15and mm15inmm thickness. FromFrom the analysis result, Figure 8 shows the the relationship between the y-axis direction of the steel plate and the attractive force fy acting on the the relationship between the y-axis direction ofsteel the steel andattractive the attractive fy acting onsteel the relationship between the y-axis direction of the plateplate and the forceforce fy acting on the steel plate. As shown in Figure 8, fy has the maximum attractive force acting when the distance from steel As shown in Figure 8, fy the hasmaximum the maximum attractive acting when the distance from plate.plate. As shown in Figure 8, fy has attractive forceforce acting when the distance from the the steel plate is 25 mm, and then the attractive force tends to decrease from 30 mm to 70 mm. the plate 25 mm, the attractive tends to decrease 30 mm 70 mm. steelsteel plate is 25ismm, andand thenthen the attractive forceforce tends to decrease fromfrom 30 mm to 70tomm.
Attractive Attractiveforce forcefyfy(N) (N)
0.25 0.25 0.2 0.2
0.15 0.15 0.1 0.1
0.05 0.05 0 0
0 0
10 10
20 20
30 30
40 40
Gap (mm) (mm) Gap
50 50
60 60
70 70
Figure 8. Relationship attractive force f y of the permanent magnet and gap. Figure 8. Relationship attractive force fyy of the permanent magnet and gap.
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5.2. Permanent Magnet Arrangement Using ANSYS To conduct a running experiment with permanent magnets in combination, we need to consider 5.2. Permanent Magnet Arrangement Using ANSYS the arrangement of permanent magnets for vibration suppression. The running steel plate vibrates in To conduct a running withmode permanent in combination, we to consider a specific frequency band. experiment If the vibration shape magnets of the frequency band can beneed confirmed, it is the arrangement of permanent magnets for vibration suppression. The running steel plate vibrates in possible to suppress the vibration by performing control considering the mode shape. The tension a specific frequency the vibration mode shape is ofcalculated the frequency canfriction be confirmed, it is generated in the steelband. plate Ifduring continuous running fromband Euler’s theory [12]. possible to suppress the vibration by performing control considering the mode shape. The tension The calculated tension is applied to the shape of the steel plate obtained from MBD. At this time, we generated in theplate steelact plate during continuous running is calculated from Euler’s made the steel as loose side tension and pull side tension. From Euler’s friction friction theory theory,[12]. the The calculated tension is applied to the shape of the steel plate obtained from MBD. At this time, effective tension of the running steel plate is Te, the loose side tension is Ts, the pull side tension iswe Tt made theare steel plate actby as the loose side tension and pull side tension. From Euler’s friction theory, the and they expressed following equation: effective tension of the running steel plate is Te , the loose side tension is Ts , the pull side tension is Tt P and they are expressed by the following equation: Te = (8) V , P Te =μψ , (8) Te × e V 2 Tt = μψ + mV (9) e × −eµψ 1 Te Tt = µψ + mV 2 ,, (9) e −1 Te T s = μψTe + mV 2 2 (10) (10) Ts = e µψ − 1 + mV , e −1 Here, V is the steel plate with running speed (m/min), is the the motor motor output output (W), (W), μ µ is the (m/min), PP is coefficient pulley, ψ coefficient of of friction friction between between the the steel steel plate and the pulley, ψ is the winding angle (rad) of the portion where where the belt is in contact with the pulley, pulley, and and m m is is the the mass mass per per unit unit length length of of the steel plate plate (kg/m). (kg/m). In this consideration, consideration, the We applied applied loose side tension the steel steel plate plate running running speed speed is is set set to to 1000 1000 m/min. m/min. We Tss and and pull pullside sidetension tensionTT thisspeed speed along surface of the steel plate. Using Equations (9) (10), and t at this along thethe surface of the steel plate. Using Equations (9) and t at (10), TtTand Ts is obtained andN.113 N. Then, frequency response analysis and modal analysis Tt and s is obtained to 176toN176 andN113 Then, frequency response analysis and modal analysis were were performed. The mesh of model the model a shell element whose numberisis264, 264,and andthe theplace place where performed. The mesh of the waswas a shell element whose number the steel plate and the pulley contacted was fixed. ANSYS Workbench Mechanical was used for this analysis. Figure 9a shows the spectral waveform waveform at the steady current value of 0.3 A, and Figure 9b shows the result of the frequency response analysis. From the analysis result, the twist modes which include the HzHz areare confirmed. In the actual experiment, it is it considered that the vibration vibrationfrom from1.5 1.5Hz Hztoto5.05.0 confirmed. In the actual experiment, is considered these twist twist modesmodes are suppressed by edgeby control we concluded that the resonance that these are suppressed edge system. control Therefore, system. Therefore, we concluded that the of the experiment is the peak of Thisvibration. frequencyThis is in frequency the frequency resonance of the experiment is the theout-of-plane peak of thevibration. out-of-plane is inband the around 1.048 Hz.around The vibration mode of this frequency was calculated by modal analysis. frequency band 1.048 Hz. Theshape vibration mode shape band of this frequency band was calculated From the result of the From modalthe analysis, magnets were placed in consideration of theplaced vibration by modal analysis. resultpermanent of the modal analysis, permanent magnets were in mode shape of of thethe steel plate. Figure is a schematic diagram. The permanent magnets were installed consideration vibration mode10 shape of the steel plate. Figure 10 is a schematic diagram. The at two position 575 mm andinstalled 1450 mm the center575 of amm pulley diameter mm.ofTherefore, permanent magnets were atfrom two position andwith 1450amm from of the700 center a pulley four wereTherefore, installed on each side. A total of eight permanent magnets withpermanent a diametermagnets of 700 mm. four permanent magnets were installed on eachwere side.installed A total on both sides on themagnets edge portion the steel of eight permanent were of installed onplate. both sides on the edge portion of the steel plate. 1 0.8
0.8
Translation (mm)
Translation (mm)
1
0.6 0.4 0.2 0
0
1
2
3
Frequency (Hz)
(a)
4
5
0.6 0.4 0.2 0 0
1
2
3
4
Frequency (Hz)
(b)
Figure 9.9.Frequency Frequency response analysis of continuous plate. (a) Experiment Figure response analysis of continuous steel plate. steel (a) Experiment result; (b) Analysisresult result.; (b) Analysis result.
5
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Figure 10. 10. Mode Mode shape shape and and arrangement arrangement of of permanent permanent magnets. magnets. Figure
6. Steel Steel Plate Plate Traveling Traveling Experience Experience 6. From the theresults resultsinin Section we could estimate the placement position of the permanent From Section 5.2,5.2, we could estimate the placement position of the permanent magnet. magnet. This arrangement was confirmed of the vibration suppression performance of steady This arrangement was confirmed of the vibration suppression performance of steady currentcurrent value value 0.3 A by conducting the experiments. Based on the previous studies, we have confirmed the 0.3 A by conducting the experiments. Based on the previous studies, we have confirmed the vibration vibration amplitude of the traveling steel plate without control was approximately 40 mm in amplitude of the traveling steel plate without control was approximately 40 mm in y direction andy direction andz direction 20 mm in[6].the z direction [6]. Therefore, experiment waselectromagnet conducted with the 20 mm in the Therefore, the experiment wasthe conducted with the control electromagnet control each condition. the result of the basic experiment, theout experiment in each condition. Frominthe result of basic From experiment, when experiment waswhen carried with gap was carried out with gap 25 mm which the attractive force was maximum, the steel plate and the 25 mm which the attractive force was maximum, the steel plate and the permanent magnet were in permanent magnet were in contact with each other, making measurement impossible. Therefore, the contact with each other, making measurement impossible. Therefore, the experiments were carried experiments carried the gap 30 mm which can be speed maintained. with the gap were 30 mm whichwith the positioning control canthe be positioning maintained.control The running was setThe to running speed was set to 1000 m/min, similar to that in the experiment in Section 4. The sensor ◦ at 1000 m/min, similar to that in the experiment in Section 4. The sensor was placed from 0◦ to 180was placed from to 180° at intervals of shaped 15° on the steel loop shaped The displacement was ◦ on intervals of 150° the steel plate loop part. Theplate displacement waspart. measured in the z direction, measured in the z direction, and the displacement was evaluated by using RMS. Figure 11 shows the and the displacement was evaluated by using RMS. Figure 11 shows the measurement results of the measurement results of the displacement RMS of displacement obtained from the experiment. From displacement RMS of displacement obtained from the experiment. From this result, the symbol [ ] in this result, the symbol [●] in Figure the measurement the guideway steady current Figure 11 shows the measurement of11 theshows guideway with a steadyof current value of with 0.7 Aaand the symbol value of 0.7 A and the symbol [■] shows the measurement of the guideway with a steady current [] shows the measurement of the guideway with a steady current value of 0.3 A with the permanent value of The 0.3 Asymbol with the magnet. Thethe symbol [▲] inwith Figure 11 compares guideway magnet. [Npermanent ] in Figure 11 compares guideway a steady currentthe value of 0.3 Awith and a steady current value of 0.3 A and the guideway on which the permanent magnet is placed. In the the guideway on which the permanent magnet is placed. In the case where the steady current value is casesame, where the steady current value is the same, a decrease in the RMS Thus, can be confirmed at all the a decrease in the RMS can be confirmed at all measurement points. it can be confirmed measurement Thus, it can be confirmedwithout that vibration can bemagnet suppressed. Comparing without that vibration points. can be suppressed. Comparing permanent with steady current 0.7 A permanent magnet with steady current 0.7 A and with permanent magnet with steady current and with permanent magnet with steady current 0.3 A, vibration can be suppressed more than 0.3 0.7 A, A vibration can be suppressed more than 0.7 A by disposing a permanent magnet. by disposing a permanent magnet. As aa result, result, it it was was confirmed confirmed that that it it exceeds exceeds the the vibration vibration suppression suppression performance performance of of the the steady steady As current value of 0.7 A. At 90° of the measurement point in both the figures, it is considered that the ◦ current value of 0.7 A. At 90 of the measurement point in both the figures, it is considered that the vibration reduction amount decreases because the lowest point of the steel plate corresponds to the vibration reduction amount decreases because the lowest point of the steel plate corresponds to the part where where vibration vibration hardly hardly changes. changes. part
RSM of displacement z (mm)
Actuators 2018, 7, 47 Actuators 2018, 7, x FOR PEER REVIEW
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without permanemt magnet 0.7 A without permanemt magnet 0.3 A with permanent magnet gap30 mm 0.3 A
1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0
15
30
45
60
75
90 105 120 135 150 165 180 θ (°)
Figure Experimental results each guideway. Figure 11.11. Experimental results of of each guideway.
Conclusions 7. 7. Conclusions thisresearch, research,we we analyzed analyzed the changing thethe steady current value InIn this the vibration vibrationphenomena phenomenawhen when changing steady current of the electromagnet used for controlling the traveling steel plate. We confirmed whether value of the electromagnet used for controlling the traveling steel plate. We confirmed whetherthe permanent magnets magnetswith withlow lowsteady steadycurrent currentvalue. value. thephenomenon phenomenoncan canbe bestable stableguidance guidance by by using using permanent Owing to varying the steady current value, stable guidance was possible in the case of a high steady Owing to varying the steady current value, stable guidance was possible in the case of a high steady current value.However, However,the theelectromagnet electromagnetunder undercontrol controlhas has aa problem problem of current value. of heat heat generation generationdue dueto toan there was was aapossibility possibilitythat thatcontrol controlperformance performancecan can anincrease increasein in current current value. value. Furthermore, Furthermore, there bebe deteriorated by an increase in the resistance value. To confirm the vibration suppression performance deteriorated by an increase in the resistance value. To confirm the vibration suppression performance lowsteady steadystate state current current value wewe analyzed thethe vibration mode of the at at a alow value using usingaapermanent permanentmagnet, magnet, analyzed vibration mode plateplate by ANSYS. Permanent magnets were placed in consideration of the calculated vibration of steel the steel by ANSYS. Permanent magnets were placed in consideration of the calculated mode shape and experiments were conducted. As a result, it was found that vibration canbebe vibration mode shape and experiments were conducted. As a result, it was found that vibration can suppressed measurement points compared with the guideway strong steady current value. suppressed at at allall measurement points asas compared with the guideway ofof strong steady current value. The arrangement of the permanent magnets exhibited the damping effect even at the same steady The arrangement of the permanent magnets exhibited the damping effect even at the same steady current value. Therefore, it was clarified that performance vibration suppression performance current value. Therefore, it was clarified that thethe performance of of vibration suppression performance improved arranging permanent magnets with low steady current value. future, will is is improved byby arranging permanent magnets with thethe low steady current value. InIn future, wewe will investigate the optimum gap of permanent magnets to confirm further vibration suppression effects. investigate the optimum gap of permanent magnets to confirm further vibration suppression effects. Author Contributions: T.B., Y.N., wrote the paper; T.B.,Y.N., S.I., R.Y. Y.N.,carried R.Y. carried an experiments; Author Contributions: T.B., S.I.,S.I., Y.N., R.Y.,R.Y., wrote the paper; T.B., S.I., out an out experiments; T.B., S.I., Y.N.,S.I., R.Y.Y.N., contributed the analyses the data; of andthe T.N. andand H.K.T.N. contributed thecontributed design of thethe controller T.B., R.Y. contributed theofanalyses data; and H.K. design and of the electric circuit. controller and electric circuit. Funding: This research received no external funding. Funding: This research received no external funding. Conflicts of Interest: The authors declare no conflict of interest. The founding sponsors had no role in the design The authors declare no conflict of interest. founding had no role and in theindesign of Conflicts the study;ofinInterest: the collection, analyses, or interpretation of data;The in the writingsponsors of the manuscript, the of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the decision to publish the results. decision to publish the results.
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