Proc. Schl. Eng. okai okai Univ., Univ., Ser. ESer. E 4(15 - (15 77-8 Bending Magnetic Levitation Control Appling the Continuous Model of Fleile Steel Plate Hikaru YONEZAWA *1, Hiroki MARUMORI *1, akaoshi NARIA * and Hideaki KAO *3 (Received on Mar. 31, 15 and accepted on Jul. 15, 15 Astract We have proposed the levitation of an ultrathin steel plate that has een ent to an etent that has not induced plastic deformation. It has een confirmed that virations are suppressed and levitation performance is improved. However, when the steel plate is ent and levitated, the elastic viration occurs. In practice, since the steel plate is ver thin, elastic viration cannot e sufficientl suppressed using onl a limited numer of electromagnets, and levitation performance is not alwas impeccale. o model a fleile thin steel plate, we appl a continuous model which epresses the motion of the plate. Appling the optimal control theor to the continuous model, the ending levitation eperiments were carried out. It was concluded ased upon the applied continuous model, that levitation performance ecome stale and desirale ending levitation performance was achieved. Kewords: Fleile steel plate, Electromagnetic levitation sstem, Elastic viration, Bending levitation control, Continuous model 1. Introduction hin steel plates are widel used as materials for automoiles, electric appliances, cans and other products in current industries. With various industrial demands, the surface qualit of steel plates continues to e enhanced. However, ecause a contact conveance using rollers is mainl adopted in the process of a thin-steel-plate production line, the prolem of surface qualit deterioration arises. In recent ears, as a countermeasure for this prolem, researches on the noncontact conveance sstem with the application of electromagnetic levitation technolog have ecome active 1-4. In the past, our research group have constructed an electromagnetic levitation control sstem with which the relative distance etween electromagnets and a steel plate are constantl maintained, aiming to prevent the steel plate from falling from the conveer or coming into contact with the electromagnet during electromagnetic levitation conveance 5. However, as the steel plate ecomes thinner, the viration caused minute unpredictale factors, including the *1 Graduate Student, Course of Mechanical Engineering * Assistant Professor, Department of Electrical and Electronic Engineering, oko Universit of Science, Suwa, Japan *3 Assistant Professor, Department of Prime Mover Engineering nonlinearit of the attractive force of the electromagnet and the change in resistance due to heat generation the electromagnet, makes it difficult to maintain the levitation state. Furthermore, when an ultrathin steel plate with a thickness of less than.3 mm is targeted for levitation, the levitation control ecomes difficult ecause the thin plate undergoes increased fleure. o overcome these prolems, we propose a levitation of an ultrathin-steel-plate that is ent to an etent that does not induce plastic deformation 6. It has een confirmed that virations with mainl low frequencies are generated when a steel plate is ent and levitated. In addition, the levitation performance of steel plate is markedl improved 7. However, when the steel plate is ent and levitated, the elastic viration arises. In practice, since the steel plate is ver thin, the elastic viration cannot e sufficientl suppressed using onl a limited numer of electromagnets, and hence it is essential to eliminate the elastic viration appling the control theor as much as possile. he control method used in the past studies of the authors had the advantage that the control sstem can e designed in a simple. However, specific virations that occur from eing a fleile steel plate were not ale to e considered this control method. In other words, it was a model for independentl feeding ack the status of each of Vol., 15 77
Hikaru YONEZAWA, Hiroki MARUMORI, akaoshi NARIA and Hideaki KAO Hikaru YONEZAWA, Hiroki MARUMORI, akaoshi NARIA and Hideaki KAO Resistance Amplifier AMP5 AMP4 AMP3 AMP AMP1 D/A converter DSP (MS3C31 4MH A/D converter Steel plate (8 mm 6 mm.7 mm Gap sensor i 1 i i 3 i 4 i 5 1 3 4 5 Electromagnet Electromagnet unit No.3 No.4 No.5 No.1 No. Steel plate (8 mm 6 mm.7 mm Fig. 1 Electromagnet control sstem θ A A' Front view Side view (A-A' section Fig. Schematic illustration of eperimental apparatus the electromagnets positions. his stud is intended to suppress the elastic 1st mode that is the most dominant in the viration generated. We applied the continuous model that can e considered integrating the state acquired at all of the electromagnets positions. In this paper, appling the optimal control theor to the continuous model, ending levitation eperiments are carried out. We eamined the levitation stailit and levitation performance using a thin steel plate with a thickness of.7 mm.. Modeling of Steel Plate 5 In eperimental apparatus, we used the same sstem as that used Marumori 8. Figure 1 shows an outline of the electromagnet control sstem. Figure shows a schematic illustration of eperimental apparatus. In the past stud, the steel plate was divided into 5 hpothetical masses and each part was modeled as a lumped constant sstem 6. he 1 degree of freedom model (1-DOF model that has een used in past studies of the authors was not ale to consider specific virations that occurred from eing a fleile steel plate. his stud is intended to suppress the elastic 1st mode that is the most dominant in the viration generated. We applied the continuous model that can e considered integrating the state acquired at all of the electromagnets positions. In a continuous model, integrated control is carried out calculating 15 values: displacement, velocit of steel plate and all the current of the electromagnet which detected at each position of the five electromagnets. In this model, the motion of the steel plate is calculated from the equations of its elastic viration. Supporting the plate the static attracting force of each magnet creates an equilirium levitation state, where the steel plate maintains a certain distance from the electromagnets. he equation of small vertical motion around the equilirium state of the steel plate sujected to magnetic forces is epressed as follows: 3 Ch 4 4 h D t 1 t 5 f cn ( t{ ( a1 n ( a1n n1 ( an ( an } (1 4 4 4 4 ( 4 4 Where C: internal damping coefficient [Ns/m ], D = Eh 3 /1(1-v [Nm], v: Poisson s ratio, f cn (t: dnamic magnetic force at the n-th coupled magnets [N], t: time [s], (, : vertical displacement [m],,, : coordinate aes indicated in Fig. 1 [m], a1n, an, a1n, an : location of the n-th coupled magnets (n = 1-5 [m], δ( : Dirac delta function [l/m]. he characteristic equations of the electromagnets can e derived as follows: Fn F f n cn sn, sn in Zn In Leff In R 1 in sn, sn in v L L Z n L Leff L Llea Zn (3 n (4 (5 Where F n : magnetic force of the coupled magnets in the equilirium state [N], I n : current of the coupled magnets in the equilirium state [A], sn, sn : position of the n-th sensor [m], v n : dnamic voltage of the n-th coupled magnets [V], i n : dnamic current of the n-th coupled magnets [A], Z n : gap etween the steel plate and electromagnet in equilirium state [m], L : inductance of one magnet coil in equilirium state [H], R : resistance of the coupled magnet coils [Ω], and L lea : leakage inductance of the one magnet coil [H]. 78 Proceedings of the School of Engineering, okai Universit, Series E
Bending Magnetic Levitation Control Appling the Continuous Model of Fleile Steel Plate Bending Magnetic Levitation Control Appling the Continuous Model of Fleile Steel Plate 1st mode 4th mode ( φ 1 = X 1 Y 1 ( φ 4 = X Y nd mode ( φ = X Y 1 3rd mode ( φ 3 = X 1 Y 5th mode (elastic 1st mode ( φ 5 = X 1 Y 3 Fig. 3 Mode shapes of the levitated steel plate State variales of the sstem are normal coordinates of vertical displacement of the plate W i (t (i = 1-5, their differential values Ẇ i (t (i = 1-5, and dnamic currents of the coupled magnet coils i n (t (n = 1-5. he control input of the sstem is the dnamic voltages of the magnets v n (t (n = l - 5. Output variales of the sstem are vertical displacements n ( sn, sn, t (n = 1-5. Using the state, control and output vectors, the forgoing eqs. (3-(6 are written as following state and output equations: W AW Bv (15 n CW (16 W 1 W5 W1 W W 5 i1 i5 (17 v v1 v 5 (18 n 1 5 1 5 i1 i5 (19 3. State Equation and Controller he vertical displacement of the plate can e epanded to an infinite series of a space-dependent eigenfunctionφ i (, as shown in Fig. 3 multiplied the time-dependent normal coordinate. he eigenfunctions of the plate are assumed to e products of the elastic eam eigenfunctions of the - and -coordinates. he function of -coordinate Y nn ( (nn = l,, satisfies the free-free oundar condition, and the function of the -coordinate is epressed in rigid modes (parallel and rotational motions X 1 (, X ( onl. In addition, since the numer of sensors used in this eperiment are 5, we selected M = 5 for the control in which consideration is given to the 5th mode (elastic 1st mode. M i i (6 i1,, W t i, X Y (mm, nn = 1,, (7 1 mm nn X 1 (8 3 X a (9 a 1 Y (1 1 3 Y (11 Ynn λnn λnn cos cosh sinλnn sinhλnn λ nn nn sin sinh cos nn cosh nn (1 cosh λ nn cos λnn 1 (13 1 λnn D f nn h (14 Here, details of matrices A, B and C are omitted due to space limitations 5. In this stud, a control sstem is constructed using a discrete time sstem; therefore, the evaluation function of a continuous sstem is digitied, and the optimal control law is otained ased on the optimal control theor of the discrete time sstem. he following discrete time sstem is here considered. d i Φ i Γv i 1 ( d d Φ ep A s (1 Γ s ep A d B ( Here, the evaluation function of the discrete time sstem is epressed as follows: J d d i Qd d i vd i rd vd i (3 i Qd1 Qd Qd Q d3 (4 Q d1 diag( q1q5 (5 Q d diag( qs1qs5 (6 Q d3 diag( qi1qi5 (7 r d r diag 1 1 1 1 1 (8 M Φ MΦ Q d 1 r Γ MΓ Γ MΦ Φ MΓ d (9 v d F d d (3 1 r d Γ MΓ Γ MΦ F d (31 Vol., 15 79
Hikaru YONEZAWA, Hiroki MARUMORI, akaoshi NARIA and Hideaki KAO Hikaru YONEZAWA, Hiroki MARUMORI, akaoshi NARIA and Hideaki KAO ale 1 Smols and values Smol Value Smol Value m 9.7 1-1 kg E 6 GPa Z n 5. 1-3 m ν.3 R Z 1. Ω L 1.41 1-1 H ρ 7.5 1 3 kg/m 3 L lea 9. 1 - H d.43 m L eff.55 1-4 H C 8. 1 7 Ns/m s1 155 mm s 645 mm s3 155 mm s4 645 mm s5 4 mm s1 85 mm s 85 mm s3 515 mm s4 515 mm s5 3 mm s 1. 1-3 s Steel plate Electromagnet 15 mm 8 mm 5 5 mm mm 7 199 mm 15 mm (a θ = ( θ = 5 9mm (c θ = 7 (d θ = 1 Fig. 4 Relationship etween tilt angle of electromagnets and shape of steel plate 1 Where Q d and r d are weighting coefficients, M is the solution of the algeraic matri Riccati equation, and s is a sampling interval. MALAB command lqrd was used to solve eq. (9 and the digital controller was designed using SIMULINK in the DSP. 4. Eperiment of Bended Levitation 4.1 Condition of eperiment ale 1 shows the specifications of the sstem. Optimal control theor (OP is applied for levitation control of the thin steel plate to compare the results under different electromagnet tilt angles. Figure 4 shows the relationship etween the tilt angle of electromagnets and shape of steel plate. In the ending levitation eperiment, the electromagnet tilt angle θ is increased at intervals of 5 from. In this stud, the standard deviation of displacement is measured. he standard deviation of displacement is measured 1 times at the electromagnet unit No. 1 for each electromagnet tilt angle, and the mean is used as the eperimental result. It is confirmed that the same tendenc is oserved in other electromagnetic units. o avoid the effect of the transient state of the thin steel plate, the measurement is started approimatel 1 s after the start of levitation. he weighting coefficients of OP (eq. (3 are set as follows: q 1 -q 5 = 1. 1 m - (3 q s1 -q s5 = 1. 1 3 (m/s - (33 q i1 -q i5 = 1 A - (34 r = 3.83 1 - V - (35 From eq. (14, natural frequenc (calculated value of elastic 1st mode = 4.3 H. 4. Levitation eperiment Figures 5 and 6 show the eperimental results otained when tilt angles of the electromagnets θ = and 15 under continuous model. In these figures, (a shows the displacement of the steel plate over time, ( shows its amplitude spectrum. Figures 7 and 8 show the eperimental results under 1-DOF model for comparison. In 1-DOF model levitation eperiments, we used the same method as that used Narita 6. When 1-DOF model is applied (Figures 7 and 8, elastic virations are ostensile on the steel plate. his is ecause the elastic viration is caused the elastic force that is applied to the steel plate as a restoring force. A peak of the amplitude spectrum is oserved at 4.3 H, which is the frequenc of the elastic 1st mode of the steel plate used on this eperiment. his is due to e a model that does not consider the virations ecited in the steel plate. herefore, it is confirmed that the viration occurs in various frequencies in Fig. 7. However, the case of ent levitation steel plate in 1-DOF model (Fig. 8, onl the viration of the elastic 1st mode is mainl generated. B ending the steel plate at θ = 7, the viration of steel plate is suppressed compared to θ =. Also, applied to continuous model, viration suppression ailit appears against the elastic 1st mode (4.3 H. Figure 9 shows the relationship etween the tilt angle of electromagnets θ and standard deviation of displacement. Standard deviation of displacement decreases with increasing electromagnet tilt angles. At a tilt angle of 7, the standard deviation of displacement is the smallest. he reason ehind the standard deviation of displacement of an increase at a tilt angle of 1 is that a tilt angle of 1 eceeds the natural deflection angle (8.5 of the steel plate with a thickness of.7 mm, leading to difficult in levitation. As a result, appling the continuous model considering the elastic viration, it is possile to suppress the elastic 1st mode of the steel plate. Moreover, ending the steel plate 8 Proceedings of the School of Engineering, okai Universit, Series E
Bending Magnetic Levitation Control Appling the Continuous Model of Fleile Steel Plate Bending Magnetic Levitation Control Appling the Continuous Model of Fleile Steel Plate Displacement [mm].4. -. -.4 1 3 4 5 ime [s] (a ime histor of displacement.15.1.5 1 3 4 5 Frequenc [H] ( Amplitude spectrum of displacement Fig. 5 Viration of a levitation steel plate for continuous model (θ = Displacement [mm].4. -. -.4 1 3 4 5 ime [s].15.1.5 (a ime histor of displacement 1 3 4 5 Frequenc [H] ( Amplitude spectrum of displacement Fig. 6 Viration of a levitation steel plate for continuous model (θ = 7.15.1.5 1 3 4 5 Frequenc [H] Fig. 7 Viration of a levitation steel plate for 1-DOF model (θ =.15.1.5 1 3 4 5 Frequenc [H] Fig. 8 Viration of a levitation steel plate for 1-DOF model (θ = 7 Standard deviation 1 [mm].6.4. Elastic 1st mode 3 6 9 1 ilt angle of electromagnets θ [ ] Fig. 9 Relationships etween tilt angle of electromagnets θ and standard deviation of displacement at sensor No. 1 at the optimal tilt angle of the steel plate, viration suppression performance was superior to 1-DOF model. From the aove, the advantage of appling the continuous model is shown without ending the steel plate. Moreover, usefulness of ending levitation can e confirmed from eperimental result with the continuous model. he model with considering elastic viration is effective to levitate thin steel plates which easil occur the viration. 5. Conclusion In this paper, appling the optimal control theor to Vol., 15 81
Hikaru YONEZAWA, Hiroki MARUMORI, akaoshi NARIA and Hideaki KAO Hikaru YONEZAWA, Hiroki MARUMORI, akaoshi NARIA and Hideaki KAO the continuous model, ending levitation eperiments were carried out. As a result, it was effective for a thin steel plate with thickness of.7 mm to e ent to an etent that did not eceed natural deflection angle and we were ale to suppress the viration as compared with the case of appling to the 1-DOF model. herefore, we confirmed the utilit of continuous model which takes account of elastic viration of steel plate. References 1 S. Matsumoto, Y. Arai,. Nakagawa: Noncontact Levitation and Conveance Characteristics of a Ver hin Steel Plate Magneticall Levitated a LIM-Driven Cart, IEEE ransactions on Magnetics, 5 No. 11, (14.. Namerikawa, D. Miutani, S. Kuroki: Roust H DIA Control of Levitated Steel Plates, IEEJ ransactions on Industr Applications, 16, No. 1 (6, 1319-134. 3 Jung Soo Choi, Yoon Su Baek: Magneticall-Levitated Steel-Plate Conveance Sstem Using Electromagnets and a Linear Induction Motor, IEEE RANSACIONS ON MAGNEICS, 44, No. 11 (8, 4171-4174. 4 H. akamine, S. orii,. Yanagida, S. Iwashita, S. odoroki: Stud of Electromagnetic Suspension Sstem Using Acceleration Signal of Electromagnet Supported with Spring, IEEJ ransactions on Industr, 133, No. 5 (8, 536-54 5 Y. Oshinoa,. Oata: Noncontact Viration Control of a Magnetic Levitated Rectangular hin Steel Plate, JSME International Journal, Series C, 45, No. 1, ( 6-69. 6. Narita, H. Marumori, S.Hasegawa, Y. Oshinoa: Basic Eperimental Considerations on Bending Levitation Control Electromagnetic Force for Fleile Steel Plate, Proseedings of the School of Engineering of okai Universit, Series E, 38 (13, 47-5. 7 H. Yoneawa, H. Marumori,. Narita, S. Hasegawa, Y. Oshinoa: Bending Magnetic Levitation Contril for hin Steel Plate (Eperimental Consideration UsingSliding Mode Control, International Power Electronics Conference, (14, 355-36. 8 H. Marumori, H. Yoneawa,. Narita, H. Kato, S. Hasegawa, Y. Oshinoa: Effective Plate hickness Range in Bending echnique for Levitation Control of Fleile Steel Plate, Proseedings of the School of Engineering of okai Universit, Series E, 39 (14, 59-66. 8 Proceedings of the School of Engineering, okai Universit, Series E