Basic study on effect of transport acceleration in electromagnetic levitation system for thin steel plate

Abstract

Thin steel plates are widely used in various industrial products. However, because of the deterioration of surface quality and metal plating that occurs during transport, some problems exist. As a solution to these problems, a noncontact transport of steel plate using electromagnetic force has been proposed. However, side slip or the dropping of the plate may occur. Therefore, we proposed the addition of electromagnetic actuators to control the horizontal motion of levitated steel plates. In this study, we examined the change in the levitation stability during noncontact transport with the addition of positioning control in the horizontal direction or changed transport conditions. As a result, the application of a magnetic field in the horizontal direction improved the levitation stability of transported thin steel plates in different transport conditions.

Keywords

Electromagnetic levitation control thin steel plate vibration control magnetic field transportation

1. Introduction

Thin steel plates are used for a wide variety of industrial products. In the manufacturing process, however, there is a large loss because of poor plating caused by contact with rollers and deterioration of surface quality. An alternative method of thin steel plate transfer by electromagnetic levitation has been examined [1–3]. However, most research on magnetic levitation technology are studies on objects that can be regarded as rigid objects, and there are few reports on the magnetic levitation of flexible bodies causing complicated deformation. In our research, we focused on thin steel plates that are thin, flexible, and difficult to control by levitation, and we proposed a system in which electromagnetic units are installed not only in the vertical direction but also in the edge part of the steel plate. By controlling the positioning of the steel plate in the horizontal direction by the electromagnet installed at the edge part, side slip and falling were prevented, and it was clarified that noncontact support conveyance is possible [4]. Furthermore, electromagnetic analysis of the magnetic field in the horizontal direction by the finite element method was performed to obtain the attraction force applied to the steel plate, and the shape of the steel plate during levitation was calculated using the finite difference method. In the levitating experiment conducted based on this, it was clarified that deflection of the flexible steel plate can be suppressed by the magnetic field from the horizontal direction, and the stability during levitating can be improved [5]. Furthermore, when assuming the actual conveying process, it is necessary to improve not only the vertical direction, which has been studied, but also the levitation stability when horizontal disturbance caused by acceleration and deceleration is input. However, the influence of the magnetic levitated steel plate on the levitating performance during transportation has not yet been sufficiently studied. Therefore, in this study, a thin steel plate with a thickness of 0.24 mm is considered, and the effect of the magnetic field from the horizontal direction on noncontact conveyance is verified experimentally.

Fig. 1.

Non-contact transport system for thin steel plate.

Fig. 2.

Electromagnetic levitation control system with horizontal positioning control.

Fig. 3.

Photograph of electromagnetic levitation system.

Fig. 4.

Configuration of electromagnet.

2. Magnetic levitation transport system

A schematic illustration of the magnetic levitation transport system is shown in Fig. 1. The system consists of a frame containing the magnetic levitation system and a linear motor for transporting the frame. During transporting, the steel plate is levitated by the magnetic levitation system, and the frame is moved together with the magnetic levitation system using a linear motor. A schematic illustration of the magnetic levitation system is shown in Fig. 2, and the photograph is shown in Fig. 3. The magnetic levitation system consists of a vertical levitation control system and a horizontal positioning control system. The levitation control system is shown at the top of Fig. 2. The levitating object is a rectangular galvanized steel plate (material SS400) of length a = 800 mm and width b = 600 mm. In the levitation control system, pairs of electromagnets are installed at five locations at the periphery and the center, and the distance from the surface of each electromagnet to the surface of the steel plate is controlled to 5 mm to magnetically levitate the steel plate. The electromagnet used in this study has enamel wire of 0.5 mm diameter wound around 1005 times. The cross sectional area of the iron core was 225 mm², and the iron core was made by processing ferrite to E type, as shown in Fig. 4. The electric circuits of the electromagnet were coupled in series and installed so that an eddy current type non-contact displacement sensor was sandwiched in the center. Furthermore, the control law was calculated by detecting the coil current of the electromagnet from the external resistance for measurement and inputting a total of 10 measurement values to a digital signal processor from an analog-to-digital converter. The thin steel plate was magnetically levitated by the control voltage output from the digital-to-analog converter to the current supply amplifier. A horizontal positioning control system is shown at the bottom of Fig. 2. Non-contact positioning control was performed by applying the electromagnet suction force to the edge of the levitating steel plate from the horizontal direction. A laser sensor (measuring the displacement by the cutoff amount of a belt-like laser beam) was used in the horizontal displacement measurement of the edge part of the steel plate. The same electromagnets were used as those in the levitation system. As shown in Fig. 1, two of these electromagnets were installed along each of the two sides of the steel plate opposite each other, and the non-contact positioning control was performed so that the distance from the surface of each electromagnet to the edge of the steel plate was 5 mm.

3. Control model

3.1. Vertical levitation control model

This study, the displacement, velocity, and current value of the electromagnet coil detected at one electromagnet and feedback to the same electromagnet. The steel plate divided into 5 parts virtually, and each part is modeled as a single-degree-of-freedom electromagnet levitation model, as shows in Fig. 5. The steel plate supported by the static attraction force from the electromagnet, there exist an equilibrium state where a certain distance is maintained. The state equations established in previous study are as follows [5]. $\begin{eqnarray}\displaystyle {\dot{z}}=A_{z}z+B_{z}v_{z} & & \displaystyle\end{eqnarray}$ (1) $\begin{eqnarray}\displaystyle \boldsymbol{z}=\left[\begin{array}{@{}ccc@{}}z & {\dot{z}} & i_{z}\end{array}\right]^{\text{T}},\quad \boldsymbol{A}_{\boldsymbol{z}}=\left[\begin{array}{@{}ccc@{}}0 & 1 & 0\\ {\displaystyle \frac{4F_{z}}{m_{z}Z_{0}}} & 0 & {\displaystyle \frac{4F_{z}}{m_{z}I_{z}}}\\ 0 & -{\displaystyle \frac{L_{eff}}{L_{z}}}\cdot {\displaystyle \frac{I_{z}}{Z_{0}^{2}}} & -{\displaystyle \frac{R_{z}}{2L_{z}}}\end{array}\right],\quad \boldsymbol{B}_{\boldsymbol{z}}=\left[\begin{array}{@{}ccc@{}}0 & 0 & {\displaystyle \frac{1}{2L_{z}}}\end{array}\right]^{\text{T}} & & \displaystyle \nonumber\end{eqnarray}$ Z₀ is the gap between the steel plate and electromagnet in the equilibrium state [m], I_z is the current of the coupled magnets in the equilibrium state [A], i_z is the dynamic current of the coupled magnets [A], L_z is the inductance of one magnet coil in the equilibrium state [H], R_z is the resistance of the coupled magnet coils [𝛺], v_z is the dynamic voltage of the coupled magnets [V], and L_eff∕Z₀ is the effective inductance of the one magnet coil [H]. v_z is obtained by feedback of the state variable z as follows. $\begin{eqnarray}\displaystyle v_{z}=-\boldsymbol{F}_{\boldsymbol{z}}\boldsymbol{z} & & \displaystyle\end{eqnarray}$ (2) $\begin{eqnarray}\displaystyle \boldsymbol{F}_{z}=\left[\begin{array}{@{}ccc@{}}f_{z1} & f_{z2} & f_{z3}\end{array}\right] & & \displaystyle \nonumber\end{eqnarray}$ where f_z1, f_z2 and f_z3 are feedback gain for displacement z, velocity ${\dot{z}}$ and current i_z.

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 Fig. 6. Therefore, the same attractive forces were generated from two electromagnets placed at one side of the steel plate. The state equations of small horizontal motion around the equilibrium state of the steel plate subjected to the same static magnetic forces from the electromagnets at two edges is expressed as $\begin{eqnarray}\displaystyle \dot{\boldsymbol{x}}=\boldsymbol{A}_{\boldsymbol{x}}∼\boldsymbol{x}+\boldsymbol{B}_{\boldsymbol{x}}∼\boldsymbol{v}_{x} & & \displaystyle\end{eqnarray}$ (3) $\begin{eqnarray}\displaystyle \boldsymbol{x}=\left[\begin{array}{@{}ccc@{}}x & {\dot{x}} & i_{x}\end{array}\right]^{\text{T}},\quad \boldsymbol{A}_{\boldsymbol{x}}=\left[\begin{array}{@{}ccc@{}}0 & 1 & 0\\ {\displaystyle \frac{4F_{x}}{m_{x}X_{0}}} & 0 & {\displaystyle \frac{4F_{x}}{m_{x}I_{x}}}\\ 0 & -{\displaystyle \frac{L_{xeff}}{L_{x}}}\cdot {\displaystyle \frac{I_{x}}{X_{0}^{2}}} & -{\displaystyle \frac{R_{x}}{2L_{x}}}\end{array}\right],\quad \boldsymbol{B}_{x}=\left[\begin{array}{@{}ccc@{}}0 & 0 & {\displaystyle \frac{1}{2L_{x}}}\end{array}\right]^{\text{T}} & & \displaystyle \nonumber\end{eqnarray}$ where F_x is the magnetic force of the coupled magnets in the equilibrium state [N], X₀ is the gap between steel plate and electromagnet in the equilibrium state [m], I_x is the current of the coupled magnets in the equilibrium state [A], i_x is the dynamic current of the coupled magnets [A], L_x is the inductance of the one magnet coil in the equilibrium state [H], R_x is the resistance of the coupled magnet coils [𝛺], v_x is the dynamic voltage of the coupled magnets [V], and L_xeff∕X₀ is the effective inductance of the one magnet coil [H]. v_x is obtained by feedback of the state variable x as follows. $\begin{eqnarray}\displaystyle v_{\text{x}}=-\boldsymbol{F}_{\text{x}}\boldsymbol{x} & & \displaystyle\end{eqnarray}$ (4) $\begin{eqnarray}\displaystyle \boldsymbol{F}_{\text{x}}=\left[\begin{array}{@{}ccc@{}}f_{x1} & f_{x2} & f_{x3}\end{array}\right] & & \displaystyle \nonumber\end{eqnarray}$ where f_x1, f_x2 and f_x3 are feedback gain for displacement x, velocity ${\dot{x}}$ and current i_x.

Fig. 5.

Theoretical model of levitation control of the steel plate.

Fig. 6.

Theoretical model of levitation control of the steel plate.

Fig. 7.

Time history of command transportation velocity.

Fig. 8.

Time histories of vertical and horizontal displacement of the steel plate. (I_x = 0.025 A).

Fig. 9.

Time histories of vertical and horizontal displacement of the steel plate. (I_x = 0.5 A).

Fig. 10.

Relationship between acceleration and horizontal displacement.

Fig. 11.

Relationship between acceleration and standard deviation of vertical displacement.

Fig. 12.

Relationship between acceleration and levitation probability.

4. Levitation experiment

4.1. Experimental condition

To evaluate the stability of the magnetically levitated steel plate when transported, a transport experiment for the magnetically levitated steel plate was performed. Figure 7 shows the time history of the speed command value of the linear motor mounted on the conveying device. The frame including the magnetically levitated steel plate travels at a constant acceleration for 1.2 s from the initial stationary state, travels at a constant speed, and then stops at a deceleration equivalent to that at the time of acceleration. In addition, to make the conveying distance constant, the time of the constant velocity section differs under each acceleration condition. The experimental conditions were: acceleration 0.39 m/s², velocity 0.5 m/s, acceleration 0.49 m/s², velocity 0.6 m/s; and acceleration 0.59 m/s², velocity 0.7 m/s. In any of the experimental conditions, the conveyance was started 2 s after the start of the measurement, and the distance to be conveyed was 1.8 m. At this time, the steady current value for the horizontal positioning control was changed in the range of I_x = 0 A(uncontrolled) to 0.5 A with respect to the steel plate levitated.

4.2. Experimental result

Figure 8 shows the time history of the displacement of the steel plate when applying the horizontal steady current values I_x = 0.025 A. Figure 8 shows the results of the acceleration of (a) 0.39, and (b) 0.59 m/s². The graph on the left side is the displacement in the horizontal direction. The graph on the right side is the displacement in the vertical direction. The result of I_x = 0.5 A is shown in Fig. 9 as well. Comparing the results in Fig. 8 and 9 shows that, by increasing the steady current value I_x in any of the transporting conditions, the maximum amplitude of the steel plate decreased, and the vibration in the vertical direction could also be suppressed. In the case of an acceleration of 0.59 m/s² with the most severe conveyance conditions, the maximum amplitude in the horizontal direction at I_x = 0.025 A was 2.3 mm, whereas, at I_x = 0.5 A, it was 0.72 mm, and the maximum amplitude could be suppressed by 68%. In any of the transporting conditions, the displacement in the horizontal direction at the constant velocity section was sufficiently controlled, so the model did not affect the conveying performance in the constant velocity section. The standard deviation in the vertical direction was 0.23 mm at I_x = 0.025 A, whereas it was 0.098 mm at I_x = 0.5 A, and the vibration could be suppressed by 57% despite all the parameters of levitation control being constant. Furthermore, when I_x = 0.025 A, the vibration oscillated at a high frequency from the state of being stationary in both the horizontal direction and the vertical direction. However, at the time of I_x = 0.5 A, the vibration of the high frequency, including the conveyance, could be suppressed. From analysis in previous study [5], when the horizontal attractive force acted on the steel plate, the bending rigidity can be assumed to increase by 3% at 0.025A and 10% at 0.5 A. It also seems that the stability was improved without depending on the vertical levitation control system, because the apparent rigidity increased when horizontal tension was applied to the steel plate. The relationship between the maximum amplitude of horizontal displacement and acceleration under each steady current value I_x condition is shown in Fig. 10. In the uncontrolled case, it cannot be transported. This is because a thin steel plate contacts the horizontal electromagnet during transportation. In any of the experimental conditions, it was possible to carry out stable conveyance by performing positioning control in the horizontal direction. Moreover, it was verified that the displacement in the horizontal direction was suppressed by increasing the steady current value. The relationship between the displacement standard deviation in the vertical direction and the acceleration is shown in Fig. 11. In this experiment, the system characteristics of levitation control were constant under all conditions. Nevertheless, the stability of the levitating direction is increased by increasing the tension by the steady urrent value of the horizontal direction control.

5. Levitation probability measurement experiment

To evaluate the influence of the magnetic field from the horizontal direction on the levitation performance of this system from the practical viewpoint, the levitating probability during transport was acquired. The transport conditions were changed within the range of I_x = 0.025 A to 0.5 A, with acceleration 0.39, 0.49, and 0.59 m/s², as in Chapter 4. The transfer experiment was conducted 50 times under each condition, and the levitation probability during transport was measured. The case where the steel plate continued to levitate for 30 s after transportation was regarded as successful. Conversely, when the steel plate touched the electromagnet of the vertical direction and the horizontal direction during the transport or fell, we regarded it as a failure. A preliminary experiment showed that it is possible for levitation to last for more than 10 min; thus, practical problems should not occur when levitation is performed for 30 s. Figure 12 shows the relationship between steady current value I_x and levitation probability under each transfer condition. As shown in the same figure, the levitation could not be maintained under any transport conditions under uncontrolled conditions, but the transport probability became 100% at I_x = 0.025 A and an acceleration of 0.39 m/s². Furthermore, the levitation probability decreases as the transport speed increases, but the levitation probability tends to rise as the steady current value increases. If the system is within I_x = 1.0 A, it is possible to levitate the steel plate for a practically sufficient time, but if I_x = 0.5 A, the levitation probability becomes almost 100% even with the most strictly set transporting condition of an acceleration of 0.59 m/s², and sufficient levitation stability was obtained.

6. Conclusion

We conducted transport experiments for a levitated steel plate (thickness: 0.24 mm) when changing the steady current value of the horizontal electromagnet and the transporting speed. By measuring the displacement of the steel plate and levitation probability, we experimentally considered levitation stability during transport. As a result, by increasing the tension applied to the steel plate by the steady current value from the horizontal direction, the displacement in the horizontal direction, which is a factor, such as sideslip, can be suppressed and the effect of reducing the vibration in the vertical direction can be obtained. This shows that the magnetic field from the horizontal direction has the effect of suppressing the vibration generated at the time of transport without depending on the levitating control system in the vertical direction. The relationship between the transporting speed and the levitation probability shows that the transporting condition that can sufficiently ensure levitation stability during transport can be ensured.

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