A basic study on levitation characteristics of metal foil by edge-supported electromagnetic levitation system

Abstract

The grasping and conveying of an object by utilizing the frictional force generated by contact are performed during various steps in the process of manufacturing industrial products. Deterioration of the surface quality due to these contacts is a problem. A noncontact way to transport steel plates using electromagnetic force has been proposed as a solution to these problems. In such applications to date, electromagnets are installed in the vertical direction. However, if the steel plate is thin and lacks sufficient flexural rigidity, it is difficult to exert enough suspension force to levitate the entire steel plate. To solve this problem, we propose an edge-supported electromagnetic levitation system suitable for flexible steel plates using electromagnets installed in the horizontal direction. In this paper, we report on levitation experiments to verify the effectiveness of the proposed system and discuss characteristics of the horizontal positioning of electromagnets and levitation suspension.

Keywords

Electromagnetic levitation metal foil noncontact support

1. Introduction

In the production line of industrial products, the process of gripping and transporting objects is performed during various steps. In many production lines, products are gripped and transported using the frictional force of contact. However, contact-based movements deteriorate the surface quality. Therefore, non-contact transport technology for production lines is required. To solve this problem, non-contact support and conveyance systems that use magnetic levitation technology are being actively studied [1–4]. One area of study is thin steel plates that are used in various products [5,6]. In recent years, the use of high-strength, thin steel plates have been encouraged to reduce environmental impact and increase efficiency. The bending stiffness of these thin steel plates is low, and conventional levitation techniques cause deflections in places where the electromagnetic force does not reach. This deflection causes complicated vibrations and makes it difficult to stabilize the levitation. Our research group has succeeded in improving the levitation stability of flexible steel plates by installing electromagnets in the horizontal direction and applying tension from the edge of the steel plate [7,8]. The attractive force from the horizontal electromagnet generates not only tension but also a supporting force in the vertical direction. Regarding this tension and supporting force, a system that both supports and stabilizes by electromagnets installed only at the edge of the steel plate would represent a more flexible magnetic levitation system. The authors confirmed that the proposed system generates enough attractive force to levitate thin steel plates from electromagnetic field analysis and experiments on the static deflection of the steel plate. Furthermore, the thinner the steel plate, the more efficiently the supporting force necessary for levitation is obtained [9]. The finite difference method is used to calculate the shape of the steel plate during levitation and analytically determine the optimal installation position of the electromagnet to suppress the deflection of the steel plate during levitation. To demonstrate the effectiveness of the proposed system, we constructed a magnetic levitation system with electromagnets installed only in the horizontal direction. In a conventional magnetic levitation system in which electromagnets are installed in the vertical direction, the performance was significantly degraded by deflection and vibration in levitation experiments that were conducted using a flexible metal foil with a thickness of 0.05 mm, which is difficult to levitate. The positioning control characteristics and levitation support characteristics of the proposed magnetic levitation system were investigated.

Fig. 1.

State of boarding ultra-compact vehicle.

2. Edge-supported levitation system

Figure 1 shows a schematic diagram of the magnetic levitation device. Figure 2 shows the arrangement of the electromagnet and sensor as seen from above. Figure 3 shows a photograph of the magnetic levitation system during levitation. The object to be levitated was a rectangular galvanized steel plate (material SS400) 400 mm long, 100 mm wide and 0.05 mm thick. As shown in Fig. 4, the magnetic levitation system has two electromagnets facing each other near the edge in the longitudinal direction of the metal foil. Perform noncontact positioning control. The electromagnets were placed in such a way that the maximum deflection calculated from the magnetic field analysis and the deflection shape analysis determined by the finite difference method was suppressed, and stable levitation was expected. Displacement was measured in the horizontal direction by a laser sensor manufactured by KEYENCE (displacement was measured as the amount of cutoff at the belt-shaped laser beam). Non-contact positioning control was performed to maintain the distance from the surface of each electromagnet to the metal foil edge at 5 mm by feedback of displacement and velocity and current. To measure the current flowing in the electromagnet, the terminal voltage of the external resistance v_Rn (n = 1–4) installed to electromagnet circuit in series was measured. Two observed values of the voltage at the external resistance and the displacement were input into 4 channels, and a total of 8 observed values from the A / D converter were input into the DSP. The current of electromagnet was calculated from the measured voltage, and the velocity was obtained by differentiating the displacement. The control output was calculated by the obtained displacement, velocity and current. The core of the electromagnet was an E type ferrite, as shown in Fig. 5. The electromagnetic coil wire has a diameter of 0.5 mm with 1005 turns. Vertical displacement was measured by a non-contact displacement sensor manufactured by Sentec to evaluate the levitated state of the metal foil.

Fig. 2.

Seat area of ultra-compact vehicle.

Fig. 3.

Active seat suspension.

When the steel plate was levitated by electromagnet in the equilibrium state, an attractive force F₀ generated on the steel plate is shown by the following equation [9]. $\begin{eqnarray}\displaystyle F_{0}=\frac{L_{\mathit{eff}}}{2}\frac{I_{x}^{2}}{{\Gamma}_{0}^{2}} & & \displaystyle\end{eqnarray}$ (1) where I_x is the current of the coupled magnets in the equilibrium state [A], 𝛤₀ is distance from center of electromagnet to edge of the steel plate [mm], L_eff∕𝛤₀ is a constant corresponding to the effective magnetic flux of the electromagnet. Establishing the equation of motion in horizontal and vertical direction, the relationship between levitation position and steady current could be obtained as follows. $\begin{eqnarray}\displaystyle 2F_{0}\frac{Z_{0}}{{\Gamma}_{0}}-mg=0 & & \displaystyle\end{eqnarray}$ (2) where m is the weight of the steel plate to be supported by a pair of electromagnets [kg], Z₀ is the levitation position of the metal foil in the vertical direction. Taking the values of 𝛤₀, Z₀, F₀ which satisfy the following equation, steel plate can be magnetically levitated.

Fig. 4.

Voice coil motor.

Fig. 5.

Electromagnet configuration.

3. Suspension force analysis of electromagnet using the finite element method

We have previously confirmed levitations of steel plates that are 0.24 mm and 0.19 mm thick with this system [9]. However, it has not known whether plates as thin as 0.05 mm, which are used in this study, can be levitated. Therefore, electromagnetic field analysis was performed before the levitation experiment. Figure 6 shows a model diagram of the electromagnetic field analysis. Figure 7 shows the relationship between the steady current value of the electromagnet obtained by electromagnetic field analysis and the bearing capacity when the metal foil is moved from the center of the electromagnet core in the z-direction. The broken line (about 0.04 N) in the figure shows the value obtained by dividing the weight of the metal foil into four parts, which indicates that the metal foil can be levitated at a steady current value corresponding to the weight.

Fig. 6.

Analysis model.

Fig. 7.

Relationship between vertical attractive force f_z for each displacement z.

Fig. 8.

Experimental model of electromagnetic suspension force.

4. Control model

The flexible metal foil generates elastic vibration in the vertical direction but can be regarded as a rigid body in motion in the horizontal plane. The proposed system is modeled as a one-degree-of-model that virtually divides the metal foil into two parts, as shown in Fig. 8, and actively controls the movement of the metal foil in the x-direction. There is an equilibrium position where the metal foil is kept a certain distance from each electromagnet by applying the same static attraction force from two opposing electromagnets installed to “sandwich” the metal foil. The displacement of the metal foil from the equilibrium position is x, and the equations of state are described as follows: $\begin{eqnarray}\displaystyle {\dot{x}}=A_{x}x+B_{x}v_{x}, & & \displaystyle\end{eqnarray}$ (3) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle x=\left[\begin{array}{@{}ccc@{}}x & {\dot{x}} & i_{x}\end{array}\right]^{\text{T}},\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle A_{x}=\left[\begin{array}{@{}ccc@{}}0 & 1 & 0\\[4.0pt] \displaystyle \frac{4F_{x}}{m_{x}X_{0}} & 0 & \displaystyle \frac{4F_{x}}{m_{x}I_{x}}\\[10.0pt] 0 & \displaystyle -\frac{L_{\mathit{xeff}}}{L_{x}}\cdot \frac{I_{x}}{X_{0}^{2}} & \displaystyle -\frac{R_{x}}{2L_{x}}\end{array}\right],\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle B_{x}=\left[\begin{array}{@{}ccc@{}}0 & 0 & \displaystyle \frac{1}{2L_{x}}\end{array}\right]^{\text{T}}\nonumber\end{eqnarray}$ where F_x is the magnetic force of the coupled magnets in the equilibrium state [N], X₀ is the gap between the metal foil 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 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 one magnet coil [H].

5. Levitation experiments

5.1. Levitation experiment condition

To evaluate the levitation characteristics of the metal foil when the steady current value I_x of the electromagnet is changed, the steady current value was varies within the range of I_x = 0.9 to 1.2 A. When levitating the metal foil, the metal foil was supported by hand and the metal foil was floated at each current value. The standard deviation was calculated for the displacement amplitude of the metal foil in the x- and z-directions. The experiment was performed five times for each condition, and the average value of the results was obtained.

Fig. 9.

Time histories of vertical and horizontal displacement of the metal foil.

5.2. Levitation experiment results

The upper part of Fig. 9 shows the time history of vertical displacement of the metal foil, and the lower part shows the time history of horizontal displacement of the metal foil. Figure 9 shows the steady current values of 0.9, 1.0, 1.1, and 1.2 A. Figure 10 summarizes the relationship between the levitation position of the metal foil in the z-direction at each steady current value and the levitation position of the metal foil obtained by electromagnetic field analysis. From Fig. 9, the standard deviation of horizontal displacement is 0.1 mm or less for all steady current values tested. The experimental values of vertical displacement are the same as the levitated position obtained by electromagnetic field analysis. These results indicate that the proposed levitation device controls the metal foil position in both x- and z-directions.

Figure 10 shows a summary of the relationship between the standard deviation of displacement in the z direction at each steady current value, the steady current value, and the levitation position of the steel plate obtained from the analysis. Figure 10 summarizes the 0.24 mm and 0.19 mm thick steel plates used so far and the 0.05 mm thick metal foil used in this report for the experiment. From Fig. 10, it was confirmed that the levitation position of the levitation target increased by increasing the steady current value at any plate thickness. Furthermore, it was confirmed that the analysis results and the experimental results agree. This indicates the possibility of controlling the vertical displacement of the metal foil by changing the steady current value.

6. Conclusion

In this study, levitation experiments were conducted to verify the effectiveness of an edge-supported magnetic levitation device for a plate thickness of 0.05 mm, which is difficult to levitate with a conventional system in which an electromagnet is installed above the levitation object. Our findings demonstrated that stable levitation of thin steel plates is possible. Furthermore, experimental measurements were conducted at a range of steady current values. The results show that, when the steady current value is increased, the levitation position of the metal foil increases and coincides with the levitation position obtained from numerical analysis. This indicates the possibility of controlling the vertical displacement of the metal foil by changing the steady current value. We conclude that thin metal foils can be levitated by the vertical supporting force generated by electromagnet positioned in the horizontal direction and that the proposed edge-supported magnetic levitation device is effective.

Fig. 10.

Time histories of vertical and horizontal displacement of the metal foil.

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