Vibration suppression effect in a maglev system for flexible steel plate with curvature

Abstract

When an ultrathin and flexible steel plate is to be levitated, levitation control becomes difficult because the ultrathin steel plate undergoes increased flexure. We have proposed the levitation of an ultrathin steel plate that is bent to an extent that does not induce plastic deformation. In this study to elucidate the levitation stability under disturbance, a random disturbance was input into the levitation system and bending levitation experiments were carried out using an ultrathin steel plate with a thickness of 0.19 mm.

Keywords

Electromagnetic levitation bending levitation control flexible steel plate

1. Introduction

Magnetic levitation technology capable of gripping and conveying objects in a non-contact manner has attracted attention, and active studies have been performed that use the characteristics of magnetic force, such as the electromagnetic suspension and induction repulsion methods [1–3]. Regarding the stability problem of the levitated object, control methods considering the mechanical characteristics of the structure have been studied [4–6]. It is expected that magnetic levitation technology can be applied to thin steel plate production processes such as cold rolling, which requires high surface quality. However, elastic vibration is induced because of the flexibility brought about by the plate thickness and area, so stability at the time of levitation is significantly impaired. In the past, our research group has constructed an electromagnetic levitation control system in which the relative distance between the electromagnet and a steel plate is constantly maintained, aiming to prevent the steel plate from falling from the conveyer or contacting the electromagnet during electromagnetic levitation conveyance. Furthermore, we have proposed a method of levitating a thin steel plate with a thickness of less than 0.3 mm by moderately bending it beforehand [7,8]. The levitation stability was improved by bending and the levitation stability could be maintained by the bending even if a disturbance was input to the control current [9]. However, considering a more practical situation, the electromagnet may be exposed to disturbances which is the vibration generated at the rail joint of the steel plate transport device. In this study, we examined the bending levitation performance experimentally in which an external disturbance was applied to a steel plate by vibrating the frame where the electromagnet unit was installed.

Fig. 1.

Electromagnetic levitation control system.

Fig. 2.

Schematic illustration of the experimental apparatus.

2. System for control experiment

Figure 1 outlines of the control system. Figure 2 shows a schematic illustration of the experimental apparatus. The object of electromagnetic levitation is a rectangular zinc-coated steel plate (SS400) with length a = 800 mm, width b = 600 mm and thickness h = 0.19 mm. To accomplish noncontact support of a rectangular ultrathin steel plate using five pairs of electromagnets (Nos. 1–5) as if the plate were hoisted by strings, the displacement of the steel plate was measured by five eddy-current gap sensors. The dimensional tolerance of the thickness of the steel plate is ±0.03 mm, the surface property Ra = 2 μm or less, the permeability 𝜇 = 2100 H/m, the saturation magnetic flux density Bm = 2.3 T, and the coercive force Hc = 80 A/m. In order to support the rectangular steel plate by the electromagnet so as to support it in a non-contact manner, the displacement of the steel plate is detected by the eddy current type non-contact displacement sensor. The electromagnet unit is composed of two electromagnets and one eddy current non-contact displacement sensor. Electromagnet units are installed at a total of five places, four at the periphery which is the minimum required to support the steel plate, and one at the center to support the center of gravity of the steel plate. Among the five pairs of electromagnets, the four pairs at the corners were inclined and a central electromagnet was moved in the vertical direction. In addition, the distance between the surfaces of the electromagnets and the steel plate was controlled at 5 mm even when θ was changed. Thus, by moving the five electromagnets, bending magnetic levitation of the steel plate was realized.

Fig. 3.

Vibrator (EMIC 513-B).

Table 1

Specifications of vibrator

Maximum excitation force	10 kgf ± 5%
Maximum acceleration	28 G ± 5%
Frequency range	3 Hz–13 kHz
Maximum input current	5.5 A ± 5%
Maximum speed	177 cm/s
Maximum amplitude	10 mm

The vibrator shown in Fig. 3 was attached below the three frames on which the electromagnet unit was installed, so that the frame could be vibrated up and down. Table 1 shows the specifications of the vibrator. In order to vibrate the vibrator, the vibration waveform was output by the signal out function of the FFT analyzer and input to the vibrator via the amplifier (371 A, EMIC company). After adjusting the amplitude and phase of the frame to be constant by using a sine wave of constant frequency, it was possible to levitate the steel plate while vibrating the frame. In each frame, an eddy-current gap sensor was installed and the displacement of the frame during excitation was measured.

3. Equation of motion

In this study, we adopted a one degree-of-freedom model. In this model, independent control is performed, in which the information on the detected values of displacement, velocity and coil current of the electromagnets under study at one position are fed back only to the same electromagnet. The steel plate is divided into five hypothetical masses and each part is modeled as a lumped constant system. In an equilibrium levitation state, the magnetic forces are determined to balance with gravity. The equation of motion around the equilibrium state of the steel plate subjected to magnetic forces is expressed as $\begin{eqnarray}\displaystyle m_{z}\frac{d^{2}}{dt^{2}}z(t)=2f_{z}(t). & & \displaystyle\end{eqnarray}$ (1) Here, m _z: mass obtained by dividing the steel plate virtually into five [kg], z: displacement from the equilibrium levitation position of the steel plate [m], and f _z: variation value of the suction force per electromagnet [N].

The equations relating to the electromagnet attraction force and the current concerning the current flowing in the electromagnet coil, which have undergone the linearization approximation, are as follows: $\begin{eqnarray}\displaystyle f_{z}(t)\text{} & =\text{} & \displaystyle \frac{F_{z}}{Z_{0}}z(t)+\frac{F_{z}}{I_{z}}i_{z}(t)\end{eqnarray}$ (2) $\begin{eqnarray}\displaystyle \frac{d}{dt}i_{z}(t)\text{} & =\text{} & \displaystyle -\frac{L_{\text{eff}}I_{z}}{L_{z}Z_{0}^{2}}\frac{d}{dt}z(t)-\frac{R_{z}}{2L_{z}}i_{z}(t)+\frac{1}{2L_{z}}v_{z}(t)\end{eqnarray}$ (3) $\begin{eqnarray}\displaystyle L_{z}\text{} & =\text{} & \displaystyle \frac{L_{\text{eff}}}{Z_{0}}+L_{\text{lea}}.\end{eqnarray}$ (4) Here, F _z: total value of the static attractive forces generated from both electromagnets pairs in the equilibrium levitation state [N], Z ₀: gap between the electromagnet surface and the steel plate surface in the equilibrium levitation state [m], I _z: steady state current for obtaining static attraction force [A], i _z: variation value of the current flowing through the electromagnet coil [A], L _eff∕Z ₀: effective inductance per electromagnet [H], L _lea: leakage inductance per electromagnet [H], L _z: inductance of the electromagnetic coil in the equilibrium levitation state [H], R _z: total resistance value of the electromagnet coil of the pair [Ω], v _z: fluctuation value from the stationary voltage applied to the electromagnet of the pair [V].

The displacement of the steel plate, speed, current flowing in the electromagnet coil, displacement of frame and speed are adopted as state variables. Subsequently, we summarize Eqs (1), (2), (3), (4) and obtain the following state equation: $\begin{eqnarray}\displaystyle \dot{\boldsymbol{z}}=\boldsymbol{A}_{z}\boldsymbol{z}+\boldsymbol{B}_{z}\boldsymbol{v}_{z}. & & \displaystyle\end{eqnarray}$ (5) However, $\begin{eqnarray}\displaystyle \boldsymbol{z}\text{} & =\text{} & \displaystyle [\begin{array}{@{}ccc@{}}z & {\dot{z}} & i_{z}\end{array}]^{T}\end{eqnarray}$ (6) $\begin{eqnarray}\displaystyle \boldsymbol{A}_{z}\text{} & =\text{} & \displaystyle \left[\begin{array}{@{}ccc@{}}0 & 1 & 0\\[12.0pt] \displaystyle \frac{2F_{z}}{m_{z}Z_{0}} & 0 & \displaystyle \frac{2F_{z}}{m_{z}I_{z}}\\[14.0pt] 0 & \displaystyle -\frac{L_{eff}}{L_{z}}\cdot \frac{I_{z}}{Z_{0}^{2}} & \displaystyle -\frac{R_{z}}{2L_{z}}\end{array}\right]\end{eqnarray}$ (7) $\begin{eqnarray}\displaystyle \boldsymbol{B}_{z}\text{} & =\text{} & \displaystyle \left[\begin{array}{@{}ccc@{}}0 & 0 & \displaystyle \frac{1}{2L_{z}}\end{array}\right]^{T}.\nonumber\end{eqnarray}$

4. Control theory

In this study, a control system is constructed using a discrete time system; therefore, the evaluation function of a continuous system is digitized, and the control law is obtained based on the optimal control of the discrete time system. $\begin{eqnarray}\displaystyle \boldsymbol{J}=(1/2)\mathop{\sum }_{k=1}^{N-1}\{\boldsymbol{z}^{\text{T}}(k)\boldsymbol{Q}\boldsymbol{z}(k)+\boldsymbol{u}^{\text{T}}(k)\boldsymbol{H}\boldsymbol{u}(k)\} & & \displaystyle\end{eqnarray}$ (8) where Q is a semi-regular matrix and H is a regular matrix. The feedback control input at this time is as follows: $\begin{eqnarray}\displaystyle \boldsymbol{v}_{z}=[f_{bz},f_{bzd},f_{bi}]\boldsymbol{z} & & \displaystyle\end{eqnarray}$ (9) where f _bz, f _bzd, and f _bi are the feedback gains for the displacement, speed, and current, respectively, in the levitation control.

5. Analytical evaluation of bending levitated steel plate

To achieve stable levitation of the 0.19 mm steel plate, it is necessary to determine the optimum angle. Therefore, the state in which the entire thin steel plate is supported by the surface is considered an optimal steel plate shape, and is defined as an optimal shape. The optimal shape is assumed to be a beam when the steel plate is viewed in the x axis, and the one-dimensional deflection shape of the beam is extruded to the size of the steel plate. To evaluate the total amount of deflection with respect to the optimal shape when the steel plate is bent, the amount of deflection of the thin steel plate by gravity is calculated, and the shape of the steel plate under bending levitation is calculated. The analysis model, method and validity were confirmed in a previous report [10]. To evaluate the total amount of deflection with respect to the ideal shape, the evaluation value j is defined as $\begin{eqnarray}\displaystyle j=\frac{\mathop{\sum }_{i=1}^{N}|z_{i}-z_{0}|}{h\cdot N} & & \displaystyle\end{eqnarray}$ (10) z _i: Displacement of the z component of the optimal shape [m], z ₀: displacement of the z component at each analysis point of the thin steel plate [m], N: total number of analysis points (N = 4941).

When the evaluation value j is small, the deflection amount of the steel plate is also small.

Furthermore, when the supporting force of the central electromagnet unit No. 5 is 0, the deflection angle at the support point when supporting the steel plate only with No. 1 to No. 4 is defined as the natural deflection angle θ_na and is expressed by $\begin{eqnarray}\displaystyle {\theta}_{\text{na}}=\frac{{\rho}gl}{2Eh^{2}}(l^{2}-6d^{2}). & & \displaystyle\end{eqnarray}$ (11)

6. Levitation experiment

A random external disturbance in a frequency between 0 and 10 Hz was added to the frame, and the levitation performance was evaluated by comparing the standard deviation of displacement for each electromagnets angle. The standard deviation of displacement was measured 10 times for each electromagnets angle, and the average value of these results was taken as the experimental value. To eliminate the influence of the transient state, measurements were performed approximately 10 s after the start of levitation.

Fig. 4.

Time histories of displacement and amplitude spectra of vibrating frames by the random disturbance.

Fig. 5.

Bending levitation result when vibrating frames by a pulse disturbance (θ = 13°).

Figure 4 shows the time histories of the displacement and amplitude spectra of the vibrating frames by the random disturbance and Fig. 5 shows the bending levitation result when the frames were vibrated by the pulse disturbance. From the spectra in Figs 4 and 5, it is shown that the disturbance input in the 0 to 10 Hz band resonates with the steel plate. Figure 6 shows the standard deviation of displacement when the frames by the random disturbance. Natural deflection angle and evaluation values of total deflection are also shown in Fig. 6. From Fig. 6, the standard deviation of displacement was increased at all electromagnet angles because of the frame vibration. In levitating without disturbance, the standard deviation of displacement was almost constant regardless of the electromagnet angle, but when the disturbance was input, the standard deviation of displacement was suppressed by tilting the electromagnet from the electromagnet angle of 0°. The standard deviation of displacement was most suppressed when the steel plate was bent particularly at 13°. In addition, when the electromagnetic angle exceeding the natural deflection angle of the 0.19 mm steel plate was 15° or more, the standard deviation of displacement tended to increase greatly by vibrating the frame. The lower the evaluation value j, the smaller the total sum of the deflections from the optimal shape. The vibration was suppressed if the tilt angle of the electromagnet was close to the optimal shape and did not exceed the natural deflection angle, so the standard deviation of displacement was reduced. Therefore, there is an optimal bending angle in an area smaller than the natural deflection angle and more inclined than the local minimum value of the evaluation value.

Fig. 6.

Standard deviation of displacement when vibrating the frames by the random disturbance.

7. Conclusion

In this study, we conducted experiments on levitation performance when the electromagnet was displaced by frame vibration in the bending levitation system. As a result, it is possible to realize a stable levitation in the face of an input external disturbance when levitating at the optimum bending angle.

In the future, we intend to conduct bending levitation experiments to verify the effectiveness of the bending levitation system when using other plate thicknesses and different control methods in the vibrating frame.

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