Metamodel for 2D magnet-rail relationship based on stepwise regression

Abstract

Almost all existing analytical formulae describing the magnet-rail relationship for maglev trains have been derived with the assumption that magnetic media operate within the linear zone of the B-H curve and, therefore, over-predict the levitation and guidance forces when the magnetic media are in their saturation states. In such cases, finite element simulations are commonly used to obtain an accurate solution of the electromagnetic force, but this process is time-consuming. In this study, metamodels suitable for saturated magnetic media are constructed based on the methods of uniform design and stepwise regression, and their performance is assessed in terms of data fitting and prediction accuracy.

Keywords

Magnetic saturation magnet-rail relationship levitation force guidance force uniform design stepwise regression metamodeling

1. Background

Owing to their low operation noise and cost of maintenance as well as small environmental footprint [1], maglev trains have attracted considerable research interest in the last three decades. Maglev trains achieve levitation and guidance through the force generated between the rail and electromagnets [2]. The effects of the current in the coil, the material properties of the magnet, and the relative poses of the electromagnet and rail in the presence of electromagnetic force are referred to as the magnet-rail relationship [3, 4]. Similar to the role of the wheel-rail relationship in the wheel train studies, the magnet-rail relationship is the foundation of the dynamical analysis of maglev trains.

Initial studies on the magnet-rail relationship were limited to one-dimensional (1D) cases [5]; that is, in addition to the current in the coil, only the vertical levitation gap was considered. Some researchers subsequently derived formulae for the levitation and guidance forces based on the flux tube to consider both the lateral offset and the vertical gap, thereby extending the magnet-rail relationship to the 2D case [6].

For soft magnetic materials, the relation between magnetic flux density and magnetic field strength can be described by B $=$ $\mu$ H [7, 8], where $\mu$ is the permeability that reflects the properties of magnetic media. Figure 1 shows the B-H curve of Q235, which is used in maglev train tracks and pole plates of the electromagnet. As shown in the figure, the B-H curve of Q235 can be divided into a linear zone and a saturation zone due to the slope. The permeability of magnetic media operating in the linear zone is fairly high, and magneto-resistance of the magnetic medium can thus be ignored, in contrast with that of the air gap. When the magnetic field strength is above 0.8 A/m, the B-H curve of Q235 gradually saturates, and the magneto-resistance of the magnetic media significantly increases and can no longer be ignored.

Figure 1.

B-H curve of Q235 steel.

Figure 2.

Guidance force prediction performance of 2D analytical formula [6].

Figure 3.

Levitation force prediction performance of 2D analytical formula [6].

To derive an analytical formula for the magnet-rail relationship, early studies relied on the assumption that magnetic media generally operates in the linear zone of the B-H curve [6, 9, 10]. However, the analytical formulae derived in the manner exhibit a substantial deviation when the working point of the electromagnet transcends the linear zone. To illustrate this problem, 400 test points uniformly sprayed in the feasible working space of the electromagnets of a maglev train were investigated. The electromagnetic forces were obtained through finite element (FE) simulations and compared with values estimated using the analytical formula in Figs 2 and 3. It was evident that at many test points, the estimated values of the levitation and guidance forces were significantly larger than those calculated through the FE simulations.

The hypothesis that electromagnets always function in the linear zone has been challenged in practical applications of maglev trains. On the one hand, in case of a relatively significant track irregularity, the working range of electromagnets often transcends the linear zone. However, controlling track irregularity at a relatively low level drastically increases the cost of track construction and maintenance. On the other hand, restricting the working scope within the linear zone significantly reduces the load ratio of electromagnets as well as that of maglev trains. Hence, it is practically significant to explore the magnet-rail relationship with consideration of magnetic saturation.

For magnetic media operating in the saturation zone, a sufficiently accurate numerical solution of the electromagnetic force can be obtained using FE simulations with intensive meshing. Nevertheless, this endeavor requires significant computation time. In this study, metamodels were therefore established to provide a fast method to calculate the electromagnetic levitation and guidance forces for the dynamic modeling and simulation of maglev trains.

2. Experiment design, modeling, and model evaluation scheme

2.1 Uniform design method

Points used for modeling must be distributed throughout the entire range of variation of experimental factors so that they can contain as much information as possible concerning the properties of the system. At the same time, the number of adopted points should be as small as possible with consideration of the computation time needed for FE simulations.

The uniform design method introduces the discrepancy criterion into the design of experiments to distribute the points uniformly in the space of the testing factors. Accordingly, various levels of each experimental factor can be reflected in the experiments, and the total number of experiments can be reduced [11, 12, 13, 14]. Therefore, this approach is extensively used [15, 16, 17, 18]. Suppose that the required number of trials is n and s factors are considered. The specific procedure of the uniform design is outlined below.

Find the following set of positive integers:

$\displaystyle A_{n,s}=\{a:a<n,\text{gcd}(a,n)=1,\text{and }a,a^{2},\ldots,a^{s% }(\text{mod }n)\;\text{are distinct}\}$ (1)

For each $a\in A_{n,s}$ , generate a U-type design, $U^{a}=(u_{a}^{ij})$ , according to the following method:

$\displaystyle u_{ij}^{a}=ia^{j-1}(\text{mod }n),i=1,\ldots,n;j=1,\ldots,s.$ (2)

Calculate the wrap-around discrepancy (WD) of each U-type design using the following method, and choose the U-type design with the minimal WD among all the $U^{a}$ as the approximate uniform design $U_{n}(n^{s})$ .

Implement first the following linear transformation on each element, $u_{a}^{ij}$ of $U_{a}(n^{s})$ :

$\displaystyle{x}_{ij}=\frac{2{u}_{ij}^{a}-1}{2n}$ (3)

Then, calculate the WD using the expression

$\displaystyle\text{WD}({U}^{a})=\sqrt{\frac{1}{n}\left({\frac{3}{2}}\right)^{s% }-\left(\frac{4}{3}\right)^{s}+\frac{2}{n^{2}}\sum_{i=1}^{n-1}\sum\limits_{j=i% +1}^{n}\prod\limits_{k=1}^{s}\left({\frac{3}{2}-\left|x_{ik}-x_{jk}\right|+% \left|x_{ik}-x_{jk}\right|^{2}}\right)}$ (4)

If the required number of levels for the $j$ th factor is $r$ , where $r<n$ , and $n$ can be divided exactly by $r$ , perform the following operations:

If $\text{mod}\left(\frac{u_{n}^{ij}}{n/r}\right)=0$ , then $u_{n}^{ij}=\frac{u_{n}^{ij}}{n/r}$ ; otherwise, $u_{n}^{ij}=\left[\frac{u_{n}^{ij}}{n/r}\right]+1$ , where, $\left[\frac{u_{n}^{ij}}{n/r}\right]$ represents the integer part of $\frac{u_{n}^{ij}}{n/r}$ .

2.2 Stepwise regression

Polynomial regression is a commonly used method for converting nonlinear regression into linear regression [19, 20, 21]. It has been widely used to model multivariate nonlinear problems. In general, a certain number of candidates are provided in polynomial regression, and the terms that should be introduced to the regression equation are determined using significance tests. Furthermore, stepwise regression also considers the change in significance of existing terms once a new term is introduced to the regression equation, thus making it a more reasonable approach. To model computer experiments, second-order polynomial models are most frequently used [14]. In general, if the nonlinearity of the problem is not adequately strong to invoke the necessity of high-order terms, the stepwise regression process automatically eliminates high-order candidate terms. In this study, considering the strong nonlinearity of the electromagnetic field, sixth-order polynomials were applied to implement stepwise regression.

Suppose there are $m$ candidates in total. The stepwise regression algorithm can then be described as follows:

Step 1:
Predetermine the level of significance needed to introduce a candidate term, $\alpha_{in}$ , and that to eliminate one, $\alpha_{\textit{out}}$ , from the regression equation.
Step 2:
Assuming p (0 $\leqq$ $p$ $\leqq$ $m$ ) terms have already been introduced to the regression equation and, thus, that there are $m-p$ terms waiting to be introduced (referred to as $x_{1},\ldots,x_{m-p}$ ), successively implement the following two operations:

Introduce variables: Introduce the $i$ th term (1 $\leqq i\leqq m-p$ ) from the remaining $m-p$ candidate terms to build a new regression equation, calculate its partial F-statistic $F_{i}$ , and obtain the corresponding level of significance of $x_{i}$ in the regression equation, $\alpha_{i}$ , using the following method:

$\displaystyle\alpha_{i}=1-\int_{0}^{F_{i}}f(t)dt$ (5)

$f(t)$ can be calculated by

$\displaystyle f(t)=\left\{{\begin{array}[]{ll}\frac{{\Gamma}\left[\frac{n1+n2}% {2}\right]}{{\Gamma}\left(\frac{n1}{2}\right){\Gamma}\left(\frac{n2}{2}\right)% }\left(\frac{n1}{2}\right)\left(\frac{n1}{n2}{t}\right)^{\frac{n1}{2}-1}\left(% 1+\frac{n1}{n2}t\right)^{-\frac{n1+n2}{2}},&t\geqslant 0\\ 0,&t<0\\ \end{array}}\right.$ (6)

where, $n_{1}=$ 1, $n_{2}=n_{d}-p-1$ , and $n_{d}$ represents the scale of experimental data. $\Gamma(\tau)$ can be calculated by

$\displaystyle{\Gamma}(\tau)=\int_{0}^{+\infty}x^{\tau}e^{-x}dx$ (7)

Repeat this operation from $i=$ 1 to $i=m-p$ . If the level of significance of the $j$ th variable (1 $\leqq j\leqq m-p$ ) is lower than that of any other $m-p-1$ variables and $\alpha_{j}<\alpha_{in}$ , introduce variable $x_{j}$ into the regression equation. At this point, $p+1$ terms have been introduced to the regression equation.

Eliminate some terms from the $p+1$ terms previously introduced in the equation. For the $p+1$ terms, calculate their respective levels of significance in the regression equation using the approach described above. If the significance of the $j$ th variable (1 $\leqq j\leqq p+1$ ) is higher than that of any other $p$ variables and $\alpha_{j}>\alpha_{\textit{out}}$ , eliminate term $x_{j}$ from the regression equation and reconstruct it. Repeat this operation until no term can be eliminated from the regression equation.

Step 3:
Repeat Step 2 until no candidate term can either be introduced into or eliminated from the regression equation.

2.3 Methods for model evaluation

Polynomial models can be evaluated using three methods: the fitting-effect test, leave-one-out cross validation (LOOCV), and case verification at untried points [14]. In the fitting-effect test, the predicted value at each test point of the final polynomial model is directly compared with the results of the FE simulation. If the prediction error meets the precision requirement, the test point is considered to have passed validation. The validation of data fitting performance can reflect the prediction effect of a metamodel at all points used for modeling; however, it cannot assess predictive performance at positions other than those used for modeling within the space for experimental factors. LOOCV [14] can be described as the following process: The $i$ th ( $1\leqslant i\leqslant n$ ) point is held out while the other $n-1$ points are used to construct the model. The constructed model is then used to predict the $i$ th point. If the prediction is satisfactory, the point is considered to have passed validation. This process must be repeated to cover all points. When the LOOCV passing rate is satisfactory, the metamodel constructed based on all points has better prediction capability at points other than those used in the modeling process. Verification at untried points is the most direct approach to evaluate the metamodel’s predictive performance. However, it requires additional experiments that are computationally expensive.

3. Model construction for the electromagnet-rail relationship of maglev trains

3.1 Description of magnet-rail system

Figure 4 shows the side view of the rail and electromagnet. All constants marked in the figure are given in Table 1. Z denotes the levitation gap between the rail and electromagnet, Y is the lateral offset, and I is the current in the coil. The ranges of variation of these three experimental factors are shown in Table 2.

Table 1
Parameters of the rail and electromagnet

Parameter	Description	Value
N	Number of turns of coil	360
L	Length of electromagnet	1343.5 m
W	Total width of rail	220 mm
w	Width of magnet pole face	28 mm
d1	Height of rail	88 mm
d2	Distance between turning point and pole face of rail	60 mm
d3	Height of electromagnet	130 mm
d4	Distance between turning point and pole face of electromagnet	70 mm

Table 2

Ranges of variation of experimental factors

Experimental factor	Range of variation
Levitation gap $Z$ (mm)	6 $\sim$ 14
Lateral offset $Y$ (mm)	0 $\sim$ 18.4
Coil current strength $I$ (A)	0 $\sim$ 45

Figure 4.

Side view of rail and electromagnet.

3.2 Predictive precision requirement

The vertical motion of the suspension module of maglev trains depends primarily on the resultant of the levitation force and the gravity of the train as well as its mass. A larger resultant force leads to higher vertical acceleration. Therefore, the metric of absolute error was used to assess the metamodel of the electromagnetic levitation force. Based on the design parameters for electromagnets, the absolute error of 900 N between the values predicted by the metamodel and those calculated by the FE simulation was used as criterion to determine whether polynomial prediction for the levitation force was satisfactory. As the lateral motion of the suspension module depends mainly on the mass of the module itself and the guidance force of the electromagnets, 200 N was used as criterion to determine whether the polynomial prediction for the guidance force was satisfactory.

Figure 5.

Prediction performance of guidance force metamodel versus the scale of experimental data. (a) LOOCV passing rate versus data scale. (b) RMSE at 20 untried points of guidance force predictions versus data scale.

Figure 6.

Prediction performance of the levitation force metamodel versus the scale of experimental data. (a) LOOCV passing rate versus data scale. (b) RMSE at 20 untried points of levitation force predictions versus data scale.

3.3 Determining the scale of experimental data

The predictive precision of the metamodel is dependent on the scale of experimental data. Under the given experimental design and modeling technique, larger scales of experimental data lead to higher precision of the metamodel that, however, incurs great amount of computation. In this paper, the scale of the experimental data was initially set to 10 and gradually increased for stepwise regression modeling under the conditions of $\alpha_{in}=$ 0.15 and $\alpha_{\textit{out}}=$ 0.15. The predicted root-mean-squared error (RMSE) at 20 untried points under different experimental data scales was calculated by

$\displaystyle\text{RMSE}=\sqrt{\sum\limits_{i=1}^{20}\frac{(x_{i}-\hat{x_{i}})% ^{2}}{20}}$ (8)

where $x_{i}$ represents the electromagnetic force value at the ith untried point and $\hat{x_{i}}$ the prediction value.

Figures 5 and 6 show the variation in the LOOCV passing rate and the predicted RMSE at 20 untried points of the guidance and the levitation forces, respectively. It is clear that the passing rate increased and the RMSE decreased with the increasing number of test points. According to the absolute error levels of the levitation and guidance forces, the scale of the experimental data used for the final model was 400 points.

3.4 Final model

Based on the 400 points of experimental data, we used the approach described above to determine metamodels for the guidance and levitation forces. The final polynomial model of the guidance force was

$\displaystyle f_{y}=0.6198-0.1761y+0.2138z-0.7183i+0.0598yz-0.3424yi{}+0.3468% zi+0.0637y^{2}-0.0739z^{2}-01914i^{2}+0.1199yzi-0.0248y^{2}z{}+0.0535y^{2}i-0.% 0656z^{2}i-0.0149z^{2}y-0.1170i^{2}y+0.0085i^{2}z-0.0121y^{3}{}+0.0077z^{3}+0.% 1904i^{3}+0.0252y^{3}z-0.0338y^{3}i+0.0387z^{3}i+0.0146z^{3}y{}+0.0438i^{3}y-0% .0919i^{3}z-0.0465z^{2}i^{2}-0.0188y^{2}zi-0.035z^{2}yi{}+0.0436i^{2}zy+0.0144% z^{5}-0.0323i^{5}-0.0154zy^{4}+0.0786iy^{4}-0.0593iz^{4}{}-0.0159y^{3}i^{2}-0.% 0114y^{2}zi^{2}+0.0073i^{2}z^{3}-0.0212yzi^{3}+0.0358yz^{3}i{}+0.0218y^{3}zi-0% .0229yz^{2}i^{2}+0.0224y^{2}zi^{2}-0.0128z^{6}+0.0107i^{6}{}-0.0088yi^{5}+0.01% 86zi^{5}-0.0447y^{5}i+0.0344z^{5}i+0.0249z^{2}i^{4}-0.0097y^{4}i^{2}{}+0.1024z% ^{4}i^{2}-0.0106y^{3}z^{3}+0.0408y^{3}i^{3}-0.1027z^{3}i^{3}+0.0187yz^{4}i{}-0% .037y^{4}zi+0.0274z^{3}y^{2}i-0.0172z^{3}i^{2}y$ (9)

The final polynomial model for the levitation force was

$\displaystyle f_{z}={}-0.5272-0.0959y-0.2946z+0.9148i+0.0765yz-0.1553yi{}-0.46% 51zi-0.0164y^{2}+0.1329z^{2}+0.2423i^{2}+0.1092\textit{yzi}+0.0104y^{2}z{}-0.0% 321y^{2}i+0.1015z^{2}i-0.0301z^{2}y-0.0149i^{2}y+0.0264i^{2}z+0.0053y^{3}{}-0.% 0129z^{3}-0.2878i^{3}-0.0121y^{3}z-0.0856z^{3}i+0.0547i^{3}y+0.1499i^{3}z{}-0.% 0073y^{2}i^{2}-0.0134y^{2}zi-0.0458z^{2}yi-0.0318i^{2}yz-0.0365z^{5}{}+0.0662i% ^{5}+0.0767z^{4}i-0.0415i^{2}z^{3}-0.0862\textit{yzi}^{3}-0.0202yz^{3}i{}+0.00% 78y^{3}zi+0.0576yz^{2}i^{2}+0.0283z^{6}-0.0222i^{6}-0.0063yz^{5}{}+0.016yi^{5}% -0.0373zi^{5}-0.1132z^{4}i^{2}+0.031y^{3}z^{3}+0.1286z^{3}i^{3}{}+0.0365y^{4}% zi+0.0358i^{3}z^{2}y$ (10)

where $y$ , $z$ , and $i$ represent the normalized values of the lateral offset Y, levitation gap Z, and coil current I, respectively. For lateral offset Y, the normalization is given by

$\displaystyle y=\frac{Y-Y_{\min}}{Y_{\max}-Y_{\min}}-1$ (11)

where $Y_{\min}$ and $Y_{\max}$ are the minimal and maximal values in the sequence of actual values of lateral offset Y, respectively. The levitation gap and current in the coil can be normalized accordingly.

4. Evaluation of metamodels

4.1 Evaluation of the guidance force metamodel

According to the criterion of the absolute prediction error of 200 N, the polynomial model of the guidance force was evaluated using the fitting-effect test, leave-one-out cross validation, and case verification at untried points.

Figure 7a shows the relationship between the values predicted by the polynomial model and the results of the FE simulation. It is evident that the values of the polynomial model were in considerably better agreement with the simulation results compared with those from the analytical formula shown in Fig. 2. Figure 7b shows the absolute error of the data fitting of the polynomial model at 400 points. The absolute prediction error at the 400 points was lower than 50 N, much smaller than the criterion. This indicates that the polynomial metamodel can perfectly fit the data from the computer simulations.

Figure 7.

Performance in predicting guidance force using the polynomial model at 400 test points. (a) Comparison between polynomial predictions of guidance force and the simulation results. (b) Absolute error in the predicted guidance force using the polynomial model.

Figure 8.

Absolute predicted error in the guidance force in leave-one-out cross-validation.

Figure 8 shows the absolute prediction error of the guidance force using LOOCV on the 400 points. The error was always lower than the passing criterion of 200 N, which was indicative of a passing rate of 100%. Therefore, the polynomial model given in Eq. (9) is accurate for predicting the electromagnetic guidance force with consideration of saturation of the magnetic medium.

Figure 9.

Performance in predicting guidance force using the polynomial model at 20 untried points. (a) Comparison between the polynomial predictions of the guidance force and the results of simulation. (b) Absolute error in the predicted guidance force using the polynomial model.

Aside from the 400 points used to construct the model, another 20 points were randomly scheduled following the uniformity principle for case verification, and are shown in Fig. 9a. It can be seen that the values predicted by the polynomial model were in good agreement with the results of the FE simulation at the 20 untried points. Figure 9b shows that the absolute prediction error was always lower than 60 N at the 20 points with a passing rate of prediction of 100%.

To illustrate the improvement in prediction performance, Table 3 gives the RMSE and the rate of satisfaction of the prediction of the guidance force using the analytical model and metamodel, which was calculated at the 400 modeling points as well as the 20 untried points. The RMSE of the metamodel was much smaller than that of the analytical formula in both cases. The rate of satisfaction of predictions by the metamodel was approximately twice that of the analytical model, which indicates that the former is more suitable to describe the guidance force of the actual magnet-rail relationship.

Table 3

Predictive performance of guidance force from analytical model and metamodel

Parameter	Analytical model	Metamodel
RMSE at 400 modeling points	862.24 N	6.8287 N
Rate of satisfaction of prediction at 400 modeling points	49.25%	100%
RMSE at 20 untried points	837.85 N	16.0348 N
Rate of satisfaction of prediction at 20 untried points	55%	100%

Table 4

Predictive performance of levitation force from analytical model and metamodel

Parameter	Analytical model	Metamodel
RMSE at 400 modeling points	4588.6 N	45.6419 N
Rate of satisfaction of prediction at 400 modeling points	53.5%	100%
RMSE at 20 untried points	3235.7 N	276.42 N
Rate of satisfaction of prediction at 20 untried points	65%	100%

Figure 10.

Performance in predicting levitation force using the polynomial model at 400 test points. (a) Comparison between the polynomial predictions of levitation force and the simulation results. (b) Absolute error in the predicted levitation force using the polynomial model.

Figure 11.

Absolute prediction error in levitation force in leave-one-out cross-validation.

Figure 12.

Performance in predicting levitation force using the polynomial model at 20 untried points. (a) Comparison between the polynomial predictions of levitation force and simulation results. (b) Absolute error in the predicted levitation force using the polynomial model.

4.2 Evaluation of the levitation force metamodel

We evaluated the final polynomial model of the levitation force, shown in Eq. (10), using the same scheme as above. Figure 10 illustrates the predicted values of the final polynomial model of the levitation force in comparison with the results of the FE simulation at the 400 modeling points in Fig. 10a and the absolute errors in Fig. 10b. It clearly indicates that Eq. (10) faithfully reproduced the 400 test points with an absolute error of less than 400 N. Figure 11 shows the absolute prediction error of levitation force using LOOCV at the 400 points. The absolute errors at these points were always smaller than 900 N for a 100% LOOCV passing rate, which indicates that the final polynomial model obtained by stepwise regression can accurately predict the levitation force even when the saturation of the magnetic media is considered. Figure 12 shows the case verification of the levitation force metamodel in Eq. (10) using 20 untried points. Although the absolute errors appear larger than those in Figs 10b and 11, they are also smaller than the prediction criterion of 900 N of the levitation force, and result in a prediction passing rate of 100%. Table 4 summarizes the RSMEs and the rates of satisfaction of prediction from the analytical model and the metamodel. It is clear that the metamodel yielded a much lower RMSE and significantly improved the rate of satisfaction of the prediction, which means that the metamodel is more suited to describe the levitation force of the actual magnet-rail relationship.

5. Conclusions

In this study, metamodels were developed for the 2D electromagnet-rail relationship to calculate the levitation and guidance forces with consideration of medium saturation based on uniform design and stepwise regression. By implementing the fitting-effect test, leave-one-out cross-validation, and case verification at untried points, it was confirmed that the metamodels can fulfill the precision requirements of dynamic simulations in engineering. Using this approach, the challenge of considering magnetic saturation in the dynamic simulations of maglev trains was successfully addressed. The proposed metamodels thus provide a more efficient means of conducting rapid computations of the electromagnetic levitation and guidance forces with consideration of magnetic saturation.

Footnotes

Acknowledgments

This work is supported by the National Key R&D Program of China (2016YFB1200600).

References

, A Study of Dynamic Modeling and Steering Capability of EMS Mid-Speed Maglev Train, Changsha: National University of Defense Technology, 2016.

Yao

and Fang

, The computation of mechanical dynamic characteristic in EMS-Maglev system by electro-mechanical equations coupled finite element analysis, International Journal of Applied Electromagnetics and Mechanics 25(1–4) (2007), 225–230.

Zhai

and Zhao

, Dynamics of Maglev Vehicle/Guideway Systems (I) – Magnet/Rail Interaction and System Stability, Chinese Journal of Mechanical Engineering 7 (2005), 1–10.

Zhao

and Zhai

, Dynamics of Maglev Vehicle/Guideway Systems (II) – Modeling and Simulation, Chinese Journal of Mechanical Engineering 8 (2005), 163–175.

Cui

Zhang

and Li

, Calculation of Electromagnetic Force and Torque of Suspension Electromagnet Based on Schwarz-Christoffel Transform, Proceedings of the CSEE 24 (2010), 129–134.

Cui

, A Study of Guidance Dynamics of Low-Speed Maglev Vehicles, National University of Defense Technology, 2010.

Zhang

, University Physics, 3

{}^{\text{rd}}

Edition A. Beijing: Tsinghua University Press, 2008.

Song

Chen

and Lin

, Characteristic and performance analysis of SRM with actual B-H curve of electrical steel, International Journal of Applied Electromagnetics and Mechanics 53 (2017), 435–449.

Binns

K.J.

and Lawrenson

P.J.

, Analysis and Computation of Electro and Magnetic Field Problems, People’s Education Press, 1980.

10.

Cao

Tao

and Cai

, Magnetic field analytical solution and electromagnetic force calculation of coreless stator axial-flux permanent-magnet machines, International Journal of Applied Electromagnetics and Mechanics 54 (2017), 93–105.

11.

Fang

, Theory, Methods, and Applications of Uniform Experiment Design-History Review, Application of Statistics and Management 3 (2004), 69–80.

12.

Fang

Liu

and Zhou

, Experimental Design and Modeling, Beijing: Higher Education Press, 2011.

13.

Fang

and Ma

, Orthogonal and Uniform Experimental Design, Beijing: Science Press, 2001.

14.

Fang

and Sudjianto

, Design and Modeling for Computer Experiments, New York: Chapman & Hall/CRC, 2006.

15.

Lü

Xiao

and Li

, Moving least squares method for reliability assessment of rock tunnel excavation considering ground-support interaction Computers and Geotechnics 84 (2017), 88–100.

16.

Jiang

and Jiang

, Establishing a Mathematical Equations and Improving the Production of l-tert-Leucine by Uniform Design and Regression Analysis, Applied Biochemistry and Biotechnology 181 (2017), 1454–1464.

17.

Xin

and Ai

, Optimization of hard modified asphalt formula for gussasphalt based on uniform experimental design, Construction and Building Materials 136 (2017), 556–564.

18.

Wang

and Zhang

, Integrating uniform design and response surface methodology to optimize thiacloprid suspension, Scientific Reports 7 (2017), 1–9.

19.

Xue

and Tang

, Design Methods of Optimization Experiments and Data Analysis, Chemical Industry Press, 2012.

20.

Wang

and Sui

, Experimental Design and MATLAB Data Analysis, Beijing: Tsinghua University Press, 2015.

21.

and Min

, Functional Regression Analysis, 2nd Ed. Beijing: Higher Education Press, 2014.

Metamodel for 2D magnet-rail relationship based on stepwise regression

Abstract

Keywords

1. Background

2.1 Uniform design method

3. Model construction for the electromagnet-rail relationship of maglev trains

3.1 Description of magnet-rail system

Table 1 Parameters of the rail and electromagnet

4.1 Evaluation of the guidance force metamodel

5. Conclusions

Footnotes

Acknowledgments

References

Table 1
Parameters of the rail and electromagnet