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This paper presents an optimum design of a lightweight vehicle levitation electromagnet, which also provides a passive guide force in a magnetic levitation system for contactless delivery applications. The split alignment of C-shaped electromagnets about C-shaped rails has a bad effect on the lateral deviation force, therefore, no-split positioning of electromagnets is better for lateral performance. This is verified by simulations and experiments. This paper presents a statistically optimized design with a high number of the design variables to reduce the weight of the electromagnet under the constraint of normal force using response surface methodology (RSM) and the kriging interpolation method. 2D and 3D magnetostatic analysis of the electromagnet are performed using ANSYS. The most effective design variables are extracted by a Pareto chart. The most desirable set is determined and the influence of each design variable on the objective function can be obtained. The generalized reduced gradient (GRG) algorithm is adopted in the kriging model. This paper’s procedure is validated by a comparison between experimental and calculation results, which shows that the predicted performance of the electromagnet designed by RSM is in good agreement with the simulation results.

Electromagnetically levitated and guided systems with linear motor propulsion are commonly used in the field of people transport vehicles, tool machines and conveyor systems because of their silent and non-contact motion [

Response Surface Methodology (RSM) is generally used with two or three design variables, however we have seven design variables, so a mixed orthogonal array table was utilized. The reason for this is that a mixed orthogonal array table is an efficient way to study the effect of several design variables simultaneously with a number small of experiments and to plan matrix simulation trials. For each design variable combination the response value is determined by the 2D and 3D Finite Element Method (FEM). In this paper we use the reduced gradient algorithm, which can lead to the selection of the most desired set of variables.

Let’s assume that the guidance forces are generated by closed-loop control of levitation electromagnets. The lateral response of the electromagnet due to the airgap control is a force increasing with the lateral offset X. This force is almost a linear function of the lateral offset of the electromagnets. Then we can imagine that the electromagnet will move laterally as a mass bound by a spring. As we know, a spring working on a mass is a mechanical resonant system, whose resonance frequency is given by:
_{guide} can be designed to keep the disturbance from causing an excessive displacement. An example of the design problem is shown in

A total mass of 100 kg is assumed to be levitated with a constant gap. The sum of lateral position stiffness of the levitation magnets is set at 10,000 N/m, 50,000 N/m and 100,000 N/m, respectively. The lateral disturbance forces are assumed to be a constant value of 100 N for about 1 second. The simulation result shows that if the lateral position stiffness remains higher than 50,000 N/m, the lateral position deviation is smaller than 5 mm under constant 100 N disturbance.

In the experiment, a total mass of 200 kg including four levitation electromagnets is levitated under small deviation of electromagnet placement conditions, as shown in

The RSM method can be readily adapted to develop an analytical model for a complex problem. With this analytical model, an objective function with constraints can be easily created and evaluated, and computation time can be saved. A polynomial approximation model is commonly used for a second-order fitted response (

The least squares method is used to estimate unknown coefficients. Matrix notations of the fitted coefficients and the fitted response model should be as shown in

Kriging is a method of interpolation named after a South African mining engineer D. G. Krige, who developed the technique while trying to increase accuracy in predicting ore reserves. In the kriging model, the global approximation model for a response y(^{2}_{total}

The unknown correlation parameters of _{1}_{2}_{n}_{i}

Optimization formulation:

Seven dimensions are selected as the design variables as shown in

_{total}_{normal}

The adjusted coefficients of the multiple determination R^{2}_{adj} for normal force and weight are _{total}_{normal}_{7}_{1}

The values of ineffective design variables are determined by RSM. _{7}_{1}

The purpose of this paper is to minimize the objective function (_{total}_{normal}_{1}_{7}_{total}_{normal}

This paper deals with optimum design of a lightweight levitation electromagnet on a vehicle, which also provides passive guide force, in a magnetic levitation system for contactless delivery applications. The optimum design procedure is introduced to design of electromagnet in the magnetic levitation system to reduce its weight and to improve the normal force of the initially designed electromagnet in the magnetic levitation system using several design variables. The most effective design variables are extracted by Pareto chart. The most desired set is determined by RSM and the kriging interpolation method and the influence of each design variables on the objective function can be obtained. This can efficiently increase the precision of the optimization and reduce the number of experiments in the optimization design using the proposed methodologies.

_{∞}control of Maglev systems

Magnetic levitation system prototype for contactless delivery application: (a) 3D modeling (b) prototype.

Optimization flow chart.

Application prototype.

Lateral position model for electromagnets under constant levitation control: (a) passive guidance model; (b) simulation result for a 100 N disturbance.

Lateral position response of magnetically sprung mass under position disturbance: (a) experimental model; (b) lateral position response (experiment); (c) lateral position stiffness response (simulation); (d) lateral force response (simulation).

Prototype of the electromagnet.

Comparison of FEM and experiment (reference model).

Design variables and flux pattern (2D and 3D FEM).

Magnetization curve of the core material used.

Pareto chart of the standardized effects (alpha=0.05) (a) Normal force response (b) Total weight response.

Response optimization.

Design variable and level.

Design variable Level | _{1} |
_{2} |
_{3} |
_{4} |
_{5} |
_{6} |
_{7} |
---|---|---|---|---|---|---|---|

−1 | 16 | 45 | 16 | 40 | 7 | 16 | 144 |

0 | 20 | 50 | 20 | 45 | 11 | 20 | 180 |

1 | 24 | 55 | 24 | 50 | 15 | 24 | 216 |

Table of mixed orthogonal array _{18}(2^{1} × 3^{7}).

_{1} |
_{2} |
_{3} |
_{4} |
_{5} |
_{6} |
_{7} |
|||
---|---|---|---|---|---|---|---|---|---|

| |||||||||

1 | 16 | 45 | 16 | 40 | 7 | 16 | 144 | 382.10 | 8.706 |

2 | 16 | 50 | 20 | 45 | 11 | 20 | 180 | 473.67 | 12.153 |

3 | 16 | 55 | 24 | 50 | 15 | 24 | 216 | 564.15 | 16.366 |

4 | 20 | 45 | 16 | 45 | 11 | 24 | 216 | 682.56 | 15.433 |

5 | 20 | 50 | 20 | 50 | 15 | 16 | 144 | 450.22 | 10.979 |

6 | 20 | 55 | 24 | 40 | 7 | 20 | 180 | 570.56 | 13.699 |

7 | 24 | 45 | 20 | 40 | 15 | 20 | 216 | 795.40 | 16.803 |

8 | 24 | 50 | 24 | 45 | 7 | 24 | 144 | 531.60 | 12.66 |

9 | 24 | 55 | 16 | 50 | 11 | 16 | 180 | 651.24 | 14.064 |

10 | 16 | 45 | 24 | 50 | 11 | 20 | 144 | 379.41 | 10.277 |

11 | 16 | 50 | 16 | 40 | 15 | 24 | 180 | 473.76 | 12.094 |

12 | 16 | 55 | 20 | 45 | 7 | 16 | 216 | 568.30 | 13.874 |

13 | 20 | 45 | 20 | 50 | 7 | 24 | 180 | 570.38 | 13.637 |

14 | 20 | 50 | 24 | 40 | 11 | 16 | 216 | 682.15 | 15.453 |

15 | 20 | 55 | 16 | 45 | 15 | 20 | 144 | 450.16 | 10.946 |

16 | 24 | 45 | 24 | 45 | 15 | 16 | 180 | 659.92 | 14.604 |

17 | 24 | 50 | 16 | 50 | 7 | 20 | 216 | 789.93 | 16.646 |

18 | 24 | 55 | 20 | 40 | 11 | 24 | 144 | 528.87 | 12.400 |

Optimum level and size.

| |||||||
---|---|---|---|---|---|---|---|

| |||||||

_{1} |
_{2} |
_{3} |
_{4} |
_{5} |
_{6} |
_{7} | |

Initial | 20 | 50 | 20 | 40 | 15 | 20 | 180 |

Optimum (RSM) | 23.877 | 45 | 16 | 40 | 7 | 16 | 166.844 |

Optimum (Kriging) | 23.877 | 45 | 16 | 40 | 7 | 16 | 166.844 |

Optimum β and correlation parameters for kriging models (N_{s}=100).

_{1}_{1}) |
_{2}_{2}) |
_{3}_{3}) |
_{4}_{4}) |
_{5}_{5}) |
_{6}_{6}) |
_{7}_{7}) |
||
---|---|---|---|---|---|---|---|---|

_{total} |
3.026e-3 | 2.304e-4 | 0.909e-3 | 0.201e-3 | 6.318e-5 | 1.195e-3 | 1.289e-2 | 16.0023 |

_{normal} |
1.319e-2 | 1.382e-5 | 3.437e-6 | 1.172e-5 | 7.819e-6 | 2.320e-6 | 4.087e-2 | 587.009 |

Comparison of initial and optimum model.

Initial | 2D FEM | 13.319 | 578.5 |

3D FEM | 13.319 | 573.48 | |

Error(2D |
0 | 0.86 | |

Optimum | 12.300 | 637 | |

12.107 | 611.68 | ||

FEM (verification) | 11.799 | 611.70 | |

Error ( |
−4.073 | −3.972 | |

Error ( |
−2.610 | 0.003 | |

Variation between initial and optimum FEM % | −11.412 | 7.754 |