Optimal design of quadratic electromagnetic exciter

2.1. Design objective function

Purpose of the quadratic electromagnetic exciter is providing sufficient vibration force to moving parts assembly in limited space. To achieve this goal, maximization of the magnetic force $F_{Magnetic}$ (Lorentz force) was set as objective function in this research. So that, whole design process tried for finding optimum values of design variables which can derive higher magnetic force in $Z$-direction.

2.2. Design variables

Fig. 4 and 5 show respectively the schematic diagram and the top view of magnetic part of the exciter excluding the plastic case, plate spring, and stinger. The whole size of magnetic part of the exciter is predetermined as$\mathrm{}$150 mm×150 mm×70 mm. There are 4 inner magnets and 4 outer magnets respectively. The electromagnetic coil is located at the center of air gap between inner magnets and outer magnets those are respectively attached to inner steel yokes and outer steel yokes. The design factors which have influence on the magnetic force of the exciter are the electric coil winding turns, the effective coil length, the input electric current and the magnetic flux density in the air gap. The specification of electric coil such as coil length coil winding turns, input electric current are predetermined as shown in Table 1. The magnetic flux density in the air gap is decided by the size of air gap, the sizes of inner magnets and inner yokes and sizes of outer permanent magnets and outer steel yokes. The size of air gap is 15 mm. And, the sizes of inner steel yokes are automatically determined by the sizes of inner permanent magnets because the predetermined size of electric coil makes the size of inner yokes be dependent on size of inner permanent magnets. However the thickness of outer yoke can be manipulated regardless of size outer magnet. In results, the thickness of inner magnet ($x_1$), the width of outer magnet ($x_2$), the thickness of outer magnet ($x_3$) and the thickness of outer yoke ($x_4$) are chosen as the design variables. The height of permanent magnet is determined as the maximum value considering the whole size of magnetic part of exciter. Table 2 shows the initial values, lower values, and upper values of the design variables.

Fig. 4Design variables and fixed constants of the exciter

Fig. 5Flux leakage areas in the exciter

2.3. Design problem formulation

The design formulation for the optimal design of the exciter is prescribed as follows:

3.1. Design of experiments

The number of design variables is four and each design variables have 5 Levels. So, we employed the orthogonal array (OA) $L$₂₅(5⁶), which is one of the DOEs for computer experiments. To perform the electromagnetic analysis of the quadratic electromagnetic exciter, we built 25 finite element exciter models according to the change in design variables. Table 5 shows 25 OA sample points.

3.2. Electromagnetic FEM (static analysis)

Magnetic force in air-gap makes coil and coil housing components which are connected with collecting plate through a shaft move in vertical direction. So this force should be predicted before design the exciter. 25 EA prototypes of the exciter model were created with the orthogonal array above, and electromagnetic static analysis was conducted with these models to derive estimated value of magnetic force in $Z\text{-direction}$ for each model. The MAXWELL, commercial electromagnetic analysis program was used to achieve this objective. Fig. 6(a) and (b) show a pre-meshed 3-dimensional finite element model of the exciter with an arrowed component that is a moving part (coil) in this device and distribution of magnetic flux density in the exciter.

Fig. 6(a) Pre-meshed 3-dimensional FE exciter model; (b) Distribution of magnetic flux density in the FE exciter model

a)

b)

MAXWELL was using DRS8 solve as an analysis problem solver, and on average, about 12.700 EA of mesh elements were created on the exciter. During the analysis process, 10 times of computation process were performed for each model on the orthogonal array. Final analysis error was about 11.37 % in the end and a Fig. 7 shows error convergence trend to that value.

Fig. 7Error convergence of FE exciter models

DOE results are shown in Table 6. The magnetic force of the quadratic electromagnetic exciter was evaluated using MAXWELL according to the change in design variables specified by the OA table. In DOE results, the magnetic force value of case No. 25 is increased by 1.68 % compared to the initial design. Although the result of case No. 25 was enhanced, we performed the metamodel-based design optimization to determine the global optimum solution that maximized the magnetic force.

3.3. Metamodeling

Metamodeling builds a metamodel that approximates the relationship between performance index (magnetic force) and design variables by using the analysis results at the sampling points specified by a DOE. To generate an appropriate metamodel, we used the Approximate Model Wizard equipped in PIAnO. The Approximate Model Wizard compared the accuracies of several popular metamodels (Polynomial Regression, RBF, Kriging, etc.) using “predicted R2” and generated the best one automatically. In metamodeling result, the “predicted R2” value of Kriging model for the magnetic force is 98.29 %.

3.4. Optimization algorithm

Since the Kriging model generated in Section 3.3 may have many local optima, an evolutionary algorithm (EA) equipped in PIAnO was used to obtain the global optimum solution. The EA, one of the global optimization methods, evolves a population (i.e., a set of design variables) generation by generation by repetitively using evolutionary operators such as selection, recombination and mutation [8-10].

3.5. Design optimization results

Using the proposed design approach, the optimal magnetic force was increased by 1.68 % compared to the initial design. Since the metamodel-based design optimization result yields predicted optimum values (Approximate Optimization pred) of performance index (magnetic force) evaluated by metamodel, the actual value (Approximate Optimization act) must be evaluated by running MAXWELL at the optimal design point. In Fig. 8, both AO_pred and AO_act are drawn along with the initial values. They are very close to each other, which confirm the accuracy of the Kriging model we employed. In Fig. 8, the objective function values are shown, which present the predicted and actual magnetic forces. The predicted and actual values were increased to 30.2 and 30.19, respectively, from 29.8.

Fig. 8Comparison of AO_pred and AO_act values

The optimum design variable values are listed along with the initial, lower and upper bound values in Table 7. The optimum values of Inner mag. T ($x_1$) and Outer mag. W ($x_2$) were found close to upper bound while the optimum and initial values of Outer mag. T ($x_3$) and Outer yoke T ($x_4$) were almost the same.