Optimization of the pole piece in coaxial magnetic gears for transfer torque ripple improvement

Abstract

In this paper, the Box-Behnken design (BBD), which is a response surface method (RSM), was used to deduce a pole piece form that is able to effectively reduce the transfer torque ripple of a coaxial magnetic gear. A 2-D numerical analysis based non-linear finite element method (FEM) was used for the analysis between the response variables and the designed variables selected by the BBD. In addition, the reaction variable estimation equation based on the design variables was determined through a multiple regression analysis and analysis of variance (ANOVA). The characteristics of the initial model and optimum model deduced through a response surface analysis were compared to prove the validity of the optimized design.

Keywords

Coaxial magnetic gear pole piece transfer torque ripple Box-Behnken design (BBD)response surface method (RSM)analysis of variance (ANOVA)finite element method (FEM)multiple linear regression analysis

1. Introduction

A coaxial magnetic gear is a non-contact machine that is used to transfer torque and to accelerate or decelerate. This type of gear has several advantages, including no mechanical loss, no required maintenance, and outstanding protection against overload [1, 2]. As a result, they are used in various applications, such as wind power generation and electric vehicles [3, 4, 5, 6, 7, 8]. However, coaxial magnetic gears have high transfer torque ripples due to the difference in the magnetic resistance between the two rotors and the pole piece, and the torque ripple of the inner rotor is higher than that of the outer rotor [9, 10, 11]. These torque ripples must be minimized during the design process because they cause vibration and noise. In order to reduce the torque ripples, the inner rotor must be designed to have multiple poles. However, when the number of inner rotor poles increases, the number of outer rotor poles also increases by the gear ratio. The increase in the number of poles causes an increase in the size of the gear and the driving frequency. Therefore, in this paper, a new pole piece form is proposed that reduces the transfer torque ripples in the coaxial magnetic gear. The response surface method (RSM) was used to investigate the relationship between the design variables used for the proposed pole piece and the response variables, such as the transfer torque and the transfer torque ripples. The Box-Behnken design (BBD) was used to establish the experimental plans, and a 2-D numerical analysis based finite element method (FEM) was used to determine the response variables. In addition, an analysis of variance (ANOVA) and regression analysis of the design variables and response variables were used to estimate the response surface equation of the response variables to the design variables, and to determine the optimum design variables for the proposed pole piece.

2. Structure of the coaxial magnetic gear

Figure 1.

Cross section of the coaxial magnetic gear with the conventional pole pieces.

Figure 1 shows the cross section and magnetic flux density distribution of a coaxial magnetic gear with conventional pole pieces. As shown in the figure, the pole pieces are placed between the two rotors and permanent magnets are attached to their surfaces and have a role in modulating the magnetic flux. There are 4 inner rotor poles and 24 outer rotor poles. An electrical steel plate was used as the material for both the back yoke and pole pieces. The number of inner and outer rotor poles determines the gear ratio and the number of pole pieces in the coaxial magnetic gear. The number of pole pieces is determined by Eq. (1) and the gear ratio is determined by Eq. (2). The gear ratio is 6 based on the number of inner and outer rotor poles.

$\displaystyle N_{S}=p_{in}+p_{out}$ (1) $\displaystyle G_{r}=\frac{p_{out}}{p_{in}}$ (2)

where $N_{s}$ is the number of pole pieces, $p_{in}$ is the number of inner rotor poles, $p_{out}$ is the number of outer rotor poles, and $G_{r}$ is the gear ratio.

Figure 2.

Design variables of pole piece.

Figure 3.

Response surface methodology (a) Central composite design, (b) Box-Behnken design.

3. Optimization of pole pieces in the coaxial magnetic gear

3.1 Response surface method and Box-Behnken design

The RSM, which is a type of design of experiment (DOE), analyses the relationships between the input variables $x_{1},\cdots x_{n}$ and result $y$ when the optimal conditions have been found through ANOVA or DOE, and is used to find the optimum response condition based on several factors. The unknown function $f$ between design variables $x_{1},\cdots x_{n}$ and dependent variable $y$ can be expressed as $y=f(x)$ , and this unknown function is called the response function. The simple linear regression model is used when there is one independent variable and one dependent variable, and the relationship between them is linear. In general, however, when explaining most industrial or social phenomena, the number of independent variables affecting dependent variables is more than two. Therefore, two or more independent variables are used, assuming a linear relationship between independent variables and dependent variables, and a multiple linear regression model is used for regression analysis. Therefore, in this paper, the multiple linear regression model is used because there are four independent variables and dependent variables. Second-order regression model is applied as the result of regression analysis by creating scatter diagram of independent variables and dependent variables. Equation (3) can be derived by assuming the reaction function $y$ follows a second-order regression model where there are $k$ number of independent variables $x$ [12].

$\displaystyle y=\beta_{0}+\sum\limits_{i=1}^{k}\beta_{i}x_{i}+\sum\limits_{i=1% }^{k}\beta_{ii}x_{ii}^{2}+\sum\limits_{i\leqslant j}^{k}\beta_{ij}x_{i}x_{j}+\varepsilon$ (3)

Table 1

High and low level of the pole piece

Level		Design variable
		A (mm)	B (mm)	C (mm)	D (mm)
Low	$-$ 1	0.2	0.2	0.2	0.2
Central	0	0.7	0.7	0.7	0.7
High	$+$ 1	1.2	1.2	1.2	1.2

Table 2

Transfer torque characteristics when only the design variable A, B are applied with fillets

Design variable A, B (mm)	Transfer torque (Nm)		Transfer torque ripple (%)
	Inner rotor	Outer rotor	Inner rotor	Outer rotor
Initial model	0.257	1.543	8.347	0.681
0.2	0.256	1.541	8.298	0.753
0.4	0.255	1.531	8.048	0.741
0.6	0.252	1.514	8.590	0.812
0.8	0.249	1.496	8.234	0.618
1.0	0.244	1.467	9.039	0.721
1.2	0.236	1.421	9.065	0.949

Table 3

Transfer torque characteristics when only the design variable C, D are applied with fillets

Design variable C, D (mm)	Transfer torque (Nm)		Transfer torque ripple (%)
	Inner rotor	Outer rotor	Inner rotor	Outer rotor
Initial model	0.257	1.543	8.347	0.681
0.2	0.256	1.541	7.328	0.810
0.4	0.255	1.533	6.331	0.796
0.6	0.253	1.525	5.478	0.881
0.8	0.252	1.518	4.015	0.804
1.0	0.251	1.507	3.032	0.881
1.2	0.249	1.497	2.209	0.781

Table 4

ANOVA results of inner rotor torque

Source	DF	SS ${}^{1)}$	MS ${}^{2)}$	F ${}^{3)}$		P ${}^{4)}$
Regression	14	0.000731	0.000052	19591.	49	0.000
Linear	4	0.000662	0.000165	62089.	74	0.000
A	1	0.000291	0.000291	109297.	10	0.000
B	1	0.000292	0.000292	109683.	92	0.000
C	1	0.000039	0.000039	14767.	57	0.000
D	1	0.000039	0.000039	14610.	37	0.000
Square	4	0.000067	0.000017	6245.	58	0.000
AA	1	0.000036	0.000036	13506.	90	0.000
BB	1	0.000036	0.000036	13441.	82	0.000
CC	1	0.000000	0.000000	82.	18	0.000
DD	1	0.000000	0.000000	87.	12	0.000
Interaction	6	0.000003	0.000000	156.	60	0.000
AB	1	0.000001	0.000001	312.	93	0.000
AC	1	0.000000	0.000000	127.	00	0.000
AD	1	0.000000	0.000000	150.	01	0.000
BC	1	0.000000	0.000000	149.	94	0.000
BD	1	0.000000	0.000000	125.	99	0.000
CD	1	0.000000	0.000000	73.	72	0.000
Residual error	12	0.000000	0.000000
Total	26	0.000731

Table 5

ANOVA results of outer rotor torque

Source	DF	SS ${}^{1)}$	MS ${}^{2)}$	F ${}^{3)}$		P ${}^{4)}$
Regression	14	0.026354	0.001882	19625.	28	0.000
Linear	4	0.023860	0.005965	62185.	94	0.000
A	1	0.010500	0.010500	109465.	56	0.000
B	1	0.010538	0.010538	109866.	79	0.000
C	1	0.001410	0.001410	14700.	35	0.000
D	1	0.001411	0.001411	14711.	05	0.000
Square	4	0.002405	0.000601	6267.	21	0.000
AA	1	0.001302	0.001302	13572.	64	0.000
BB	1	0.001293	0.001293	13478.	81	0.000
CC	1	0.000008	0.000008	85.	81	0.000
DD	1	0.000008	0.000008	85.	74	0.000
Interaction	6	0.000090	0.000015	156.	89	0.000
AB	1	0.000030	0.000030	313.	71	0.000
AC	1	0.000012	0.000030	126.	63	0.000
AD	1	0.000014	0.000014	150.	32	0.000
BC	1	0.000014	0.000014	150.	54	0.000
BD	1	0.000012	0.000012	126.	33	0.000
CD	1	0.000007	0.000007	73.	80	0.000
Residual error	12	0.000001	0.000000
Total	26	0.026356

Table 6

ANOVA results of inner rotor torque ripple

Source	DF	SS ${}^{1)}$		MS ${}^{2)}$		F ${}^{3)}$		P ${}^{4)}$
Regression	6	42.	9784	7.	1631	1101.	32	0.000
Linear	4	42.	8566	10.	7142	1647.	30	0.000
A	1	0.	2216	0.	2216	34.	07	0.000
B	1	0.	1192	0.	1192	18.	33	0.000
C	1	23.	1328	23.	1328	3556.	67	0.000
D	1	19.	3830	19.	3830	2980.	13	0.000
Square	1	0.	0466	0.	0466	7.	17	0.014
AA	1	0.	0466	0.	0466	7.	17	0.014
Interaction	1	0.	0751	0.	0751	11.	55	0.003
CD	1	0.	0751	0.	0751	11.	55	0.003
Residual error	20	0.	1301	0.	0065
Total	26	43.	1085

Table 7

ANOVA results of outer rotor torque ripple

Source	DF	SS ${}^{1)}$	MS ${}^{2)}$	F ${}^{3)}$	P ${}^{4)}$
Regression	6	0.067762	0.011294	29.86	0.000
Linear	4	0.057069	0.014267	37.72	0.000
A	1	0.035663	0.035663	94.29	0.000
B	1	0.020449	0.020449	54.07	0.000
C	1	0.000956	0.000956	2.53	0.128
D	1	0.000001	0.000001	0.00	0.965
Square	2	0.010693	0.005347	14.14	0.000
AA	1	0.004044	0.004044	10.69	0.004
BB	1	0.008577	0.008577	22.68	0.000
Residual error	20	0.007564	0.000378
Total	26	0.075326

${}^{1)}SS_{\textit{Regression}}=\sum(\mathord{\buildrel\lower 3.0pt\hbox{$% \scriptscriptstyle\frown$}\over{y}}-\bar{y})^{2},SS_{\textit{Error}}=\sum(y-% \bar{y})^{2}$ . ${}^{2)}MS_{\textit{Regression}}=SS_{\textit{Regression}}/k,MS_{\textit{Error}}% =SS_{\textit{Error}}/n-k-1$ . ${}^{3)}F=MS_{\textit{Regression}}/MS_{\textit{Error}}$ . ${}^{4)}P-\textit{value}<$ 0.05.

Figure 4.

Response surface of inner and outer rotor torque, inner and outer torque ripple.

Figure 5.

Overlaid contour plot.

where $\varepsilon$ is the vector of random errors, and a normal distribution with mean zero and variance $\sigma^{2}$ is generally assumed. Therefore, the response function estimated from the approximation function in vector form is

$\displaystyle y=X\beta+\varepsilon$ (4)

where $X$ is the matrix of the design variables, $\beta$ is the vector of regression coefficients. The least-squares method that has the minimum sum of squares of random errors was used for the vector of regression coefficients, and the minimum square estimation of the regression coefficients vector is equal to Eq. (5).

$\displaystyle\beta=(X^{\prime}X)^{-1}X^{\prime}y$ (5)

BBD is a type of RSM that is used to efficiently estimate the first-order and second-order terms when it is known that all experiments are performed in stable ranges, and when all factors are not low level or high level at the same time. There are advantages in BBD when the number of experiments is relatively less than that required for central composite design (CCD) for the same number of factors, when the experiment cost at the factorial point is too high, or when the experiments are practically impossible. On the other hand, the design at the axial point is required when using CCD. Figure 3 shows the response surface methodology. $\alpha$ is the axis point as shown in Fig. 3a.

The $\alpha$ value is typically determined to maintain the rotatability, indicating the distance from the center of the experimental area to the axis. Rotatability refers to the property that the predicted variance of two points at the same distance from the center of the experimental area has the same value, and the rotatable design means the experiment plan with such a rotatability. The predictive variance of the rotatable design is a function of distance from the center of the experimental area and not a function of direction. Therefore, the contour line of the predicted dispersion becomes concentric [12]. Rotatability is a very useful feature in response surface analysis because it is not known at which point in the experimental area the optimum value will be obtained before the experiment. The value of $\alpha$ is determined by the number of factorial test points in the central composite plan as follows [13].

$\displaystyle\alpha=2^{k/4}$ (6)

where $k$ is the number of independent variables. The $\alpha$ calculated by the Eq. (6) is 2.

In the design of the axial point, “ $\alpha$ ” has the value of $\pm$ 2 when there are three design variables in which 2 times larger and smaller experiment points are obtained compared to the initially configured design variable [13]. In this case, the range of design variables that were configured in advance increases and the proposed pole piece may not be formed and can have the design variables that are impossible to design [8]. Therefore, the BBD was used in this paper for optimization because it was difficult to determine the range of initial design variables. The design variables necessary to reduce the transfer torque ripples in the coaxial magnetic gear are shown in Table 1 and Fig. 2. As shown in Fig. 2, the four edges of the pole pieces are chamfering in order to reduce the magnitude of the transfer torque ripples.

3.2 Transfer torque and torque ripple characteristics analysis by the design variables

Before the pole piece was optimized, the transfer toque characteristics of the coaxial magnetic gear based on the selected design variables were analyzed. Table 2 shows the transfer torque and torque ripple characteristics when only the design variables A and B are applied with fillets (C and D are not applied with fillets). Table 3 shows the transfer torque and torque ripple characteristics when only the design variables C and D are applied with fillets (A and B do not apply fillet).

The reduction in the inner and outer transfer torque can be seen in Tables 2 and 3 as the area of the pole piece was reduced due to the edge chamfering. However, the minimum value of the transfer torque ripple based on the change of design variables A and B was shown to be random regardless of the pole piece area. The transfer torque ripple caused by the change of design variables C and D only decreased in the inner rotor and increased in the outer rotor. Therefore, it was shown that design variables C and D were not significant to the reduction of transfer torque ripple of the outer rotor. In addition, it was found that the four edges to which the edge chamfering process was applied each showed different optimum design variables.

3.3 Multiple linear regression analysis and analysis of variance

A multiple linear regression analysis (MLRA) includes the second- or third-order terms for the dependent variables, and interaction terms between the dependent variables. In Tables 4–7, degree of freedom (DF) represents the number of points that can freely change in the given conditions, SSR refers to the regression sum of squares, SSE is the residual sum of squares, MS shows each mean square, and the F-value is the test statistic and tests if the regression equation is statistically significant when explaining data properties. Where $y$ is result of the experiment, $\bar{y}$ is average of the $y$ values, and $\mathord{\buildrel\lower 3.0pt\hbox{$\scriptscriptstyle\frown$}\over{y}}$ is value predicted by the regression equation. The p-value represents the significant probability, and it was determined through a regression analysis that a p-value lower than 0.05 was determined to be statistically significant for analysis purposes.

However, it was considered that factors with p-values of 0.05 or higher were not significant and do not influence the response variables. Therefore, an ANOVA was performed by eliminating insignificant factors with p-values higher than 0.05 in sequence [12, 13]. Through this process, the interaction and square term factors with p-values higher than 0.05 were eliminated in each variance analysis table.

The regression analyses on the design variables A, B, C, and D of the proposed pole piece were performed using the results of the analysis conducted by the BBD.

As a result, the ANOVA of the inner rotor torque, outer rotor torque, inner rotor torque ripple, and outer rotor torque ripple are shown in Tables 4–7. As a result of the ANOVA of the inner rotor torque and outer rotor torque, the square and interaction terms were all judged as insignificant terms. As a result of the ANOVA of the inner rotor torque ripple, BB, CC, DD term of the square term, and the AB, AC, AD, BC and BD terms, the interaction terms were judged as insignificant terms and thus, were eliminated. For the outer rotor torque ripple, CC, DD term of the square term and interaction terms were all judged as insignificant terms and eliminated.

The linear terms C and D are insignificant factors that have p-values higher than 0.05 in the ANOVA table of outer rotor torque ripple, but do not have to be eliminated during regression analysis because the total linear term is significant. Therefore, the inner rotor torque, outer rotor torque, inner rotor torque ripple, and outer rotor torque ripple can be estimated as second-order models. The regression analysis estimation equation by each design variable are

$\displaystyle f_{IT}(A,B,C,D)=0.256314+0.003346A+0.003290B-0.001937C-\,0.00188% 8D-0.010392AA-0.010367BB-0.000811CC-\,0.000835DD+0.001826AB-0.001164AC+0.00126% 5AD+\,0.001264BC+0.001159BD-0.000886CD$ (7) $\displaystyle f_{OT}(A,B,C,D)=1.53920+0.02022A+0.01980B-0.01144C-\,0.01145D-0.% 062495AA-0.062279BB-0.004969CC-\,0.004967DD+0.010971AB-0.006970AC+0.007594AD+% \,0.007600BC-0.006962BD-0.005321CD$ (8) $\displaystyle f_{ITR}(A,B,C,D)=8.307-0.197A+0.1993B-2.393C-2.158D+0.335AA-\,0.% 548CD$ (9) $\displaystyle f_{OTR}(A,B,C,D)=0.6256-0.0317A-0.1224B+0.0179C-0.0005D+0.1006AA% +\,0.1464BB$ (10)

where $f_{IT}(A,B,C,D)$ and $f_{OT}(A,B,C,D)$ are the torque estimation equations of the inner rotor and outer rotor, respectively, and $f_{ITR}(A,B,C,D)$ and $f_{OTR}(A,B,C,D)$ are the transfer torque ripple estimation equations of the inner rotor and outer rotor, respectively.

Figure 6.

Magnetic flux density of optimized model.

Figure 7.

Torque waveforms of inner and outer rotor.

3.4 Multiple linear regression analysis and analysis of variance

Considering the transfer torque and torque ripple due to pole piece design optimization, the multi-objective function optimization model of the coaxial magnetic gear can be defined as follows.

$\displaystyle\textit{max}:\{f_{IT}(A,B,C,D),f_{OT}(A,B,C,D)\}$ (11) $\displaystyle\textit{min}:\{f_{ITR}(A,B,C,D),f_{OTR}(A,B,C,D)\}$ (12)

A MLRA was performed using the experimental results according to the BBD and the response surface plots of the inner rotor torque, outer rotor torque, inner rotor torque ripple, and outer rotor torque ripple are shown in Fig. 4. Numerous surface plots must be used for analysis because a response surface analysis is performed using two fixed factors, and each surface plot can observe the response surfaces of only two design variables at the same time. Therefore, only one surface plot was indicated by setting the optimum design variable to a fixed value. The overlaid contour plot of the design variables of the proposed pole piece is shown in Fig. 5. Generally, one design variable is placed on the x-axis and the other variable is placed on the y-axis in overlaid contour plots, and three or more variables must be fixed in a specific level. Contour lines of each response in the overlaid contour plot are overlapped in one graph and each contour line shows the boundary of the response function value set in the legend. The white region is the optimal region that satisfies all ranges shown in the legend. Therefore, the optimal design variables A, B, C, and D were determined to be 0.2 mm, 0.4 mm, 1.2 mm and 1.2 mm, respectively, through the response surface analysis and overlaid contour plot. The magnetic flux density of the optimal model is shown in Fig. 6. The torque waveforms of the inner rotor and outer rotor are shown in Fig. 7. As shown in Fig. 7, inner and outer transfer torques each decreased by 3.5% compared to the initial model and transfer torque ripple decreased by 73.2% and 2.9% by optimization. Figure 8 shows the torque harmonics. It was found that the 29th harmonic component was the main cause of high transfer torque ripple. It was found that the proposed design method was a very effective method to offset the 29th harmonic component. The optimization results are shown in Table 8.

Table 8

Optimization results

		Initial model	Optimized model
Inner rotor	Torque (Nm)	0.257	0.247
	Torque ripple (%)	8.347	2.237
Outer rotor	Torque (Nm)	1.543	1.488
	Torque ripple (%)	2.76	2.66

Figure 8.

Torque harmonics of inner and outer rotor.

4. Conclusion

In this paper, the BBD, which is a RSM, was used to deduce a pole piece form that was able to effectively reduce the transfer torque ripple in a coaxial magnetic gear. An analysis between the response variables and the design variables selected by the BBD was performed using a 2-D numerical analysis based non-linear FEM. In addition, a MLRA and ANOVA were used on the regression equation of the response variables. As a result, the inner and outer rotor torque ripples were each reduced by 73.2% and 2.9%, respectively, compared to the initial model.

Footnotes

Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2015R1D1A3A01017536) and the Technology Innovation Program (or Industrial Strategic Technology Development Program) (No. 10076577) funded by the Ministry of Trade, Industry & Energy (MOTIE, Korea).

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