Design and analysis of a high speed double-sided axial flux permanent magnet motor suspended vertically by magnetic bearings

Abstract

This paper presents the electromagnetic and mechanical analysis of an axial flux permanent magnet (AFPM) motor for high speed (12000 rpm) rotor which is vertically suspended by magnetic bearings. In the analysis, a prototype AFPM motor with a double-sided rotor and a coreless stator between the rotors are considered. Firstly, electromagnetic analysis of the motor is carried out by using magnetic equivalent circuit method. Then, the rotor disk thickness is determined based on a rotor axial displacement due to the attractive force between the permanent magnets placed on opposite rotor disks. Hereafter, an analytical solution is carried out to determine the natural frequencies of the rotor-shaft system. Finally, 3D finite element analysis (FEA) is carried out to verify the analytical results and some experimental results are given to verify the analytical and numerical results and prove the stable high-speed operation.

Keywords

Electromagnetic modeling high speed axial flux permanent magnet motor mechanical stress analysis rotor vibration analysis

1. Introduction

In the last two decades, studies about AFPM machines have intensified due to their most highlighted advantages: high power density, high efficiency and high torque-to-volume ratio. The compact and robust structure of the AFPM machines makes them very suitable for electrical energy generating applications. Especially, those working at low speeds are widely used in wind energy applications, flywheel energy storage systems (FESS) and electrical vehicles [1–6].

With suspending the AFPM motor by magnetic bearings, the friction between contacted surfaces is removed, so, bearing and thermal losses at high speeds are significantly reduced. Moreover, the AFPM motor is enabled to operate at very high speed due to non-contact bearings, with the option of being used for direct drive applications. With eliminating gear mechanism would lead to reduced transmission losses that bring high-power density and higher efficiency [7–10].

There are very small airgaps between the stator and permanent magnets (PMs) mounted on the rotor disks that produce a very high attractive force between the PMs on opposite rotor disks. These forces may cause to bend rotor disks from the outer edges. Since the rotor disks of the motor represent around 60% of the total active mass of the machine, the mechanical design of such a machine should carefully be reviewed. Power to weight ratio of AFPM machine can be increased by optimizing the rotor disk thickness. Mueller et al. investigated the inactive mass in the design of AFPM machines for direct-drive low-speed wind turbines [6]. Fei et al. studied a mechanical analysis and design optimization of the rotor discs for a high-speed air-cored AFPM generator; however mechanical vibrations of the machine were not considered [7,8]. Sadeghierad et al. presented designing procedure of back-iron thickness of the AFPM generator considering the optimization of terminal voltage, output power, and efficiency [9]. Kumar et al. investigated the sinusoidal back EMF considering the mechanical stress analysis for an AFPM machine for high-speed energy storage application. So far, most studies focus on the analysis and design for AFPM machines from the point of view of electromagnetic aspects. Although some examined studies have addressed mechanical issues, they are dealt with as supplementary to the electro-magnetic design and most of them are considered for low-speed applications [11–18].

This study contributes a systematic analysis and design methodology for an AFPM motor with a double-sided rotor and a coreless stator between the rotors from the point view of estimating mechanical stress resulted from the electromagnetic origin for high-speed operations. A unique structure is proposed for the AFPM machine suspension that integrated with radial and axial magnetic bearings allowing the rotor to reach high-speeds. Hence, employing such configuration for high-speed flywheel energy storage systems supported by magnetic bearings may eliminate flywheel disk and twin-rotor disks of the AFPM can function as flywheel disk. If it is mentioned that the motor rotates at high speed, the vibration modes that will occur should also be examined. However, there is almost no study on this subject for AFPM in the literature; we carried out a detailed analysis on this issue. While the design provides a considerable shortening of axial length of shaft when compared with the horizontally suspended counterparts; it also increases the gyroscopic rate and reduces the gyroscopic effects that may occur at high speeds.

The paper is organized as following; at first, electromagnetic analysis methodology using magnetic equivalent circuit approach is outlined. The same model used in electromagnetic analysis is also utilized in 3-D finite element analysis to verify correctness of the analytical results of magnetic equivalent method approach. Then, mechanical stress and vibration analysis of the AFPM are performed by using electromagnetic analysis results. Effectiveness of the analytical approach is confirmed by 3-D finite element analysis both mechanical stress and vibration analysis. At last, the constructed prototype is operated at high-speeds, the simulation results are verified by performing the hammer test of the rotor and examining the mechanical vibration values at high speeds.

2. Electromagnetic and structural analysis

The AFPM motor considered in this study has double-sided rotor disks with a coreless stator between them. Figure 1 show the produced rotor disk and stator before assembled. The coreless stator coils are housed in a non-magnetic epoxy-resin plate. The rotor disks are constructed from structural steel and PMs are mounted on them. The dimensions of the constructed prototype are also given in Fig. 1.

Fig. 1.

Prototype AFPM motor: one of the rotor disks with PMs (left), Stator (right).

2.1. Electromagnetic analysis of the AFPM motor

Double sided rotor with internal coreless stator topology used for low and medium power generators and have various advantages, including the absence of cogging torque, as well as their linear torque – current characteristics, high power density, and compact construction [11,12,16,19–21]. Also, by removing the core losses, this type of generator has a higher efficiency compared to the conventional generators. Considering the number of poles should be low to reach high speeds, it is decided to design the motor with 2 pole pairs. The coils in the stator are designed with a single layer concentrate winding form with 15 turns per coil. After assembling the AFPM motor, the rotor disks are integrated with a shaft suspended by magnetic bearings both radial and axial directions. The total system is shown in Fig. 2.

Fig. 2.

AFPM motor vertically suspended by magnetic bearings.

The magnetic equivalent circuit for a pole associated with the motor topology is shown in Fig. 3. In this circuit, R_gap, R_PM, R_lk and E_PM are the air-gap reluctance, the permanent magnet reluctance, the leakage flux path reluctance and magneto motive force due to coercivity of the PMs, respectively. The flux travels axially from one rotor disc to the other and completes its path by returning circumferentially around the rotor disc to the next magnet. We can define the flux by assuming that all reluctance are collected by R_total; $$\begin{eqnarray} \displaystyle {\Phi}={\displaystyle \frac{E_{PM}}{R_{\mathit{total}}}}. & & \displaystyle \end{eqnarray}$$(1) Reluctances embedded in R_total; R_PM, R_lk and R_gap can be defined as: $$\begin{eqnarray}\displaystyle R_{PM}={\displaystyle \frac{l_{PM}}{{\mu}_{0}{\mu}_{r}A_{PM}}},\quad R_{lk}={\displaystyle \frac{l_{lk}}{{\mu}_{0}A_{lk}}},\quad R_{gap}={\displaystyle \frac{l_{gap}}{{\mu}_{0}A_{gap}}}. & & \displaystyle\end{eqnarray}$$(2) Calculation of the flux flowing through associated air-gap reluctance can be handled by adopting either node or mesh analysis of circuit analysis. Here we used node analysis rather than the mesh. To validate the flux analysis, a 3-D finite element analysis is employed by Maxwell tool of ANSYS package. As shown in Figs 4 and 5 the maximum flux density of airgap by FEM is 0.54 Tesla and that of calculated by the electromagnetic model is 0.59 Tesla and this confirms effectiveness of magnetic equivalent (analytical) method. Available maximum flux is directly related by the output power of the AFPM and treated as machine parameter [1].

Fig. 3.

Magnetic equivalent circuit of the AFPM for a specified pole cross section.

Fig. 4.

Magnetic flux density distribution in the air-gap of the AFPM. (a) FEM representation of the flux profile for the specified pole region. (b) Magnetic flux distribution motor structure.

Fig. 5.

The magnetic flux density profile through a specified pole cross section.

2.2. Mechanical stress and vibration analysis of the system

The rotor disks of the motor are constructed of two 8 mm-thick steel disks with 7 mm-thick permanent magnets which represent around 60% of the total active weight of the machine. Here, to determine the thickness of the rotor disks, bending values are examined as a result of the attraction forces between the magnets. Well-known analysis of magnetic equivalent circuits can be used to determine the airgap flux density and hence the Maxwell stress is given by $\begin{eqnarray}\displaystyle q={\displaystyle \frac{B_{g}^{2}}{2{\mu}_{0}}} & & \displaystyle\end{eqnarray}$ (3) where B_g is the airgap flux density and μ₀ is the permeability of free space.

Magnetic flux density vs. airgap curve was calculated using ANSYS Maxwell to decide the air gap between stator and PM’s mounted on rotor discs. In the literature, the magnetic flux density value is found around 0.6 T for similar sizes AFPM’s. Then it is decided that the operating airgap point should be 2 mm on both sides when the mechanical limits are considered, as can be seen from the Fig. 6.

Fig. 6.

Magnetic flux density vs. airgap between the opposite poles.

Hereafter, Fig. 7 shows the attraction force value if a distance of 15 mm is found between opposite magnets; the difference between the results doesn’t exceed 10%. Although the difference between the results is minimal at the nominal operating point, the error increases due to flux leakages that is considered in FEA. So that, FEA results are taken into account to see the magnetic pull force which corresponds to a magnitude of 500 N per magnet and a total of 12 magnets which corresponds to 6000 N. These values will be used to find deflection and stress results of different thickness rotor discs.

Fig. 7.

Magnetic attraction force between the opposite magnets.

In order to calculate the bending and stress values, the rotor disk with geometric details is first modeled analytically. Circular plate theory can be used to model the rotor disk as a flat circular plate of constant thickness with its outer edge free and the inner edge fixed. The geometrical details of the rotor-magnets system can be seen in Fig. 1. Circular plate equation is written as follows by taking the center of the plate in polar coordinates [23]: $\begin{eqnarray}\displaystyle {\nabla}^{2}{\nabla}^{2}w=\left({\displaystyle \frac{\partial ^{2}}{\partial r^{2}}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial r}}+{\displaystyle \frac{1}{r^{2}}}{\displaystyle \frac{\partial ^{2}}{\partial {\theta}^{2}}}\right)\left({\displaystyle \frac{\partial ^{2}w}{\partial r^{2}}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial w}{\partial r}}+{\displaystyle \frac{1}{r^{2}}}{\displaystyle \frac{\partial ^{2}w}{\partial {\theta}^{2}}}\right)={\displaystyle \frac{q}{D}} & & \displaystyle\end{eqnarray}$ (4)D, the bending stiffness of the rotor disk. For the case with the symmetrical axis, the plate equation is as follows: $\begin{eqnarray}\displaystyle {\nabla}^{2}{\nabla}^{2}w=\left({\displaystyle \frac{\partial ^{2}}{\partial r^{2}}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial r}}\right)\left({\displaystyle \frac{\partial ^{2}w}{\partial r^{2}}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial w}{\partial r}}\right)={\displaystyle \frac{q}{D}}. & & \displaystyle\end{eqnarray}$ (5) The lateral bending function of the rotor disk (w) can be found by solving (5) using the disk boundary conditions: $\begin{eqnarray}\displaystyle w={\displaystyle \frac{qr^{4}}{64D}}+C_{1}+C_{2}\ln r+C_{3}r^{2}+C_{4}r^{2}\ln r & & \displaystyle\end{eqnarray}$ (6) where C₁, C₂, C₃ and C₄, are integral constants. The appendix for the calculation of these constants is available in the reference [6]. For bending the disk, bending moments per unit length is expressed as follows: $\begin{eqnarray}\displaystyle \begin{array}{@{}c@{}}\displaystyle M_{r}=-D\left({\displaystyle \frac{d^{2}w}{dr^{2}}}+{\displaystyle \frac{v}{r}}{\displaystyle \frac{dw}{dr}}\right)\\ \displaystyle D=\frac{Eh^{3}}{12(1-v^{2})}\end{array} & & \displaystyle\end{eqnarray}$ (7)h, is the thickness of the rotor disk. By differentiating w and using boundary conditions, moment equations can be obtained. The maximum stress on the disk can be written using the moment equation: $\begin{eqnarray}\displaystyle ({\sigma}_{r})_{\text{max}}={\displaystyle \frac{6M_{r}}{h^{2}}}. & & \displaystyle\end{eqnarray}$ (8) The mechanical vibrations of the rotor-disk system used in many engineering applications play a crucial role in system life. Therefore, the bending and torsional modes of the rotor must be calculated and the design must be made considering these conditions. The modal analysis of rotor-disk systems takes into account thin and hard disks without flexibility. It is assumed that the shaft has a negligible mass and is very elastic. There are two rotor disks and a thrust disk on the elastic shaft in the system. As a result, the system consists of three disks shown in Fig. 2.

For a three-disk rotor-disk system, the equations of motion in matrix form can be written as: $\begin{eqnarray}\displaystyle \left[\begin{array}{@{}ccc@{}}I_{p1} & 0 & 0\\ 0 & I_{p2} & 0\\ 0 & 0 & I_{p3}\end{array}\right]\left[\begin{array}{@{}c@{}}\ddot{{\varphi}}_{1}\\ \ddot{{\varphi}}_{2}\\ \ddot{{\varphi}}_{3}\end{array}\right]+\left[\begin{array}{@{}ccc@{}}k_{1} & -k_{1} & 0\\ -k_{1} & k_{1}+k_{2} & -k_{2}\\ 0 & -k_{2} & k_{2}\end{array}\right]\left[\begin{array}{@{}c@{}}{\varphi}_{1}\\ {\varphi}_{2}\\ {\varphi}_{3}\end{array}\right]=\left[\begin{array}{@{}c@{}}0\\ 0\\ 0\end{array}\right] & & \displaystyle\end{eqnarray}$ (9)I_p1, I_p2 and I_p3 are the inertia of disks; φ₁, φ₂ and φ₃ are angular displacement of disks. k₁ and k₂ are torsional stiffness of shaft parts. The following equation is used to obtain the natural frequencies of torsional vibration with simple harmonic motion and the determinant is equalized to zero to find the corresponding natural frequencies [22,24]: $\begin{eqnarray}\displaystyle -{\omega}_{n}^{2}[M]+[K]\{{\varphi}\}=0. & & \displaystyle\end{eqnarray}$ (10) Here, the gyroscopic effects are neglected, as the shaft is thick enough and relatively short. Besides, conical rigid modes of the rotor are investigated using FEM analysis but analytical solution for this analysis is not given here due to the complexity. Moreover, bending vibrations are studied analytically which is also not given here since the shaft is vertically suspended and assuming there is no imbalance.

3. Results and discussion

3.1. Numerical analysis

In this section, a comparison between analytical model results and FEA will be explained. In order to verify the analytical results, finite element analyses are performed by using Modal and Structural Analysis sections of ANSYS Workbench; electromagnetic force analysis is performed by using ANSYS Maxwell. Electromagnetic and mechanical properties of the materials used in FEA is tabulated in Table 1. The analytical approach of electromagnetic analysis based on magnetic equivalent modelling of machine components is developed by considering flux leakage effect between adjacent poles. The model is quite simple and logical to determine the air gap flux density of the machine. It is found that flux distribution shape is identical and amplitude values compatible. As can be seen in Fig. 5, the highest flux amplitude error is less than 10%. Hereafter, the electromagnetic force and mechanical stress analysis results are demonstrated. Figure 7 shows the attraction force analyses between the rotor disks. The result of this analysis is used in structural analysis. To simplify the analysis it is assumed that the stress q spread across the area of the magnet. The Maxwell stress is thus modified by multiplying it by the magnet cross-sectional area, so that the overall force acting on the rotor disk remains constant.

Table 1
Mechanical properties of the materials used in FEA

Part Material 𝜌 (kg/m³) E (GPa) v

Rotor disks Carbon steel (1045) 7800 200 0.31

Permanent magnets Sintered NeFeB – N42 7500 160 0.3

RMB rings & thrust disk Soft magnetic composite (SMC) 7300 100 0.23

Sensor disk Silicon iron 7100 120 0.31

Part	Material	𝜌 (kg/m³)	E (GPa)	v
Rotor disks	Carbon steel (1045)	7800	200	0.31
Permanent magnets	Sintered NeFeB – N42	7500	160	0.3
RMB rings & thrust disk	Soft magnetic composite (SMC)	7300	100	0.23
Sensor disk	Silicon iron	7100	120	0.31

Since the attractive forces of PMs are high and the airgap between the stator and rotor disks are small, the deflection of the rotor disks must be limited to prevent contact PMs from stator surface. The results of the mechanical stress analysis for different rotor thickness values are tabulated in Table 2. Considering maximum allowable stress for the used material and assuming that the motor will operated at reasonably constant environment conditions subjected to load and stresses that can be determined readily, the safety factor of 2 is selected and 8 mm thickness value is determined for the rotor disks. Analytical solution is carried out only for 8 mm thickness value. Figures 8 and 9 show the stress and bending values obtained as a result of the analysis. The comparative results of the mechanical stress analysis comply with each other and show that the bending values of the outer edges of rotor disks are much lower than the critical limits.

Table 2

Mechanical stress analysis comparison

Parameter	Thickness 6 mm	Thickness 7 mm	Thickness 8 mm	Analytical for 8 mm	Thickness 9 mm	Thickness 10 mm
Deflection (um)	90.019	64.372	47.866	47.756	36.545	28.647
Eq. Stress (MPa)	75.219	62.574	56.753	56.695	48.906	42.351

Fig. 8.

Stress analysis of rotor disk using FEA.

Fig. 9.

Deformation analysis of rotor disk using FEA.

Then, rotor mechanical vibration analysis is carried out; conical, bending and torsional vibration modes can be seen in Fig. 10. In the vibration analysis of the rotor, the radial magnetic bearing points are assumed to have stiffness value of 1000 N/mm. In addition, the axial magnetic bearing point is considered as fixed point assuming that the rotor does not move axial direction. While the mechanical design is performed, it is assumed that the shaft will rotate at a speed of 12000 rpm. As seen from the analysis results, given in Table 3, the results obtained by the analytical method and the results of numerical analysis overlap each other. In addition, it is sufficient as a design parameter that the 1st and 2nd conical mode is lower than the rotational speed of the rotor. Because rotor can be accelerated as fast enough to pass these critical frequencies and operating speed is far away from the 1st bending mode.

Fig. 10.

Vibration mode shapes of the shaft using FEA.

Table 3

Natural frequency analysis comparison

Mode #	Analytical solution	FEA	Hammer test
1. Rigid mode	–	64 Hz	64 Hz
2. Rigid mode	–	146 Hz	146 Hz
1. Bending mode	472 Hz	468 Hz	468 Hz
2. Bending mode	1018 Hz	1014 Hz	–
1. Torsional mode	1558.5 Hz	1560.6 Hz	–
2. Torsional mode	8388.4 Hz	8377.2 Hz	–

3.2. Experimental verification

A prototype motor has been produced and integrated into the magnetic bearing system which shown in Fig. 11. The motor can be directly drive by using the AB50A200I AxCent^{^TM} panel mount servo drives that are targeted for centralized control and offers high bandwidth and quick response times. The AB50A200I features built-in isolation between high and low power signals. Motor speed can be controlled by a closed-loop system. The PC sends analog control signals (via MicroLabBox) produced by drive logic based on hall-effect sensors’ feedback. The motor has been tested under the no-load condition and mid-speed increases over 6000 rpm with the input power at 2.7 kW which shown in Fig. 12. The test has been repeated several times to operate the rotor with varying rates of acceleration and deceleration. In addition, the motor was operated at maximum speed for 2 hours; thus, it has been confirmed that the rotor is mechanically robust. Also, hammer test is applied to the rotor using Bruel & Kjaer equipment, the results of which are shown in Fig. 13. Both analytical and FEA results verified via hammer test results that there is no vibration mode frequency near the operating speed which can be interpreted as shaft can be rotated safely.

Fig. 11.

Experimental setup (1 – Upper RMB, 2 – Axial HMB and Thrust disk, 3 – AFPM Rotor disks and Stator, 4 – Lower RMB).

Fig. 12.

Variation of the input power versus speed.

Fig. 13.

Hammer test frequency response function graph.

4. Conclusion

Since it has the advantage of disregarding necessity of flywheel in a FESS an AFPM may have great potential usage in such systems. Moreover, it may eliminate attraction force between the stator and rotor core due to non-ferromagnetic core usage possibility leading to a coreless structure. In this paper, to exploit the outlined advantages of AFPM machines electromagnetic, mechanical stress and vibration analysis were presented. The analytical approach of electromagnetic analysis based on magnetic equivalent modelling of machine components was developed by considering flux leakage effect between adjacent poles. The model is quite simple and logical to determine the air gap flux density of the machine. The effectiveness of the magnetic equivalent model is tested by a comparative study of 3-D FEM analysis of the same model. For the mechanical stress and vibration analysis, an analytical model employing the findings of electromagnetic analysis (especially attraction force between the rotor disks and sizing of rotor disks) was developed. Furthermore, to validate the effectiveness of the analytical approach of mechanical stress and vibration analysis, for the same model a series of 3-D FEM analyses were carried out. It was seen that the discrepancy between analytical model and FEM analysis can be negligible since it is less than 8%. The approach followed by the paper is effective and systematic to analyze electromagnetic and mechanical properties of the AFPM for possible use in FESS supported by magnetic bearing. In addition, models developed for electromagnetic and mechanical analysis can be used safely in the dimensioning and optimization of AFPM machines for high-speed operations.

Footnotes

Acknowledgements

This work is supported by the Scientific and Technological Research Council of Turkey under Project 217M073.

References

Gieras

Wang

and Kamper

, Axial Flux Permanent Magnet brushless Machines, Springer, New York, USA, 2008.

Bumby

J.R.

and Martin

, Axial-flux permanent-magnet air-cored generator for small-scale wind turbines, IEE Proc. – Electric Power Applications152(5) (2005), 1065–1075, doi:10.1049/ip-epa:20050094.

Lee

S.G.

and Cho

Y.H.

, Structure analysis of axial flux permanent magnet synchronous machine for wind generators, International Journal of Applied Electromagnetics & Mechanics39(1) (2012), 1027–1033, doi:10.3233/JAE-2012-1574.

Zhao

Han

Dai

and Hua

, Study on the electromagnetic design and analysis of axial flux permanent magnet synchronous motors for electric vehicles, Energies12(18) (2019), doi:10.3390/en12183451.

Hosseini

S.M.

Mirsalim

and Mirzaei

, Design, prototyping, and analysis of a low cost axial-flux coreless permanent-magnet generator, IEEE Transactions on Magnetics44(1) (2008), 75–80, doi:10.1109/TMAG.2007.909563.

Mueller

M.A.

McDonald

A.S.

and Macpherson

D.E.

, Structural analysis of low-speed axial-flux permanent-magnet machines, IEE Proceedings-Electric Power Applications152(6) (2005), 1417–1426, doi:10.1049/ip-epa:20050227.

Fei

Luk

P.C.K.

and El-Hasan

T.S.

, Rotor integrity design for a high-speed modular axial-flux permanent-magnet generator, IEEE Transactions on Industrial Electronics58(9) (2011), 3848–3858, doi:10.1109/TIE.2011.2106097.

Fei

Luk

P.C.K.

and Jinupun

, Design and analysis of high-speed coreless axial flux permanent magnet generator with circular magnets and coils, IET- Electric Power Applications4 (2010), 739–747, doi:10.1049/iet-epa.2010.0081.

Sadeghierad

Darabi

Lesani

and Monsef

, Rotor yoke thickness of coreless high-speed axial-flux permanent magnet generator, IEEE Transactions on Magnetics45(4) (2009), 2032–2037, doi:10.1109/TMAG.2008.2010452.

10.

Kumar

Lipo

T.A.

and Kwon

, A 32,000 rev/min axial flux permanent magnet machine for energy storage with mechanical stress analysis, IEEE Transactions on Magnetics52(7) (2016), 1–4.

11.

Cao

Tao

Cai

Jia

and Zhang

, Magnetic field analytical solution and electromagnetic force calculation of coreless stator axial-flux permanent-magnet machines, International Journal of Applied Electromagnetics and Mechanics54(1) (2017), 93–105, doi:10.3233/JAE-160112.

12.

Minaz

M.R.

and Çelebi

, Design and analysis of a new axial flux coreless PMSG with three rotors and double stators, Results in Physics7 (2017), 183–188, doi:10.1016/j.rinp.2016.10.026.

13.

Syed

Q.A.

You

Y.M.

and Kwon

B.I.

, Design and comparative analysis of single and multi-stack axial flux permanent magnet generator, Int. Journal of Applied Electromagnetics & Mechanics39(1) (2012), 865–872, doi:10.3233/JAE-2006.

14.

Cvetkovski

and Petkovska

, Multi-objective approach of design optimization of axial flux permanent magnet motor, International Journal of Applied Electromagnetics and Mechanics51(s1) (2016), 115–123, doi:10.3233/JAE-2006.

15.

Wang

Zhou

Wen

Liu

Zhang

and Guo

, Sensitivity analysis and optimal design of a stator coreless axial flux permanent magnet synchronous generator, Energy Sustainability11(5) (2019), doi:10.3390/su11051414.

16.

Taran

Rallabandi

Heins

and Ionel

D.M.

, Coreless and conventional axial flux permanent magnet motors for solar cars, IEEE Transactions on Industry Applications54(6) (2018), 5907–5917, doi:10.1109/TIA.2018.2855123.

17.

Daghigh

Javadi

and Javadi

, Improved analytical modeling of permanent magnet leakage flux in design of the coreless axial flux permanent magnet generator, Canadian Journal of Electrical and Computer Engineering40(1) (2017), 3–11, doi:10.1109/CJECE.2016.2550667.

18.

Daghigh

Javadi

and Torkaman

, Optimal design of coreless axial flux permanent magnet synchronous generator with reduced cost considering improved PM leakage flux model, Electric Power Components and Systems (2017), 264–278, doi:10.1080/15325008.2016.1248248.

19.

Khan

Bukhari

S.S.H.

and Ro

, Design and analysis of a 4-kW two-stack coreless axial flux permanent magnet synchronous machine for low-speed applications, IEEE Access7 (2019), 173848–173854, doi:10.1109/ACCESS.2019.2957046.

20.

Wojciechowski

R.M.

, Analysis and optimization of an axial flux permanent magnet coreless motor based on the field model using the superposition principle and genetic algorithm, Archives of Electrical Engineering65(3) (2016), 601–611, doi:10.1515/aee-2016-0043.

21.

Pranjić

and Virtič

, Rotor mechanical stress analysis of a double-sided axial flux permanent magnet machine, Journal of Energy Technology10(4) (2017), 57–70.

22.

Witczak

and Swiniarski

, Vibration analysis of permanent magnet motor with concentrated windings, Int. Journal of Applied Electromagnetics and Mechanics46(2) (2014), 419–425, doi:10.3233/JAE-141953.

23.

Roark

R.J.

, Formula for Stress and Strain, McGraw-Hill, New York, USA, 1990.

24.

Rao

S.S.

, Mechanical Vibrations, Addison Wesley, Boston, USA, 1995.