Mechanical Stress in Rotors of Permanent Magnet Machines—Comparison of Different Determination Methods

Abstract

In this work, different analytical methods for calculating the mechanical stresses in the rotors of permanent magnet machines are presented. The focus is on interior permanent magnet machines. First, an overview of eight different methods from the literature is given. Specific differences are pointed out, and a brief summary of the analytical approach for each method is provided. For reference purposes, a finite element model is created and simulated for each rotor geometry studied. A total of seven rotors rom representative automotive powertrains are considered in their specific speed range. The analytical methods are used to determine the maximum mechanical stress concentration factors for the seven rotor geometries, in which we are determined to find maximum mechanical stress as a final step of the analytical process. For each geometry and each respective operating speed range, the deviations from the finite element reference are determined. In addition, the error in the selected geometry variations is evaluated. A recommendation for the method with the lowest error considering all cases studied is given specifically for the stress in the airgap bridge and the central bridge.

1. Introduction

Mechanical stress analysis for rotor geometries is particularly important when considering the ongoing trend toward higher speeds in electric machines for traction applications. An overview of this is given in [1] for commercially available battery or hybrid electric vehicles.

The traction machines of the Toyota Prius across the generations can serve as an example. Here, it can be seen that with approximately the same rotor diameter range, the maximum speed has increased from 6000 rpm in the first Prius generation introduced in 1997 to 17,000 rpm in the fourth generation introduced in 2015 [2,3]. With increased speed and the same radial rotor dimensions, the resulting centrifugal forces increase accordingly. These centrifugal forces act on the rotor and cause high mechanical stresses in certain areas, depending on the rotor pole geometry [4,5].

The maximum mechanical stresses (MMS) in the rotor are limited by the yield strength of the materials. The commonly used electric steel materials have yield strength values between 320 and 500 MPa [6,7], and the electrical machines on the market for traction applications have MMS values mainly in that range. As an example, Figure 1 shows finite element analysis (FEA) results of the MMS of a rotor based on the electric machine of the Toyota Prius III at its maximum speed of 13,500 rpm.

The MMS estimation in a rotor configuration is typically carried out by static mechanical FEA [7,8,9,10,11]. However, the convergence of the solution strongly depends on the mesh size and quality. The time for re-meshing and solving after each iteration, as well as extra computational effort for MMS estimation, must be considered [12,13]. To reduce time and computational cost, several analytical methods using simplified equivalent models to determine the average mechanical stress in these rotors have been proposed in the literature.

To obtain a meaningful value for MMS based on average mechanical stress, a stress concentration factor (SCF) should be determined at the intersections where the geometric cross-section changes drastically [12,13,14,15,16,17,18]. Figure 2 shows a generic rotor pole of an interior permanent magnet synchronous machine (IPMSM). The areas where MMS mainly occurs are the intersection points of the air gap bridges and the central bridges (circled).

The approach of using equivalent mechanical models for the stress analysis in electrical machine rotor configurations is relatively new.

Table 1. Analytical methods proposed in the literature for determining the maximum mechanical stress of an IPMSM.

Analytical Methods	Year of Publication	Author	Applied on
ERM1	2002–2006	Schätzer [19]–Binder et al. [20]	Air gap bridges
ERM2	2014	Li Yi et al. [21]	Air gap bridges
ERM3	2016	Chai et al. [12]	Air gap bridges
ERM4	2018	Chu et al. [15]	Air gap bridges
ERM5	2020	Chu et al. [13]	Air gap bridges
CFM	2020	Chu et al. [13]	Air gap bridges
BTM	2018	Kleilat et al. [18]	Air gap and central bridges
CBM	2016	Chai et al. [12]	Central bridges

shows the methods studied with their publication date. The equivalent ring method (ERM), the centrifugal force method (CFM) and the beam theory method (BTM) are proposed to determine MMS on the air gap bridges using different approaches. The ERM method has evolved throughout the years with improvements and simplifications in the calculations. Therefore, there are five different ERM methods [12,13,15,19,20,21]. In this paper, the different ERM methods are chronologically referred to as ERM1 to ERM5. The central bridge method (CBM) and BTM method determine the MMS at the central bridges of V-shaped rotor geometries using different approaches [12,18,22].

Chu et al. [15] proposed an SCF for the air gap bridges on flat rotor geometries by using the polynomial surface fitting technique. Chai et al. [12] and Bremner et al. [17] proposed two different references: Peterson’s stress concentration [16] and Roark’s formulas for stress and strain [23] to determine SCF at points where MMS occur.

The listed analytical methods have been developed recently and are validated in the literature only for basic rotor geometries. Consequently, the comparison of accuracy and stability of these proposed methods needs to be further investigated. The evaluation of the most accurate and stable analytical methods is performed by applying the methods to current geometries of IPMSM traction drives available on the market. The results are compared to detailed FEA simulations in Ansys^® Mechanical, Release 18.1. The most suitable method can finally be included in the pre-design processes and can significantly reduce the computational effort.

2. Materials and Methods

2.1. Analytical Methods to Determine the Maximum Mechanical Stress at the Air Gap Bridge and the Central Bridge

The process of each method and their main differences are shown in Figure 3 and Figure 4.

The most relevant parameters of an IPMSM are defined in Figure 5, the nomenclature of the regions is shown in Figure 1. The nomenclature and parameter definition for a V-shaped rotor geometry are analogous to the flat-shaped rotor geometries. ${\mathrm{R}}_0$ is the outer radius of the rotor, d is the length of the airgap bridge, r_a is the outer radius, r_c is the inner radius of the magnet pocket, h is the magnet height, t is the length of central separation of magnet pockets, θ is the opening angle of the V-shape of the magnet, and α is the angle between the position of r_a to the center of the magnet pole.

Figure 6 shows the basic equivalent geometry approaches of each proposed method in the literature. In general, all methods define a simplified geometry, with an equivalent mass based on geometry and materials under investigation. In methods ERM2 to ERM5, the magnet-surrounding core material and the magnet are considered. For the ERM1 method only, the core material is considered. In ERM methods, only one ring is positioned at the airgap bridge representing the simplified geometry, as shown in Figure 6. With ERM1 and ERM2, no centroid radius of magnets and rotor poles are used, only the cross-sectional areas of rotor pole and ribs. With ERM3 to ERM5, these geometric quantities are considered, resulting in a more realistic representation of the model. All ERM methods use a sole ring at the outer radius of the rotor, which also limits the applicability of the stress estimation on the stress in the airgap bridge. This limitation must be considered when the maximal mechanical stress occurs in other regions of the rotor. Although a more detailed equivalent geometry is used in the CFM method, a similar limitation to the ERM method applies since no central bridge is represented. The stress in the central bridge can only be captured by the BTM and CBM methods mentioned above. Both methods also consider the central bridge by a beam or spoke, respectively, cf. Figure 6c,d. In the following, all mentioned methods are briefly recapitulated, and the equations of equivalent stress and maximum stress are given.

First, the ERM1 method [19,20] is considered, which approximates the rotor pole and magnets as an equivalent ring model with additional mass density. The thickness of the ring is equal to the narrowest thickness of the air gap bridges. The authors introduced the equivalent density ${\mathsf{\rho }}_{\mathrm{eq},\mathrm{erm}1}$ as:

$${\mathsf{\rho }}_{\mathrm{eq},\mathrm{erm}1}{=\mathsf{\rho }}_{\mathrm{Fe}}\cdot \frac{{\mathrm{A}}_{\mathrm{Fe}}{+\mathrm{A}}_{\mathrm{m}}}{{\mathrm{A}}_{\mathrm{eq}}},$$

(1)

where ${\mathsf{\rho }}_{\mathrm{Fe}}$ is the mass density of electric steel in kg/m³. A_Fe, A_m and A_eq represent the cross-sectional area of the rotor pole, the permanent magnets and the equivalent ring, respectively, as shown in Figure 6a. It can be observed that only the density of the electric steel is assumed for the whole considered region.

The tangential stresses ${\mathsf{\sigma }}_{\mathrm{eq},\mathrm{erm}1}$ and MMS ${\mathsf{\sigma }}_{\mathrm{eq},\max 1}$ inside the equivalent ring are calculated as:

$${\mathsf{\sigma }}_{\mathrm{eq},\mathrm{erm}1}={\left(\frac{{\mathrm{R}}_{\mathrm{eqo}}{+\mathrm{R}}_{\mathrm{eqi}}}{2}\right)}^2\cdot {\mathsf{\omega }}^2\cdot {\mathsf{\rho }}_{\mathrm{eq},\mathrm{erm}1},$$

(2)

$${\mathsf{\sigma }}_{\mathrm{eq},\max 1}{=\mathrm{k}}_{\mathrm{air}\mathrm{gap}\mathrm{bridge}}\cdot {\mathsf{\sigma }}_{\mathrm{eq},\mathrm{erm}1},$$

(3)

Here, ${\mathrm{R}}_{\mathrm{eqo}}$ and ${\mathrm{R}}_{\mathrm{eqi}}$ represent the outer and inner radius of the equivalent ring, respectively, as shown in Figure 6a. ω is the rotational speed in rad/s, and ${\mathrm{k}}_{\mathrm{air}\mathrm{gap}\mathrm{bridge}}$ is the SCF at the air gap bridges, which is introduced in Section 2.2 Determination of Stress Concentration Factors.

Equation (1) is improved in the ERM2 method [21] by also considering the mass density of magnets:

$${\mathsf{\rho }}_{\mathrm{eq},\mathrm{erm}2}=\frac{{\mathsf{\rho }}_{\mathrm{Fe}}{\mathrm{A}}_{\mathrm{Fe}}{+\mathsf{\rho }}_{\mathrm{m}}{\mathrm{A}}_{\mathrm{m}}}{{\mathrm{A}}_{\mathrm{eq}}},$$

(4)

where ${\mathsf{\rho }}_{\mathrm{m}}$ is the mass density of the magnet material.

ERM3 method uses Equation (4) and is further developed by taking the centroid radii of magnets and the rotor pole into account with:

$${\mathsf{\rho }}_{\mathrm{eq},\mathrm{erm}3}=\frac{\left({\mathrm{R}}_{\mathrm{Fe}}{\mathsf{\rho }}_{\mathrm{Fe}}{\mathrm{A}}_{\mathrm{Fe}}{+\mathrm{R}}_{\mathrm{m}}{\mathsf{\rho }}_{\mathrm{m}}{\mathrm{A}}_{\mathrm{m}}\right)\frac{1-\cos \left(\mathsf{\theta }\right)}{2}}{{\mathrm{R}}_0{\mathrm{A}}_{\mathrm{eq}}},$$

(5)

where ${\mathrm{R}}_{\mathrm{Fe}}$ and ${\mathrm{R}}_{\mathrm{m}}$ stand for the distance between the rotor center and the center of gravity of rotor pole ${\mathrm{G}}_{\mathrm{Fe}}$ and center of gravity of the permanent magnets ${\mathrm{G}}_{\mathrm{m}}$, respectively. ${\mathsf{\rho }}_{\mathrm{m}}$ is the mass density of the permanent magnet in kg/m³. $\mathsf{\theta }$ represents the angle between two magnets of a single pole in rad (see Figure 5). The centroid radius of the equivalent ring ${\mathrm{R}}_0$ is calculated as:

$${\mathrm{R}}_0=\left(\frac{{\mathrm{R}}_{\mathrm{eqo}}{+\mathrm{R}}_{\mathrm{eqi}}}{2}\right),$$

(6)

The calculations in Equation (5) are simplified in the ERM4 method [15], and Equation (2) is replaced by an adopted ${\mathsf{\sigma }}_{\mathrm{eq},\mathrm{erm}4}$ as:

$${\mathsf{\sigma }}_{\mathrm{eq},\mathrm{erm}4}=\frac{{\mathsf{\rho }}_{\mathrm{eq},\mathrm{erm}4}{\mathsf{\omega }}^2}{8}[\left(3+\mathsf{\nu }\right)\left({\mathrm{R}}_{\mathrm{eqo}}^2{(1+\mathrm{R}}_{\mathrm{eqi}}{)+\mathrm{R}}_{\mathrm{eqi}}^2\right)-\left(1+3\mathsf{\nu }\right){\mathrm{R}}_{\mathrm{eqi}}^2],$$

(7)

where $\mathsf{\nu }$ is the Poisson’s ratio of electric steel, and ${\mathsf{\rho }}_{\mathrm{eq},\mathrm{erm}4}$ is the equival ${\mathsf{\rho }}_{\mathrm{eq},\mathrm{erm}4}$ is the equivalent density for the ERM4 method as:

$${\mathsf{\rho }}_{\mathrm{eq},\mathrm{erm}4}=\frac{{\mathrm{R}}_{\mathrm{m}}{\mathsf{\rho }}_{\mathrm{m}}{\mathrm{A}}_{\mathrm{m}}+{\mathrm{R}}_{\mathrm{Fe}}{\mathsf{\rho }}_{\mathrm{Fe}}{\mathrm{A}}_{\mathrm{Fe}}}{{\mathrm{R}}_0{\mathrm{A}}_{\mathrm{eq}}},$$

(8)

In the ERM5 method [14], the simplifications in the ERM4 method are used, but instead of using Equation (7) to determine the tangential stresses, the initial approach, Equation (2), is used again.

The CFM method considers rotor pole and magnets as a rigid body [13]. By performing a force analysis on that rigid body, average stresses at the air gap bridges ${\mathsf{\sigma }}_{\mathrm{cfm}}$ are determined as:

$${\mathsf{\sigma }}_{\mathrm{cfm}}=\frac{{(\mathsf{\rho }}_{\mathrm{m}}{\mathrm{A}}_{\mathrm{m}}{+\mathsf{\rho }}_{\mathrm{Fe}}{\mathrm{A}}_{\mathrm{Fe}}{)\mathsf{\omega }}^2{\mathrm{R}}_{\mathrm{c}}}{2\sin (\frac{\mathsf{\alpha }}{2})\mathrm{d}}$$

(9)

where ${\mathrm{R}}_{\mathrm{c}}$ is the distance between the rotor center and the center of gravity of magnets with rotor pole ${\mathrm{G}}_{\mathrm{c}}$ as shown in Figure 6b. $\mathrm{d}$ is the air gap bridge thickness and $\mathsf{\alpha }$ is half pole arc in rad as shown in Figure 5.

The BTM method simplifies the air gap bridges and central bridges to simple rectangular beams and considers the rotor pole with magnets as a rigid body, as can be seen in Figure 6c [18,22]. Average mechanical stresses at the air gap bridges ${\mathsf{\sigma }}_{\mathrm{btm},\mathrm{air}\mathrm{gap}\mathrm{bridge}}$ and central bridges ${\mathsf{\sigma }}_{\mathrm{btm},\mathrm{central}\mathrm{bridge}}$ are given as:

$${\mathsf{\sigma }}_{\mathrm{btm},\mathrm{air}\mathrm{gap}\mathrm{bridge}}=\left(\frac{\sqrt{{\mathrm{c}2}^2+{\left(6\cdot \frac{\mathrm{c}3}{\mathrm{d}}\right)}^2}{\mathsf{\omega }}^2}{{\mathrm{l}}_{\mathrm{stack}}}\right)\frac{{\mathrm{m}}_{\mathrm{total}}{\mathrm{R}}_{\mathrm{c}}}{\mathrm{d}}$$

(10)

$${\mathsf{\sigma }}_{\mathrm{btm},\mathrm{central}\mathrm{bridge}}=\left(\frac{\mathrm{c}1\cdot {\mathsf{\omega }}^2}{{\mathrm{l}}_{\mathrm{stack}}}\right)\frac{{\mathrm{m}}_{\mathrm{total}}{\mathrm{R}}_{\mathrm{c}}}{\mathrm{t}}$$

(11)

where c1, $\mathrm{c}2$ and $\mathrm{c}3$ represent constant coefficients that quantify the relationship between centrifugal force, normal force and bending moment, respectively. The coefficients must be fitted to the FEA results. After the initial FEA calculation, these coefficients must be adjusted so that the FEA result matches the analytical result. ${\mathrm{l}}_{\mathrm{stack}}$ and $\mathrm{t}$ are the stack length and the central bridge thickness, respectively. The total mass of the rotor pole with magnets ${\mathrm{m}}_{\mathrm{total}}$ is expressed as:

$${\mathrm{m}}_{\mathrm{total}}=\left({\mathsf{\rho }}_{\mathrm{Fe}}{\mathrm{A}}_{\mathrm{Fe}}{+\mathsf{\rho }}_{\mathrm{m}}{\mathrm{A}}_{\mathrm{m}}\right)\cdot {\mathrm{l}}_{\mathrm{stack}}$$

(12)

In the CBM method, central bridges are equated with spokes, and the rotor pole with magnets and the rotor core are equated with an outer equivalent ring and an inner equivalent ring as in the conventional ERM methods [12]. The mechanical stresses at the central bridge ${\mathsf{\sigma }}_{\mathrm{cbm}}$ are determined as:

$${\mathsf{\sigma }}_{\mathrm{cbm}}=\frac{\mathrm{Y}}{{\mathrm{A}}_{\mathrm{sr}}}$$

(13)

where ${\mathrm{A}}_{\mathrm{sr}}$ is the radial cross-sectional area of the spokes, which is equal to the multiplication of the stack length and the thickness of the central bridge. The uniformly distributed tension on the inner ring $\mathrm{Y}$ is calculated as:

$$\mathrm{Y}=\mathrm{X}+\frac{{\mathrm{q}}_{\mathrm{s}}{\mathsf{\omega }}^2}{2\mathrm{g}}{(\mathrm{R}}_2^2{-\mathrm{R}}_1^2)$$

(14)

where ${\mathrm{R}}_1$ and ${\mathrm{R}}_2$ are the inner and outer radii of the spokes. ${\mathrm{q}}_{\mathrm{s}}$ is the spokes average weight per unit.

The tension between the spokes and the outer ring $\mathrm{X}$ is recalculated due to a probable mistake in the formula given in [13] as:

$$\mathrm{X}=\frac{\frac{{4\mathsf{\pi }\mathrm{q}}_0{\mathrm{R}}_0^2}{{\mathrm{EA}}_0}-\frac{{\mathrm{nR}}_{\mathrm{v}}{\mathrm{q}}_{\mathrm{s}}{\mathsf{\omega }}^2{(\mathrm{R}}_2^2{-\mathrm{R}}_1^2)}{{4\mathsf{\pi }\mathrm{EA}}_{\mathrm{v}}\mathrm{g}}-\frac{{(\mathrm{R}}_2{-\mathrm{R}}_1{)\mathrm{q}}_{\mathrm{s}}{\omega }^2}{{2\mathrm{A}}_{\mathrm{s}}\mathrm{gE}}\cdot \left({\mathrm{R}}_2^2-\frac{{\mathrm{R}}_2^2{+\mathrm{R}}_2{\mathrm{R}}_1{+\mathrm{R}}_1^2}{3}\right)}{\frac{{\mathrm{R}}_2{-\mathrm{R}}_1}{{\mathrm{A}}_{\mathrm{s}}\mathrm{E}}{+\mathsf{\lambda }}_0+\frac{{\mathrm{nR}}_{\mathrm{v}}}{{2\mathsf{\pi }\mathrm{EA}}_{\mathrm{v}}}}$$

(15)

where ${\mathrm{q}}_0$ is the uniformly distributed inertial load, $\mathrm{n}$ is the number of spokes, $\mathrm{E}$ is the Young’s modulus of electric steel, $\mathrm{g}$ is the gravitational acceleration, ${\mathsf{\lambda }}_0$ is the flexibility coefficient of the outer ring, which express how the displacement along the centerline of the outer ring is influenced by the tension between the spokes and the outer ring, and ${\mathrm{R}}_{\mathrm{v}}$ is the centerline radius of the inner ring. ${\mathrm{A}}_0$, ${\mathrm{A}}_{\mathrm{v}}$, ${\mathrm{A}}_{\mathrm{s}}$ represent the cross-sectional areas of the outer ring, the inner ring and the spokes. The equations for the defined symbols can be taken from [12].

2.2. Determination of Stress Concentration Factors

Chu et al. [13,14,15] proposed several methods to determine an accurate stress concentration factor (SCF) at the air gap bridges. The method proposed in [15] is considered in this study because it is applicable to a rotor geometry without considering a rotor pole number. The SCF at the air gap bridge k_{air gap bridge} is determined as:

$${\mathrm{k}}_{\mathrm{air}\mathrm{gap}\mathrm{bridge}}=1.389-0.2374\cdot \frac{\mathrm{d}}{{\mathrm{r}}_{\mathrm{a}}}+0.007181\cdot \mathsf{\alpha }+0.03567\cdot {\left(\frac{\mathrm{d}}{{\mathrm{r}}_{\mathrm{a}}}\right)}^2$$

$$0.02148\cdot \frac{\mathrm{d}}{{\mathrm{r}}_{\mathrm{a}}}\cdot \mathsf{\alpha }+0.000745\cdot {\left(\frac{\mathrm{d}}{{\mathrm{r}}_{\mathrm{a}}}\right)}^3-0.001721\cdot {\left(\frac{\mathrm{d}}{{\mathrm{r}}_{\mathrm{a}}}\right)}^2\cdot \mathsf{\alpha }$$

(16)

where r_a represents the notch radius at the air gap bridge as shown in Figure 5. The case of a stepped flat bar under tensile stress from [16] proposed by [12] is taken as an equivalent case to determine SCF at the central bridge k_{central bridge} as:

$$\begin{gathered} {\mathrm{k}}_{\mathrm{central}\mathrm{bridge}}=1.97-0.384\cdot \left(\frac{{2\mathrm{r}}_{\mathrm{c}}}{{\mathrm{t}+2\mathrm{r}}_{\mathrm{c}}}\right)-1.018\cdot {\left(\frac{{2\mathrm{r}}_{\mathrm{c}}}{{\mathrm{t}+2\mathrm{r}}_{\mathrm{c}}}\right)}^2 \\ +0.43\cdot {\left(\frac{{2\mathrm{r}}_{\mathrm{c}}}{{\mathrm{t}+2\mathrm{r}}_{\mathrm{c}}}\right)}^3 \end{gathered}$$

(17)

where r_c is the notch radius on the central bridge as shown in Figure 5, and the constant numbers are presented in Reference [16] to determine the SCF.

2.3. Initial Conditions and Assumptions

There are many factors such as thermal stress, rotor vibration, and unstable operation that affect the mechanical strength of a rotor. In this study, simplifications and initial conditions are assumed when evaluating the mechanical strength of IPMSMs in FEA. These assumptions can be considered feasible since the analytical approaches do not account for any of the simplified or neglected effects:

• Constant speed in steady state operation is assumed.
• Maximum deformation and stresses are mainly caused by centrifugal forces. The effect of electromagnetic forces between rotor and stator, and attraction forces between the permanent magnets and the rotor are considered negligible compared to centrifugal forces.
• Thermal effects are neglected.
• One pole in a two-dimensional model is used.
• The rotor core is assumed as an entity, and the lamination effects are considered negligible.
• Yield indicated by planar von Mises stress.
• The eccentricity of the rotor, the vibration, and the dynamic forces of the shaft are neglected.
• The analytical and numerical methods are based on the mono dimensional elastic theory. The inaccuracy of the methods proposed in this study increases with the mechanical stress above the yield point.

2.4. Investigated Machines

Seven rotor geometries with different rotor topologies, including those of the e-machines of the Nissan Leaf vehicle (rotor 3) and the Toyota Prius III vehicle (rotor 4), are used to investigate the stability of the analytical methods [1]. The main parameters of investigated geometries are presented in

Table 2. Geometries of the investigated rotors.

Rotor Geometry	Geometry Definition/Unit
	Outer Diameter Ro/mm	Max. Speed/rpm	Air Gap Bridge Thickness d/mm	Central Bridge Thickness t/mm	Angle between Magnets θ/degree
Rotor 1 flat shape	238	8000	1.26	-	-
Rotor 2 flat shape	206	11,200	1.8	-	-
Rotor 3 delta shape	130	10,400	0.7	-	-
Rotor 4 V-shape	160	13,500	1.94	1.8	146
Rotor 5 V-shape	206	11,200	2	2	170
Rotor 6 V-shape	146	18,100	2.3	3.4	137
Rotor 7 double V-shape	124	11,200	0.9	1.45	126

. To verify the general applicability of the analytical methods, different rotor topologies such as flat shape, V-shape, double V-shape and delta shape, different sizes, and different speed ranges are considered. One pole of each rotor topology is shown in Figure 7.

To determine the strength of the rotor geometries, FEA analyses were first performed to determine the location of the MMS and to visualize the stress distribution. Then, MMS at the air gap bridges for rotor 1, rotor 2 and rotor 3, and MMS at the central bridges for rotor 4, rotor 5, rotor 6 and rotor 7 were determined using the different analytical methods. To investigate the validity of these methods, the FEA results were taken as reference and the percentage difference between the FEA results and the analytical methods was determined.

The validity of the analytical methods was investigated over the speed range as well as through geometric design adjustments. To investigate the validity over the speed range, the speeds were changed from 1000 to 10,000 rpm. The design parameters shown in Figure 5 such as air gap bridge thickness (Figure 5d), magnet thickness (Figure 5h), notch radius on the air gap bridge r_a, notch radius on the central bridge r_c, and central bridge thickness were investigated by changing only the investigated parameter and keeping all other parameters constant. In addition, the entire rotor geometry was scaled to study the change in the stability of the analytical methods. The studied design parameters were changed in a reasonable range from half of the initial value to double the reference value.

The FEA analysis was performed using static structural analysis in ANSYS Workbench. Figure 8a shows the boundary conditions applied on a single rotor pole based on the assumptions and conditions given in reference papers [8,12,15,21]. A frictionless support was assumed on the symmetry axes to satisfy the symmetry condition. A cylindrical support on the inner surface of the rotor allowed the chosen nodes to move freely in radial direction, while preventing them to move on the tangential axis. Figure 8b shows the increased mesh density at the intersection points of air gap and central bridges where MMS can occur to achieve a converged solution. A rough contact between magnets and rotor core was assumed. Plane stress theory was assumed for the analysis of the rotor [11,12].

3. Performance Review of the Methods and Results

For comparison, the results obtained with the different methods are presented in relative deviation (given as error in the following) to the FEA results. The FEA is taken as a reference, since experimental validation is out of the scope of this study, at this point. The relative percentage error between the analytical and FEA results is defined as:

$$\mathsf{\Delta }\mathsf{\sigma }=\frac{{\mathsf{\sigma }}_{\max ,\mathrm{FEA}}-{\mathsf{\sigma }}_{\max ,\mathrm{analytical}}}{{\mathsf{\sigma }}_{\max ,\mathrm{FEA}}}$$

(18)

where σ_max,FEA is the maximal stress determined in FEA and σ_{max,analytical} is the corresponding maximum stress of the respective method considered. The method that has the least error is considered to be the most suitable.

Since rotors 4 to 7 exhibit the maximum stress in the central bridge rather than in the air gap bridges, only BTM and CBM can be applied. Rotors 1, 2 and 3 do not have central bridges, the seven rotor geometries will therefore be split into variants with MMS in airgap bridges (see

Table 3. Evaluation of analytical methods to determine the maximum mechanical stress at the air gap bridges.

Analytical Methods	Rotor Geometries
	Rotor 1	Rotor 2	Rotor 3
ERM1	−18.5%	6.3%	55.6%
ERM2	−17.1%	6.6%	56.2%
ERM3	26.2%	59.3%	39.9%
ERM4	19.8%	35.1%	31.7%
ERM5	−25.1%	−1.7%	−9.7%
CFM	−25.5%	−2.3%	−10.7%
BTM	0%	−20%	−60%
FEA results	348 MPa	752 MPa	350 MPa

) and variants with MMS in central bridges, as shown in

Table 4. Evaluation of the analytical methods to determine maximum mechanical stress at the central bridges.

Analytical Methods	Rotor Geometries
	Rotor 4	Rotor 5	Rotor 6	Rotor 7
BTM	0%	46.9%	−0.2%	−29%
CBM	−11%	45.5%	−4.6%	−22%
FEA results	341 MPa	614 MPa	487 MPa	121 MPa

. Table 3 shows Δσ at maximum speed for rotors 1 to 3, and from the table, it can be seen that errors for the ERM1 and ERM2 methods are close to each other, and they change about 70% for the selected geometries. This is relatively high compared to other methods, since a maximum spread of 33% is shown for the ERM 3 method, and a minimum of 15.5% for the CFM method is observable. Another drawback of ERM1 and ERM2 methods is shown by the high error for rotor 3. The reason is probably the different rotor topologies (rotor 3 is delta shaped). Here, the simplicity of ERM1 and ERM2, which only consider cross-sectional areas and do not consider the centroid radii, is shown to be a downside, as the stress for rotor 3 is overestimated by 55.6% and 56.2%, respectively. Considering that most traction machines usually have a V-shaped or delta-shaped rotor pole geometry, cf. (1), the ERM1 and ERM2 methods are not considered further in this study and are not further recommended to use for any delta-shaped geometry. Although the ERM3 and ERM4 methods take more information of the rotor geometry and material data into account, in this investigation, the resulting error to the FEA, ranging between 19.8% and 59.3%, is not satisfactory, especially since the analytical methods show an underestimation of the MMS.

The ERM5 and CFM methods achieve an error of less than 2.5%, with ERM5 showing the lowest error compared to the FEA results. It ranges from 1.7% to 25.1% and gives overestimations and thus a worst-case estimation for the rotor geometries studied. The ERM5 method is chosen for further stability investigations for the MMS at the air gap bridges, since it shows the least error, considering all investigated rotor geometries, and uses a simple equivalent model. This gives the benefit that possible future error and discrepancies can be corrected more easily.

The BTM method was initially applied to the geometries of rotor 1 and rotor 4 for the air gap bridges and the central bridges, respectively. As mentioned earlier, the BTM method requires an initial fitting with a first rotor geometry. This was conducted for rotor 1 and rotor 4. After fitting, naturally, the error for the fitted rotors is 0. For the other rotors, such a fitting was not performed. Considering a comparative study with multiple rotors, or a development process with significant rotor geometry changes, a repeated fitting process with FEA is not very realistic. Table 3 shows the high error for BTM for stress in the airgap bridge of up to −60%. Table 4 also shows a high deviation for BTM. However, for stresses in the central bridge, the error is in general lower and closer to the error of the CBM method. The unstable behavior after any significant change in the rotor geometries, despite the initial fitting of the BTM method for stress in the airgap bridge, leads to the conclusion that the BTM method is not recommend for MMS estimation in the airgap bridge and will be not considered in further stability investigations.

The CBM method is selected as the most suitable method for determining the MMS at the central bridges, especially because the method does not require an initial fitting compared to the BTM method. The MMS is overestimated for rotors 4, 6 and 7. The highest deviation is rotor 7 with −29%. Rotor 7 shows the highest deviation for BTM and CBM (excluding rotor 5), which is likely due to its double V-shaped geometry. As can be seen in Table 4, the CBM as well as BTM method show a relatively high error for rotor 5. The reason for that is explained in [12]. It is concluded that when the angle between two magnets is close to 180 degrees, the method does not provide an MMS estimate that is close to the actual expected values due to the uniform distribution of mass density at the rotor pole. Therefore, Rotor 5 is not considered in further investigations.

In order to investigate the behavior of the method over the entire speed range of the machines, Figure 9 and Figure 10 show the MMS at the air gap bridge and the central bridge, respectively. The determined MMS in FEA and the selected analytical methods in the particular speed range are shown for rotors 1 to 3 and for rotors 4, 6 and 7. Additionally the error of each method is shown. It can be observed that for chosen methods, ERM 5 and CBM methods, the resulting error is stable over the whole speed range and is close to the given numeric value in Table 3.

For development processes, the MMS could be investigated for geometry variations in a given rotor design. For this approach, a stable behavior of the method is required for specific geometry changes. The stability of selected rotor geometries under geometric design variations is presented in Figure 11 and Figure 12. In this study, universal rotor geometry parameters are chosen, namely airgap bridge thickness, magnet thickness, notch radius and a general geometry scaling. The range of the parameters was chosen arbitrarily from 50% to 200% of the original value. This is a rather wide variation range especially for airgap and central bridge thickness, and the upper and lower limits might not be a typical rotor geometry, which could also affect the results of the analytical approaches and FEA simulations. The stability of the notch radius is only investigated for rotor 1 and rotor 4 geometries. Since the SCF is only dependent on the notch radius, but not dependent on the analytical methods, it is expected to have the same stability for different geometries. For the ERM5 method, magnet and core area will definitely be considered. However, the airgap bridge thickness will affect the actual stress in the rotor. This does not seem to be captured by the ERM5 method, as can be seen in Figure 11, where the error for an increased airgap bridge is increased for all rotor geometries (maximal by 25% and around 12% for rotor 2 and rotor 3). The geometry scaling seems to have the least effect on the resulting error; here, a slightly higher effect on the ERM5 is visible (below 10%) compared to the CBM (below 5%). In terms of overall error, geometry scaling can be performed accurately with ERM and CBM. For ERM5, a reduced magnet thickness compared to the original geometry shows a significant underestimation of the MMS, especially for rotor 3 (delta-shaped rotor). This again indicates that the more complex rotor geometry must be treated with caution using the simple ERM model. The variation of the central bridge thickness with regard to CBM is comparable to the variation of the airgap bridge thickness with ERM5. Examination of the CBM error with variation of the central bridge thickness shows the highest change in error (around 40% for rotor 6 and rotor 7), while variations in magnet thickness and geometry scaling are well below 10%.

4. Conclusions

The maximum speed of electrical machines used as traction drives is increasing due to the requirements for higher power density. In order to ensure the mechanical stability of the electric machines, simulations or estimations must be carried out in the design process, in particular, for the mechanical loads due to the centrifugal forces acting on the rotor. Appropriate analytical methods have been developed to achieve acceptable values for computation times and manpower requirements. Published approaches were used for electrical machines available on the market, and the resulting accuracy (compared to finite element analyses as reference) was verified. In addition, the stability of these methods was investigated with variations in operating speed and rotor dimensions. The methods that show the highest accuracy for the maximum mechanical stress at airgap bridges are equivalent ring method 5, and for central bridges the central bridge method. These two methods show errors of less than 25.1 and 22%, respectively, for the selected rotor geometries. Parameters such as magnet thickness, air gap bridge thickness, central bridge thickness, notch radius and geometry scale were studied for the methods with the highest accuracy to verify the stability of these methods.

For future work, especially for early development stages, the equivalent ring method 5 can be used for fast evaluation process, since it was found that the estimates are feasible even if the geometry is significantly changed. Additionally, stress concentration factor for the central bridge will be identified for a more general use. For an improved and more holistic approach, a merged method combining equivalent ring method 5 and the central bridge method will be developed and benchmarked with even more recent traction motor geometries.