Strength Analysis and Design of a Multi-Bridge V-Shaped Rotor for High-Speed Interior Permanent Magnet Synchronous Motors

Abstract

High-speed operation is a crucial approach for achieving high power density of drive motors for new energy vehicles. However, mechanical strength of the rotor has become the primary bottleneck in the development of high-speed drive motors. Adopting a multi-bridge structure can effectively enhance the mechanical strength of the V-shaped rotors widely used in interior permanent magnet synchronous motors (IPMSMs). Firstly, based on the equivalent centroid principle, the centrifugal forces generated by the rotor’s pole shoes and permanent magnets are calculated. An improved centrifugal force method is proposed to establish an analytical mechanical model of the multi-bridge V-shaped rotor structure. This method comprehensively considers the force conditions, deformation constraints, and material properties of the magnetic bridges. Additionally, stress concentration is taken into account to ensure the accuracy of the model. The effects of various structural parameters on the maximum mechanical stress and deformation are then analyzed. These parameters include the V-angle, pole shoe angle, and the dimensions of three types of magnetic bridges, namely, the central bridge, air-gap bridge, and middle bridge. Finally, recommendations for selecting structural parameters in the mechanical strength design of multi-bridge V-shaped rotors are summarized. The effectiveness of the proposed rotor structure is verified through finite element method and experiments.

1. Introduction

In recent years, the rapid development of the new energy vehicle industry has made high-speed operation a key technical pathway for improving power density and achieving lightweight designs in electric drive systems. This trend has become a major focus in the advancement of drive motors [1,2]. Compared to surface-mounted permanent magnet synchronous motors, interior permanent magnet synchronous motors (IPMSMs) are gaining widespread attention in new energy vehicles. The structural characteristics of embedded permanent magnets within the rotor laminations make the manufacturing process relatively simple. IPMSMs eliminate the need for additional protective sleeves during high-speed operation, feature higher saliency ratios, and offer greater overload capacity. While SPMSMs are generally recognized for higher power density, IPMSMs can achieve improved power density in specific operating conditions, such as during flux-weakening operations, by utilizing reluctance torque to enhance the overall performance. Furthermore, IPMSMs are particularly effective for flux-weakening operations, enabling a wider range of speeds [3,4,5]. Among them, the V-shaped IPMSM stands out as a prominent choice due to its superior magnetic concentration capability and higher saliency ratio, which enable effective utilization of reluctance torque to enhance torque density [6].

However, the V-shaped rotor faces significant challenges during high-speed operation, particularly the weak magnetic bridge structures, which endure substantial centrifugal forces. This can lead to structural damage and increase the risk of motor failure, potentially causing severe safety incidents. Consequently, in-depth analysis and optimization of the mechanical strength of IPMSM rotor structures have become focal points of interest for both academia and industry.

Existing research predominantly focuses on relatively simple radial interior rotors. Studies such as those in [7,8] analyzing mechanical stress in these structures have developed analytical models using the centrifugal force method, which simplifies the rotor components as rigid bodies, calculates the centrifugal force acting on the bilateral bridges, and estimates the maximum mechanical stress (MMS) as the ratio of the centrifugal force to the cross-sectional area of the bilateral bridge. These studies suggest that increasing the number of magnetic bridges can help distribute mechanical stress and explore how the number and size of magnetic bridges influence rotor strength and leakage characteristics. Binder et al. [9] employed the equivalent ring method to establish a mechanical model for the MMS of radial rotors, and confirmed the accuracy of the model with finite element method (FEM). Similarly, Zhang et al. [10] used the equivalent ring method to calculate the strength of radial rotors, analyzed the maximum stress variations of magnetic bridges at different rotational speeds, and determined the rotor structure’s maximum allowable operating speed. Additionally, Chu et al. [11,12] introduced a new stress concentration factor function to precisely calculate the MMS of radial rotors, and validated it through FEM and plastic deformation experiments.

In contrast, V-shaped rotors are structurally more complex. In addition to the two air-gap bridges found in radial rotors, V-shaped rotors feature an additional central bridge located along the magnetic pole’s symmetry line. This design broadens the q-axis magnetic path and increases the proportion of reluctance torque, leading to improved electromagnetic performance. However, existing mechanical analysis methods, such as the equivalent ring method, face significant limitations when applied to V-shaped IPMSMs. To address this issue, Chai et al. [13] proposed an improved equivalent ring method combined with a spoke-type rotor model, establishing a mechanical model for V-shaped IPMSM rotors and analyzing the stress variations of magnetic bridges with changing the V-shaped angle. Kleilat et al. [14] proposed the beam theory method to build a mechanical model of the V-shaped rotor, and verified the effectiveness of the method by experimental results. Monissen et al. [15] explored the differences among various analysis methods across different speed ranges, compared them with FEM results, and recommended the optimal approach for calculating stress in air-gap and central bridges.

Although significant progress has been made in the mechanical strength analysis of radial IPMSM rotors, major gaps still remain in the study of more complex V-shaped rotor structures, particularly those incorporating multiple bridges. For multi-bridge V-shaped rotor structures, the highly complex stress and deformation behavior under high-speed operation presents a critical challenge. This problem is closely related to structural parameter selection, as improper design can lead to excessive stress, deformation, or even mechanical failure, which limits the safe operating speed and power density of the motor. Addressing these challenges requires a comprehensive understanding of the relationships between structural parameters, mechanical stress, and deformation, which has not been systematically investigated in the literature. Moreover, accurately predicting rotor deformation is crucial for ensuring motor safety during high-speed operation. Therefore, further theoretical exploration is needed to provide valuable guidance for optimizing rotor designs to achieve both higher reliability and better performance.

In this manuscript, an improved centrifugal force method is proposed to investigate the mechanical strength of a multi-bridge V-shaped IPMSM rotor structure. This method considers the force balance equation, deformation compatibility equation, and material physical equation of the rotor magnetic bridges to establish a mechanical model for the multi-bridge V-shaped rotor structure. The MMS and deformation of each magnetic bridge are analyzed with different rotor-structural parameters. Based on these findings, the rotor structure is optimized, and the theoretical model is validated through FEM and experimental tests. Finally, design recommendations are proposed for high-speed multi-bridge V-shaped IPMSM rotors.

2. Multi-Bridge V-Shaped Rotor Structure and Its Basic Assumptions

2.1. Multi-Bridge V-Shaped Rotor Topology

Figure 1a illustrates a conventional V-shaped interior rotor topology, which primarily consists of a rotor core and two permanent magnets (PMs). In the rotor core structure, to reduce magnetic flux leakage and improve the utilization rate of PMs as well as maintaining the structural integrity of the rotor, two air-gap bridges are typically set near the outer surface of the rotor adjacent to the PMs. Additionally, a central bridge is positioned between the two PMs in each magnetic pole.

To further enhance the mechanical strength of the rotor, and inspired by references [7,8], a multi-bridge V-shaped rotor structure, as shown in Figure 1b, can be designed by appropriately increasing the number of magnetic bridges. This structure aims to raise the motor’s maximum allowable operating speed, thereby achieving higher power density. The multi-bridge V-shaped rotor structure is realized by dividing the PMs in the V-shaped rotor into multiple segments, with additional middle bridges inserted between adjacent smaller PMs. By increasing the number of middle bridges, the centrifugal forces generated by the PMs and pole shoe are more evenly distributed, significantly enhancing the rotor’s mechanical strength.

However, while increasing the width and number of magnetic bridges can improve the mechanical strength of the rotor, it may also result in greater magnetic flux leakage to reduce the electromagnetic performance of the motor. Therefore, there is a contradictory relationship between the mechanical properties of the rotor and the electromagnetic properties of the motor. An effective way to solve this problem is to reduce the size and number of magnetic bridges as much as possible while ensuring that the rotor structure meets the minimum mechanical strength requirements. Given the similarities in strength analysis and structural design of multi-bridge V-shaped rotors, and considering the space constraints for PMs in the rotor core, this study focuses on the five-bridge rotor structure shown in Figure 1b for analysis.

2.2. Basic Assumptions for Analysis

The mechanical stress and deformation of the rotor structure are influenced by various factors. To facilitate theoretical analysis, the following assumptions are made:

(1)

During high-speed operation, the rotor is subjected to electromagnetic forces, centrifugal forces, and disturbance forces. However, since the impact of centrifugal forces on rotor strength is significantly greater than that of other forces, only centrifugal forces are considered.

(2)

The analysis focuses on the rotor structure under high-speed steady-state conditions, excluding the effect of transient conditions.

(3)

The influence of multi-physical fields, such as temperature and electromagnetic fields, on the stress and deformation of the rotor is neglected.

(4)

It is assumed that during high-speed rotation, the central bridge experiences tensile deformation, the air-gap bridges undergo bending deformation, and the middle bridges are subjected to a combination of tensile and bending deformations.

(5)

The MMS in the rotor core is assumed to remain within the elastic limit, that is, the relationship between stress and strain in the rotor structure obeys Hooke’s Law.

(6)

The deformation of the pole shoe, permanent magnets, and other parts of the rotor is considered negligible compared to the deformation of the magnetic bridges. Therefore, all other parts of the rotor, except the magnetic bridges, are treated as a rigid body.

3. Analytical Mechanical Model

In this Section, an analytical mechanical model is developed to investigate the stress distribution and deformation of the rotor. First, the equivalent centroid principle is employed to calculate the high-speed centrifugal forces generated by the pole shoe and permanent magnets of the rotor. Then, an improved centrifugal force method is proposed by integrating the force conditions, deformation compatibility, and materials’ physical properties of the magnetic bridges. This leads to the development of an analytical mechanical model for the multi-bridge V-shaped rotor, which provides reference values for stress and deformation in each magnetic bridge. Finally, by considering the stress concentration factors and reference stresses, the MMS within the rotor structure is determined.

3.1. Centrifugal Force Calculation

To facilitate the establishment of an analytical mechanical model for the multi-bridge V-shaped rotor, the key structural parameters of the rotor are defined as shown in Figure 2. In this Figure, it is important to note that the angle between the centerline of the middle bridge width and the normal direction of the rotor pole shoe surface at that position is defined as β, describing the direction angle of middle bridge. During high-speed operation of the rotor, the direction of the centrifugal force generated by any mass point is along the line connecting the point and the geometric center of the rotor and away from the center. Due to the structural symmetry of the rotor, the centrifugal force produced by any mass point in the region of the pole shoe and permanent magnets offsets each other along the direction perpendicular to the symmetry axis, that is, the y-axis, so only the component of the centrifugal force along the y-axis direction needs to be calculated. In this study, the equivalent centroid principle is employed to calculate the centrifugal forces generated by the rotor’s pole shoes and permanent magnets.

The centroid formula for a homogeneous object of uniform thickness is given as follows:

$$y_{\mathrm{C}}={\displaystyle \frac{{\displaystyle \sum _n{\displaystyle \underset{F_n}{∯}y\mathrm{d}{\sigma }_n}}}{{\displaystyle \sum _n{A}_{F_n}}}}$$

(1)

where A_Fn is the cross-sectional area of the nth region F_n. Due to the complex shape of the pole shoe region, it is divided into an annular region F₁, an arc-shaped region F₂, and a trapezoidal region F₃, as shown in Figure 2. Based on this division, the centroid position of the pole shoe region can be expressed as follows:

$$y_{\mathrm{F}}={\displaystyle \frac{{\displaystyle \underset{F_1}{∯}y\mathrm{d}{\sigma }_{F_1}}+{\displaystyle \underset{F_2}{∯}y\mathrm{d}{\sigma }_{F_2}}+{\displaystyle \underset{F_3}{∯}y\mathrm{d}{\sigma }_{F_3}}}{A_{F_1}+A_{F_2}+A_{F_3}}}$$

(2)

where

$${\displaystyle \underset{F_1}{∯}y\mathrm{d}{\sigma }_{F_1}}={\displaystyle \frac{2}{3}}\left(R_{\mathrm{o}}^3-R_{\mathrm{i}}^3\right)\sin {\displaystyle \frac{\alpha }{2}}$$

(3)

$${\displaystyle \underset{F_2}{∯}y\mathrm{d}{\sigma }_{F_2}}={\displaystyle \frac{2}{3}}{R}_{\mathrm{i}}^3{\sin }^3{\displaystyle \frac{\alpha }{2}}$$

(4)

$${\displaystyle \underset{F_3}{∯}y\mathrm{d}{\sigma }_{F_3}}=l_{\mathrm{c}}\cos {\displaystyle \frac{\theta }{2}}\left(R_{\mathrm{i}}^2\sin \alpha -R_{\mathrm{i}}{l}_{\mathrm{c}}\sin {\displaystyle \frac{\alpha +\theta }{2}}+{\displaystyle \frac{1}{3}}{l}_{\mathrm{c}}^2\sin \theta \right)$$

(5)

$$A_{F_1}={\displaystyle \frac{\alpha }{2}}\left(R_{\mathrm{o}}^2-R_{\mathrm{i}}^2\right)$$

(6)

$$A_{F_2}={\displaystyle \frac{1}{2}}{R}_{\mathrm{i}}^2\left(\alpha -\sin \alpha \right)$$

(7)

$$A_{F_3}=l_{\mathrm{c}}\cos {\displaystyle \frac{\theta }{2}}\left(2R_{\mathrm{i}}\sin {\displaystyle \frac{\alpha }{2}}-l_{\mathrm{c}}\sin {\displaystyle \frac{\theta }{2}}\right)$$

(8)

The centroid of the permanent magnets is expressed as follows:

$$y_{\mathrm{P}}={\displaystyle \frac{{\displaystyle \underset{P_1}{∯}y\mathrm{d}{A}_{P_1}}+{\displaystyle \underset{P_1}{∯}y\mathrm{d}{A}_{P_1}}}{A_{P_1}+A_{P_2}}}$$

(9)

$$A_{P_1}=A_{P_2}=h_{\mathrm{p}}\left(b_{{\mathrm{p}}_1}+b_{{\mathrm{p}}_2}\right)$$

(10)

$${\displaystyle \underset{P_1}{∯}y\mathrm{d}{A}_{P_1}}=h_{\mathrm{p}}{b}_{{\mathrm{p}}_1}[R_{\mathrm{i}}\cos {\displaystyle \frac{\alpha }{2}}-{\displaystyle \frac{1}{2}}{h}_{\mathrm{p}}\sin {\displaystyle \frac{\theta }{2}}-\left({\displaystyle \frac{1}{2}}{b}_{{\mathrm{p}}_1}+c_1\right)\cos {\displaystyle \frac{\theta }{2}}]$$

(11)

$${\displaystyle \underset{P_2}{∯}y\mathrm{d}{A}_{P_2}}=h_{\mathrm{p}}{b}_{{\mathrm{p}}_2}[R_{\mathrm{i}}\cos {\displaystyle \frac{\alpha }{2}}-{\displaystyle \frac{1}{2}}{h}_{\mathrm{p}}\sin {\displaystyle \frac{\theta }{2}}-\left(b_{{\mathrm{p}}_1}+{\displaystyle \frac{1}{2}}{b}_{{\mathrm{p}}_2}+c_1+c_2\right)\cos {\displaystyle \frac{\theta }{2}}]$$

(12)

Thus, the total centrifugal force F generated by the pole shoe and permanent magnets is given by the following:

$$F=F_{\mathrm{F}}+F_{\mathrm{P}}$$

(13)

$$F_{\mathrm{F}}={\rho }_{\mathrm{F}}{L}_{\mathrm{a}}{y}_{\mathrm{F}}{\omega }^2\left(A_{F_1}+A_{F_2}+A_{F_3}\right)$$

(14)

$$F_{\mathrm{P}}=2{\rho }_{\mathrm{P}}{L}_{\mathrm{a}}{y}_{\mathrm{P}}{\omega }^2\left(A_{P_1}+A_{P_2}\right)$$

(15)

where ρ_F and ρ_P are the densities of the pole shoe and permanent magnets, respectively. L_a is the effective length of the rotor.

3.2. Improved Centrifugal Force Method Based Mechanical Model

The pole shoe and permanent magnets are isolated from the rotor structure and analyzed as independent objects for force analysis. In addition to the centrifugal force F, these objects are subjected to forces F_a from the central bridge, F_b from the air-gap bridges, and F_c from the middle bridges. It is assumed that the direction of F_c is along the outer normal direction of the pole shoe surface where the middle bridge is located, as illustrated in Figure 2.

According to the equilibrium principle of the force system, the following relationship stands:

$$F_{\mathrm{a}}+2F_{\mathrm{b}}\cos {\displaystyle \frac{\alpha }{2}}+2F_{\mathrm{c}}\cos {\displaystyle \frac{\phi }{2}}=F$$

(16)

where φ = π − θ. Under the action of forces, the magnetic bridges will undergo elastic deformation, as depicted in Figure 3. According to the theory of material mechanics, the deformation of the three magnetic bridges must be coordinated with each other, that is, to meet the geometric compatibility condition. Therefore, from the geometric relationship of each magnetic bridge before and after deformation in Figure 3, it can be seen that the deformations of the central magnetic bridge ω_a, the air gap magnetic bridge ω_b, and the middle bridge ω_c satisfies the following:

$${\omega }_{\mathrm{b}}={\omega }_{\mathrm{a}}\cos {\displaystyle \frac{\alpha }{2}}, {\omega }_{\mathrm{c}}={\omega }_{\mathrm{a}}\cos {\displaystyle \frac{\phi }{2}}$$

(17)

According to the basic hypothesis 5 in Section 2.2, the central bridge is subjected to tensile load, and the air-gap bridge is subjected to bending load. According to the theory of material mechanics, ω_a and ω_b are, respectively, expressed by the following:

$${\omega }_{\mathrm{a}}={\displaystyle \frac{F_{\mathrm{a}}{b}_1}{A_{\mathrm{a}}E}}, {\omega }_{\mathrm{b}}={\displaystyle \frac{F_{\mathrm{b}}{b}_2^3}{3E{I}_{\mathrm{b}}}}$$

(18)

where E is the elastic modulus of the rotor core, A_a and I_b are the cross-sectional area of the central bridge and the moment of inertia of the air-gap bridge, b₁ and b₂ represent the thicknesses of the central bridge and bilateral bridges, respectively. For the load F_c acting on the middle bridge, it can be decomposed into two perpendicular components, namely F_ct along the width direction and F_cb along the thickness direction. F_ct causes tensile deformation ω_ct along the width direction, while F_cb induces bending deformation ω_cb along the thickness direction. As a result, ω_ct and ω_cb are orthogonal, and they are, respectively, depicted as follows:

$${\omega }_{\mathrm{ct}}={\displaystyle \frac{F_{\mathrm{c}}{b}_3\cos \beta }{A_{\mathrm{c}}E}}, {\omega }_{\mathrm{cb}}={\displaystyle \frac{F_{\mathrm{c}}{b}_3^3\sin \beta }{3E{I}_{\mathrm{c}}}}$$

(19)

where b₃, A_c, and I_c are the thicknesses of the middle bridges, cross-sectional area, and moment of inertia of the middle bridges, respectively. Thus, the total deformation ω_c of the middle bridges can be expressed as follows:

$${\omega }_{\mathrm{c}}=\sqrt{{\omega }_{\mathrm{ct}}^2+{\omega }_{\mathrm{cb}}^2}=k_{\mathrm{c}}{F}_{\mathrm{c}}$$

(20)

where the coefficient k_c depends on the geometric characteristics, elastic modulus, and direction angle of the middle bridge, and it can be expressed as follows:

$$k_{\mathrm{c}}={\displaystyle \frac{b_3}{E}}\sqrt{{\displaystyle \frac{{\cos }^2\beta }{A_{\mathrm{c}}^2}}+{\displaystyle \frac{b_3^4{\sin }^2\beta }{9I_{\mathrm{c}}^2}}}$$

(21)

By solving Equations (16)–(18) and (20) simultaneously, the forces on each magnetic bridge can be determined as follows:

$$F_{\mathrm{a}}={\displaystyle \frac{F}{1+k_1+k_2}}$$

(22)

$$F_{\mathrm{b}}={\displaystyle \frac{3b_1{I}_{\mathrm{b}}F\cos \frac{\alpha }{2}}{A_{\mathrm{a}}{b}_2^3\left(1+k_1+k_2\right)}}$$

(23)

$$F_{\mathrm{c}}={\displaystyle \frac{b_{1}F\cos \frac{\phi }{2}}{A_{\mathrm{a}}E{k}_{\mathrm{c}}\left(1+k_1+k_2\right)}}$$

(24)

where the coefficients k₁ and k₂ are as follows:

$$k_1={\displaystyle \frac{6b_1{I}_{\mathrm{b}}}{A_{\mathrm{a}}{b}_2^3}}{\cos }^2\left({\displaystyle \frac{\alpha }{2}}\right)$$

(25)

$$k_2={\displaystyle \frac{2b_1}{A_{\mathrm{a}}E{k}_{\mathrm{c}}}}{\cos }^2\left({\displaystyle \frac{\phi }{2}}\right)$$

(26)

By substituting Equations (22)–(24) into Equations (18)–(20), the deformation of each magnetic bridge can be obtained. The reference stresses σ_a, σ_b, and σ_c of the central bridge, air-gap bridges, and middle bridges are as follows:

$${\sigma }_{\mathrm{a}}={\displaystyle \frac{F_{\mathrm{a}}}{L_a{a}_1}}$$

(27)

$${\sigma }_{\mathrm{b}}={\displaystyle \frac{a_2{b}_2{F}_{\mathrm{b}}}{2I_{\mathrm{b}}}}$$

(28)

$${\sigma }_{\mathrm{c}}=F_{\mathrm{c}}\left({\displaystyle \frac{\cos \beta }{L_a{a}_3}}+{\displaystyle \frac{a_3{b}_3\sin \beta }{2I_{\mathrm{c}}}}\right)$$

(29)

3.3. Stress Concentration Factor

The sharp change in the cross-section between the magnetic bridge and the rotor pole shoe will lead to stress concentration, which will result in local high stress in the connection part. In practical engineering applications, the stress concentration factor is usually adopted to quantitatively describe the stress concentration phenomenon. In the interior rotor structure, the stress concentration factor not only depends on the geometry of the connecting part but also is closely related to the number of poles of the motor [11].

Reference [11] gave a method to accurately calculate the stress concentration factor function. In this function, the stress concentration factor S is composed of two parts, namely, the part S_g related to the geometric shape of the connection part and the part S_m related to the overall structure of the motor. For the six-pole IPMSM here, the stress concentration factor S can be expressed as follows:

$$S=S_{\mathrm{g}}+S_{\mathrm{m}}$$

(30)

$$S_{\mathrm{g}}={\displaystyle \frac{2\left(\frac{a}{r}+1\right)\sqrt{\frac{a}{r}}}{\left(\frac{a}{r}+1\right){\tan }^{-1}\sqrt{\frac{a}{r}}+\sqrt{\frac{a}{r}}}}$$

(31)

$$S_{\mathrm{m}}=1.803r_{\mathrm{cp}}-1.531$$

(32)

where a and r represent the thickness of the magnetic bridge and the fillet radius of the connection part, respectively. r_cp is the ratio of the pole arc to the pole pitch of the motor. The stress concentration factors of the magnetic bridges are determined by the Formulas (30)–(32).

Thus, the MMS of each bridge is obtained by multiplying the reference stress by the stress concentration factor, and the highest value among the three bridges is taken as the MMS σ_max of the rotor structure, namely:

$${\sigma }_{\max }=\max \left\{S_{\mathrm{a}}{\sigma }_{\mathrm{a}}, S_{\mathrm{b}}{\sigma }_{\mathrm{b}}, S_{\mathrm{c}}{\sigma }_{\mathrm{c}}\right\}$$

(33)

where S_a, S_b, and S_c are the stress concentration factors of the central bridge, air-gap bridges, and middle bridges, respectively.

4. Influence of Structural Parameters on Strength Characteristics

The stress and deformation of the magnetic bridges are closely related to the structural parameters of the rotor. The influence of the structural parameters of the rotor on the stress and deformation of each magnetic bridge is analyzed in this section with rotor speed 20,000 r/min. The material physical properties and the main structural parameters of the rotor are listed in

Table 1. Physical characteristics of rotor materials.

Physical Properties	Rotor Core	Permanent Magnet
Material	20SW1200	NdFeB
Density ρ (Kg/m3)	7850	7450
Young modulus E (GPa)	200	150
Poisson ratio μ	0.29	0.24
Yield strength σs (MPa)	420	/
Ultimate tensile strength σu (MPa)	540	80

and

Table 2. Main structural parameters of the rotor.

Structural Parameters	Value	Structural Parameters	Value
Air-gap length g (mm)	0.8	Air-gap bridge thickness a2 (mm) (mm)	1.0
Rotor radius Ro (mm)	49.4	Air-gap bridge width b2 (mm)	4.5
Inner radius of annular region Ri (mm)	48.4	Middle bridge thickness a3 (mm)	1.0
Width of permanent magnet 1 bp1 (mm)	7.5	Middle bridge width b3 (mm)	6.2
Width of permanent magnet 2 bp2 (mm)	7.5	Direction angle β (°)	15
Thickness of permanent magnets hp (mm)	6.0	V-shaped angle θ (°)	116
Width of permanent magnet slot lc (mm)	17.8	Pole shoe angle α (°)	44
Positioning distance 1 c1 (mm)	1.8	Filet radius of a central bridge ra (mm)	1.5
Positioning distance 2 c2 (mm)	2.6	Filet radius of an air-gap bridge rb (mm)	0.8
Central bridge thickness a1 (mm)	2	Filet radius of a middle bridge rc (mm)	1
Central bridge width b1 (mm)	5.5	——	——

.

4.1. Influence of Structural Angles

The V-shaped angle and pole shoe angle are critical parameters in the design of V-type interior rotor structures, significantly influencing the forces, MMS, and deformations of the magnetic bridges, as illustrated in Figure 4 and Figure 5. In these Figures, CB, AB, and MB represent the central bridge, air gap bridge, and middle bridge, respectively, while S and D denote stress and deformation. The meanings of the legends in the following Figures are the same as here.

As the V-shaped angle increases, the forces on the central and air gap bridges decrease, while the forces on the middle bridges initially increase and then decrease. This is due to the reduction in pole shoe area, leading to lower total centrifugal force. The trends in MMS and deformation across all magnetic bridges closely follow the force variations. Conversely, increasing the pole shoe angle enlarges the pole shoe area and the total centrifugal force, resulting in higher MMS and deformation across all magnetic bridges. Therefore, appropriately increasing the V-shaped angle while reducing the pole shoe angle can effectively improve the mechanical strength of the rotor structure.

4.2. Influence of Central Bridge Structural Parameters

The influence of the central bridge structural parameters on the force, MMS, and deformation of each magnetic bridge is shown in Figure 6 and Figure 7. As the thickness of the central bridge increases, the total centrifugal force remains unchanged, and the cross-sectional area of the central bridge increases. According to Equations (22)–(24), the force on the central bridge increases slightly, while the forces on the air-gap and middle bridges decrease, as shown in Figure 6a. Although the force of the central bridge increases, its increment ratio is obviously smaller than the increment ratio of the cross-sectional area, so the MMS and deformation of the central bridge decrease. Consistent with the force trend, the MMS and deformation of the air-gap middle bridges decrease synchronously.

Conversely, as the central bridge width increases, its force gradually decreases, while the forces on the air-gap and middle bridges increase, as shown in Figure 7a. The MMS of the central bridge follows the same trend in the force, indicating that width has a minimal impact on its MMS. However, due to the increase in width being much larger than the decrease in force, the central bridge deformation increases significantly, as shown in Figure 7b. Similarly, the MMS and deformation of the air-gap and middle bridges increase in line with the force changes.

It is evident that increasing the central bridge thickness effectively reduces the MMS and deformation of the magnetic bridges, significantly enhancing the mechanical strength of the V-shaped rotor. Reducing the central bridge width only lowers the deformation but makes little contribution to decreasing the rotor MMS.

4.3. Influence of Air-Gap Bridge Structural Parameters

The influence of the structural parameters of the air gap bridge on the force, MMS, and deformation of each magnetic bridge is shown in Figure 8 and Figure 9. In Figure 8a, as the thickness of the air-gap bridge increases, its moment of inertia grows, leading to a gradual decrease in the forces on the central and middle bridges, while the force on the air-gap bridge increases significantly. In Figure 8b, the MMS of all magnetic bridges follows the same trend as the force, indicating that force is a key factor in determining MMS. The deformation of all magnetic bridges aligns with the force trend in the central bridge. When the air-gap bridge thickness reaches 1.7 mm, the MMS of the central and air-gap bridges are equal. The opposite trend between the deformation and force of the air-gap bridge is due to the relationship where deformation is proportional to the force and inversely proportional to the cube of its thickness. Since the increase ratio in force is smaller than the increase ratio in the cube of thickness, the deformation decreases as thickness increases. Due to geometric compatibility, the deformation of all magnetic bridges must be coordinated, namely, increase or decrease at the same time. Thus, the deformation trends in all magnetic bridges are identical.

In Figure 9a, the force change trends in the magnetic bridges with the air-gap bridge width are opposite to those of the thickness. As the air-gap bridge width increases, the forces on the central and middle bridges increase, while the forces on the air-gap bridges decrease. Similarly, the MMS change trends in all magnetic bridges are consistent with those of forces. The deformations of all magnetic bridges gradually increase, as shown in Figure 9b. The reason for the increase in air-gap bridge deformation is that its deformation is proportional to both the force and the cube of its width.

It can be seen from the above analysis that changing the structural parameters of the air-gap bridge cannot effectively reduce the MMS or deformation of the V-shaped rotor. Therefore, the thickness and width of the air-gap bridge can be appropriately reduced and increased, respectively, to reduce the magnetic flux leakage and improve the electromagnetic performance of the motor.

4.4. Influence of Middle Bridge Structural Parameters

The influence of the structural parameters of the intermediate bridge on the force, MMS, and deformation of the magnetic bridge is shown in Figure 10, Figure 11 and Figure 12. As the middle bridge thickness increases, PM 2 moves closer to the rotor center, reducing the total centrifugal force. Meanwhile, the cross-sectional area and moment of inertia of the middle bridge increase, leading to a slight decrease in the forces on the central and air-gap bridges, while the force on the middle bridge increases obviously, as shown in Figure 10a. In Figure 10b, the MMS of all magnetic bridges follows the same trend as those of forces. As the middle bridge thickness increases, the deformation of all magnetic bridges gradually decreases. The reduction in middle bridge deformation is due to the fact that, according to Equations (19) and (20), although tensile deformation increases and bending deformation decreases with thickness, the reduction in bending deformation outweighs the increase in tensile deformation.

In Figure 11, with the increase in the middle bridge width, the forces on the central and air-gap bridges increase slowly, while the forces on the middle bridges decrease significantly. The MMS change trends in all magnetic bridges are consistent with those of the forces. The deformations of all magnetic bridges increase slightly.

In Figure 12a, as the middle bridge direction angle decreases, the total centrifugal force remains nearly constant, while the forces on the central and air-gap bridges decrease, and the forces on the middle bridges increase significantly. When the direction angle is zero, the force on the middle bridges reaches 3.29 kN, while the force on the central bridge is 8.46 kN. The force on the middle bridges accounts for 38.89% of the central bridge force and 23.25% of the total centrifugal force. This indicates that at this point, the middle bridge can effectively share the total centrifugal force, significantly reducing the load on the central bridge which bears most of the total centrifugal force. This greatly enhances the mechanical strength of the rotor structure. The main reason is that the load-bearing mode of the middle bridge is closely related to its direction angle. As the direction angle decreases, the load-bearing mode transitions from simultaneously bearing tensile and bending loads to predominantly bearing tensile loads with bending loads playing a secondary role.

When the direction angle is zero, the force direction of the middle bridge aligns with its width, resulting in the middle bridge only bearing tensile loads, as shown in Figure 13a. Bearing only tensile loads in the middle bridge not only significantly reduces the force on the central bridge but is also highly beneficial for reducing rotor deformation. Under the same load, for magnetic bridges subjected only to bending or tensile deformation, the bending deformation is much greater than the tensile deformation. For the middle bridge subjected to both tensile and bending loads, the variations in tensile and bending deformation with the direction angle are shown in Figure 13b, where the deformation ratio is defined as the ratio of the bending deformation to the tensile deformation at the same orientation angle. It can be observed that as the direction angle decreases, the tensile deformation increases, the bending deformation decreases, and the deformation ratio decreases nearly linearly. When the direction angle is reduced to zero, the total deformation of the middle bridge reaches its minimum, while the force on it reaches its maximum, representing the optimal design value for the direction angle.

In Figure 12b, the MMS trends in magnetic bridges remain consistent with those of forces. When the middle bridge direction angle is zero, the MMS of the entire rotor structure reaches its minimum value of 245 MPa, which is well below the yield strength 420 MPa of the rotor core material, fully meeting the strength requirements at 20,000 r/min. The deformations of the magnetic bridges decrease as the middle bridge direction angle decreases. The reduction in middle bridge deformation is due to the change in its load-bearing mode, shifting to primarily tensile loads with bending loads playing a secondary role as the direction angle decreases.

The above analysis shows that only when the middle bridge is subjected exclusively to tensile loads can it effectively share the load on the central bridge. This is also the primary reason why adjusting parameters such as the width and thickness of the air-gap and middle bridges cannot significantly reduce rotor MMS. It is evident that the key to the design of a multi-bridge V-shaped rotor structure lies in ensuring that the direction angle of the middle bridge is zero, namely, the width of the middle bridge must align with the normal direction of the rotor pole shoe surface at its location.

5. Structural Design and Its Validations

5.1. Structural Design

Based on the discussion and analysis in the previous Section, the key considerations for the mechanical strength design of multi-bridge V-shaped IPMSM rotor structures are as follows:

(1)

Increasing the V-shaped angle, reducing the pole shoe angle or increasing the central bridge thickness all contribute to reducing the MMS and deformation of the rotor. Reducing the central bridge width mainly reduces rotor deformation.

(2)

Changing the thickness and width of the air-gap and middle bridges has a limited effect on the improvement of the rotor’s mechanical strength. Therefore, on the premise of meeting the mechanical strength requirement, a smaller thickness and a larger width can be selected to reduce the magnetic flux leakage of the permanent magnet and improve the electromagnetic performance of the motor.

(3)

The most critical principle in designing a multi-bridge V-shaped rotor structure is to ensure that the centerline of the middle bridge width aligns with the normal direction of the adjacent pole shoe surface. This alignment maximizes the middle bridge’s ability to bear centrifugal loads, significantly enhancing the rotor’s mechanical strength.

Based on key structural design principles, three different rotor topologies are developed. Scheme 1 is a conventional V-shaped rotor, Scheme 2 is a multi-bridge V-shaped rotor with the middle bridge direction angle set at 15°, and Scheme 3 is a multi-bridge V-shaped rotor with the middle bridge direction angle set at 0°. In all three schemes, the V-angle angle, the pole shoe angle, the structural parameters of the central and air-gap bridges, as well as the total width and thickness of the permanent magnets are identical. The only major difference between Schemes 2 and 3 is the middle bridge direction angle.

With the same primary structural parameters, the forces, MMS, and deformation of the magnetic bridges in three rotor structures vary with rotational speed as shown in Figure 14 and Figure 15. In Figure 14, the forces on magnetic bridges increase with speed. Although two additional middle bridges are included in Scheme 2, their inappropriate direction angle results in primarily bending deformation, preventing them from effectively sharing the overall centrifugal load. Thus, in both Schemes 1 and 2, the central bridge remains the main load-bearing structure, experiencing similar forces at different speeds. In contrast, the middle bridge direction angle in Scheme 3 is 0°, and the middle bridge only bears the tensile load, which can share 21% of the total centrifugal force, thus effectively reducing the force on the central bridge.

In Figure 15, the dashed line indicates the yield strength 420 MPa of the rotor core material. The MMS and deformations of all magnetic bridges increase as the rotational speed rises. To facilitate comparison among the three structural schemes, the speed at which the MMS of the rotor core material reaches its yield strength is defined as the yield speed. Since the rotor will undergo irreversible plastic deformation if the rotational speed exceeds this yield speed, the yield speed represents the maximum allowable operating speed of the rotor. Obviously, the speed corresponding to the intersection of the central bridge MMS and the dotted line is the yield speed of each rotor structure. The theoretical yield speeds for Schemes 1, 2, and 3 are 20,861 r/min, 20,887 r/min, and 26,862 r/min, respectively. The yield speed of Scheme 2 is only 0.12% higher than that of Scheme 1, while the yield speed of Scheme 3 is 28.77% and 28.61% higher than those of Schemes 1 and 2, respectively. Interestingly, at their respective yield speeds, the maximum deformations of all magnetic bridges in the three schemes are nearly identical, being 7.26 μm, 7.34 μm, and 7.34 μm, respectively. This shows that ensuring the middle bridge direction angle of a multi-bridge V-shaped rotor structure is 0° can significantly increase the rotor’s maximum allowable operating speed and effectively enhance the motor’s power density.

Given the complexity of the multi-bridge rotor structure and some simplifications and assumptions in the analytical model, theoretical results require validation. This paper verifies the effectiveness of the multi-bridge V-shaped rotor design method through finite element analysis and experiments.

5.2. Numerical Verification

FEM is commonly used to validate the effectiveness of theoretical analysis methods. With the structural parameters remaining consistent with those in Section 5.1, Figure 16 and Figure 17 primarily present the stress and deformation distributions of the rotor structure in Scheme 3 at a rotational speed of 20,000 r/min.

Table 3. MMS of magnetic bridges in three rotor structure schemes.

Magnetic Bridge	MMS in Scheme 1			MMS in Scheme 2			MMS in Scheme 3
	ICFM (MPa)	FEM (MPa)	Relative Error (MPa)	ICFM (MPa)	FEM (MPa)	Relative Error (MPa)	ICFM (MPa)	FEM (MPa)	Relative Error (MPa)
Central bridge	386.03	414.75	6.92%	385.08	406.46	5.26%	232.82	235.08	0.96%
Air-gap bridge	261.55	279.44	6.40%	193.84	191.84	1.04%	117.20	138.14	15.06%
Middle bridge	——	——	——	57.40	65.51	12.38%	135.28	146.62	7.73%

and

Table 4. Deformations of magnetic bridges in three rotor structure schemes.

Magnetic Bridge	Deformation in Scheme 1			Deformation in Scheme 2			Deformation in Scheme 3
	ICFM (μm)	FEM (μm)	Relative Error (μm)	ICFM (μm)	FEM (μm)	Relative Error (μm)	ICFM (μm)	FEM (μm)	Relative Error (μm)
Central bridge	6.67	5.27	26.57%	6.73	4.78	40.79%	4.07	2.78	46.40%
Air-gap bridge	6.10	5.81	4.99%	6.15	4.68	31.41%	3.72	3.17	17.35%
Middle bridge	——	——	——	6.50	5.35	21.50%	3.93	3.59	9.47%

compare the MMS and deformation of each magnetic bridge across the three schemes.

In Figure 16, the MMS of the air-gap bridge occurs on the side of its fixed end near the rotor center, which aligns with the response characteristics of the air-gap bridge under the bending load. The stress distribution in the central bridge is relatively uniform in the middle region, indicating that the central bridge experiences the tensile load during high-speed rotation. The middle bridge also exhibits a uniform stress distribution, suggesting that when the direction angle is 0°, the middle bridge is subjected only to the tensile load, effectively enhancing its ability to bear centrifugal forces. This validates the effectiveness of the middle bridge structural design method in the multi-bridge V-shaped rotor. Compared with Figure 16d, since the middle bridge direction angle in Scheme b is designed to be 15°, the middle bridge is subjected to both tensile and bending loads, and the stress distribution is not uniform, as shown in Figure 16e. These observations strongly confirm that the basic assumption (4) in Section 2.2 is reasonable and effective. As seen in Table 3, the ICFM values of the MMS for the magnetic bridges in the V-shaped interior rotor structure are in good agreement with the FEM values, indicating that the proposed mechanical model can accurately predict the MMS of the multi-bridge V-shaped interior.

Figure 17 presents the deformation contour map of the multi-bridge V-shaped rotor in the z-axis direction, with units in millimeters. According to the geometric relationship between the difference in the deformation displacement along the z-axis at the two ends of the magnetic bridge and the assumed deformation, the FEM values of the magnetic bridge deformation are obtained, as shown in Table 3. It can be observed that while the ICFM can quantitatively calculate the magnetic bridge deformations, there is a significant error between the ICFM and FEM values for some bridges, particularly the central bridge. The main reasons include the exclusion of deformation in the rotor structures outside the magnetic bridges during the theoretical modeling, and the fact that due to the fillet structure, the magnetic bridges cannot deform strictly as assumed. Additionally, the actual force conditions of the magnetic bridges cannot fully align with the assumptions due to the complexity of the rotor structure. However, it is fortunate that the theoretical model overestimates the deformation of the magnetic bridges in all three rotor structure schemes. In mechanical strength design, this conservative estimate actually provides a greater safety margin, helping to avoid potential overload or excessive deformation in the rotor structure during actual use, thereby ensuring the long-term safe and reliable operation of the rotor.

5.3. Experimental Verification

By skillfully using the tensile testing machine and designing an appropriate tooling, a rotor strength testing device can be built, as shown in Figure 18a. The basic process of the strength test is as follows: Firstly, the total centrifugal force generated by the rotor pole shoe and permanent magnet at a given speed is calculated according to Equation (13). This force is then converted into an equivalent tensile force, which is gradually increased from zero to the predetermined value, held constant for approximately 20 s, and subsequently unloaded back to zero. This tensile force serves as the input signal for the test device. Then, the measured rotor core is fixed by the upper and lower clamping devices of the tensile testing machine and tooling to ensure that the loading-force direction is along the width direction of the central bridge. The lug structure in Figure 18b is designed for this purpose. Subsequently, the loading command is issued by the console to test the rotor core and record the test data. Finally, the test results are analyzed according to the theory of material mechanics. The primary objective of the experiment is to replicate the equivalent centrifugal force conditions and validate the mechanical response of the magnetic bridges under these conditions, consistent with the analytical and numerical results. It should be noted that the lifting lug shown in Figure 18b is an auxiliary structure specifically designed to facilitate the application of tensile loads in the static experiment. Under static conditions, the lifting lug does not generate any additional forces, and its presence does not affect the accuracy of the load applied to the magnetic bridges.

The strength test result of the multi-bridge V-shaped interior rotor core is shown in Figure 19. The force-deformation curve of the measured rotor structure conforms to the typical mechanical characteristics of the metal material tensile test, that is, the elastic stage, the yield stage, the strengthening stage, the necking stage and the fracture stage. The fracture location of the multi-bridge V-shaped structure is shown in Figure 18c. The blue pentagram in Figure 19 represents the yield point of the core material, and the corresponding loading force is 10.248 kN. According to Equation (13), the test value of the rotor yield speed is 25,653 r/min, and the relative error between the theoretical yield speed 26,862 r/min and the experimental yield speed is 4.71%. The reason why the test value is lower than the theoretical value is that the direction of the loading force is not well maintained along the width direction of the central bridge during the actual test process, which can be seen from the fact that one middle bridge does not fracture with the central bridge and another middle bridge simultaneously in Figure 18c.

It should be noted that the deformation shown in Figure 19 represents the overall deformation of half of the rotor structure in the direction of the applied tensile force, rather than the deformation of individual magnetic bridges. Although direct comparisons of deformation and stress between experimental and analytical or numerical results are not feasible due to the thinness of the magnetic bridges, the experiment clearly shows that the first failure occurs at the magnetic bridges. Furthermore, the yielding phase of the force–deformation curve reflects the dynamic response of these critical elements, aligning well with the analytical results and validating the proposed approach.

In summary, the theoretical yield speed agrees well with the experimental yield speed, confirming the accuracy and effectiveness of the proposed analysis and design method for the multi-bridge V-shaped rotor structure. Despite the limitations in direct comparisons, the experimental results validate the analytical predictions regarding the failure behavior of the magnetic bridges.

6. Conclusions

In this research, an improved centrifugal force method is proposed to establish an analytical mechanical model of the multi-bridge V-shaped rotor structure. Compared to traditional methods, this approach incorporates force conditions, deformation constraints, material properties, and stress concentration effects. These enhancements enable a more accurate analysis of the mechanical behavior of complex rotor structures. The trends in the MMS and deformation of the rotor under different structural parameters are analyzed. Based on these analyses, the structure of the multi-bridge V-shaped rotor is designed, and the effectiveness of the mechanical model and design method is verified by FEM and experimentation. The main conclusions are as follows:

(1)

The mechanical model based on the improved centrifugal force method can accurately calculate the MMS of the multi-bridge V-shaped rotor structure and effectively predict the upper limit of the magnetic bridge deformation. Therefore, the proposed analysis method for the mechanical strength of the multi-bridge V-shaped rotor structure is effective.

(2)

Appropriately increasing the V-angle, increasing the central bridge thickness, or reducing the pole shoe angle can contribute to reducing the rotor’s maximum MMS and deformation. However, reducing the central bridge width only reduces the rotor’s deformation.

(3)

To effectively reduce the MMS and deformation of a conventional V-shaped rotor structure through a multi-bridge design, it is essential to ensure that the width direction of the middle bridge aligns with the normal direction of the outer surface of the adjacent pole shoe.

Based on the findings of this work, several directions for future research can be identified. First, future work could explore the multi-physics coupling effects of magnetic, mechanical, and thermal fields on rotor performance, providing a more comprehensive understanding of the interactions between these factors. Second, the current study focused on static equivalent loading conditions; therefore, future experiments under high-speed dynamic loading conditions could better simulate real-world operational scenarios. Finally, the proposed method could be further extended to optimize the design of other complex rotor structures, such as those with asymmetric magnetic bridges or hybrid magnetic circuits, enhancing its generality and applicability in various motor designs.