Study on Multi-Parameter Collaborative Optimization of Motor-Pump Stator Slotting for Cogging Torque and Noise Suppression Mechanism

Abstract

As a highly integrated and compact power unit, the motor-pump finds critical applications in emerging electric vehicle (EV) domains such as electro-hydraulic braking and steering systems, where its vibration and noise performance directly impacts cabin comfort. A key factor limiting its NVH (Noise, Vibration, and Harshness) performance is the electromagnetic vibration and noise induced by the cogging torque of the built-in brushless DC motor (BLDCM). Traditional suppression methods that rely on stator auxiliary slots exhibit certain limitations. To address this issue, this paper proposes a collaborative optimization method integrating multi-parameter scanning and response surface methodology (RSM) for the design of auxiliary slots on the motor-pump’s stator teeth. The approach begins with a multi-parameter scanning phase to identify a promising region for global optimization. Subsequently, an accurate RSM-based prediction model is established to enable refined parameter tuning. Results demonstrate that the optimized stator structure achieves a 91.2% reduction in cogging torque amplitude for the motor-pump. Furthermore, this structure effectively suppresses radial electromagnetic force, leading to a 5.1% decrease in the overall sound pressure level. This work provides a valuable theoretical foundation and a systematic design methodology for cogging torque mitigation and low-noise design in motor-pumps.

1. Introduction

With the rapid development of electric vehicle technology, the industry has set higher demands for the power density, NVH performance, and system integration of drive and auxiliary systems. Particularly during vehicle operation, cabin smoothness and quietness have become important indicators for measuring driving and riding quality. Compared to traditional internal combustion engine vehicles, EVs lack the masking effect of engine noise, making the noise generated by the motor, gears, and various auxiliary systems more prominent [1]. Against this backdrop, the highly integrated motor-pump system centered on the Brushless DC Motor, thanks to its outstanding advantages such as compact structure and high energy transfer efficiency, is gradually becoming a key technological development direction in fields like electric vehicle braking and steering [2].

However, the inherent cogging torque of BLDCM—periodic torque fluctuations arising from the interaction between the stator core and permanent magnets—induces significant torque variations. Within motor-pump systems, these fluctuations not only degrade the control precision of servo systems but also exacerbate the transmission of vibration and noise [3]. Electromagnetic excitation readily propagates through rigidly connected structures, amplifying vibrations and directly compromising the reliability of the entire hydraulic system. Therefore, suppressing cogging torque has become a critical technical challenge, as it is not only essential for advancing motor-pump systems toward higher power density and lower noise levels but also directly affects the overall performance of electric vehicles. Reducing the electromagnetic excitation at its source is one of the most effective strategies. This study focuses on suppressing cogging torque and its induced vibration and noise, with the motor-pump serving as a representative case of a compact, noise-sensitive integrated motor system.

Existing suppression methods primarily include pole arc coefficient optimization [4], segmented rotor skewing [5], eccentric stator teeth [6], and stator slotting [7]. For instance, Reference [8] proposed an improved segmented skewed-pole rotor design, which reduced cogging torque from 2.55 N·m to 158 mN·m in an 8-pole 48-slot motor—a decrease of 93.8%—while effectively mitigating axial unbalanced magnetic forces. This demonstrates that systematic rotor modifications can achieve substantial cogging torque suppression. Reference [9] developed a synergistic approach integrating segmented skewing and pole arc coefficient design, enhanced by response surface methodology and particle swarm optimization for parameter tuning. This methodology achieved 96.452% cogging torque suppression, with experimental validation confirming its efficacy. Reference [10] systematically investigates the influence of unequal tooth widths on cogging torque. The study demonstrates that a rational allocation of tooth widths not only optimizes the flux density distribution in the teeth and alleviates saturation but also suppresses the amplitude of cogging torque to a certain extent.

In contrast, stator slotting technology has attracted considerable attention owing to its simple manufacturing process, low cost, and minimal impact on the main magnetic circuit. For example, Reference [11] examined the individual effects of slot depth, width, and number on cogging torque, identifying optimal configurations under single parameter variations. While providing fundamental insights, this study did not account for parameter interactions. Subsequent work progressed to dual parameter analyses: Reference [12] investigated the influence of single and dual parameter variations on cogging torque, revealing preliminary coupling effects. Reference [13] adopted a more integrated approach, employing a co simulation strategy that combined auxiliary slots and pole arc coefficients to simultaneously minimize cogging torque. Further advancing the methodology, Reference [14] applied a Kriging response surface model coupled with a multi-objective genetic algorithm to optimize five geometric parameters (radius and spacing of circular auxiliary slots) for a 2 pole 6 slot high speed permanent magnet motor, achieving an 80.04% reduction in cogging torque, which was experimentally validated. Reference [15] combined rotor V skewing with stator auxiliary slots; by evaluating six slot size combinations, a 6.4% reduction in the A weighted sound pressure level was attained, confirming the acoustic benefits of auxiliary slots. Reference [16] achieved a 91.3% reduction in cogging torque for an interior U type permanent magnet synchronous motor through multi parameter co optimization combining Halbach magnetization, U- and I- type pole structures, and elliptical rotor auxiliary slots, while also reducing the maximum sound pressure level of the motor by 9 dB (corresponding to a 10.1% decrease).

Traditional research has predominantly been limited to single- or dual-parameter optimization, lacking systematic global optimization that accounts for the complex inter-actions among multiple parameters, and generally treating cogging torque minimization as the sole objective. Concurrently, the complex nonlinear coupling between cogging torque and noise is often overlooked, leading to designs optimized purely for cogging torque reduction potentially aggravating vibration and noise in practical operation. To address these research gaps, this study focuses on the 8-pole 12-slot BLDCM within the motor-pump system, particularly the double-arc helical gear motor-pump proposed by the research team. A multi-parameter collaborative optimization method is introduced for the design of stator auxiliary slots in brushless DC motors. The core contributions of this study are as follows: An optimization strategy is introduced that integrates initial multi-parameter global scanning with subsequent RSM, systematically considering the complex interactions among multiple parameters to achieve cogging torque minimization. Building on electromagnetic optimization, an electromagnetic–mechanical–acoustic coupled simulation analysis is incorporated to systematically evaluate the actual improvement in vibration and noise performance achieved by the optimized design, ensuring an overall enhancement of NVH performance.

2. Motor-Pump System Structure

The motor-pump structure designed by the research team is shown in Figure 1. Its core power source is a BLDCM, with the motor rotor coaxially connected to the drive shaft of the double-circular helical gear pump. In the cooling channel design, oil entering the system through the inlet does not directly enter the pump chamber but instead flows through the annular channel on the outer side of the motor’s inner casing, forming an effective cooling circulation. This cooling oil continuously dissipates heat generated during motor and pump operation, enabling active cooling of critical components and ensuring stable performance under high-temperature conditions.

Regarding hydraulic performance, this design employs double-arc helical gears as the core transmission elements. This gear configuration enables smooth power transmission during meshing, fundamentally eliminating the oil trapping phenomenon common in traditional gear pumps. It significantly reduces flow and pressure pulsations, effectively suppressing system vibration and noise while enhancing the overall NVH performance of the unit.

3. Cogging Torque Analysis and Finite Element Modeling Method

3.1. Cogging Torque Generation Principle

The cogging torque in permanent magnet motors arises from the interaction between the permanent magnets and the slot structure of the motor core. It is defined as the negative derivative of magnetic field energy with respect to the position angle when the armature windings are de-energized, namely:

$$T_{\mathrm{cog}}=-{\displaystyle \frac{\partial W}{\partial \alpha }}$$

(1)

In the formula: $W$ is the total magnetic field energy of the motor; $\alpha$ is the relative angular position between the stator and rotor; $T_{\mathrm{cog}}$ is the cogging torque of the permanent magnet motor.

Compared to air and permanent magnets, the variation in magnetic field energy stored within the motor core is negligible. Therefore, the stored energy within the motor can be expressed as:

$$W \approx W_{air} +W_{PM}= {\displaystyle \frac{1}{2{\mu }_0}}{\displaystyle {\int }_v{B}^2(\theta ,\alpha )\mathrm{d}v}$$

(2)

In the formula: $W_{air}$ represents the magnetic field energy in the air gap; $W_{PM}$ denotes the magnetic field energy of the permanent magnet; ${\mu }_0$ is the relative magnetic permeability; $B$ is the air gap flux density; $v$ is the integration region for the permanent magnet and air gap, $\theta$ is the mechanical angle at the standstill position.

$$B(\theta ,\alpha )=B_r(\theta ){\displaystyle \frac{h_m(\theta )}{h_m(\theta )+\delta (\theta ,\alpha )}}$$

(3)

In the formula: $B_r$ represents the remanence of the permanent magnet; $h_m$ is the length of the magnetization direction of the permanent magnet; $\delta$ indicates the effective air gap length.

Substituting Formula (2) into Formula (3) yields:

$$W = {\displaystyle \frac{1}{2{\mu }_0}}{[B_r^2 ( \theta ) {\displaystyle \frac{h_m( \theta )}{h_m(\theta ) +\delta (\theta ,\alpha )}}]}^2 \mathrm{d}v$$

(4)

By integrating the function over [0, 2π], the analytical expression for cogging torque $T_{\mathrm{cog}}$ is obtained as:

$$T_{cog} = {\displaystyle \frac{\pi z{L}_a}{4{\mu }_0}}( R_2^2 - R_1^2 ) {\displaystyle \sum _{n = 1}^{\infty }n}{G}_n{B}_{r{\displaystyle \frac{nz}{2p}}}\sin nz\alpha$$

(5)

In the formula: $z$ denotes the number of stator slots; $L_a$ represents the axial length of the stator core; $R_1$ is the inner radius of the stator; $R_2$ is the outer radius of the stator; $G_n$ is the Fourier decomposition coefficient of the square of the relative air gap magnetic permeability; $n$ represents an integer that makes ${\displaystyle \frac{nz}{2p}}$ an integer; $p$ is the number of pole pairs.

3.2. Finite Element Model

Auxiliary slots are machined into the stator teeth to analyze the effects of slot quantity, slot position angle, slot width, and slot depth on the motor’s cogging torque. The stator tooth structure is shown in Figure 2, where β represents the auxiliary slot position angle (the angle between the auxiliary slot centerline and the stator tooth centerline), w denotes the auxiliary slot width, and d indicates the auxiliary slot depth.

This study investigates an 8-pole, 12-slot surface-mount BLDCM. A two-dimensional simulation model was established using ANSYS Maxwell software, version 2022 R2, as shown in Figure 3. The primary structural parameters are listed in

Table 1. Structural Parameters of BLDCM.

Parameters	Parameter Value	Parameters	Parameter Value
Number of stator slots	12	Stator Outer Diameter/mm	120
pole number	8	Stator Inner Diameter/mm	71
Permanent magnet thickness/mm	4	Stator Slot Depth/mm	15
Rated Power/kW	5	Stator Tooth Width/mm	9
Rated speed n1/rpm	6000	Rotor Outer Diameter/mm	69
Winding Layers	2	Rotor Inner Diameter/mm	14
Conductors per Slot	12	Auxiliary Slot Position Angle β/(°)	4.2
Parallel Branches	1	Auxiliary Slot Width w/mm	3.6
Air gap length/mm	1	Auxiliary Slot Depth d/mm	0.2
Permanent Magnet Model	NdFeB35	Armature length/mm	156

.

(1)

Time Step Convergence Analys

In the cogging torque simulation, the motor speed is set to rotate at a constant angular velocity of 1°/s, with the armature winding current set to zero. The peak value of cogging torque within one electrical cycle (60/n₁/p) is used as the evaluation metric.

The choice of time step size directly impacts computational accuracy and stability. In this study, the motor speed was set to 6000 rpm, corresponding to an electrical period of 2.5 ms. To determine an appropriate time step size, simulations were conducted using four different step sizes: 0.25 ms, 0.05 ms, 0.025 ms, and 0.0125 ms. The rate of change in the peak cogging torque amplitude was used as the convergence evaluation criterion. As summarized in

Table 2. Time step size analysis.

Time Step Size/ms	Cogging Torque Amplitude/N·m	Rate of Change (%)
0.25	1.174	-
0.05	1.145	2.5
0.025	1.128	1.5
0.0125	1.123	0.4

, when the time step size was reduced from 0.25 ms to 0.05 ms, the rate of change in cogging torque amplitude was 2.5%. Further reduction to 0.025 ms decreased the rate of change to 1.5%. When the time step size was ≤0.025 ms, the rate of change in cogging torque amplitude fell below 2%, indicating convergence on. Therefore, a time step of 0.025 ms was selected, balancing computational accuracy and efficiency.

(2)

Mesh Convergence Analysis

To enhance computational accuracy while maintaining efficiency, this study employs a region-based mesh refinement strategy tailored to the physical field requirements. Given the significant impact of air gap mesh refinement on simulation reliability, multiple local meshes were implemented in this region: four layers of dense mesh elements were distributed circumferentially around the air gap, as shown in Figure 4.

For regions such as the stator, rotor, and permanent magnets, the “Surface Approximation” method was applied for processing. To eliminate the randomness of simulation results caused by mesh density, a mesh independence verification was conducted. Four different mesh densities were compared and analyzed, as detailed in

Table 3. Mesh convergence analysis.

Mesh Refinement Level	Number of Mesh Elements	Cogging Torque Amplitude/N·m	Rate of Change (%)
Coarse mesh	15,840	1.187	-
Medium mesh	21,273	1.146	3.5
Fine mesh	35,170	1.128	1.6
Extra-fine mesh	69,174	1.121	0.6

. After two consecutive mesh refinements, the rate of change in the cogging torque amplitude remained below 2%, indicating good mesh convergence. Therefore, the finally adopted mesh model consists of approximately 35,170 elements, effectively balancing computational accuracy and efficiency.

4. Multi-Parameter Co-Optimization and Performance Analysis of Stator Auxiliary Slots

4.1. Analysis of the Influence of Single-Parameter Variations on Cogging Torque

4.1.1. Impact of Auxiliary Slot Quantity on Cogging Torque

The introduction of auxiliary slots can effectively reduce cogging torque, and its mechanism primarily involves two aspects. On one hand, the incorporation of auxiliary slots effectively increases the fundamental periodicity of the cogging torque. The newly generated cogging torque waveform can counteract that produced by the original slot openings, thereby reducing the amplitude of the total cogging torque. On the other hand, adding auxiliary slots in the teeth increases the equivalent air gap length, which further contributes to the suppression of cogging torque.

As mentioned earlier, for an 8-pole 12-slot motor, the number of auxiliary slots N_n should avoid the following parameters:

$$N_n+1=k{N}_p$$

(6)

In the formula: $N_n$ denotes the number of fundamental cogging torque cycles per slot pitch; $N_p$ denotes the number of auxiliary slots, $k=1,2,3\dots \dots$.

Otherwise, cogging torque may actually increase due to the addition of auxiliary slots.

$$q={\displaystyle \frac{z}{2mp}}={\displaystyle \frac{c}{d}}$$

(7)

$$N_p=d=={\displaystyle \frac{2p}{N_m}}$$

(8)

In the formula: $q$ denotes the number of slots per pole per phase, representing an irreducible fraction; $c$ denotes the numerator of the simplified fraction; $d$ denotes the denominator of the simplified fraction; $m$ denotes the number of phases in the motor; $N_m$ denotes the greatest common divisor of $z$ and $2p$.

Therefore, for this motor, the fundamental cogging torque cycle count averaged per slot pitch is $N_p$ = 2. Substituting into formula 6, the number of auxiliary slots $N_p$ should avoid values of 1, 3, 5…, constrained by the stator tooth space. The number of auxiliary slots $N_p$ is set to 2.

Initial settings: Two auxiliary slots are opened, with auxiliary slot position angle β = 5°, auxiliary slot width w = 2.6 mm, and auxiliary slot depth d = 0.5 mm. A single-variable control experiment investigates the effects of slot position angle, slot width, and slot depth on the cogging torque of the motor.

4.1.2. Influence of Auxiliary Slot Position Angle on Cogging Torque

Considering the spatial constraints on the motor stator teeth, cutting auxiliary slots would compromise their mechanical strength. Therefore, the position angle β of the auxiliary slots was set within the range of 3° to 8°, with the cogging torque calculated at 1° intervals. The variation trend of the cogging torque amplitude with β is shown in Figure 5. It can be observed that as β increases from 3° to 4°, the cogging torque amplitude decreases correspondingly, reaching a minimum value of 0.92 N·m at β = 4°. Subsequently, as β continues to increase, the cogging torque amplitude gradually rises, peaking at 7°, before declining again.

4.1.3. Influence of Auxiliary Slot Width on Cogging Torque

The auxiliary slot depth w was varied from 0.8 mm to 4 mm in 0.4 mm increments to calculate the cogging torque of the motor. The variation trend of the cogging torque amplitude with respect to w is shown in Figure 6. It can be observed that when w is within the range of 0.8 mm to 2 mm, the amplitude of cogging torque shows no significant change, reaching a minimum value of 0.6 N·m at w = 1.2 mm. As w continues to increase beyond 2 mm, the amplitude of cogging torque rises markedly.

4.1.4. Influence of Auxiliary Slot Depth on Cogging Torque

The range of auxiliary slot depth d is set from 0.1 mm to 1 mm, with cogging torque calculated at 0.1 mm intervals. The variation trend of cogging torque amplitude with d is shown in Figure 7. It can be observed that as d gradually increases from 0.1 mm, the cogging torque amplitude correspondingly decreases, reaching a minimum value of 0.34 N·m at d = 0.2 mm. When d continues to increase beyond 0.2 mm, the cogging torque amplitude then rises significantly.

Based on the results of the parametric analysis of the auxiliary slot position angle β, slot depth d, and slot width w, the values corresponding to the minimum cogging torque amplitude for each parameter were selected. This determines a set of structural parameter combinations for the motor stator auxiliary slots, namely Combination 1: (β = 4°, w = 1.2 mm, d = 0.2 mm). It should be noted that Combination 1 is solely based on results obtained through single-parameter scanning and does not account for potential coupling effects on cogging torque arising from interactions among multiple parameters.

4.2. System Analysis and Preliminary Optimization of Multi-Parameter Interactions

Considering both manufacturing feasibility and the mechanical strength of the stator teeth, the auxiliary slot parameters were set as follows: Number of auxiliary slots: 2; Slot position angle β: 3° to 8°, with 1° increments; Slot width w: 0.8 mm to 4 mm, with 0.4 mm increments; Slot depth d: 0.1 mm to 1 mm, with 0.1 mm increments This generated a total of 540 parameter combinations. Leveraging the parametric sweep capability of Maxwell, batch finite element calculations were performed for all combinations. Through systematic analysis of these 540 sets of simulation data, the patterns and trends of cogging torque amplitude variation with β, w, and d were obtained, as shown in Figure 8.

Parameter combinations yielding cogging torque amplitudes below 0.8 N·m are primarily concentrated within the range of slot position angles 3–6°, slot widths 3–4 mm, and slot depths 0.2–0.4 mm. Simulation results indicate that when the auxiliary slot parameters are set as follows: (β = 4°, w = 3.6 mm, d = 0.3 mm), the cogging torque amplitude reaches its minimum. This is designated as Combination 2.

4.3. Multi-Parameter Cooperative Optimization Based on Response Surface Methodology

4.3.1. Determination of Response Surface Experiment Factors

(1)

Experimental Factors

To effectively suppress the cogging torque of the motor, this study adopts the approach of opening rectangular auxiliary slots in the stator teeth and selects the auxiliary slot position angle β, slot width w, and slot depth d as the optimization design variables. This selection is based on the following considerations:

These three geometric parameters are the most fundamental elements defining the shape and position of the auxiliary slots, together constituting its complete geometric definition, each being indispensable. They directly affect the distribution path of magnetic flux lines in the teeth and the local magnetic reluctance: the β determines the relative phase between the slot and the main magnetic field, influencing the cancellation effect of torque harmonics; The w and d collectively determine the cross-sectional area of the slot, controlling the intensity of magnetic flux diversion and magnetic field modulation.

Adjusting any one of these parameters will directly and significantly alter the amplitude and waveform of the cogging torque. Therefore, it is essential to systematically and collaboratively optimize β, w, and d to fully exploit the potential of auxiliary slots in suppressing cogging torque.

In addition to the aforementioned three parameters, factors such as the shape of the auxiliary slot and the chamfer angle of the slot edges may also influence cogging torque and noise. However, this study focuses on optimizing the basic dimensions of rectangular slots. Future research may build upon this foundation to conduct further optimization.

(2)

Experimental Center Point and Ranges

As indicated in Section 4.2, the results of the global parameter scan show that parameter combinations with cogging torque amplitudes below 0.8 N·m are primarily concentrated in the following region: β ranging from 3° to 6°, w from 3 to 4 mm, and d from 0.2 to 0.4 mm. To achieve finer parameter optimization, this study adopts the following strategy:

First, the point with the lowest cogging torque within this region (β = 4°, w = 3.6 mm, d = 0.3 mm) is selected as the center point for the response surface design. This ensures that the optimization starts from a region of high performance.

Subsequently, based on the principles of the Box-Behnken experimental design, three key influencing factors—auxiliary slot position angle (X₁), auxiliary slot width (X₂), and auxiliary slot depth (X₃)—were set at three levels within their adjacent ranges as shown in

Table 4. Response Surface Design Factors and Levels.

Factor	Level
	−1	0	1
Auxiliary Slot Position Angle β/(°)	3	4	5
Auxiliary slot width w/mm	3.2	3.6	4
Auxiliary slot depth d/mm	0.2	0.3	0.4

. This design not only covers the high-performance parameter regions identified in the preliminary screening but also accurately captures the parameter interaction effects within the local range. As a result, it ensures both the predictive accuracy and optimization effectiveness of the model while significantly improving design efficiency.

4.3.2. Selection of the Optimization Objective for Response Surface Methodology

In this study, the suppression of cogging torque is selected as the primary optimization objective, mainly based on the following three considerations:

Cogging torque is the direct source of electromagnetic vibration. In highly integrated electromechanical systems such as motor-pumps, the periodic torque fluctuations caused by cogging torque are directly transmitted to the hydraulic pump body through rigid connection structures, becoming the main excitation source for system vibration and noise. At the electromagnetic design stage, directly suppressing this excitation source is more efficient compared to mitigating the final noise through structural or acoustic means.

There exists a strong correlation, albeit nonlinear, between cogging torque and noise. In the field of motor design and NVH optimization, using key electromagnetic indicators such as cogging torque and radial electromagnetic force as proxy objectives for noise is a widely validated engineering practice.

The “segmented optimization-systematic verification” strategy adopted in this study: After determining the optimal electromagnetic parameters, a systematic verification is conducted to assess the actual improvement effects of the proposed solution on radial electromagnetic force harmonics, structural vibration, and final sound pressure level, thereby ensuring the practical contribution of the optimization results to the system’s NVH objectives.

If sound pressure level were directly used as an optimization objective, a more complex model would need to be established, significantly increasing computational costs. Moreover, the reliability of such a model would heavily depend on the accuracy and sample size of multi-physics simulations. In future work, we will consider simultaneously using cogging torque and sound pressure level as objective functions to provide more comprehensive parameter design guidance for low-noise motor-pump systems.

4.3.3. Regression Model Building and Significance Testing

Based on the response surface design scheme outlined in the table, a total of 17 experimental points were conducted, including 5 central points. The response surface design and results are presented in

Table 5. Box-Behnken Response Surface Design and Results Analysis.

Design Point	β/(°)	w/(mm)	d/(mm)	Cogging Torque Amplitude
1	3	3.2	0.3	1.448173467
2	5	3.2	0.3	0.985552974
3	3	4	0.3	1.806316154
4	5	4	0.3	1.465824537
5	3	3.6	0.2	1.484889146
6	5	3.6	0.2	0.514881858
7	3	3.6	0.4	1.845997937
8	5	3.6	0.4	1.949926025
9	4	3.2	0.2	0.352774102
10	4	4	0.2	0.202377387
11	4	3.2	0.4	0.525525105
12	4	4	0.4	0.779341669
13	4	3.6	0.3	0.189880401
14	4	3.6	0.3	0.189880401
15	4	3.6	0.3	0.189880401
16	4	3.6	0.3	0.189880401
17	4	3.6	0.3	0.189880401

.

Using Design-Expert 13.0 software, a second-order multiple regression analysis was performed on the data in the table, yielding the regression model equation shown below. A significance analysis was conducted on the model, with the results shown in

Table 6. Significance Analysis.

Source	Sum of Squares	df	Mean Square	F-Value	p-Value	Significance
Model	7.13688	9	0.792987	27.09059	0.000125	**
β	0.348275	1	0.348275	11.89802	0.010701	*
w	0.110881	1	0.110881	3.788012	0.092674
d	0.810181	1	0.810181	27.67798	0.001172	**
βw	0.003729	1	0.003729	0.127388	0.731673
βd	0.288334	1	0.288334	9.850286	0.016413	*
wd	0.040847	1	0.040847	1.395448	0.276061
β2	5.190153	1	5.190153	177.3098	3.15E-06	**
w2	0.067201	1	0.067201	2.295761	0.173503
d2	0.093215	1	0.093215	3.184488	0.117517
Residual	0.204902	7	0.029272
Lack of Fit	0.204902	3	0.068301
Pure Error	0	4	0
Cor Total	7.341782	16

Note: * Significant (p < 0.05), ** Highly significant (p < 0.01).

.

$$\begin{array}{c}Y=35.3958-10.1709X_1-6.4539X_2-25.5793X_3+0.0763X_1{X}_2+2.6848X_1{X}_3+2.5263X_2{X}_3\hfill \\ \hspace{1em}+1.1103X_1^2+0.7896X_2^2+14.8791X_3^2\hfill \end{array}$$

(9)

The p-value of the regression model is less than 0.01, confirming that the model is highly statistically significant. Lack of fit is not paramount as it is desired [17]. The model’s goodness of fit is evaluated using the coefficient of determination, where R² = 0.9721 and the adjusted R²adj = 0.9362. This indicates that the regression equation adequately explains the variation in cogging torque, demonstrating excellent fitting performance.

The F-values for each factor in the regression model indicate the strength of influence on the cogging torque amplitude. A higher F-value signifies a stronger impact. Therefore, the order of influence of individual factors on the cogging torque amplitude is: auxiliary slot depth (d) > auxiliary slot position angle (β) > auxiliary slot width (w).

4.3.4. Response Surface Interaction Analysis

Response surfaces were generated to analyze the effects of slot position angle (X₁), slot width (X₂), and slot depth (X₃) on the amplitude of the cogging torque. The results are shown in Figure 9.

The steeper the slope of the response surface, the greater the influence of that factor on the amplitude of the cogging torque. Within the test range, the interaction between the slot position angle and slot depth exerts a more significant effect on the cogging torque.

Using Design-Expert 13.0 software to solve the above quadratic regression equation, the optimal auxiliary slot combination for minimizing the cogging torque amplitude is determined as: (β = 4.24098°, w = 3.64939 mm, d = 0.206267 mm). This combination is designated as Combination 3, with the software predicting a cogging torque amplitude of 0.0185 N·m.

Simulation verification of this parameter combination using Maxwell software yielded yielding a cogging torque amplitude of 0.03 N·m. The predicted amplitude closely matches the simulated result, indicating the model’s strong predictive capability and reliability. This confirms the feasibility of the established cogging torque amplitude regression model.

4.3.5. Computational Burden and Efficiency Analysis

(1)

Computational Burden of the Proposed Method

The proposed method consists of two stages: Stage 1: Multi-Parameter Scanning. Three key parameters—slot position angle β, slot width w, and slot depth d—were discretized within reasonable ranges, generating a total of 540 parameter combinations. Stage 2: Response surface optimization based on the Box-Behnken design, three factors and three levels were selected to construct a response surface model, requiring 17 simulations.

In total, 557 sets of finite element simulations were performed using batch processing in ANSYS Maxwell. Each simulation took approximately 2 min. By utilizing the software’s parallel computing capability on the workstation used in this study (supporting up to 8 concurrent simulation tasks), the tasks were executed in parallel, effectively reducing the overall computational cycle. Statistically, the total time required to complete all simulations was approximately 2 h.

(2)

Comparison with Benchmark Search Strategies

To validate the efficiency of the proposed method, it was compared with two traditional strategies:

Single-Parameter Scanning Method: Parameters are optimized sequentially while ignoring coupling effects among them. If each parameter is sampled at 10 levels, 30 simulations would be required, with a total time of about 10 min. However, this approach is prone to falling into local optima.

Full Factorial Search Method: If each of the three parameters is sampled at 10 levels, 1000 simulations would be needed, with a total time of about 4 h. Although theoretically exhaustive, this method incurs significantly higher computational costs than the proposed approach and lacks the refined modeling and precise optimization capabilities offered by the response surface methodology.

The proposed method first roughly locates the optimal region through multi-parameter scanning and then performs precise optimization using a response surface model. This approach not only ensures global search capability but also significantly reduces computational burden. Compared with the full factorial search, the computation time is reduced by approximately 50%, while avoiding the local optimum issue commonly associated with traditional single-parameter optimization methods

4.4. Comparison of Results from Different Optimization Schemes

4.4.1. Comparison of Cogging Torque Results

Based on the analysis results above, comparing the cogging torque waveforms of the three optimized schemes with the motor without auxiliary slots reveals: As shown in Figure 10, without auxiliary slots, the cogging torque amplitude is 1.13 N·m. After adopting Combination 1 auxiliary slots, the amplitude decreases to 0.65 N·m, representing a reduction of 42.5%. With Combination 2 auxiliary slots, the cogging torque further decreased to 0.19 N·m, representing an 83.2% reduction; With Combination 3 auxiliary slots, the cogging torque amplitude decreased to 0.03 N·m, achieving an overall reduction of 97.3%.

The findings indicate that when employing stator slotting technology to suppress cogging torque, optimizing only a single or a few parameters often leads to local optima, making it difficult to achieve the system’s global optimum solution. To address this, this paper proposes a multi-parameter-response surface collaborative optimization strategy. By conducting multi-parameter global optimization to preliminarily locate the optimal solution region and constructing an accurate model based on the response surface method for precise solution, it effectively overcomes the limitations of traditional methods. This approach comprehensively accounts for complex parameter interactions, enabling accurate and efficient acquisition of the global optimum. It provides a reliable pathway for determining the optimal auxiliary slot parameters for cogging torque suppression.

Although Combination 3 theoretically achieves the minimum cogging torque amplitude of 0.03 N·m, the corresponding parameter values (β = 4.24098°, w = 3.64939 mm, d = 0.206267 mm) present substantial manufacturing challenges in practical engineering applications. Specifically:

Manufacturability constraints: The required precision exceeds the typical capabilities of conventional machining processes, imposing excessively high demands on equipment and process control.

Production cost: Such high-precision parameters are difficult to maintain consistently in mass production, resulting in low feasibility for industrialization.

Therefore, based on considerations of engineering feasibility and production economics, the parameters were rounded to values that are easy to machine and measure. The final combination is: β = 4.2°, w = 3.6 mm, d = 0.2 mm. Simulation verification shows that this solution still significantly reduces the cogging torque amplitude to 0.10 N·m (a reduction of 91.2%). With minimal performance loss (compared to Combination 3), this approach greatly enhances the manufacturability and overall cost-effectiveness of the design.

4.4.2. Comparative Analysis of Major Electromagnetic Properties

To analyze whether the stator auxiliary slots would affect other motor performance characteristics, key electromagnetic performance indicators of three configurations—without auxiliary slots, Combination 3 (theoretical optimum), and the final combination—were compared and analyzed under rated operating conditions. The simulation results are summarized in

Table 7. Motor Performance Comparison Table.

Performance Indicators	Without Auxiliary Slots	Combination 3	Final Combination
Average Torque Tavg/N·m	7.47	7.26	7.27
Torque Ripple Ratio/%	24	12	12
Phase Back EMF Amplitude/V	222.2	220.8	220.2
Losses/W	438.2	415.5	417.4

, and the relevant waveform comparisons are shown in Figure 11.

As shown in Table 7: Torque: Compared to the configuration without auxiliary slots, the average torque of both auxiliary slot structures decreased by approximately 0.2 N·m, representing a reduction of only about 2.8%, which is within an acceptable range. However, the torque ripple ratio decreased by 12% compared to the configuration without auxiliary slots, indicating that the auxiliary slots not only suppress no-load cogging torque but also contribute to smoother operation under load.

Back EMF: The amplitudes of the no-load back electromotive force (back EMF) are essentially consistent across all three configurations.

Losses: The total losses for Combination 3 and the final rounded combination are 415.5 W and 417.4 W, respectively, representing reductions of approximately 5.1% and 4.7% compared to the loss of 438.2 W for the configuration without auxiliary slots. This indicates an improvement in overall system efficiency.

After introducing stator auxiliary slots, the motor maintains its average torque and back EMF performance while reducing torque ripple and total system losses. This demonstrates that the auxiliary slots do not adversely affect the motor’s core performance. Instead, they achieve comprehensive performance optimization while preserving output capability. Furthermore, the final rounded combination exhibits highly consistent performance with Combination 3 across all key indicators, validating the rationality of the parameter rounding. Therefore, the final combination: β = 4.2°, w = 3.6 mm, d = 0.2 mm—not only fully inherits the benefits of the theoretical optimization but also offers greater manufacturability and practical application value.

4.4.3. Comparative Analysis with Pole Arc Coefficient Optimization Method

To comprehensively evaluate the superiority of the multi-parameter-response surface collaborative optimization method proposed in this paper, a comparative analysis was conducted using a widely applied cogging torque suppression method—pole arc coefficient optimization—based on the same 8-pole 12-slot BLDCM model. To ensure fairness and consistency in the comparison, all simulations were performed using the same Maxwell 2D motor model and adhered to the meshing strategy and mesh independence verification criteria described in Section 3.2.

In the pole arc coefficient optimization method, the pole arc coefficient α was varied from 0.7 to 1 with a step size of 0.01, resulting in a total of 31 parametric finite element simulations to identify the minimum cogging torque amplitude. The simulation results indicated that the minimum cogging torque amplitude of 0.613 N·m was achieved at α = 0.83, representing a 45.8% reduction compared to the unoptimized baseline value of 1.13 N·m. In contrast, with the final parameter combination proposed in this study, the cogging torque amplitude was further reduced to 0.10 N·m, achieving an overall reduction of 91.2%.

The pole arc coefficient optimization method, due to its single-variable nature, offers a clear advantage in computational efficiency, requiring approximately 10 min for simulations. However, its suppression effect on cogging torque is significantly lower than that achieved by the multi-parameter collaborative optimization method proposed in this study. This highlights the inherent limitations of single-parameter optimization when addressing complex electromagnetic–structural coupling problems. Although the proposed method consumes more computational resources, requiring approximately 2 h for simulations, the two-stage strategy of “global parameter scanning for region identification + response surface modeling for precise optimization” systematically explores multiple design variables and their interactions, ultimately yielding a globally superior solution.

Furthermore, as indicated in Reference [8], rotor skewing, while capable of substantially reducing cogging torque, introduces issues such as increased process complexity, higher manufacturing costs, and the introduction of axial magnetic pull. Considering all factors, stator slotting technology demonstrates more significant effectiveness in suppressing cogging torque and offers greater engineering practicality and comprehensive advantages. Therefore, it is more suitable for adoption in practical design applications.

4.5. Robustness and Sensitivity Analysis

This study conducted optimization design and performance analysis based on finite element simulations. Considering that manufacturing tolerances, eccentricity, and material variability in actual production and assembly may affect motor performance, this section provides a preliminary discussion on related uncertainty factors. The discussion focuses on five key variables: slot position angle β, slot width w, slot depth d, rotor static eccentricity ε, and permanent magnet remanence Br. The impact of fluctuations in these variables within a certain range on motor performance is analyzed.

In Maxwell, variables such as β, w, d, ε, and Br were set as input variables, with maximum cogging torque defined as the output. Using the initial values β = 4.2°, w = 3.6 mm, d = 0.2 mm, ε = 0, and Br =1.09978, as the baseline, each parameter was independently varied within a ±5% range. Subsequently, a sensitivity analysis component was added in Workbench 2022 R2 via the optiSLang 2022 R2 module, employing an advanced Latin Hypercube Sampling method with a sample size of 200.

The sensitivity coefficients for the resulting cogging torque amplitude are shown in Figure 12. A positive value indicates a positive correlation between the parameter and the cogging torque, while a negative value indicates a negative correlation. The analysis reveals the following order of influence (from highest to lowest):slot depth d > rotor static eccentricity ε > slot position angle β > permanent magnet remanence Br > slot width w.

Figure 13 illustrates the influence of fluctuations in key parameters within a ±5% range on the distribution of cogging torque amplitude, with the horizontal axis representing the cogging torque amplitude and the vertical axis representing the probability density.

Under independent random variations of the aforementioned parameters, there is an 87% probability that the cogging torque amplitude remains below 0.2 N·m. This indicates that even when accounting for manufacturing tolerances, assembly eccentricities, and material parameter variations, the optimized auxiliary slot structure can maintain a relatively low cogging torque amplitude. Although experimental measurement data are not yet available in this study, the robustness analysis has preliminarily validated the engineering feasibility of the proposed optimization solution.

5. Analysis of Electromagnetic Vibration and Noise in the BLDCM

The electromagnetic excitation sources on the stator tooth surfaces primarily include radial electromagnetic forces and tangential electromagnetic forces [18]. Previous studies have generally regarded radial electromagnetic forces as the main source of electromagnetic vibration and noise [19,20]. Therefore, this study focuses on analyzing the radial electromagnetic force under conditions where the stator has an open auxiliary slot with the following parameters: β = 4.2°, w = 3.6 mm, and d = 0.2 mm. The objective is to provide a basis for reducing the motor’s sound pressure level.

5.1. Stator Structural Resonance Analysis

Stator slotting alters structural characteristics, necessitating modal simulation of the stator to determine whether resonance will occur. Resonance occurs when the natural frequency of a stator mode aligns with the time frequency of the corresponding radial electromagnetic force order [21]. This resonance phenomenon directly impacts the vibration and noise performance of motor-pumps.

5.1.1. Analysis of Radial Electromagnetic Force

The results of the 2D FFT decomposition of the radial electromagnetic force under load [22,23], obtained from the Maxwell transient analysis, are presented in Figure 14 shows the 2D-FFT results of radial electromagnetic force under load from transient analysis, with negative spatial orders indicating a force direction opposite to the motor’s rotation.

Figure 14 reveals that the electromagnetic force magnitudes are relatively large at even spatial orders such as 0, ±8, and ±16. The radial electromagnetic force reaches its maximum magnitude at the (0, 0) spatiotemporal order. Since the magnitude of the motor’s radial electromagnetic force is approximately inversely proportional to the fourth power of its order, lower-order electromagnetic forces exert a greater influence on motor vibration and noise.

However, the radial electromagnetic force of order (0, 0) does not cause vibration, whereas low-order harmonics such as (−4, 2), (4, 4), and (8, ±2) can induce resonance when their frequencies approach the natural frequencies of the corresponding structural modes in the motor.

5.1.2. Modal Analysis

Using Workbench finite element software to analyze the stator modes, the modal analysis results are shown in Figure 15. At the rated speed of 6000 rpm, the fundamental electrical frequency f1 of the radial electromagnetic force is 400 Hz.

The modal shapes and natural frequencies of the stator are analyzed:

(a) represents the breathing mode. It manifests as uniform expansion and contraction of the stator structure. This mode is not excited by electromagnetic forces and therefore contributes negligibly to noise.

(b) represents the second-order mode, with a natural frequency of approximately 2054 Hz. It exhibits an elliptical deformation with two nodal diameters. This mode can potentially be excited by electromagnetic forces of spatial order 2.

(c) represents the third-order mode, with a natural frequency of approximately 2789 Hz. It shows a triangular deformation with three nodal diameters. Theoretically, it can be excited by spatial electromagnetic force harmonics of order 3.

(d) represents the fourth-order mode, with a natural frequency of approximately 3539 Hz. It displays a quadrilateral deformation with four nodal diameters. The spatial order of this mode corresponds to the lowest non-zero order (fourth order) of the radial electromagnetic force. Nevertheless, as can be seen from Figure 14, the temporal frequencies of the fourth-order electromagnetic force are 800 Hz, 1600 Hz, 3200 Hz, 4000 Hz, etc., which differ significantly from the natural frequency of this mode (3539 Hz). Therefore, resonance between them does not occur, thereby avoiding intense vibration and noise at these frequencies.

5.2. Vibration and Noise Analysis

5.2.1. Vibration and Noise Analysis Under Rated Load Conditions

Following established practices in motor NVH analysis [24], a multi-physics finite element model integrating electromagnetic, structural, and acoustic domains was established. Within the Maxwell 2D electromagnetic simulation, the rotational speed is varied from 400 to 6000 rpm at intervals of 800 rpm, and the electromagnetic forces on each stator tooth are extracted under rated load conditions. These forces were then applied as the excitation source for the structural vibration analysis by mapping them onto the 3D mechanical model for harmonic response analysis.

Furthermore, a cylindrical acoustic field solution domain with a diameter of 1 m was established. The outer circular surface was defined as a radiation boundary condition, while the inner circular surface was in contact with the outer surface of the motor stator. The harmonic response analysis results were applied as boundary conditions within the acoustic solution domain to complete the acoustic field calculation.

Finally, the far-field sound pressure level (SPL) waterfall plot of the motor across different rotational speeds is obtained, as shown in Figure 16, while the sound pressure level comparison at the rated speed is presented in Figure 17.

As can be seen from Figure 16, with frequency on the horizontal axis and motor speed on the vertical axis, clearly illustrates the variation in the motor’s sound pressure level across different rotational speeds and frequencies. The peak sound pressure level of the motor occurs at 6000 rpm and 6400 Hz (corresponding to the 16th harmonic of the fundamental frequency), regardless of whether auxiliary slots are added to the stator. This observation aligns with theoretical predictions. Under this operating condition, the peak sound pressure level with auxiliary slots measures 51.547 dB, which is 9.8% lower than the 57.156 dB recorded without slots.

Figure 17 indicates that with the implementation of auxiliary slots in the stator, the motor’s sound pressure level increases at the 2nd, 4th, and 14th harmonics of the fundamental frequency, while it decreases at the 6th, 8th, 10th, 12th, and 16th harmonics. The peak sound pressure level for both configurations occurs at the 16th harmonic, measuring 48.587 dB without auxiliary slots and 46.117 dB with them, corresponding to a reduction of approximately 5.1%.

5.2.2. Vibration and Noise Analysis Under Different Load Conditions

To verify the robustness and applicability of the optimized stator auxiliary slot structure under different operating conditions, this study compares and analyzes the sound pressure level at the 16th harmonic frequency of the motor under rated load for both the configuration without auxiliary slots and the configuration with auxiliary slots. The results are presented in

Table 8. Sound pressure level under different operating conditions.

Structure	25% Rated Load	50% Rated Load	75% Rated Load	100% Rated Load
Without auxiliary slots	46.43	46.93	47.62	48.58
With auxiliary slots	44.98	45.51	45.68	46.11
change rate/%	3.1	3	4.1	5.1

Note: The unit is dB.

.

The results indicate that, although the sound pressure level slightly increases under electrostatic load conditions due to enhanced electromagnetic coupling, the optimized motor maintains effective noise reduction across all tested load scenarios.

5.2.3. Discussion on the Impact on NVH Performance

The results above demonstrate that optimizing cogging torque can effectively suppress the main radial electromagnetic force harmonics (such as the 8th and 16th orders), thereby reducing the overall sound pressure level. However, Figure 14 also indicates a slight increase in sound pressure level at minor harmonics, such as the 2nd, 4th, and 14th orders. This further illustrates that minimizing cogging torque does not equate to minimizing the sound pressure level across all frequency components, as there exists a complex and nonlinear transmission relationship between the two.

Nevertheless, due to the significant reduction in the main excitation sources, the overall noise performance has been markedly improved. This observation further suggests that if system-level NVH performance is the ultimate goal, using the sound pressure level itself or its key components as direct optimization targets may lead to more targeted and effective design outcomes.

The optimized stator auxiliary slot structure provides a key foundation for improving the overall NVH performance of the motor-pump system by significantly suppressing cogging torque and weakening the key-order radial electromagnetic forces. In the context of actual electric vehicle operating conditions, its mechanism of action is primarily manifested in two aspects:

Weakening of Vibration Transmission Path:

Cogging torque fluctuations are directly transmitted to the hydraulic pump and mounting frame through the rigid connection of the motor rotor and pump drive shaft. Reducing the amplitude of the cogging torque means a significant attenuation of the excitation intensity at the source. This directly lowers the excitation forces transmitted to the pump body and its connected structures, thereby effectively suppressing the structural vibration of the entire power unit. In electric vehicle electro-hydraulic braking or steering systems, this vibration suppression helps reduce the vibration and noise transmitted into the passenger cabin through the vehicle body structure, thus enhancing ride comfort.

Source Control of Electromagnetic Noise:

The optimized structure effectively suppresses key radial electromagnetic force harmonics, such as the 8th and 16th orders, thereby reducing the excited radial vibration modal response of the stator. This not only lowers the overall noise level but also specifically attenuates noise components at specific frequencies, which holds significant value for improving cabin quietness and ride comfort.

6. Conclusions

The multi-parameter and response surface collaborative optimization strategy proposed in this study effectively overcomes the limitation of traditional single- and dual-parameter optimization in stator auxiliary slot design, which is prone to falling into local optima. By integrating global parameter scanning with precise modeling, it successfully achieves the global optimal solution for auxiliary slot parameters. This method provides a systematic and reliable solution for suppressing cogging torque, offering a theoretical basis for the low-noise and high-performance design of motor-pumps. Furthermore, this method exhibits excellent generalizability and can be extended to the design and optimization of permanent magnet motors, including traction motors and various auxiliary motors in electric vehicles.

The stator auxiliary slot structure (β = 4.2°, w = 3.6 mm, d = 0.2 mm) obtained through this collaborative optimization method reduces the cogging torque amplitude of the motor by 91.2% while ensuring machining feasibility. Electromagnetic–mechanical–acoustic multiphysics coupling simulation analysis demonstrates that the optimized structure effectively suppresses the main radial electromagnetic force harmonics, leading to an approximately 5.1% reduction in the overall sound pressure level of the motor-pump system under rated operating conditions.

Based on the objective of minimizing cogging torque, this study systematically analyzes the vibration and noise response characteristics of the motor. The research reveals that minimizing cogging torque does not always directly correspond to the lowest sound pressure level, as a complex coupling relationship exists between the two. Therefore, future research could adopt the minimization of sound pressure level as the direct optimization objective, conducting an in-depth investigation into the influence patterns of auxiliary slot parameters on the noise of BLDCM. This would facilitate the development of a systematic parameter optimization framework for low-noise motor-pump systems and even electric vehicle motor systems.

Research Limitations and Future Work

The multi-parameter collaborative optimization method proposed in this study has demonstrated significant effects at the simulation level, yet it still has the following limitations, which need to be addressed in future work:

Lack of Experimental Validation: The conclusions of this study are based on finite element simulations and response surface models. Although mesh independence validation and robustness analysis were conducted, experimental test data for the prototype of this design are lacking. Therefore, the practical effectiveness of the proposed method requires further confirmation.

Indirect Nature of Optimization Objectives: This study focused on minimizing cogging torque as the primary optimization goal, with noise reduction as an indirect outcome. Although the overall sound pressure level was reduced, noise minimization was not directly targeted, suggesting potential for further noise reduction.

Based on the above limitations, future research will focus on the following directions:

Experimental Validation and Performance Confirmation: Prototypes of the stator with optimized parameters will be manufactured, and an NVH test platform for the motor-pump system will be established. By measuring cogging torque, vibration spectra, and noise sound pressure levels, the accuracy of the simulation model will be validated, and the engineering effectiveness of the optimization scheme will be confirmed.

Multi-objective collaborative optimization design: A multi-objective optimization model is constructed with cogging torque, radial electromagnetic force harmonics, and overall sound pressure level as the core objectives. Advanced algorithms are employed to seek the globally optimal balance between electromagnetic performance and NVH performance.