In the rapidly evolving landscape of electric vehicle technology, the drive system stands as a critical component, directly influencing the overall performance, efficiency, and durability of the vehicle. As a key element in the motor and transmission systems, drive bearings must withstand high-speed operations, frequent start-stop cycles, and complex vibrational environments, which are particularly pronounced in electric vehicles compared to traditional internal combustion engine vehicles. The growing adoption of electric vehicles, especially in markets like China EV, underscores the need for advanced optimization techniques to enhance bearing design. This research addresses these challenges by proposing a multi-strategy improved particle swarm optimization algorithm, integrating elements from particle swarm optimization and genetic algorithms to optimize bearing structural parameters for better performance in electric vehicle applications.
The drive bearings in electric vehicles are subjected to unique operational conditions, including significant temperature fluctuations and complex vibration spectra, which demand robust design solutions. Traditional single-objective optimization methods often fall short in handling the multi-faceted nature of these problems, where multiple parameters must be balanced to achieve optimal outcomes. For instance, in China EV markets, the push for higher efficiency and longer lifespan necessitates innovations in bearing technology. Our approach combines the strengths of multi-objective particle swarm optimization (MOPSO) and non-dominated sorting genetic algorithm (NSGA-II) to tackle these issues, incorporating penalty functions for constraint handling and dynamic velocity control to avoid local optima. This method not only improves the convergence and diversity of solutions but also ensures that the optimized bearings meet the rigorous demands of electric vehicle drive systems.
To elaborate, the multi-strategy algorithm begins by initializing a population and applying a penalty function to transform constrained optimization problems into unconstrained ones. The penalty function is defined as follows: $$ R(x) = f(x) + \sum_{i=1}^{n} a_i P_i(x) + \sum_{i=1}^{n} b_i Q_i(x) $$ where \( R(x) \) is the penalty function, \( f(x) \) is the original objective function, \( P_i(x) \) and \( Q_i(x) \) represent constraint conditions, and \( a_i \), \( b_i \) are penalty factors. As iterations progress, the difference between the penalty function and the objective function diminishes, guiding the solution toward the global optimum. This is particularly relevant for electric vehicle components, where constraints on dimensions and material properties must be strictly adhered to.
Velocity control is another crucial aspect of our algorithm, preventing premature convergence by regulating particle speeds. The velocity update formula is given by: $$ v_{jd}^n = w_j v_{jd}^{n-1} + \gamma (p_{\text{best},j}^{n-1} – x_{jd}^{n-1}) + \gamma (q_{\text{best}}^{n-1} – x_{jd}^{n-1}) $$ where \( v_{jd}^n \) denotes the velocity of particle \( j \) in dimension \( d \) at iteration \( n \), \( w_j \) is the inertia weight, \( p_{\text{best},j}^{n-1} \) and \( q_{\text{best}}^{n-1} \) are the personal and global best positions from the previous iteration, \( x_{jd}^{n-1} \) is the particle’s position, and \( \gamma \) is a convergence factor. To enforce stability, the velocity is clamped within thresholds: $$ v_{jd}^n = \begin{cases} v_{\min} & \text{if } v_{jd}^n < v_{\min} \\ v_{\max} & \text{if } v_{jd}^n > v_{\max} \\ v_{jd}^n & \text{otherwise} \end{cases} $$ where \( v_{\min} \) and \( v_{\max} \) are predefined limits. This dynamic control is essential for handling the high-speed demands of electric vehicle bearings, ensuring that the algorithm explores the solution space effectively without oscillating excessively.
Furthermore, the algorithm incorporates genetic operations such as crossover and mutation to enhance diversity and escape local optima. The crossover operation generates new individuals as follows: $$ c_1 = \alpha x_1 + (1 – \alpha) x_2, \quad c_2 = (1 – \alpha) x_1 + \alpha x_2 $$ where \( x_1 \) and \( x_2 \) are parent individuals, \( c_1 \) and \( c_2 \) are offspring, and \( \alpha \) is a random number controlling the distance between parents and offspring. The value of \( \alpha \) is determined by: $$ \alpha = \begin{cases} (2 \cdot \text{rand})^{1/(\eta+1)} & \text{if rand} \leq 0.5 \\ 1/(2 – 2 \cdot \text{rand})^{1/(\eta+1)} & \text{otherwise} \end{cases} $$ where \( \eta \) is the distribution index for the crossover operator, and rand is a random number in [0,1]. This combination of PSO and GA strategies allows for a more comprehensive search, which is vital for optimizing the complex parameters of electric vehicle drive bearings.
In applying this algorithm to cylindrical roller bearings commonly used in electric vehicles, we define the optimization objectives as maximizing the rated static load \( C_r \), maximizing the rated dynamic load \( C_{or} \), and minimizing the frictional heat generation rate \( H \). The design variables include parameters such as the curvature radii of the inner and outer rings, roller diameter, number of rollers, and pitch diameter. Constraints are imposed to ensure practical feasibility; for example, the pitch diameter \( D_p \) must satisfy: $$ D_p \geq \frac{D + d}{2} + h (D – d) $$ where \( D \) and \( d \) are the outer and inner diameters of the bearing, respectively, and \( h \) is the clearance coefficient set to 0.1. This constraint accounts for the thickness of the bearing rings, which is critical for maintaining structural integrity under the high loads typical in electric vehicle applications.
The number of rollers \( N \) is constrained by the assembly angle to prevent interference: $$ N \leq \frac{\pi}{\theta_{\max}} \cdot \frac{D_p}{D_g} $$ where \( \theta_{\max} \) is the maximum assembly angle and \( D_g \) is the roller diameter. Additionally, the roller diameter must fall within practical limits: $$ D_{\min} \leq D_g \leq D_{\max} $$ where \( D_{\min} \) and \( D_{\max} \) are empirical constants derived from typical bearing designs. These constraints ensure that the optimized bearing can be manufactured and assembled reliably, which is a key consideration for mass production in the electric vehicle industry, including China EV markets.
To handle the multi-objective nature of the problem, we employ the analytic hierarchy process (AHP) to assign weights to the objectives, facilitating a balanced evaluation of the Pareto front solutions. The overall evaluation function is defined as: $$ G = 0.49 A_1 + 0.33 A_2 – 0.18 A_3 $$ where \( A_1 \), \( A_2 \), and \( A_3 \) represent the normalized values for rated static load, rated dynamic load, and frictional heat generation rate, respectively. This weighted sum approach allows us to select the best compromise solution from the Pareto set, ensuring that the optimized bearing meets the multifaceted requirements of electric vehicle drive systems.
To validate the performance of our multi-strategy algorithm, we conducted comparative tests against other multi-objective optimization algorithms, including multi-ant colony optimization, non-dominated sorting genetic algorithm, multi-objective immune algorithm, and standard multi-objective particle swarm optimization. The algorithms were evaluated on benchmark test functions with a maximum of 200 iterations, a population size of 200, and velocity thresholds of [-1, 1]. Each algorithm was run 10 times to account for random variations, and the results were averaged for fairness.
| Algorithm | Metric | Hypervolume Indicator | Inverted Generational Distance | Spacing |
|---|---|---|---|---|
| Multi-Ant Colony Optimization | Mean | 0.7193 | 0.0011 | 0.0058 |
| Standard Deviation | 0.0012 | 0.0007 | 0.0022 | |
| Non-Dominated Sorting Genetic Algorithm | Mean | 0.7156 | 0.0008 | 0.0085 |
| Standard Deviation | 0.0025 | 0.0010 | 0.0035 | |
| Multi-Objective Immune Algorithm | Mean | 0.7178 | 0.0035 | 0.0093 |
| Standard Deviation | 0.0003 | 0.0011 | 0.0004 | |
| Multi-Objective Particle Swarm Optimization | Mean | 0.7169 | 0.0045 | 0.0053 |
| Standard Deviation | 0.0003 | 0.0007 | 0.0012 | |
| Proposed Multi-Strategy Algorithm | Mean | 0.7209 | 0.0001 | 0.0048 |
| Standard Deviation | 0.0002 | 0.0001 | 0.0002 |
As shown in Table 1, the proposed algorithm achieves a higher hypervolume indicator (0.7209 on average) compared to the others, indicating better convergence and diversity of solutions. The inverted generational distance and spacing metrics are also lower (0.0001 and 0.0048, respectively), demonstrating superior performance in terms of solution quality and distribution. These results highlight the effectiveness of our approach in handling multi-objective optimization problems, which is essential for developing reliable components for electric vehicles, particularly in the competitive China EV sector.
For the practical application, we focused on optimizing a NU202EM-type cylindrical roller bearing using finite element analysis in Abaqus. The bearing material was set as GCr15 steel, and the operating conditions simulated a speed of 15,000 rpm with varying radial loads. The optimized parameters, derived from the multi-strategy algorithm, included an inner ring curvature radius coefficient of 0.6255 mm, an outer ring curvature radius coefficient of 0.6482 mm, a roller diameter of 13.8662 mm, 12 rollers, and a pitch diameter of 51.5877 mm. These values were selected based on the highest overall evaluation score \( G \) from the Pareto front, ensuring a balance between load capacity and thermal performance.
The simulation results revealed significant improvements in bearing performance after optimization. For instance, the temperature rise under different radial loads was substantially reduced. Prior to optimization, the bearing temperature increased steadily with operation time, reaching high values under heavy loads. After optimization, the maximum temperature at a radial load of 20 kN dropped by 17.77%, from an initial 68.52°C to 56.37°C after 140 minutes of operation. Similarly, at 12 kN, the temperature decrease was 11.57%, enhancing the thermal stability crucial for electric vehicle applications where overheating can lead to efficiency losses and premature failure.
Contact stress and deformation analyses further validated the optimization outcomes. The contact stress, which escalates with increasing radial load, was reduced by 30.83% at 24 kN, from 4635.28 MPa to 3205.62 MPa. This reduction is critical for extending the fatigue life of the bearing, as lower stress levels minimize the risk of surface damage and pitting. The contact deformation also showed a notable decrease of 26.41% at the same load, from 42.15 μm to 31.02 μm, indicating improved structural integrity and reduced wear. These enhancements are particularly beneficial for electric vehicles, where high reliability and longevity are demanded in dynamic driving conditions.
The mathematical models used in the optimization process incorporate key performance indicators. The rated static load \( C_r \) for cylindrical roller bearings can be expressed as: $$ C_r = f_s \cdot Z \cdot D_g \cdot \cos(\alpha) $$ where \( f_s \) is a factor depending on the bearing geometry, \( Z \) is the number of rollers, \( D_g \) is the roller diameter, and \( \alpha \) is the contact angle. Similarly, the rated dynamic load \( C_{or} \) is given by: $$ C_{or} = f_c \cdot (Z \cdot D_g)^{2/3} $$ where \( f_c \) is a material and geometry-dependent constant. The frictional heat generation rate \( H \) is modeled as: $$ H = \mu \cdot F_r \cdot \omega \cdot r $$ where \( \mu \) is the friction coefficient, \( F_r \) is the radial load, \( \omega \) is the angular velocity, and \( r \) is the bearing radius. By optimizing these parameters simultaneously, our algorithm ensures a holistic improvement in bearing performance, addressing the core challenges in electric vehicle drive systems.
In conclusion, the multi-strategy improved particle swarm optimization algorithm presented in this research offers a robust framework for optimizing drive bearings in electric vehicles. By integrating MOPSO and NSGA-II with penalty functions and dynamic velocity control, the algorithm achieves superior convergence and diversity in solving multi-objective problems. The application to cylindrical roller bearings results in significant reductions in temperature, contact stress, and deformation, thereby enhancing fatigue life and operational efficiency. This approach is especially relevant for the advancing electric vehicle industry, including China EV, where innovation in component design is pivotal for market competitiveness. Future work will involve experimental validation under real-world conditions and extension to other bearing types to further solidify the algorithm’s applicability.
The implications of this research extend beyond immediate performance gains; it contributes to the sustainable development of electric vehicles by improving energy efficiency and reducing maintenance needs. As the demand for electric vehicles grows globally, optimized components like drive bearings will play a crucial role in meeting environmental and economic goals. The continuous refinement of such algorithms will undoubtedly support the evolution of electric vehicle technology, making it more accessible and reliable for consumers worldwide.