In recent years, electric vehicles have attracted significant attention due to their advantages in terms of energy conservation and environmental protection. However, compared to traditional fuel vehicles, the components of the electric vehicle transmission system often operate under dynamic loads of high frequency, strong impact, and long cycles, which makes the transmission gears more prone to contact fatigue failure. Therefore, accurately calculating the dynamic load of high-speed helical gears in electric vehicles under actual working conditions and obtaining their dynamic load spectra is crucial for the fatigue life prediction and reliability analysis of the electric vehicle transmission system.
Introduction to Electric Vehicle Transmission Systems
Electric vehicles share some similarities with traditional fuel vehicles in terms of body structure, but there are notable differences in the structure and operating environment of the power transmission system. The transmission system of electric vehicles typically omits the torque converter, clutch, and other torsional damping components, resulting in an underdamped system. Additionally, the transmission system adopts a multi-stage reduction and fewer gear positions, with a shorter power transmission path and a significant increase in the number of cycles.
The new characteristics of the electric vehicle transmission system bring about new theoretical and technical issues, and the prediction of the transmission system’s lifespan and reliability is a bottleneck that restricts its further performance improvement. This article focuses on a high-speed helical gear in a fixed transmission ratio electric vehicle, establishes a control model for the vehicle’s permanent magnet synchronous motor, and simulates the model based on the cyclic driving conditions (UDDS) to obtain the dynamic output torque of the motor. The dynamic torque of the motor is used as the driving torque of the gear to obtain the contact stress spectrum of the helical gear pair under the cyclic conditions. The rainflow counting method is used to count the gear contact stress spectrum and obtain the relationship between the contact stress amplitude and frequency of the high-speed helical gear under the cyclic conditions.
Transmission System Structure and Transmission Loading Analysis
The basic structure of a fixed transmission ratio system for an electric vehicle is shown in Figure 1. The motor has a large starting torque, can achieve a constant torque at low speeds and a constant power at high speeds, and is easy to achieve stepless speed regulation. To analyze the load on the transmission, it is necessary to study the driving force and driving resistance of the car.
During the driving of the car, the wheels are subjected to the torque transmitted by the engine, and the torque exerts a force on the ground. The ground, in turn, exerts a driving force Ft on the wheels. The relationship between Ft and the input torque of the gearbox is given by Equation (1): Ft = Ttq * ig * i0 * η / r, where Ttq represents the input torque of the gearbox, ig represents the transmission ratio, i0 represents the main reduction ratio, η represents the efficiency of the transmission system, and r represents the wheel radius.
The resistance encountered by the car during driving is divided into four parts: rolling resistance, air resistance, acceleration resistance, and gradient resistance, which constitute the driving resistance of the car, as shown in Equation (2): ∑Fk = Ff + Fw + Fi + Fj (k = f, w, i, j), where Ff represents the rolling resistance, Fw represents the air resistance, Fi represents the gradient resistance, and Fj represents the acceleration resistance. The specific expressions can be found in the textbook on automotive theory.
Therefore, the driving equation of the car is as follows: Ft = ∑Fk (k = f, w, i, j), as shown in Equation (3). From the driving equation, the load torque TL of the motor can be calculated as follows: TL = ∑Fk / (ig * i0 * η), as shown in Equation (4).
Here is a summary table of the transmission system structure and loading analysis:
| Parameter | Equation | Description |
|---|---|---|
| Driving Force | Ft = Ttq * ig * i0 * η / r | Relationship between the driving force and the input torque of the gearbox |
| Driving Resistance | ∑Fk = Ff + Fw + Fi + Fj (k = f, w, i, j) | Composition of the driving resistance |
| Car Driving Equation | Ft = ∑Fk (k = f, w, i, j) | Equation representing the balance between the driving force and the driving resistance |
| Motor Load Torque | TL = ∑Fk / (ig * i0 * η) | Calculation of the load torque on the motor based on the driving resistance |
Vehicle Motor Model and Simulation
The mathematical model of the permanent magnet synchronous motor (PMSM) in the d – q axis rotating coordinate system is shown in Figure 2. When the PMSM adopts the vector control (id = 0) strategy, the steady-state voltage equations are: ud = -ωe * Lq * iq and uq = Rs * iq + ωe * ψf, where ud and uq represent the d – axis and q – axis stator voltages, respectively, ωe represents the electrical angular velocity of the motor, Lq represents the q – axis inductance, iq represents the q – axis stator current, Rs represents the stator resistance, and ψf represents the flux linkage of the permanent magnet.
In the vector space, the magnetic field of the permanent magnet is orthogonal to the magnetic potential of the stator, and the transverse current component that generates the torque and the current component that generates the magnetic flux do not affect each other, thereby achieving the decoupling of the torque and the magnetic flux. At this time, the torque equation is Te = 3/2 * np * ψf * iq, where Te represents the electromagnetic torque and np represents the number of pole pairs.
Based on the vector control principle of the PMSM, the schematic diagram of the vector control system for the PMSM is constructed, as shown in Figure 3.
The simulation model of the PMSM is built using MATLAB / Simulink based on the mathematical model and control strategy. The PMSM module in the Simulink toolbox is already encapsulated and can be used directly after setting the parameters. The control strategy is built according to Figure 3 and the mathematical model of the PMSM. The simulation the urban driving cycle (UDDS) developed by the US Environmental Protection Agency (EPA). The theoretical load of the PMSM is calculated according to Equation (4). The dynamic output torque curve of the motor is shown in Figure 4. It can be seen that the actual output torque of the motor has high-frequency fluctuations and strong impacts.
Here is a summary table of the vehicle motor model and simulation:
| Parameter | Description |
|---|---|
| PMSM Mathematical Model | Equations and parameters of the PMSM in the d – q axis rotating coordinate system, including voltage equations and torque equation |
| Vector Control Strategy | Description of the vector control (id = 0) strategy, including the steady-state voltage equations and the torque equation |
| Vector Control System Schematic | Diagram of the vector control system for the PMSM, including the components and their connections |
| Simulation Model | Description of the simulation model built using MATLAB / Simulink, including the use of the PMSM module, the selection of the simulation 工况 (UDDS), and the calculation of the motor’s load |
Load History Calculation and Cycle Counting
The contact stress of the transmission gear is calculated by treating the helical gear as an equivalent spur gear. The maximum contact stress σH on the tooth surface of the helical gear occurs on the pinion. Using the dynamic output torque of the permanent magnet synchronous motor as the driving torque of the high-speed helical gear, the formula for calculating the contact stress of the spur gear is used, and the formula is as follows: σH = ZE * ZH * Ze * Zβ * √(2 * K * T1 / (b * d1²) * (u + 1) / u), where K represents the load coefficient, T1 represents the nominal torque of the pinion, b represents the meshing tooth width, d1 represents the pitch diameter of the pinion, u represents the gear ratio (u = z2 / z1, where z2 is the number of teeth of the large gear and z1 is the number of teeth of the pinion), ZE represents the elastic coefficient, ZH represents the node area coefficient, Ze represents the coincidence degree coefficient (Ze = 1), and Zβ represents the helix angle coefficient (Zβ = √(cosβ)).
The contact stress of the driving gear is calculated to obtain the contact stress spectrum of the helical gear under the cyclic conditions, as shown in Figure 5.
The rainflow counting method is used to count the cycles of the gear contact stress spectrum. The mean – frequency relationship of the contact stress amplitude of the high-speed helical gear of the electric vehicle transmission counted by the rainflow counting method is shown in Figure 6. The statistical analysis and K – S hypothesis test of the counting results show that the load mean follows a normal distribution, and the load amplitude follows a Weibull distribution. The mean and standard deviation of the mean distribution of the gear contact stress are 508 MPa and 82.3 MPa, respectively.
Here is a summary table of the load history calculation and cycle counting:
| Parameter | Description |
|---|---|
| Gear Contact Stress Calculation | Formula and parameters for calculating the contact stress of the gear, treating the helical gear as an equivalent spur gear |
| Contact Stress Spectrum | The contact stress spectrum of the helical gear obtained by calculating the contact stress of the driving gear |
| Rainflow Counting Method | Introduction to the rainflow counting method and its application in counting the gear contact stress spectrum |
| Contact Stress Amplitude Mean – Frequency Relationship | The mean – frequency relationship of the contact stress amplitude of the high-speed helical gear obtained by the rainflow counting method |
Conclusion
Based on the dynamic control model of the vehicle permanent magnet synchronous motor and the cyclic driving conditions, this article uses computer simulation to obtain the contact stress spectrum of the gear under the cyclic conditions. By using the rainflow counting method for counting and statistical analysis, a more accurate distribution law of the dynamic contact stress of the gear is obtained, which overcomes the defect of low calculation accuracy caused by assuming the load distribution law based on experience and improves the accuracy of the calculation of the system reliability. This provides a basis for further establishing the dynamic reliability model of the electric vehicle transmission system.
In the future, further research can be conducted in the following aspects: exploring more accurate models and algorithms for calculating the dynamic load of gears, considering the influence of factors such as temperature, lubrication, and material properties on the fatigue life of gears, and conducting more in-depth experimental studies to validate the theoretical predictions. This will help to further improve the reliability and performance of the electric vehicle transmission system.
Overall, the research on the dynamic load spectrum of high-speed helical gears in electric vehicle transmission systems is of great significance for the development and optimization of electric vehicles. By accurately understanding the load characteristics of the gears, we can better predict their fatigue life and improve the reliability of the transmission system, which will contribute to the wider application and promotion of electric vehicles.