The transmission shaft is a critical component connecting the powertrain to the drive axle in a battery EV car, responsible for transmitting driving torque. However, during rapid acceleration or high-speed operation, the transmission shaft may enter an overspeed rotation state, where its rotational speed exceeds the designed safe operating range. This condition can induce severe low-frequency vibrations, leading to accelerated component wear, driving instability, and even safety hazards. A significant challenge during rapid acceleration in a battery EV car is the phenomenon of acceleration torque loss, which is intrinsically linked to the dynamic excitations generated by the shaft under overspeed conditions. This article addresses this issue by proposing a novel control strategy based on a reference torque framework to mitigate overspeed rotation and its detrimental effects.
The core of the problem lies in the coupling relationship between the shaft’s structural vibrations and the transmitted torque. When a battery EV car accelerates sharply, the driveline experiences transient loads that can excite the natural frequencies of the transmission shaft. If the excitation frequency, particularly from second-order engine or motor excitations, coincides with these natural frequencies, it can lead to resonance, manifesting as pronounced low-frequency vibration and a measurable drop in usable acceleration torque—the so-called torque loss. Therefore, accurately modeling this dynamic coupling and preemptively identifying the onset of overspeed conditions are paramount for effective control.
Dynamic Characteristics and Finite Element Modeling of the Transmission Shaft
The first step involves a precise understanding of the transmission shaft’s dynamic behavior. A finite element model is constructed using ABAQUS software, incorporating the precise geometry, material properties (such as Young’s modulus, Poisson’s ratio, and density), and boundary conditions of a typical battery EV car transmission shaft. The model is constrained appropriately to simulate real mounting conditions. Modal analysis is then performed to extract the shaft’s natural frequencies and mode shapes, which are fundamental to identifying potential resonance points.
The low-frequency vibration of the transmission shaft in a battery EV car can be described by the differential equation of motion:
$$ \mathbf{Q}\ddot{\mathbf{x}} + \mathbf{C}\dot{\mathbf{x}} + \mathbf{T}\mathbf{x} = \mathbf{A}(t) $$
Where:
- $\mathbf{x}$ is the displacement vector.
- $\dot{\mathbf{x}}$ and $\ddot{\mathbf{x}}$ are the velocity and acceleration vectors, respectively.
- $\mathbf{Q}$ is the mass matrix.
- $\mathbf{C}$ is the damping matrix.
- $\mathbf{T}$ is the stiffness matrix related to torque and system stiffness.
- $\mathbf{A}(t)$ is the excitation force vector, often related to motor or gear meshing forces.
For a battery EV car driveline, significant excitation often comes from the second-order component of the motor/engine force. Ignoring damping for natural frequency calculation, the free-vibration equation is $\mathbf{Q}\ddot{\mathbf{x}} + \mathbf{T}\mathbf{x} = 0$. Solving this eigenvalue problem yields the natural frequencies $\xi_i$. For a shaft with symmetric joints, one complete revolution produces two excitation cycles, leading to a prominent 2nd-order excitation. The coupling relationship between this 2nd-order low-frequency vibration and the acceleration torque can be derived, considering the shaft’s critical speed $v_l$. The critical speed is calculated as:
$$ v_l = 1.2 \times 10^8 \frac{\sqrt{R^2 + r^2}}{t^2} $$
Where $R$ is the outer radius, $r$ is the inner radius of the tubular shaft, and $t$ is a time constant related to the system. The coupling constraint under 2nd-order excitation is expressed as:
$$ (\mathbf{T} – \xi^2 \mathbf{Q}) \mathbf{x} = \mathbf{Q}\ddot{\mathbf{x}} + \mathbf{C}\dot{\mathbf{x}} \cdot \frac{z}{v_l} $$
Here, $z$ is a safety factor defined as the ratio of the critical speed $v_l$ to the maximum allowable speed $v_{max}$ for the battery EV car transmission shaft. This equation encapsulates the dynamic interaction that leads to torque loss during aggressive acceleration in a battery EV car.
Intelligent Overspeed Condition Identification Using a Convolutional Recurrent Neural Network (CRNN)
Recognizing the onset of the hazardous coupling state is crucial for timely intervention. A Convolutional Recurrent Neural Network (CRNN) is employed for this purpose. This hybrid network architecture is ideally suited for processing the time-series vibration data from the battery EV car’s transmission shaft. The vibration characteristics (acceleration spectra, time-domain signals) are used as input features. The network consists of three main components:
- Convolutional Layers: Multiple one-dimensional convolutional layers extract local and hierarchical spatial patterns from the vibration signal segments. The operation for layer $u$ is:
$$ E^u_o = g\left( \sum_{o \in M^{u-1}} E^{u-1}_o * \omega^u_o + n^u_o \right) $$
Where $E$ is the feature map, $*$ denotes the convolution operation, $\omega$ are the filter weights, $n$ is the bias, and $g$ is the activation function (e.g., ReLU). - Recurrent Layers (LSTM): The extracted spatial features are fed into Long Short-Term Memory (LSTM) cells. These cells capture temporal dependencies and long-term patterns in the sequential data, which is essential for identifying trends leading to overspeed. The LSTM cell’s hidden state $j(t)$ at time $t$ is updated as:
$$ j(t) = g[\omega x(t) + j(t-1) + n] $$
Where $x(t)$ is the input at time $t$. - Output Layers: A fully connected layer integrates the high-level features from the LSTM. To optimize for deployment in a battery EV car’s real-time controller, this layer undergoes pruning (removing insignificant weights) and quantization (converting weights to lower precision). Finally, a Softmax layer classifies the shaft’s state into “Normal” (0) or “Overspeed/Coupling Condition” (1):
$$ U = \text{Softmax}(\mathbf{E}_q) $$
Where $\mathbf{E}_q$ is the output of the quantized fully connected layer.
The CRNN is trained on historical and simulated data from the battery EV car driveline, learning to accurately identify the precursor patterns to overspeed and torque-loss conditions.
Reference Torque-Based Control Strategy with Fuzzy Logic Compensation
When the CRNN identifies an impending or ongoing overspeed coupling condition, the control system is activated. The foundation of this control is the Reference Torque, which serves as a stable benchmark for desired operation. For a battery EV car, the driver’s demand torque $T_w$ is derived from the accelerator pedal position:
$$ T_w = T_{min} + X_1 (T_{max} – T_{min}) – T_b $$
Where $T_{min}$ and $T_{max}$ are the minimum and maximum achievable torques for the current vehicle speed, $X_1$ is the pedal position ratio, and $T_b$ is any braking torque. The maximum available torque $T_{max}$ is constrained by tire-road adhesion, varying for dry, wet, and icy conditions for the battery EV car:
$$ T_{max} = \begin{cases}
T_1 & (T_2 \leq T_3) \\
\frac{T_p}{9550} \cdot \frac{n}{T_2} & (T_2 > T_3)
\end{cases} $$
Here, $T_1$, $T_2$, $T_3$ represent friction limits on different surfaces, $T_p$ is the peak motor power, and $n$ is motor speed. The reference torque is selected from the motor’s external characteristic curve, which defines constant-torque and constant-power regions.
A fuzzy logic controller is designed to compute the necessary compensation torque. It uses two inputs:
- Accelerator Pedal Change Rate ($\dot{k}_a$): $\dot{k}_a = k_a(t) – k_a(t-1)$.
- Motor/Transmission Shaft Speed.
The output is the Compensation Torque. To ensure driving comfort and mechanical safety in the battery EV car, the rate of change of drive torque is limited by the vehicle’s jerk $J$:
$$ J = \frac{d a_0}{d t} \approx \frac{q r_1}{i \eta} \frac{d M_q}{dt} $$
Where $a_0$ is vehicle acceleration, $q$ is vehicle mass, $r_1$ is wheel radius, $i$ is total gear ratio, $\eta$ is efficiency, and $M_q$ is drive torque. Limiting jerk ensures smooth operation:
$$ \frac{d M_q}{dt} \leq 10 \cdot \frac{q r_1}{i \eta} $$
The fuzzy sets for inputs and output are defined as {NQ (Negative Large), ND (Negative Small), ZP (Zero), PD (Positive Small), PQ (Positive Large)}. Membership functions are triangular or trapezoidal. A rule base is constructed, for example: “IF Pedal Change Rate is PQ AND Motor Speed is PQ, THEN Compensation Torque is NQ” to reduce torque and curb overspeed in the battery EV car. The de-fuzzified output provides the final compensation torque value, which is applied to counter the torque loss and stabilize the shaft rotation.
Simulation Setup and Comparative Analysis
The proposed integrated methodology—finite element analysis, CRNN identification, and fuzzy compensation control—was simulated using ADVISOR software in a Python environment. The battery EV car and driveline parameters were configured for the simulation.
| Parameter | Low Gear | High Gear |
|---|---|---|
| Speed Range (km/h) | 0 – 40 | 0 – 80 |
| Motor Speed Range (rpm) | 0 – 3,000 | 0 – 5,000 |
| Load (Nm) | 100 | 200 |
| Torque Range (Nm) | 50 – 70 | 70 – 100 |
The finite element modal analysis confirmed natural frequencies, and the coupling relationship under 2nd-order excitation was established, clearly showing the correlated drop in acceleration torque with the onset of specific vibrations.
| Algorithm Component | Parameter | Value |
|---|---|---|
| ABAQUS Model | Shaft Length (mm) | 1000 |
| Shaft Diameter (mm) | 50 | |
| Elastic Modulus (GPa) | 210 | |
| Poisson’s Ratio | 0.3 | |
| Density (kg/m³) | 7850 | |
| CRNN Network | Convolutional Layers | 3 |
| Convolution Kernel Size | 3×3 | |
| LSTM Hidden Units | 128 | |
| Learning Rate | 0.002 | |
| Control | Reference Torque (Nm) | 200 |
| Overspeed Threshold (Nm) | 240 | |
| Max Allowable Torque Change Rate (Nm/s) | 7.2 |
Simulation Results and Performance Evaluation
The proposed method was compared against two other contemporary strategies: a trajectory-based model predictive control and a conventional adaptive controller. The comparison focused on key performance metrics for the battery EV car application.
1. Overspeed Condition Identification Accuracy: The CRNN-based identifier achieved an accuracy of 93.3%, correctly classifying shaft states. This high accuracy is attributed to the model’s ability to learn the complex spatiotemporal patterns in vibration data preceding overspeed in a battery EV car.
2. Control Precision (Torque Error): The proposed method demonstrated superior torque-tracking performance. The compensation control minimized the deviation between the actual and desired shaft torque.
| Control Method | Torque Error Range (Nm) |
|---|---|
| Trajectory-based MPC | 5.5 – 9.0 |
| Adaptive Controller | 4.0 – 5.0 |
| Proposed Method | 1.0 – 2.0 |
3. System Response Time: The time taken from condition identification to applying effective compensation was measured. A shorter response time is critical for preventing instability in a battery EV car.
| Simulation Run | Trajectory-based MPC | Adaptive Controller | Proposed Method |
|---|---|---|---|
| 1 | 65.6 | 62.5 | 38.6 |
| 2 | 68.5 | 59.6 | 43.8 |
| 3 | 62.5 | 70.8 | 36.8 |
| 4 | 75.4 | 64.8 | 42.6 |
| 5 | 76.2 | 71.5 | 37.9 |
| Average | ~68.2 | ~66.8 | ~41.2 |
4. Reduction in Acceleration Torque Loss: The primary goal was to mitigate the torque loss during aggressive acceleration of the battery EV car. The proposed method successfully reduced this loss, bringing the acceleration torque close to the ideal demanded level, unlike the comparison methods which showed losses up to 15 Nm.
5. Vibration Suppression: Effective control of overspeed rotation also dampens excessive vibration. The proposed method maintained post-control vibration levels below 1.2 m/s², with significantly less fluctuation compared to other methods, enhancing comfort and component life in the battery EV car.
Conclusion and Future Work
This article presented a comprehensive strategy for controlling overspeed rotation in a battery EV car transmission shaft. The method integrates high-fidelity finite element modeling to understand dynamic coupling, a sophisticated CRNN for accurate and early condition identification, and a fuzzy logic-based compensation controller anchored by a reference torque. Simulations confirm its effectiveness, showing high identification accuracy, precise torque control with minimal error, fast response, significant reduction in acceleration torque loss, and effective vibration suppression.
The proposed framework offers a robust solution to a key drivability and durability challenge in electric vehicles. Future work will focus on multi-objective optimization balancing control performance with energy efficiency and component lifespan. Furthermore, leveraging larger datasets from fleet operations and employing advanced deep reinforcement learning could lead to even more adaptive and predictive control systems for the next generation of battery EV car drivetrains.