The proliferation of battery EV cars represents a pivotal shift in global transportation, aligning with strategic decarbonization goals. The reliability of the drivetrain, particularly the gearbox, is paramount for the safety and performance of these vehicles. However, incipient gear faults in a battery EV car gearbox are often masked by strong background noise, random impulses, and complex transfer paths within the compact and high-speed electric drivetrain environment. This makes the extraction of weak fault features a significant challenge, necessitating advanced signal processing techniques for effective early-stage diagnosis.
Traditional methods like envelope analysis often fail under low signal-to-noise ratio (SNR) conditions prevalent in a battery EV car. While band-pass filtering can isolate frequency bands, it requires prior knowledge of fault characteristics. Adaptive filters need a clean noise reference, which is impractical. Therefore, a method that can blindly enhance periodic fault impulses from noisy observations is crucial for maintaining the health of a battery EV car’s transmission system.
Proposed Methodology: An Integrated SSA and Adaptive CYCBD Approach
To address the challenge of diagnosing gear faults in a battery EV car gearbox, a novel two-stage methodology is proposed. The core idea is to first perform preliminary noise reduction and fault frequency estimation, then use this information to adaptively configure a powerful blind deconvolution algorithm for precise fault impulse recovery.
Stage 1: Signal Denoising and Preliminary Frequency Estimation
The first stage employs Singular Spectrum Analysis (SSA) for initial signal cleansing. SSA is a non-parametric technique that decomposes a time series into a set of independent and interpretable components such as trends, oscillatory modes, and noise without requiring a priori frequency information—a key advantage when dealing with the unknown fault signatures in a battery EV car.
The SSA procedure for a signal \(x(n)\) of length \(N\) involves four steps:
1. Embedding: The one-dimensional time series is mapped into a trajectory matrix \(\mathbf{X}\) using a window length \(L\).
$$ \mathbf{X} = \begin{bmatrix}
x_1 & x_2 & \dots & x_{K} \\
x_2 & x_3 & \dots & x_{K+1} \\
\vdots & \vdots & \ddots & \vdots \\
x_L & x_{L+1} & \dots & x_N
\end{bmatrix}, \quad K = N – L + 1 $$
2. Singular Value Decomposition (SVD): The trajectory matrix is decomposed via SVD: \(\mathbf{X} = \mathbf{U} \mathbf{\Sigma} \mathbf{V}^T\). The diagonal elements \(\sigma_i\) of \(\mathbf{\Sigma}\) are the singular values, and the vectors \(\mathbf{U}_i\) and \(\mathbf{V}_i\) are the associated left and right singular vectors. The scree plot of singular values helps separate signal-dominated components from noise-dominated ones.
3. Grouping: Components corresponding to distinct frequency bands (e.g., fault-related modulations) are identified and grouped. Let \(I = \{i_1, i_2, …, i_p\}\) be the set of indices corresponding to the fault-sensitive components. The grouped matrix is \(\mathbf{X}_I = \mathbf{X}_{i_1} + \mathbf{X}_{i_2} + … + \mathbf{X}_{i_p}\).
4. Reconstruction: The grouped matrix \(\mathbf{X}_I\) is transformed back into a one-dimensional time series \(x_{rec}(n)\) via diagonal averaging. This reconstructed signal \(x_{rec}(n)\) contains the enhanced fault-related information with reduced broadband noise.
Subsequently, the Envelope Harmonic Product Spectrum (EHPS) is applied to the denoised signal \(x_{rec}(n)\) to estimate the fundamental fault characteristic frequency \(f_c\). The EHPS amplifies harmonic-related spectral structures (HRSS) in the envelope spectrum \(E(f)\):
$$ \text{EHPS}(f) = \prod_{m=1}^{M} |E(m \cdot f)| $$
where \(M\) is the number of harmonics considered. The frequency \(f\) that maximizes \(\text{EHPS}(f)\) provides a robust estimate of the true fault frequency \(\hat{f}_c\), which is critical for configuring the next stage. This step is vital for the adaptive diagnosis of a battery EV car gearbox where exact rotational speeds might vary.
Stage 2: Adaptive Maximum Second-Order Cyclostationary Blind Deconvolution (CYCBD)
The second stage uses the CYCBD algorithm to perform high-resolution blind deconvolution. The observed vibration signal \(y(n)\) is modeled as the convolution of a periodic fault impulse train \(s(n)\) with the system’s impulse response \(g(n)\), plus noise \(v(n)\):
$$ y(n) = s(n) * g(n) + v(n) $$
CYCBD aims to find a Finite Impulse Response (FIR) filter \(\mathbf{h}\) of length \(L_f\) that recovers an estimate of the fault impulses \(\hat{s}(n)\):
$$ \hat{s}(n) = \mathbf{h} * y(n) = \mathbf{h}^T \mathbf{Y}_n $$
where \(\mathbf{Y}_n\) is a vector of recent observations. The filter is designed by maximizing the Second-Order Cyclostationarity Indicator (\(ICS_2\)), which measures the strength of periodic modulations in the squared envelope of the signal. For a discrete signal \(\hat{s}(n)\), the \(ICS_2\) is defined as:
$$ ICS_2 = \frac{\sum_{k=1}^{K} |c_k^{\hat{s}}|^2}{|c_0^{\hat{s}}|^2} $$
with cyclic cumulants \(c_k^{\hat{s}}\) calculated at the cyclic frequency \(\alpha = k / T_s\), where \(T_s = 1/\hat{f}_c\) is the estimated fault period from Stage 1:
$$ c_k^{\hat{s}} = \frac{1}{N} \sum_{n=0}^{N-1} |\hat{s}(n)|^2 e^{-j2\pi (k \hat{f}_c / f_s) n} $$
Maximizing \(ICS_2\) leads to a generalized eigenvalue problem:
$$ \mathbf{R}_{YY}^W \mathbf{h} = \lambda \mathbf{R}_{YY} \mathbf{h} $$
where \(\mathbf{R}_{YY}\) is the autocorrelation matrix of the input signal and \(\mathbf{R}_{YY}^W\) is a weighted correlation matrix. The optimal filter \(\mathbf{h}_{opt}\) is the eigenvector corresponding to the largest eigenvalue \(\lambda_{max}\).
The Adaptive Mechanism: Determining the Optimal Filter Length
The performance of CYCBD is highly sensitive to the filter length \(L_f\). A short filter may not capture the full impulse response, while an excessively long filter increases computational cost with diminishing returns. To automate this critical parameter selection for the robust diagnosis of a battery EV car, an Efficiency Assessment Index based on Autocorrelation Energy (EAE) is proposed.
The procedure is as follows:
1. For a range of candidate filter lengths \(L_f^{(i)}\), apply CYCBD using the estimated \(\hat{f}_c\).
2. For each filtered output \(\hat{s}_i(n)\), compute its autocorrelation function \(R_i(\tau)\).
3. Calculate the energy \(E_i\) of the autocorrelation function over a significant lag range.
4. Record the computational time \(T_i\) for each filter length.
5. Compute the EAE index, which balances the de-noising performance (autocorrelation energy) against computational efficiency:
$$ \text{EAE}(L_f^{(i)}) = w_1 \cdot T_i + w_2 \cdot E_i $$
where \(w_1\) and \(w_2\) are weighting coefficients (\(w_1 > w_2\) to prioritize efficiency). The filter length \(L_f^*\) that minimizes the EAE index is selected as optimal.
| Parameter | Symbol | Role in Methodology |
|---|---|---|
| SSA Window Length | \(L\) | Governs the resolution of the decomposition. Typically set below \(N/2\). |
| SSA Grouping Set | \(I\) | Selects components associated with fault modulations for reconstruction. |
| EHPS Harmonic Count | \(M\) | Number of harmonics multiplied; amplifies the fundamental fault frequency peak. |
| Estimated Fault Frequency | \(\hat{f}_c\) | Used as the prior cycle frequency \(\alpha\) for CYCBD. |
| CYCBD Filter Length | \(L_f^*\) | Adaptively chosen via the EAE index. Critical for impulse shape matching. |
| EAE Weighting Coefficients | \(w_1, w_2\) | Balance computational time and autocorrelation energy (e.g., \(w_1=2, w_2=1\)). |
Simulation Analysis for a Battery EV Car Drivetrain Scenario
To validate the method, a simulated vibration signal of a battery EV car gearbox with a localized gear tooth fault is constructed. The model includes deterministic gear meshing components, periodic fault impacts, and significant Gaussian noise to mimic real operational conditions of a battery EV car.
The composite signal \(x_{total}(t)\) is given by:
$$ x_{total}(t) = x_{mesh}(t) \cdot [1 + A(t)] + x_{impact}(t) + v(t) $$
where \(x_{mesh}(t) = \sum_{h} A_h \cos(2\pi h f_m t + \phi_h)\) represents gear meshing vibrations, \(A(t)\) is the amplitude modulation due to the fault at frequency \(f_c\), \(x_{impact}(t)\) is a series of periodic impulses at \(1/f_c\) intervals convolved with a system resonance, and \(v(t)\) is additive white Gaussian noise.
The simulation parameters for a typical battery EV car reduction gearbox are summarized below:
| Parameter | Value | Description |
|---|---|---|
| Shaft Rotational Frequency | 60.25 Hz | Corresponding to a high-speed motor in a battery EV car. |
| Gear Fault Frequency (\(f_c\)) | 26.6 Hz | Characteristic frequency of the faulty gear. |
| Meshing Frequency (\(f_m\)) | 2048 Hz | Primary gear mesh frequency. |
| Sampling Frequency (\(f_s\)) | 51200 Hz | High sampling rate typical for battery EV car NVH analysis. |
| Signal-to-Noise Ratio (SNR) | -18 dB | Simulates a severe noise environment. |
Applying the proposed method: First, SSA effectively separates the fault-modulated component from the noisy raw signal. The EHPS applied to the SSA output shows a distinct peak at 26.6 Hz, accurately identifying \(f_c\). Using this frequency, the EAE index is calculated for filter lengths from 50 to 800. The index reaches a minimum at \(L_f^* = 300\), which is then used for CYCBD processing. The final envelope spectrum of the CYCBD output reveals a clear spectral structure with the fundamental fault frequency \(\hat{f}_c\) and its multiple harmonics, starkly contrasting with the noisy, feature-poor spectrum of the original signal. This simulation proves the method’s capability to extract weak cyclic impulses, a common requirement in the prognosis of a battery EV car gearbox.
Experimental Validation on a Battery EV Car Drivetrain Test Bench
The methodology was further tested using experimental data from a dedicated battery EV car drivetrain fault test bench. The setup includes a permanent magnet synchronous motor, a single-speed reduction gearbox, a differential, half-shafts, and a loaded wheel. An accelerometer was mounted on the gearbox casing near the bearing supporting the shaft with the intentionally damaged gear (chipped tooth).
Operating conditions simulated a cruising battery EV car: a motor speed of 3615 RPM (input shaft frequency \(f_{in} \approx 60.25\) Hz) with applied braking torque. The measured vibration signal was heavily contaminated by electromagnetic noise from the motor and structural vibrations, representative of the challenges in monitoring a battery EV car.
The raw time-domain signal showed no obvious periodic impacts. Direct envelope analysis yielded a cluttered spectrum where the suspected fault frequency (26.6 Hz) was barely discernible among numerous other spectral lines. Processing with the proposed adaptive SSA-CYCBD framework changed the diagnostic outcome markedly. The SSA stage successfully isolated the frequency band carrying the fault-induced modulations. The EHPS provided a clear estimate of the fault periodicity. The adaptive EAE criterion selected an optimal filter length of \(L_f^* = 400\) for the CYCBD stage. The final envelope spectrum exhibited dominant peaks precisely at 26.6 Hz and its higher-order harmonics (2x, 3x, 4x), with a dramatic suppression of unrelated noise components. This provides conclusive evidence of a localized gear tooth fault, demonstrating the method’s practical efficacy for the condition monitoring of a battery EV car’s transmission system.
Performance Comparison and Discussion
The superiority of the proposed adaptive SSA-CYCBD method is highlighted by comparing its performance against other established adaptive feature extraction techniques commonly considered for battery EV car diagnostics, such as Time Synchronous Averaging (TSA), Variational Mode Decomposition (VMD), and the standard CYCBD with parameters set by other means (e.g., Fast Spectral Correlation).
| Method | Fault Frequency Clarity | Noise Suppression | Parameter Adaptation | Suitability for Battery EV Car |
|---|---|---|---|---|
| Proposed SSA-CYCBD-Adaptive | Excellent. Clear fundamental and multiple harmonics. | Very High. Unrelated spectral components are effectively removed. | Full. Automatically estimates \(f_c\) and selects \(L_f^*\). | Ideal. Handles high noise and unknown parameters autonomously. |
| Time Synchronous Averaging (TSA) | Moderate. Fundamental may be visible, but harmonics are often weak. | Moderate. Reduces asynchronous noise but sensitive to speed fluctuations. | Requires a precise tachometer signal for synchronization. | Good if precise tacho exists, but adds sensor complexity. |
| Variational Mode Decomposition (VMD) | Variable. Depends on correct selection of mode number and penalty factor. | Moderate. May leave residual noise or split fault energy across modes. | Requires pre-setting of key parameters (K, α). | Challenging due to the need for parameter tuning for different battery EV car operating states. |
| CYCBD with pre-set parameters | Poor to Good. Highly sensitive to the accuracy of the pre-set cycle frequency and filter length. | Can be high if parameters are perfectly set, otherwise poor. | None. Relies entirely on prior knowledge or manual tuning. | Impractical for widespread deployment in varying battery EV car applications. |
The key advantages of the proposed method for battery EV car gearbox diagnosis are threefold:
1. Blind Frequency Estimation: The combination of SSA and EHPS reliably identifies the fault characteristic frequency without any tachometer or prior geometric information, which is crucial when dealing with proprietary or complex gearboxes in a battery EV car.
2. Optimized Impulse Recovery: The EAE-based adaptive selection of the CYCBD filter length ensures an optimal balance between fault feature enhancement and computational efficiency. This makes the algorithm robust and suitable for potential on-board or edge-computing applications in a battery EV car.
3. Strong Noise Immunity: The sequential denoising (SSA) and high-gain, cycle-aware deconvolution (CYCBD) provide a dual layer of noise resistance, making it effective even in the low-SNR environment typical of a battery EV car drivetrain.
Conclusion and Future Perspectives
This article has presented a comprehensive, adaptive framework for the diagnosis of incipient gear faults in battery EV car gearboxes. The integration of Singular Spectrum Analysis for preliminary denoising, Envelope Harmonic Product Spectrum for blind fault frequency estimation, and a novel efficiency-guided adaptive Maximum Second-Order Cyclostationary Blind Deconvolution algorithm forms a powerful tool. It successfully overcomes the limitations of strong background noise and the lack of prior knowledge, autonomously extracting weak periodic fault impulses characteristic of gear damage. Validation through both simulated signals and experimental data from a battery EV car drivetrain test bench confirms the method’s effectiveness and superior performance compared to other adaptive techniques.
Future work will focus on extending this framework to address more complex scenarios prevalent in battery EV cars. This includes the diagnosis of concurrent compound faults (e.g., bearing and gear faults) by enhancing the frequency estimation stage to identify multiple characteristic frequencies. Furthermore, research will investigate the integration of this diagnostic method with cloud-based fleet management systems for battery EV cars, enabling predictive maintenance strategies and improving overall vehicle reliability and safety.