Spiral bevel gears transmit torque between intersecting shafts through curved teeth that engage gradually and carry combined radial and axial loads. In high-speed, heavy-load drivetrains—automotive differentials, helicopter tail drives, rolling mills, and industrial reducers—the gearbox is a critical node. A tooth root crack, a missing tooth, or a wear patch can alter the vibration signature long before it becomes a visible failure. Condition monitoring therefore has to separate weak, impulsive fault transients from meshing harmonics, shaft harmonics, bearing noise, and structural resonances.
Failure Modes and Signal Physics
Spiral bevel gear meshing is not a simple one-dimensional impact process. The contact ratio, tooth-flank curvature, mounting error, and axial thrust all modulate the vibration. A root crack reduces local stiffness and produces impacts once per revolution. A missing tooth creates a strong once-per-revolution impulse and amplitude modulation. Surface wear changes the contact pattern and raises distributed high-frequency energy. At high speed, these events overlap in time and frequency, so a classifier must learn both local impulse shape and longer-range temporal context.
A convolutional neural network is effective at learning local waveform filters, but it does not explicitly model long-range order. It can detect an impulse, yet the sequence of impulses across a revolution and the phase relationship to the shaft are better captured by a recurrent model. The hybrid design used here combines a convolutional front end for local morphological features with a bidirectional long short-term memory network for forward and backward temporal dependencies.
Signal Conditioning and Fault-Sensitive Mode Selection
Raw acceleration is first denoised by wavelet thresholding. The reported configuration uses a Daubechies db1 wavelet, one decomposition level, soft thresholding, and reconstruction. The purpose is not to remove all noise but to suppress broadband spikes that would otherwise dominate kurtosis.
Empirical mode decomposition (EMD) then decomposes the denoised signal into intrinsic mode functions (IMFs). The first 10 IMFs are retained. Each IMF must satisfy two conditions: the number of extrema and zero crossings must differ by at most one, and the mean of the upper and lower envelopes must be zero at any point. The highest-kurtosis IMF is selected and used to reconstruct a new vibration signal. Kurtosis is defined as
$$
\mathrm{Kurtosis} = \frac{\sum_{i=1}^{N}(x_i-\bar{x})^4}{N\sigma^4}
$$
where \(x_i\) is an IMF sample, \(\bar{x}\) is the sample mean, \(\sigma\) is the sample standard deviation, and \(N\) is the number of samples. High kurtosis indicates a heavy-tailed, impulsive waveform. In a gearbox with a cracked or missing tooth, fault-induced impacts raise kurtosis, so the selected IMF is assumed to carry more fault-related transient energy. This is a heuristic: kurtosis also responds to isolated outliers, and a single IMF may discard useful modulation in other bands. The wavelet denoising step is therefore important to the logic.
Overlap sampling expands the sample set. The window must span at least one revolution:
$$
X_w \ge \frac{60F_s}{n}
$$
and the number of windows is bounded by
$$
N \le \frac{L_t-X_w}{L_s}+1
$$
where \(X_w\) is the window length in samples, \(F_s\) is the sampling frequency in Hz, \(n\) is the shaft speed in r/min, \(L_t\) is the total signal length, and \(L_s\) is the step size. At 1,200 r/min and 5,000 Hz, one revolution is 250 samples; the 1,024-point window spans about 4.1 revolutions. A 60 s record contains 300,000 samples. Using a 128-point step yields 2,240 windows per health state.
Depthwise Separable CNN and Bi-LSTM
The DCNN front end uses depthwise separable convolution. A depthwise convolution applies a separate spatial filter to each input channel; a pointwise 1×1 convolution then mixes channels. This reduces parameters and computation relative to a standard convolution. The block also uses Inception-style parallel branches and kernel factorization: 3×3 kernels are split into 1×3 and 3×1 pairs, and 5×5 kernels into 1×5 and 5×1 pairs. The branches are concatenated, giving the network multiple receptive fields at the same stage.
Batch normalization stabilizes layer inputs, permits a higher learning rate, and reduces overfitting. Residual connections provide identity paths around convolutional blocks, so gradients can bypass deep stacks and earlier features can be reused. Max pooling retains the strongest local responses. The complete front end has four convolutional stages before the sequence model.
The Bi-LSTM has two layers and reads the feature sequence in both directions. Forward and backward states are combined, so the classifier sees past and future context at each time step. The LSTM gates use sigmoid activation. After the recurrent layers, fully connected layers with dropout of 0.3 map the features to seven class scores, and softmax produces the diagnosis. Training used 30 epochs and a learning rate of 0.001. The input is a 1,024-point vibration window; the output is one of seven health states.
Experimental Setup and Dataset
The fault-injection experiments were performed at the fault-diagnosis laboratory of Jinzhong University. The rig included a drive motor, ICP (integrated circuit piezoelectric) accelerometers, the spiral bevel gearbox, a planetary gearbox, a magnetic powder brake, a torque sensor, and a DASP acquisition system. The base plate measured 1,800 mm × 900 mm; the gearbox measured 350 mm × 250 mm. A triaxial ICP accelerometer used channels 1–3 for the X, Y, and Z axes; a uniaxial ICP accelerometer used channel 4 for the Z axis. Sensors were mounted on the upper covers at the large- and small-gear axes. The motor ran at 1,200 r/min, the sampling frequency was 5,000 Hz, and each condition was recorded for 60 s.
| Parameter | Value |
|---|---|
| Motor speed | 1,200 r/min |
| Sampling frequency | 5,000 Hz |
| Record length per condition | 60 s |
| Base plate dimensions | 1,800 mm × 900 mm |
| Gearbox dimensions | 350 mm × 250 mm |
| Accelerometer channels | Triaxial X/Y/Z on channels 1–3; uniaxial Z on channel 4 |
| Acquisition system | DASP |
Seven health states were tested: large-gear root crack, large-gear tooth-surface wear, large-gear missing tooth, pinion root crack, pinion missing tooth, pinion tooth-surface wear, and normal. Each state contributed 2,240 windows of 1,024 points with a 128-point step, giving 15,680 windows. An 80/20 split gives 12,544 training windows and 3,136 test windows, or 1,792 and 448 per class.
| Health state | Class samples | Training samples | Test samples |
|---|---|---|---|
| Large-gear root crack | 2,240 | 1,792 | 448 |
| Large-gear tooth-surface wear | 2,240 | 1,792 | 448 |
| Large-gear missing tooth | 2,240 | 1,792 | 448 |
| Pinion root crack | 2,240 | 1,792 | 448 |
| Pinion missing tooth | 2,240 | 1,792 | 448 |
| Pinion tooth-surface wear | 2,240 | 1,792 | 448 |
| Normal | 2,240 | 1,792 | 448 |
| Total | 15,680 | 12,544 | 3,136 |
Reported Diagnostic Performance
On the clean data, the accuracy curve reached 100% and the loss fell below 1% after about 10 epochs. In the displayed test confusion matrix, all 3,136 test windows were classified correctly. This is a strong laboratory result for a balanced seven-class problem, but it is a closed-set result: the classifier was not asked to reject unknown faults or normal data outside the seven labels.
When 10 dB noise was added to the raw signal and the full preprocessing chain was repeated, convergence slowed to about 15 epochs and the reported accuracy was 98.8%. In one displayed confusion matrix, two pinion missing-tooth windows were classified as normal; the remaining windows were correct. The 98.8% figure and the two-error matrix do not reconcile arithmetically for a single 3,136-window test set, so they should be treated as separate reported outcomes rather than one exact count.
| Test condition | Reported accuracy | Convergence | Confusion-matrix observation |
|---|---|---|---|
| Clean signal | 100% | About 10 epochs | All 3,136 test windows correct |
| 10 dB added noise | 98.8% | About 15 epochs | Two pinion missing-tooth windows classified as normal in one run |
Five repeated runs were used for comparison with CooAtten-Resnet, CNN-GRU, and IEWT-DELM. The reported curves place DCNN-Bi-LSTM highest in accuracy and fastest in convergence on clean data. Under 10 dB noise, all models degrade, but the proposed model retains the best accuracy and convergence in the comparison. Exact baseline accuracies are not tabulated, so only the ranking and qualitative behaviour can be stated.
What the Evidence Supports
The reported results support three limited claims. First, a hybrid convolutional-recurrent model can learn the seven laboratory health states from 1,024-point vibration windows with high accuracy. Second, wavelet denoising plus kurtosis-based IMF selection is compatible with strong noise robustness in the tested 10 dB additive-noise case. Third, depthwise separable convolution, batch normalization, and residual connections keep the network trainable at the reported depth and convergence rate.
The results do not establish field reliability. The test matrix uses one motor speed, 1,200 r/min, on a laboratory rig with balanced classes. Load levels are not itemised in the result set, and cross-speed, cross-load, and cross-machine transfer are untested. The 1,024-point windows overlap by 896 points; if training and test windows were split randomly, near-duplicate samples could inflate accuracy. The split is described only by ratio, not by grouping. EMD can suffer mode mixing and end effects, and selecting only the highest-kurtosis IMF discards information in other modes. Kurtosis is also sensitive to outliers, so the denoising step carries part of the burden. No confidence intervals or run-to-run variance are reported for the five comparison runs.
The method is a data-driven classifier, not a physics-based fault-size estimator. It assigns a health state; it does not estimate crack length, remaining useful life, or load capacity.
Practical Implications for Spiral Bevel Gearbox Monitoring
For industrial condition monitoring, the value of the pipeline is its separation of concerns: wavelet denoising and IMF selection reduce impulsive noise, the convolutional front end learns local waveform shape, and the Bi-LSTM captures sequence context. This is relevant to spiral bevel gearboxes in automotive axles, helicopter drives, rolling mills, and other intersecting-shaft applications where accelerometers can be mounted on the housing.
Deployment would require more than a high test accuracy. A production system needs validation across speed and load, sensor placement, lubricant and temperature variation, and unknown fault modes. The model should be trained or fine-tuned on target-machine data, and the window-level split should be replaced by a run-level or asset-level split to avoid leakage. EMD and Bi-LSTM also add computational cost relative to a plain CNN, so edge deployment would need pruning, quantization, or an offline diagnostic workflow.
Used within those limits, the DCNN-Bi-LSTM approach is a credible laboratory blueprint for diagnosing spiral bevel gearbox faults under noise, but not yet a field-qualified prognostic system.