Spiral bevel gears carry torque between intersecting shafts. Their teeth are cut along a curved path on a conical pitch surface, so contact begins at one end of the tooth and rolls gradually across the flank. The resulting contact ratio and smoothness explain their dominance in vehicle final drives, rotorcraft transmissions and heavily loaded industrial reducers, where straight bevel or spur stages would be noisier and weaker.
The same geometry that makes the mesh quiet makes its faults difficult to detect. Mesh harmonics are modulated by transmission error and load, while the evidence of a root fracture or a broken tooth appears as short impulses that the housing attenuates and bearing noise and neighbouring gear stages bury. Diagnosis therefore fails less often because the classifier is shallow than because the impulse-to-noise ratio at the sensor is low, and because a one-dimensional vibration record holds two kinds of information simultaneously: local waveform morphology, which convolutional layers exploit, and dependence between successive revolutions, which a feed-forward network ignores. A practical architecture has to address both, and has to do so with enough depth to represent the fault physics without overfitting a few thousand correlated samples.
Signal conditioning: wavelet thresholding, EMD, and kurtosis-based mode selection
Conditioning precedes learning. The raw acceleration record is first denoised by wavelet thresholding: the signal is decomposed with a single-level db1 (Haar) wavelet, the detail coefficients are soft-thresholded, and the signal is reconstructed. Thresholding in the coefficient domain removes broadband noise while preserving impulsive transients better than a fixed-band low-pass filter, because small coefficients are attenuated without a hard frequency cut-off.
The denoised signal is then decomposed by empirical mode decomposition (EMD), introduced by Huang and co-workers in 1998 as an adaptive, data-driven alternative to fixed-basis transforms. EMD iteratively extracts intrinsic mode functions (IMFs), each satisfying two conditions: over the data span the number of extrema and zero crossings differ by at most one, and the mean of the upper and lower envelopes defined by the local extrema is zero at every point. The sifting procedure locates extrema, interpolates the envelopes, subtracts their mean, tests the IMF conditions, and repeats on the residual until the residual becomes monotone or a set number of modes has been extracted. No basis function is chosen in advance, which suits the frequency modulation produced by speed and load variation.
The first ten IMFs are retained, and each is scored by kurtosis, the fourth standardized moment:
$$\mathrm{Kurtosis}=\frac{\sum_{i=1}^{N}\left(x_i-\bar{x}\right)^4}{N\sigma^4}$$
where $x_i$ are the sample values of the IMF, $\bar{x}$ the mean, $\sigma$ the standard deviation and $N$ the number of samples. A Gaussian record scores about 3; isolated impacts and sharp transients inflate the value. A cracked or missing tooth generates periodic impacts at the shaft rate and its harmonics, so the IMF carrying those impacts scores highest and is taken as the fault-bearing component. That single IMF, rather than the full ten-mode decomposition, becomes the model input.
The selection rule embeds an assumption worth stating: impulsiveness is treated as a proxy for fault energy. This is reasonable for fracture and missing-tooth classes, whose signatures are impulsive by nature, and less certain for distributed surface wear, whose evidence is broadband and may be spread over several modes. Kurtosis is also inflated by isolated sensor spikes and random impacts, so a noisy channel can nominate the wrong mode. EMD itself is susceptible to mode mixing, and discarding the neighbouring IMFs discards potentially complementary information.
Sample construction for a convolutional model
Convolutional training needs many labelled segments, so the conditioned signal is cut into overlapping windows. The window must span at least one revolution of the gear:
$$X_w \ge \frac{60F_s}{n}$$
with $n$ the shaft speed in r/min and $F_s$ the sampling frequency in Hz. At $n=1{,}200$ r/min and $F_s=5{,}000$ Hz, one revolution corresponds to 250 samples, so the window actually used, $X_w=1{,}024$ samples, covers about 4.1 revolutions. The number of windows obtainable from a record of length $L$ with step $L_s$ is
$$N \le \frac{L-X_w}{L_s}+1,\qquad L_s \le X_w$$
A step of 128 samples gives 87.5% overlap. Each 60 s recording at 5 kHz contains 300,000 samples, which yields roughly 2,336 admissible windows; 2,240 were kept per condition. The overlap is a standard augmentation for vibration data, but it has a consequence for how the reported accuracy should be read: adjacent windows share most of their samples, so the effective number of independent observations is far smaller than the nominal count, and a random train/test split lets nearly identical segments appear on both sides.
Architecture: separable convolution for space, bidirectional LSTM for time
Deep separable convolution replaces the standard convolution kernel with a depthwise convolution followed by a pointwise one, and factorizes square kernels into asymmetric pairs. For a kernel of width $k$ acting on one channel, factorization into $1\times k$ and $k\times 1$ stages reduces the multiplications from $k^2$ to $2k$ — from 9 to 6 for a $3\times3$ kernel and from 25 to 10 for a $5\times5$ kernel. Inception-style branches run several such kernels in parallel and concatenate their outputs, so the network sees several receptive-field scales at the same depth instead of buying scale with extra layers. Batch normalization after each convolution keeps the input distribution of every layer stable, which permits a larger learning rate and acts as a mild regularizer. Residual identity shortcuts route earlier feature maps around the block, countering the degradation that appears when plain stacks are deepened.
The spatial features then enter a two-layer bidirectional LSTM. Gated recurrent units with sigmoid gating learn which information to retain across a sequence, and the bidirectional arrangement processes the segment forwards and backwards, concatenating both hidden states so that each time step is described by its past and future context. Two Bi-LSTM layers were used, followed by fully connected layers with dropout of 0.3 and a softmax output over the seven classes.
| Stage | Configuration | Function |
|---|---|---|
| Input | 1-D segment, 1,024 samples, kurtosis-selected IMF | Fault-bearing waveform |
| Spatial extraction | 4 blocks: depthwise-separable convolution with Inception-style multi-branch kernels, batch normalization, max pooling | Multi-scale local morphology, reduced parameter count |
| Temporal modelling | 2-layer Bi-LSTM, forward and backward passes | Long-range dependence within and across revolutions |
| Regularization | Dropout 0.3 between fully connected layers | Overfitting control |
| Output | Softmax, 7 classes | Fault classification |
| Training | 30 epochs, learning rate 0.001, 8:2 train/test split | Optimization and hold-out evaluation |
Test rig, dataset and results
Fault-simulation experiments were run on a dedicated test bench at the fault diagnosis laboratory of Jinzhong University. The bench comprises a drive motor, a spiral bevel gearbox (350 mm × 250 mm) followed by a planetary gearbox, a magnetic powder brake, a torque transducer, a guard, a DASP acquisition system and a baseplate of 1,800 mm × 900 mm. A three-axis ICP accelerometer fed channels 1–3 (X, Y, Z) and a single-axis ICP accelerometer fed channel 4, both mounted on the housing covers at the large- and small-gear axes. Shaft speed was 1,200 r/min and the sampling frequency 5,000 Hz, with 60 s recorded per condition after the gears had been replaced and the signal had stabilized. Seven conditions were produced: root fracture, tooth-surface wear and missing tooth on the bull gear, the same three faults on the pinion, and a healthy reference.
| Label | Component | Condition | Samples |
|---|---|---|---|
| 0 | Bull gear | Root fracture | 2,240 |
| 1 | Bull gear | Tooth-surface wear | 2,240 |
| 2 | Bull gear | Missing tooth | 2,240 |
| 3 | Pinion | Root fracture | 2,240 |
| 4 | Pinion | Missing tooth | 2,240 |
| 5 | Pinion | Tooth-surface wear | 2,240 |
| 6 | — | Healthy | 2,240 |
| Total | 15,680 | ||
The 8:2 split allocated 12,544 segments to training and 3,136 to testing. Training converged quickly: accuracy started low and loss high, but by roughly the tenth epoch both curves had flattened, with test accuracy reaching 100% and loss falling below 1%. The confusion matrix for a representative run classified all 3,136 test segments correctly.
With additive noise at 10 dB SNR injected into the raw vibration signal, early epochs fluctuated as the noise masked fault features and convergence slowed, but the curves stabilized after about 15 epochs at 98.8% accuracy. Two pinion missing-tooth segments were assigned to the healthy class; every other segment was correct. The pinion classes rotate faster than the bull-gear classes and produce impacts at higher repetition rates with lower energy per impact, which makes them the plausible first candidates for confusion with a healthy mesh under noise.
Comparison with other architectures
Three representative methods were re-trained on the same data and each configuration was run five times: a continuous-wavelet-transform image pipeline feeding a coordinate-attention residual network (CooAtten-Resnet), a dual-channel feature-fusion CNN-GRU, and an improved empirical wavelet transform combined with a deep extreme learning machine (IEWT-DELM). Across the five repeats the proposed model showed the highest accuracy, the smallest run-to-run scatter and the fastest convergence; the baselines were both less accurate and visibly less stable. Under 10 dB added noise every model degraded, but the deep separable CNN–Bi-LSTM retained the best accuracy and still converged faster than the alternatives.
What the evidence does and does not establish
Several boundaries are worth making explicit. All results come from one rig, one shaft speed and one gear set; load levels from the magnetic powder brake are not reported, and no cross-speed or cross-load transfer test was performed, so the invariance of the learned features to operating point is untested. The 100% figure applies to a closed seven-class problem in which overlapping windows from the same recordings populate both training and test sets — a setting that flatters any deep model. The noise study used synthetically added noise at a single moderate SNR; real machinery noise is coloured, impulsive and non-stationary, and performance at 0 dB or below is unknown.
The signal-processing front end also carries costs. A single-level Haar decomposition with soft thresholding is a coarse denoiser, and mode mixing in EMD plus single-IMF selection discards information that neighbouring modes may hold. Most importantly, the reported comparisons involve other authors’ full pipelines, not ablations of this one. Without a matched-parameter 1-D CNN baseline and a variant without the Bi-LSTM stage, the improvement cannot be attributed specifically to separable convolution, to residual connections, or to bidirectional temporal modelling; the contribution of each element remains unquantified.
Implications for condition monitoring practice
Three design choices transfer beyond this dataset. First, the conditioning chain is physics-guided rather than learned: selecting the mode with the highest kurtosis encodes the expectation that a cracked or broken tooth radiates impacts, and it does so at negligible computational cost compared with adding capacity to the network. Second, separable convolution with residual shortcuts is a practical route to depth on vibration datasets of a few thousand segments, where parameter efficiency matters more than raw representational power. Third, bidirectional temporal modelling suits recordings made in steady state, where the whole segment is available at once; a streaming monitor would need a causal, unidirectional recurrent stage, and the accuracy reported here would not transfer unchanged.
Taken together, the results indicate that a two-stage pipeline — impulsive-mode selection followed by a hybrid spatial–temporal network — can separate seven spiral bevel gearbox conditions with few errors even at 10 dB SNR, which is the range where a screening layer ahead of human review becomes useful. Establishing that the same pipeline holds across speeds, loads and machines is the next evidentiary step, and it is a substantially harder one than improving the classifier on a fixed, closed dataset.