Spiral bevel gears transmit torque between intersecting shafts through teeth cut along a curve rather than a straight taper. Contact begins at one end of the tooth and migrates gradually across the flank, so load is shared over several teeth at once. The resulting transmission is smoother and quieter than a straight bevel stage of comparable size, which is why curved-tooth bevel sets appear in automotive differentials, helicopter tail drives, machine-tool spindles and heavy industrial reducers. The same geometry that makes the gear quiet also makes it unforgiving: correct meshing depends on precise mounting distance and backlash, and small misalignments shift the contact patch toward the tooth edge, concentrating stress exactly where root bending stress is already highest.
Diagnosing such a gearbox from vibration alone is difficult for three reasons that compound one another. First, the meshing frequency and its harmonics are modulated by shaft rotation, so a local defect such as a chipped tooth or a root crack does not produce a fixed spectral line but a family of sidebands whose spacing and amplitude change with load. Second, the impulses generated by a crack are short, weak and non-stationary; they are easily buried under the meshing energy of healthy teeth. Third, plant environments superimpose broadband random excitation and occasional impulsive transients from adjacent equipment, so any classifier must survive a signal-to-noise ratio that is neither constant nor known in advance.
Two Limitations of Conventional Diagnostic Pipelines
Signal-decomposition front ends — empirical mode decomposition, variational mode decomposition and their multi-channel variants — improve the separability of weak impulses from harmonic interference, and sparse-decomposition formulations that exploit periodicity in the time domain jointly with harmonic sparsity in the frequency domain have pushed this line of work further. Kernel-based and shallow classifiers such as support vector machines and kernel extreme learning machines then operate on hand-engineered indicators. These pipelines remain effective when the fault signature is known in advance, but the feature set must be redesigned for each new gearbox, and the hand-off between decomposition and classification discards information that no downstream model can recover.
Deep architectures remove the manual feature stage, and convolutional networks, deep belief networks and recurrent variants have all been applied to gearbox data. Two gaps persist. Most designs commit to a single representation: either a time–frequency image processed by a convolutional backbone, or a raw one-dimensional sequence processed by a recurrent network. A time–frequency image preserves the localisation of energy in time and frequency but blurs fine waveform detail through the analysis window; a raw sequence retains that detail but gives a recurrent layer no explicit frequency structure to exploit. Few architectures use both representations and fuse them. Separately, attention is usually applied as a two-dimensional spatial weight map, which indicates where in a feature map to look but not which positions inside a convolutional kernel matter for the pattern being detected. The approach described here addresses both points: a dual-channel network in which a residual backbone with receptive-field attention processes short-time Fourier transform images while a bidirectional gated recurrent unit with self-attention processes the raw sequence, with the two feature sets concatenated before classification.
Signal Preparation: Overlapping Sampling and Time–Frequency Imaging
Deep models trained on a few long recordings overfit. Overlapping sampling enlarges the training set by exploiting the continuity of a vibration record: a window of fixed width is slid along the signal with a step smaller than the window, so one recording yields many partially correlated samples. If the shaft turns at n r/min and the sampling frequency is Fs, the window must span at least one full revolution so that every tooth on the gear passes the sensor within each sample:
$$X_w \ge \frac{60}{n}\,F_s$$
For a record of total length Lt, window width Xw and step Ls, the number of extractable samples is bounded by
$$N \le \frac{L_t – X_w}{L_s} + 1$$
The overlapping step is a deliberate trade: a small step maximises sample count but raises correlation between neighbouring samples, so the effective size of the training set grows more slowly than the raw count suggests.
The spatial channel receives a short-time Fourier transform of each window. A window function w(t) is applied to successive frames, and each frame is transformed to give a local spectrum:
$$x_m(t) = x(t)\,w(t – m L_s)$$
$$\mathrm{STFT}(m,k) = \sum_{t=0}^{L_w-1} x_m(t)\,e^{-j 2\pi k t / L_w}$$
Rows of the resulting matrix index time frames and columns index frequency bins, with magnitude giving the energy at each time–frequency point. Window length controls the resolution split, and is set from the lowest feature frequency fmin that must remain resolvable:
$$L_w = \alpha\,\frac{F_s}{f_{\min}}$$
with the scaling factor α typically between 2 and 5. A short window sharpens time localisation at the cost of frequency resolution; a long window does the reverse. The transform produces 256 × 256 RGB time–frequency maps, while the identical windows are retained as one-dimensional sequences for the temporal channel, so no information is discarded at the input stage.
Spatial Branch: Residual Learning with Receptive-Field Attention
The image branch is a residual network whose basic block contains two 3 × 3 convolutions, each followed by batch normalisation and a ReLU activation, with a 1 × 1 convolution on the shortcut path to reconcile channel counts. The identity path allows gradients to bypass the block, which is what makes deeper stacks trainable without the degradation that affects plain convolutional stacks.
Receptive-field attention (RFA) is inserted at several depths of this backbone. Its distinguishing feature is the shape of the attention it produces. A spatial attention module such as CBAM generates a two-dimensional weight map of size h × w, reweighting locations in the feature map. RFA instead generates a four-dimensional tensor of size c × k × k per sample, assigning an independent weight to every tap of every convolutional kernel. The mechanism therefore modulates the convolution itself rather than its output map.
Generation begins with global average pooling, which compresses each channel to a context vector:
$$Z = \frac{1}{h\,w}\sum_{i=1}^{h}\sum_{j=1}^{w} X_c(i,j)$$
A two-layer bottleneck with a nonlinearity maps this vector to a raw attention vector whose length equals the total number of kernel parameters c·k·k:
$$A_{\text{raw}} = W_2\,\delta(W_1 Z + b_1) + b_2$$
$$A^{k\times k} = \mathrm{Sigmoid}\left(\mathrm{Reshape}(A_{\text{raw}})\right)$$
Each element of Ak×k expresses how important the corresponding tap of the corresponding output channel is for the current input. The tensor is then multiplied element-wise with the original depthwise kernels to form dynamic kernels, applied at three scales so that local, intermediate and broad context are captured simultaneously:
$$F_{3\times3} = X * (K_{3\times3} \odot A^{3\times3})$$
$$F_{5\times5} = X * (K_{5\times5} \odot A^{5\times5})$$
$$F_{7\times7} = X * (K_{7\times7} \odot A^{7\times7})$$
Scale mixing is input-dependent. The same global context vector Z is passed through a linear layer and a softmax to yield three normalised weights, and the branch output is their weighted sum:
$$[w_1, w_2, w_3] = \mathrm{Softmax}(W_f Z + b_f)$$
$$Y = w_1 F_{3\times3} + w_2 F_{5\times5} + w_3 F_{7\times7}$$
In a time–frequency image, the practical effect is that the network can allocate kernel capacity to the narrowband ridges associated with meshing harmonics while suppressing diffuse background texture, and can do so differently for different fault classes. The branch finally emits a 512-dimensional feature vector.
Temporal Branch: Bidirectional Gating with Self-Attention
The sequence branch is built on the gated recurrent unit, which controls information flow through an update gate and a reset gate and thereby mitigates the vanishing-gradient problem of a plain recurrent layer:
$$z_t = \sigma(W_z x_t + U_z h_{t-1} + b_z)$$
$$r_t = \sigma(W_r x_t + U_r h_{t-1} + b_r)$$
$$\tilde{h}_t = \tanh\!\left(W_h x_t + r_t \odot (U_h h_{t-1}) + b_h\right)$$
$$h_t = (1 – z_t)\odot h_{t-1} + z_t \odot \tilde{h}_t$$
A single direction can only condition on past samples, which is a poor match for a diagnosis problem where the relevance of an impulse depends on what follows it — the decay pattern of a resonance, for instance, arrives after the excitation. The bidirectional arrangement runs one GRU over the sequence in forward order and another in reverse order, and combines the two hidden states, so the representation at each time step is informed by both its past and its future context.
Bidirectionality supplies context but not selectivity: every time step contributes to the pooled representation with roughly equal authority. A self-attention layer placed after the BiGRU reweights the hidden states. Queries, keys and values are linear projections of the sequence,
$$Q = XW_Q, \qquad K = XW_K, \qquad V = XW_V$$
and the scaled dot-product attention assigns each step a weight derived from its similarity to all other steps:
$$\mathrm{Attention}(Q,K,V) = \mathrm{Softmax}\!\left(\frac{QK^{\top}}{\sqrt{d_k}}\right)V$$
The scaling by √dk keeps the softmax input in a range where gradients remain well behaved. In diagnostic terms the layer learns which moments in a 1024-point window carry class evidence — the impact transient of a chipped tooth or the modulation cycle of a worn flank — and attenuates the rest. The branch outputs a 256-dimensional vector.
Fusing the Branches
The 512-dimensional image feature and the 256-dimensional sequence feature are concatenated into a single representation, passed through a fully connected layer for dimensionality reduction, activated by ReLU and regularised by dropout before the final softmax classifier. Training used an initial learning rate of 0.001 with dynamic scheduling, a dropout rate of 0.5 and 30 epochs. The division of labour is clean: the convolutional path resolves local spectral structure in the time–frequency plane, the recurrent path models how that structure evolves sample by sample, and the attention layers in each path determine which regions and which moments matter.
Test Rig and Dataset
Fault data were acquired on a spiral bevel gearbox test stand at the fault-diagnosis laboratory of Jinzhong University. The drivetrain comprised a drive motor, a torque sensor, the bevel gearbox, a planetary reduction stage and a magnetic-particle brake, with speed and load controlled through an acquisition and control system. Four ICP accelerometer channels were used: a three-axis sensor whose X, Y and Z axes mapped to channels 1–3, and a single-axis sensor whose Z axis mapped to channel 4, all mounted on the housing cover above the gear axis. Signals were recorded with a DASP acquisition system at 1 000 r/min and a sampling frequency of 5 000 Hz. Seven conditions were produced: healthy operation, broken tooth on the large gear, broken tooth on the pinion, root crack on the large gear, root crack on the pinion, tooth-surface wear on the large gear, and tooth-surface wear on the pinion. Each condition was recorded for 60 s.
| Parameter | Value |
|---|---|
| Speed / sampling frequency | 1 000 r/min / 5 000 Hz |
| Fault classes | 7 (healthy, broken tooth ×2, root crack ×2, wear ×2) |
| Record length per class | 60 s |
| Window length / sliding step | 1 024 points / 128 points |
| Samples per class / total | 1 120 / 7 840 |
| Training / test split (8:2) | 6 272 / 1 568 |
| Time–frequency image size | 256 × 256, RGB |
With n = 1 000 r/min and Fs = 5 000 Hz, one revolution corresponds to 300 samples, so a 1 024-point window covers roughly 3.4 shaft revolutions — comfortably above the one-revolution floor of the window-length criterion. The 128-point step gives a 12.5 % overlap per shift, producing 1 120 windows from each 60 s record.
Diagnostic Performance
Convergence on noise-free data
Over 30 epochs, training and test accuracy rose together and training and test loss fell together, both flattening at the end of training with only a narrow gap between the two loss curves. The narrow gap matters more than the absolute value: it indicates that the error reduction generalised to samples the model never saw, rather than being absorbed by memorisation of the training windows. On the held-out test set of 1 568 samples, every sample was assigned to its correct class, and the noise-free confusion matrix is strictly diagonal with no off-diagonal entries.
Mixed-noise testing
Robustness was probed by adding Gaussian–impulse mixed noise at signal-to-noise ratios of 10 dB and 20 dB to the raw vibration signals before transformation, with the impulse component occurring at a 5 % probability. The mixture is a deliberate choice: Gaussian noise stands in for broadband plant interference, while impulse noise represents the abrupt transients produced by adjacent machinery.
At 10 dB, training and test accuracy climbed in step, converging close to 100 % after roughly 20 epochs, with both losses settling near 0.2 and remaining nearly coincident throughout. At 20 dB, the test-accuracy curve fluctuated noticeably — including a marked dip around epoch 10 — convergence was slower, the test loss showed a corresponding spike, and the final loss plateau sat higher than in the 10 dB case. This ordering runs opposite to what the signal-to-noise ratio alone would predict, since the 10 dB condition injects ten times the noise power of the 20 dB condition. The observation is best read as an artefact of the specific mixture and of a single training trajectory rather than as a general property of the model: with only 5 % of samples carrying impulsive contamination, the realised noise statistics depend on where impulses fall within the windows, and one run per condition cannot separate that variance from a genuine trend. Both conditions nevertheless ended with high training and test accuracy and a small generalisation gap, and the confusion matrices confirm the pattern — at 10 dB most classes remain correctly separated with a small number of misassignments, while at 20 dB the misassignments increase both in count and in the number of classes affected. A per-SNR accuracy table, averaged over multiple runs, would be needed to quantify robustness as a function of SNR rather than to describe it qualitatively.
Stability and statistical checks
Because a single train–test split can flatter a model, three additional checks were run. Thirty repetitions with randomised splits produced a mean accuracy of 99.93 % with a standard deviation of 0.12 %. An averaged confusion matrix over those runs placed all diagonal entries above 99.9 %, with per-class accuracy between 99.85 % and 99.92 % and every off-diagonal element below 0.04 %. Five-fold cross-validation returned accuracies above 99.8 % in every fold, a mean of 99.92 % and a standard deviation of 0.000 4, indicating that the result is not sensitive to how the data are partitioned.
| Check | Result |
|---|---|
| 30 runs, randomised splits | 99.93 % ± 0.12 % |
| 5-fold cross-validation | 99.92 % mean, std 0.000 4, all folds > 99.8 % |
| Per-class accuracy | 99.85 % – 99.92 % |
| Off-diagonal (misclassification) elements | < 0.04 % |
| Parameters / memory / latency | 18.7 M / ≈320 MB / 12.3 ms (RTX 3080) |
Comparison with alternative architectures
The fused model was benchmarked against two recent sequence architectures, Transformer and Mamba, and two established classifiers, a support vector machine and a deep belief network, under identical data and split conditions.
| Model | Accuracy (%) | Observation |
|---|---|---|
| ResNet-RFA + BiGRU-SATT (proposed) | ≈100 | Highest accuracy and most stable convergence |
| Mamba | 96 | Best of the comparison models |
| SVM | 94 | Strong for a shallow classifier |
| Transformer | 91 | Below SVM on this dataset size |
| DBN | 89 | Weakest of the group |
The ranking is informative beyond the headline number. A state-space model such as Mamba performs well because it handles long sequences efficiently, but it operates on one representation only. The Transformer trails a classical SVM, which is consistent with the data budget: self-attention over a modest number of samples has little redundancy to exploit and lacks the convolutional inductive bias that helps when training data are limited. The gap between the fused model and the best single-representation alternative — four percentage points over Mamba — is attributable to the combination of the two input views and to the parameter-level reweighting inside the convolutional kernels, though the ablation needed to separate those two contributions is not part of this evaluation.
Cost, Deployment and Scope of Validity
On an NVIDIA RTX 3080, a single inference takes about 12.3 ms, with 18.7 M parameters and roughly 320 MB of memory in use. That is adequate for offline analysis and for many online monitoring schemes at typical gear-meshing rates, but the parameter count and latency are marginal for resource-constrained edge hardware. Practical deployment would likely require downsampling the time–frequency input, or applying pruning and quantisation, with a re-check of accuracy after compression. The self-attention term also scales with sequence length, so lengthening the analysis window to capture slower modulation phenomena raises cost non-linearly.
The results should be read within their experimental envelope. All data come from a single test stand at one speed (1 000 r/min) and one load path, with seven artificially seeded conditions on one gear pair. Faults induced by machining or seeding differ from faults that develop in service, where crack propagation, pitting and lubricant degradation overlap and produce compound signatures rather than the single-label classes used here. Cross-machine transfer, varying speed and load, and compound-fault labelling remain open questions, and the near-perfect accuracy on this dataset should be treated as evidence that the dual-representation design separates these seven conditions cleanly, not as a guarantee of the same margin on an unfamiliar gearbox. Under severe noise, a denoising front end would still be advisable before the network, since the attention mechanisms reweight evidence that is present rather than recover evidence that has been submerged.
What the architecture establishes is a general principle for rotating-machinery diagnosis: when a signal carries both local spectral structure and long-range temporal dependence, giving each aspect its own pathway — and letting attention operate at the level of kernel parameters in one path and at the level of time steps in the other — extracts more than forcing a single backbone to represent both. The overlapping-sampling scheme and the window-length criterion tie the data pipeline to the physical rotation period, which keeps the learned features anchored to tooth-passing events rather than to arbitrary segment boundaries.