Spiral bevel gears carry the first stage of power transfer in helicopter main reducers and in many other compact, high-ratio transmissions. Their teeth mesh with a sliding-rolling contact that is deliberately offset and curved, which spreads load over several teeth and keeps the drive quiet, but it also makes the resulting vibration complicated to interpret. An input-stage gear runs at high speed under heavy torque, and the measured signal is further shaped by amplitude and frequency modulation from tooth-meshing stiffness variation, from the relative motion of bearing elements, and from the changing position of the sensor relative to the meshing zone. Load fluctuations under heavy duty add a directly non-stationary component, and speed fluctuations at high rotation rates intensify the modulation sidebands. Extracting a feature set that still separates health states under these conditions is the central difficulty in diagnosing this class of gear.
From handcrafted indicators to learned representations
Classical gear diagnostics relies on indicators computed from the time and frequency domains — sideband ratios around the meshing frequency, kurtosis, envelope-spectrum peaks at characteristic defect orders. These indicators encode expert knowledge about which modulation pattern corresponds to which defect, and they degrade when the modulation is strong, when several defects coexist, or when the operating point drifts. A representation learned directly from the measured signal removes the need to specify those patterns in advance.
Among the deep architectures applied to rotating machinery, three families dominate. Stacked autoencoders learn a compact code by reconstructing their own input; deep belief networks stack restricted Boltzmann machines and are trained layer by layer; convolutional networks exploit local correlation in the input through shared filters. Reported variants have added denoising objectives to improve robustness under strong background noise, applied deep networks to planetary gearboxes with unsupervised pretraining followed by supervised fine-tuning, imposed sparsity constraints so that features are learned from raw vibration without labels, built hierarchical diagnosis networks that quantify both fault type and damage severity, and combined convolutional structures with compressed sensing or with dropout and batch normalization to speed convergence and tolerate variable load. The common thread is that the discriminative information is discovered from data rather than prescribed.
Autoencoder fundamentals
An autoencoder is a symmetric three-layer network that maps an input vector X to a hidden representation H and then reconstructs the input as Y from that representation. The encoder performs a nonlinear transformation
$$H = f_{\theta}(X) = \sigma(WX + b)$$
where $\sigma$ is the activation function, $W$ is the weight matrix and $b$ is the hidden-layer bias, with $\theta = \{W, b\}$. The decoder reconstructs through
$$Y = g_{\theta’}(H) = \sigma(W’H + b’)$$
with $\theta’ = \{W’, b’\}$, $W’$ the decoding weight matrix and $b’$ the output-layer bias. Training minimizes the mean squared reconstruction error between input and output,
$$J(W,b) = \sum L(x,y) = \sum \left\| y – x \right\|_2^{2}$$
and the hidden vector H obtained at the minimum is the learned code. Because the hidden layer is narrower than the input, this code is a compression: it retains what the network needs to rebuild the signal and discards the rest.
Sparsity and layer-wise stacking
Stacking several autoencoders of decreasing width produces a deep network whose lower layers hold increasingly abstract descriptions of the input. Training such a stack directly by backpropagation invites vanishing gradients. Three countermeasures are standard: linear activation of the neurons, a pretraining-plus-fine-tuning schedule, and hidden layers whose outputs are kept sparse.
A sparse autoencoder adds the third of these. By penalizing the average activation of each hidden unit against a small target value, the network is forced to represent each input with few active units, which typically yields features with better expressive power than an unconstrained bottleneck. Stacking sparse autoencoders — the output of one encoder becoming the input of the next — gives a stacked sparse autoencoder (SSAE) in which each hidden layer is the feature set pretrained by one sparse autoencoder, and a classifier forms the output layer.
Training proceeds greedily, one layer at a time. At step n, the output of the already-trained n−1 layers becomes the input of layer n, and the weights so obtained initialize the full stack. All layers are then unrolled and the weights are optimized jointly on labelled samples, a fine-tuning stage that adapts the unsupervised representation to the discrimination task. The result is a hierarchy in which vibration signals are compressed layer by layer into a low-dimensional fault signature.
Test bench and fault modes
The experiments were carried out on a dedicated bevel-gear fault-simulation bench comprising a variable-speed drive motor, a bevel-gear test gearbox, a speed controller and a magnetic powder brake that sets the load. The gearbox contains two bevel-gear stages, and the fault specimens were installed in the first stage, i.e. the input stage. Four fault modes were introduced on the first-stage pinion and on the input bearing: tooth pitting of the bevel gear, tooth cracking of the bevel gear, bearing inner-race defect and bearing outer-race defect. Combining single and compound damage with the healthy condition gave seven states in total. Each state was recorded under four operating conditions, with 360 samples per state.
Signal preprocessing and network configuration
Setting the input-layer width equal to the raw sample length would make training impractically slow, so each time-domain record was transformed by a fast Fourier transform and only the first 2 000 points of the spectrum were retained as the network input. The SSAE was configured as 2 000–1 000–500–200–8–7: an input layer, four hidden layers and an output layer of seven nodes, one per health state. Each hidden layer was first pretrained by encoding and decoding, and the whole stack was then fine-tuned on the training samples. The 8-dimensional output of the innermost hidden layer served as the fault feature vector for the downstream classifier comparison.
Feature extraction over the seven states produced 2 520 eight-dimensional feature vectors in total, 360 per state. For each state, 270 samples were drawn at random for training and the remaining 90 for testing.
| Health state | Samples | Training | Testing | Operating conditions |
|---|---|---|---|---|
| Healthy | 360 | 270 | 90 | 4 |
| Gear tooth crack | 360 | 270 | 90 | 4 |
| Gear tooth pitting | 360 | 270 | 90 | 4 |
| Bearing inner-race defect | 360 | 270 | 90 | 4 |
| Bearing outer-race defect | 360 | 270 | 90 | 4 |
| Tooth crack + inner-race defect | 360 | 270 | 90 | 4 |
| Tooth pitting + outer-race defect | 360 | 270 | 90 | 4 |
| Total | 2 520 | 1 890 | 630 | — |
Class separability of the learned features
Two-dimensional embeddings of the 8-dimensional features produced by t-distributed stochastic neighbour embedding show the seven states forming compact, well-separated clusters. Samples of the same state concentrate closely, and the distance between cluster centres is large relative to the within-cluster spread, with the clusters arranged in a regular regional pattern. In other words, the compression performed by the SSAE preserves — and in this projection, makes explicit — the distinctions that the classifiers subsequently exploit. This is the qualitative property that matters for a gearbox diagnosis chain: the representation itself, not the classifier, does most of the discrimination.
Classification results
Three classifiers were applied to the same SSAE features: a Softmax layer, an extreme learning machine (ELM), and a support vector machine whose penalty and kernel parameters were tuned by particle swarm optimization (PSO-SVM).
| Health state | Softmax | ELM | PSO-SVM |
|---|---|---|---|
| Healthy | 100 | 100 | 100 |
| Gear tooth crack | 100 | 100 | 100 |
| Gear tooth pitting | 100 | 100 | 100 |
| Bearing inner-race defect | 96.7 | 97.8 | 100 |
| Bearing outer-race defect | 100 | 100 | 100 |
| Tooth crack + inner-race defect | 100 | 100 | 100 |
| Tooth pitting + outer-race defect | 100 | 100 | 100 |
| Overall | 99.5 | 99.7 | 100 |
All three classifiers recognize the gear-origin states perfectly, including the two compound states in which a gear defect coexists with a bearing defect. The only errors are confined to the bearing inner-race class: 96.7% for Softmax and 97.8% for ELM correspond to three and two misclassified samples out of 90, and in both cases the errors are inner-race samples assigned to the outer-race class. The confusion is therefore a specific pairwise overlap in the learned feature space rather than a diffuse degradation. Because the two bearing defects differ mainly in the modulation imposed by whether the fault zone passes through the loaded region, a compact 8-dimensional code can plausibly lose part of that distinction; the stronger margin-based decision boundary of PSO-SVM apparently recovers it, giving 100% on this test set.
Softmax and ELM perform almost identically in accuracy, but ELM reaches its decision with a much cheaper training procedure, since its hidden-layer weights are set without iterative backpropagation. That matters when the model must be retrained as operating conditions accumulate. PSO-SVM is the most accurate but pays for it with an optimization loop over the classifier hyperparameters.
Interpretation and scope
Three conclusions follow from the results. First, layer-wise stacking of sparse autoencoders yields a representation in which seven health states of an input-stage spiral bevel gear — healthy, two gear defects, two bearing defects and two compound defects — are separable across four operating conditions, without any hand-designed fault indicator and without prior knowledge of the defect orders. Second, the choice of classifier on top of that representation accounts for a residual difference of at most half a percentage point; the representation, not the classifier, carries the diagnosis. Third, the confusion that remains is structurally interpretable, concentrated between the two bearing raceways, which suggests where additional feature capacity or a dedicated binary discriminator would be best spent.
The stated recognition rates apply to a closed test set of 630 samples drawn from the same bench and the same four operating conditions as the training data, with a 3:1 train–test split. They describe separability within that envelope and should not be read as a guarantee for gears of other geometry, for untried speed and torque combinations, or for signals recorded with different sensor mounting. Extending the validation to varying speed–load trajectories and to a held-out gear set is the natural next step before the method is used as a maintenance decision tool on an actual helicopter transmission.