Severity, not mere presence, is the difficult target
Spiral bevel gears carry the final right-angle reduction in marine propulsion, helicopter rotor drives, aero-engines and automotive drivelines because they combine high torque capacity, a large transmission ratio, smooth meshing and a compact envelope. Their contact geometry, however, is intrinsically time-varying: the contact path sweeps across the flank while the number of engaged teeth and the local mesh stiffness change continuously, so the measured casing acceleration is already a strongly modulated, non-stationary signal before any defect exists. Tooth-surface damage modifies that modulation pattern rather than adding an isolated impulse, and it does so in the presence of broadband interference from bearings, shafts and structure.
Discriminating a lightly damaged tooth from a moderately damaged one is consequently harder than deciding whether damage is present at all. The two conditions differ mainly in the strength and distribution of modulation components around the mesh harmonics, and a classifier fed raw time samples must recover that weak difference directly from a noisy waveform. Yet it is precisely the severity level that determines the maintenance decision: a light defect may justify continued operation under increased monitoring, whereas a moderate defect calls for scheduled replacement before secondary damage propagates into the bearings and the shafting.
Two ingredients are therefore needed. The signal representation must concentrate the small modulation signatures and suppress the noise that masks them, and the classifier must map that representation onto severity classes without hand-crafted feature selection, since manually designed indicators have repeatedly failed to capture the complex relationship between damage state and measured vibration.
Modulation signal bispectrum as a demodulating transform
The modulation signal bispectrum (MSB) is a higher-order spectral quantity built specifically to detect amplitude and phase modulation of a carrier by a lower-frequency process. For a discrete vibration signal $x(t)$ with Fourier transform
$$X(f)=\sum_{t=-\infty}^{\infty}x(t)\,e^{-j2\pi ft},$$
the MSB is defined as the expectation of a four-term product in which two sideband components straddle a carrier:
$$B_{MS}(f_c,f_x)=E\left[X(f_c+f_x)\,X(f_c-f_x)\,X^{*}(f_c)\,X^{*}(f_c)\right],$$
where $f_c$ is the carrier frequency, taken at a mesh harmonic or shaft harmonic, $f_x$ is the modulation frequency, $f_c+f_x$ and $f_c-f_x$ are the upper and lower sidebands, $X^{*}$ denotes the complex conjugate, and $E[\cdot]$ is the mathematical expectation. A component at $f_c$ survives the product only if symmetric sidebands exist at $f_c\pm f_x$ with a consistent phase relationship; the phase of the MSB is obtained from the sideband phases with the carrier phase removed, so segments whose modulation phase is unstable are attenuated. Normalising by the carrier power gives the sideband estimate
$$B_{MS}^{SE}(f_c,f_x)=E\left[\frac{X(f_c+f_x)\,X(f_c-f_x)\,X^{*}(f_c)\,X^{*}(f_c)}{\left|X(f_c)\right|^{2}}\right],$$
which removes the influence of carrier amplitude and makes sideband strength comparable across operating conditions. Because the estimator is a higher-order moment, stationary Gaussian noise averages toward zero, phase-random non-Gaussian interference is likewise suppressed, and the retained energy is concentrated in the deterministic modulation produced by the gear mesh and its interaction with damage.
The practical consequence is a two-dimensional map indexed by carrier frequency $f_c$ and modulation frequency $f_x$. Fault-related modulation appears as structured ridges in that plane, at positions determined by the mesh and shaft kinematics rather than by the absolute noise level, which is the property that makes the map suitable as an image-like input to a convolutional network.
From 1024-sample segments to 128 × 128 image samples
Each recorded signal is cut into segments of 1 024 samples, and the MSB is computed for every segment. The default isometric rendering of the resulting surface is unsuitable for classification because the perspective hides part of the feature plane, so the surface is rotated to a plan view that exposes the full $(f_c,f_x)$ distribution. The plan view is then rendered as an RGB bitmap, converted to a single-channel image and resized to 128 × 128 pixels, giving one sample per segment. Because the transform is applied segment by segment, the sample set inherits the operating condition of the recording and no information is shared between training and test segments.
This step is where most of the noise rejection occurs. A raw acceleration segment at 3.2 kHz mixes mesh harmonics, shaft orders, bearing tones and broadband excitation in a single vector, and a network must learn band-selective filters before it can see the damage signature. The MSB map has already performed that selection analytically, so the convolutional filters operate on a representation in which the discriminant information is expressed as spatial structure rather than as a small amplitude difference buried in a waveform.
Network architecture and training settings
The classifier is a compact convolutional neural network (CNN): one input layer, three convolutional layers, two max-pooling layers, a flatten layer, two fully connected layers and a softmax output over the three gear states. Rectified linear units are used throughout the convolutional and fully connected stages. Batch normalisation follows each convolutional layer to accelerate convergence and stabilise training, and dropout is applied in the fully connected stage to limit overfitting on a data set of moderate size. Max pooling is used rather than average pooling to shorten training. The layer configuration is summarised in Table 1.
The convolution output at layer $l$ for neuron $j$ is
$$a_j^{l}=\sigma\left(\sum_{i\in M_j}a_i^{l-1}\,\omega_{ij}^{l}+b_j^{l}\right),$$
where $a_i^{l-1}$ is the input from the previous layer, $M_j$ the input feature map, $\omega_{ij}^{l}$ the kernel weight, $b_j^{l}$ the bias and $\sigma$ the activation function. The fully connected stage computes
$$h(X)=\sigma’\left(b’+\omega’X\right),$$
with $X$ the flattened feature vector, $\omega’$ the weight matrix, $b’$ the bias vector and $\sigma’$ the activation. Key hyperparameters were selected by hierarchical optimisation rather than exhaustive search: 90 epochs, learning rate $1\times10^{-4}$, mini-batch of 50 segments, 3 × 3 kernels and 2 × 2 pooling windows.
| Layer | Filters / channels | Kernel (pool) size | Stride | Activation |
|---|---|---|---|---|
| Input | 1 (128 × 128) | — | — | — |
| Convolution 1 | 8 | 3 × 3 | 1 × 1 | ReLU |
| Max pooling 1 | 8 | 2 × 2 | 1 × 1 | — |
| Convolution 2 | 16 | 3 × 3 | 1 × 1 | ReLU |
| Max pooling 2 | 16 | 2 × 2 | 1 × 1 | — |
| Convolution 3 | 32 | 3 × 3 | 1 × 1 | ReLU |
| Flatten | — | — | — | — |
| Fully connected 1 | — | — | — | ReLU |
| Fully connected 2 | — | — | — | ReLU |
| Output | 3 | — | — | Softmax |
Test bench, gear sets and data set
Vibration was acquired on a spiral bevel gear fault-simulation bench, with the damaged gear mounted on the input shaft and a single accelerometer on the same shaft so that the fault path lies between the source and the sensor. Three gear sets were used: an undamaged reference, a lightly damaged set and a moderately damaged set, the latter two produced by introducing progressive tooth damage. Signals were recorded with a Brüel & Kjær data-acquisition system at a sampling frequency of 3.2 kHz and a shaft speed of 900 r/min, which places the mesh harmonics well inside the analysed band while keeping the record length short enough for segment-wise processing.
Each condition contributed 500 segments, giving 1 500 samples in total. For every condition, 150 segments were drawn at random for training and the remaining 350 for testing, so the model was trained on 450 samples and evaluated on 1 050. Classes are labelled 0, 1 and 2 for the undamaged, lightly damaged and moderately damaged gears.
Recognition results
Two comparisons isolate the contribution of the representation and the contribution of the classifier. In the first, the same CNN was trained on raw one-dimensional vibration segments and on MSB plan-view images. To remove run-to-run variability, each configuration was repeated 100 times and the reported figure is the mean recognition accuracy. In the second, the MSB representation was held fixed and the classifier was varied between the CNN, a support vector machine with an RBF kernel (kernel parameter $\delta=3$, penalty factor $C=4$) and a back-propagation network with 32 hidden nodes and ReLU activation.
| Input to CNN | Test samples | Mean accuracy (%) | Training time (s) |
|---|---|---|---|
| MSB plan-view images | 1 050 | 99.91 | 67 |
| Raw vibration segments | 1 050 | 93.53 | 107 |
The representation change moves accuracy from 93.53 % to 99.91 % and shortens training from 107 s to 67 s: the MSB map both removes interference, so fewer epochs are needed to reach a decision boundary, and reduces the burden on the first convolutional layer. The confusion matrices for the two inputs show the same effect in per-class terms — the raw-signal model leaves a visible off-diagonal population, whereas the MSB model is almost entirely diagonal, with the residual errors distributed across the three classes rather than concentrated in a single confused pair.
| Method | Test samples | Mean accuracy (%) | Training time (s) |
|---|---|---|---|
| MSB + CNN | 1 050 | 99.91 | 97 |
| MSB + SVM (RBF) | 1 050 | 99.42 | 2 374 |
| MSB + BP network | 1 050 | 89.37 | 621 |
Holding the representation fixed confirms that the gain is not only a preprocessing effect. The SVM reaches 99.42 % on the same images but requires 2 374 s to train, more than an order of magnitude above the CNN, because a multi-class SVM is built from a set of binary classifiers and each of them is fitted on its own label subset. The BP network, which is shallow, has far less capacity to exploit the spatial arrangement of the MSB map; its accuracy falls to 89.37 %. The CNN combines depth with weight sharing, so its parameter count stays manageable while its nonlinear mapping is richer, and it delivers the highest accuracy at the lowest training cost. The two comparison tables list 67 s and 97 s for the same MSB–CNN pipeline; the difference reflects separate training runs and timing scopes rather than a change of configuration, and either value remains far below the SVM figure.
Engineering interpretation and limits of validity
For condition-based maintenance, the relevant output is not a binary alarm but a graded state that can be mapped onto inspection intervals and spares planning. A pipeline that returns a severity class from a 1 024-sample window is compatible with online monitoring, since the MSB of a short segment is cheap to compute relative to the network inference, and the accuracy gap between adjacent severity classes is what limits the usefulness of simpler detectors.
Several constraints bound the result. The data set covers a single shaft speed of 900 r/min, one gear pair geometry and three discrete states, and the load level of the bench is not part of the reported configuration, so generalisation across speed, torque and tooth geometry remains unverified; mesh harmonics shift with speed, and the $(f_c,f_x)$ regions that carry the discriminant information would move accordingly. Damage is represented by two severity levels only, so the method has been shown to separate classes, not to estimate remaining life or continuous damage depth. The three classes also differ substantially in physical appearance, which makes the near-ceiling accuracy partly a measure of task separability; performance on early-stage pitting or on wear that has not yet broken the tooth surface would be a harder test. Finally, image-based inputs impose a fixed 128 × 128 resolution and a plan-view rendering step, so attenuation of small sidebands by resampling needs to be checked when the analysis band or the segment length is changed.
The main design lesson is that analytical demodulation and learned classification are complementary rather than competing. The MSB encodes the physical structure of mesh modulation — symmetric sidebands around a carrier with stable phase — and the CNN learns which parts of that structure separate one severity level from the next without hand-specified indicators. Extending the same scheme to variable speed and load, and to a finer grading of damage progression, is the natural next step for turning a three-class demonstration into a monitoring tool.