Spiral bevel gears are widely used where torque must be transmitted between intersecting shafts with smooth motion, a high transmission ratio and a compact layout. Their curved, inclined teeth engage gradually, which raises the contact ratio and reduces impact compared with straight bevel teeth. These properties explain their presence in ship propulsion systems, helicopter drivetrains, aeroengines and automotive axle drives. The same heavy and variable loading that makes the gear type useful also makes tooth damage a maintenance concern: as damage progresses, vibration and transmission error increase, and a severe fault can threaten the whole drivetrain.
The monitoring question is therefore not only whether a gear is damaged, but how far the damage has progressed. A classifier that separates normal, lightly damaged and moderately damaged gears can support inspection planning and spare-part decisions. The signal problem is difficult: meshing vibration is modulated by shaft rotation and load, the background is noisy, and the waveform is nonlinear and nonstationary. Time-domain statistics or a single spectrum can smear the weak damage signature. A two-dimensional representation that preserves modulation coherence, followed by a learner that can exploit spatial patterns, offers a practical alternative.
Modulation signal bispectrum as a physics-guided image transform
Gear vibration is not a pure tone. A carrier component around a mesh-related frequency is amplitude- and phase-modulated by shaft rotation, load fluctuation and local tooth defects. A defect that repeats once per revolution or once per tooth engagement generates sidebands at fc + fx and fc − fx, where fc is the carrier frequency and fx is the modulation frequency. The modulation signal bispectrum (MSB) multiplies the upper and lower sidebands with the conjugate carrier and averages the product. A coherent sideband pair survives; random noise tends to average out.
For a discrete vibration block x[n], the Fourier transform is
$$X(f)=\sum_{n=0}^{N-1}x[n]e^{-j2\pi fn/N}$$
and the MSB is defined as
$$B_{MS}(f_c,f_x)=E\left[X(f_c+f_x)X(f_c-f_x)\left(X^*(f_c)\right)^2\right]$$
where E[·] denotes the expectation over the segment ensemble and * denotes complex conjugation. The amplitude and phase of the MSB are
$$A_{MS}(f_c,f_x)=\left|B_{MS}(f_c,f_x)\right|$$
$$\phi_{MS}(f_c,f_x)=\phi(f_c+f_x)+\phi(f_c-f_x)-2\phi(f_c)$$
The amplitude map is the image presented to the classifier; the phase relation explains why a peak in that map represents a phase-coherent sideband pair rather than an isolated spectral line. Stationary Gaussian noise has random phase and is suppressed by the averaging. Broadband nonstationary disturbances that do not maintain the sideband relation are also reduced relative to coherent modulation. This property is important for gear diagnostics, where the damage signature is often weak compared with mesh harmonics and background vibration.
To construct an image sample, each vibration block is transformed and the MSB is evaluated on the carrier–modulation plane. The default isometric surface view can hide weak peaks behind stronger ones, so a top view is used to project the amplitude onto the plane and retain all visible components. The projected map is resized to 128 × 128 and stored as a single-channel image. The MSB stage is a fixed transform; the learned part of the pipeline is the convolutional neural network (CNN) that maps the image to a severity class.
Test bench and sample construction
The validation used a spiral bevel gear system fault simulation bench. The gearbox was fitted with three input-shaft gear sets representing normal, light damage and moderate damage. An accelerometer on the input shaft measured vibration acceleration, and a BK test system recorded the signals at a sampling frequency of 3.2 kHz while the shaft ran at 900 r/min, corresponding to a 15 Hz shaft rate. The damage labels were defined by the physical gear sets, not by an automated threshold on the vibration signal.
Each record was split into 1024-point blocks, equal to 0.32 s at 3.2 kHz. Every block produced one MSB amplitude map, which was converted to a top-view image, resized to 128 × 128 × 1 and normalized. The balanced data set contained 500 images per class. For each class, 150 images were randomly selected for training and the remaining 350 for testing. The resulting split contained 450 training images and 1050 test images across the three conditions.
CNN classifier and training protocol
The classifier is a compact CNN with one input layer, three convolution layers, two max-pooling layers, a flatten layer, two fully connected layers and a three-unit softmax output. The input is a 128 × 128 × 1 MSB map. The convolution layers use 8, 16 and 32 filters of size 3 × 3 with stride 1, followed by ReLU activation. Batch normalization is applied after each convolution to improve convergence and stability. Two 2 × 2 max-pooling layers follow the first two convolutions, and dropout is used in the fully connected layers to reduce overfitting. Weight sharing in the convolution layers keeps the number of trainable parameters modest relative to a fully connected image network.
| Layer | Channels or units | Kernel or pool size | Activation |
|---|---|---|---|
| Input | 1 | – | – |
| Convolution 1 | 8 | 3 × 3 | ReLU |
| Max pooling 1 | 8 | 2 × 2 | – |
| Convolution 2 | 16 | 3 × 3 | ReLU |
| Max pooling 2 | 16 | 2 × 2 | – |
| Convolution 3 | 32 | 3 × 3 | ReLU |
| Flatten | – | – | – |
| Fully connected 1 | – | – | ReLU |
| Fully connected 2 | – | – | ReLU |
| Output | 3 | – | Softmax |
Batch normalization follows convolutions 1–3; dropout is applied in the fully connected layers. Hyperparameters were selected through a hierarchical search, giving 90 training epochs, a learning rate of 0.000 1, a mini-batch size of 50, 3 × 3 convolution kernels and 2 × 2 pooling. Each configuration was repeated 100 times, and the recognition accuracy was averaged over the repeats. Training times are recorded values from the corresponding runs rather than a fixed hardware benchmark.
Recognition results
The MSB-CNN confusion matrix was strongly diagonal: most test blocks were assigned to the correct severity class, with few off-diagonal entries. The comparison with a raw-vibration input isolates the practical value of the MSB image pipeline. With the same CNN classifier family, replacing raw vibration by MSB maps raised the mean accuracy from 93.53% to 99.91%, an increase of 6.38 percentage points, and reduced the recorded training time from 107 s to 67 s in that batch.
| Input to the CNN | Test samples | Mean accuracy (%) | Training time (s) |
|---|---|---|---|
| MSB amplitude maps | 1050 | 99.91 | 67 |
| Raw vibration | 1050 | 93.53 | 107 |
The raw signal is one-dimensional and noise-rich, so the network must learn both damage cues and nuisance variation from the same sequence. The MSB map presents coherent sideband structure as local texture, which convolution filters can exploit more directly. Because the raw-signal branch uses a one-dimensional input, this comparison evaluates the complete MSB-image pipeline rather than the transform in isolation.
The MSB representation was also tested with two alternative classifiers: a support vector machine with an RBF kernel and a back-propagation (BP) neural network. The RBF-SVM used a kernel parameter δ = 3 and a penalty factor C = 4. The BP network used 32 hidden-layer nodes and ReLU activation. The CNN matched the best SVM accuracy within 0.49 percentage points while cutting training time by more than an order of magnitude. Against the BP network, the CNN improved accuracy by 10.54 percentage points and trained about 6.4 times faster.
| Classifier | Test samples | Mean accuracy (%) | Training time (s) |
|---|---|---|---|
| CNN | 1050 | 99.91 | 97 |
| RBF-SVM | 1050 | 99.42 | 2374 |
| BP network | 1050 | 89.37 | 621 |
The CNN training time differed between the two comparison batches, 67 s and 97 s. This variation is small relative to the gap between the CNN and the SVM or BP network and does not change the ranking. The SVM result also shows that MSB maps are informative even for a kernel method: its accuracy was close to that of the CNN. The main practical advantage of the CNN in that comparison was therefore computational, not a large accuracy margin. The BP network’s lower accuracy is consistent with its shallow structure, which limits its ability to build hierarchical image features from the MSB maps.
Interpretation and limits
The result supports a representation argument more than a classifier argument. MSB does not create new information; it rearranges phase-coherent modulation into a domain where local filters can detect it. The CNN then learns the mapping from sideband patterns to severity. Convolution weight sharing reduces sensitivity to small translations, pooling compresses the feature maps, and batch normalization and dropout stabilize training. The MSB stage remains a fixed, interpretable transform, while the CNN supplies the decision rule that would otherwise require manual feature engineering.
The validation is limited to one test bench, one shaft speed of 900 r/min and three discrete conditions. It does not establish performance under variable load, different speeds, other damage modes or very early damage. The reported 99.91% is an average over repeated runs on this laboratory data set, not a field guarantee. Cross-unit generalization was not evaluated, and the output is a severity class rather than a continuous remaining-life estimate. Deployment on a target gearbox would require validation or retraining on the relevant operating profile and damage population.
Conclusion
MSB amplitude maps combined with a compact CNN provide a direct route from short vibration blocks to spiral bevel gear damage severity. In the three-class test, the MSB-CNN model reached a mean accuracy of 99.91%, compared with 93.53% for raw vibration with a CNN classifier. It also trained faster than the MSB-SVM and MSB-BP alternatives while outperforming the BP network and matching the SVM closely in accuracy. Within the tested speed and damage range, the evidence supports MSB maps as an effective input representation for CNN-based severity recognition of spiral bevel gears.