Spiral bevel gears transmit motion between intersecting shafts and are widely used in vehicle drivelines, aircraft accessory drives, and marine propulsion. Their curved teeth and gradual contact pattern distribute load across the tooth surface, giving high load capacity and a stable transmission ratio. The same geometry makes vibration-based condition monitoring difficult. Mesh force travels across the tooth face, several teeth contribute to the measured signal at once, and the vibration record contains mesh harmonics, structural resonances, and broadband excitation. A broken tooth or root crack adds weak, nonlinear impulses that are easily buried in this background.
Shallow classifiers such as back-propagation (BP) networks, support vector machines (SVM), and nearest-neighbor methods can separate faults when handcrafted features are informative. They do not, however, learn hierarchical representations directly from image-like data. A convolutional neural network (CNN) can learn such representations, but only if the input preserves fault information and keeps the learning problem tractable. The design question is therefore not simply whether to use a CNN, but what representation to feed it.
The diagnostic problem in spiral bevel gears
Spiral bevel gear faults often develop under high load and in enclosed transmission housings. A root crack changes local tooth stiffness and modifies the contact pattern; a fractured tooth removes part of the load-bearing surface and introduces intermittent impact. These changes are nonlinear and nonstationary, and they occur against a strong narrowband mesh background. Conventional time-domain statistics or spectral peaks may shift with load and speed, while features selected for one gear set may not transfer to another.
A useful input representation should therefore satisfy three conditions: it should retain amplitude and phase information, it should suppress the influence of Gaussian background noise, and it should be compact enough for a CNN to train on a limited number of labeled examples. The bispectrum meets the first two requirements; cropping the bispectrum plane addresses the third.
Bispectrum: a third-order representation that preserves phase
For a zero-mean, real, stationary vibration segment \(x(n)\), the third-order moment is
$$R(\tau_1,\tau_2)=E\left\{x(n)x(n+\tau_1)x(n+\tau_2)\right\}$$
where \(E\{\cdot\}\) denotes expectation. For zero-mean signals, this moment is equivalent to the third-order cumulant. The bispectrum is its two-dimensional Fourier transform:
$$B(\omega_1,\omega_2)=\sum_{\tau_1=-\infty}^{\infty}\sum_{\tau_2=-\infty}^{\infty}R(\tau_1,\tau_2)e^{-j(\omega_1\tau_1+\omega_2\tau_2)}$$
A direct estimate from the discrete Fourier transform \(X(\omega)\) is
$$\hat{B}(\omega_1,\omega_2)=X(\omega_1)X(\omega_2)X^{*}(\omega_1+\omega_2)$$
where the asterisk denotes complex conjugation. Unlike the power spectrum, which is real and phase-blind, the bispectrum is complex and preserves phase relationships among frequency components. It is sensitive to quadratic phase coupling, a signature of nonlinear interactions such as intermittent tooth contact or a breathing crack.
The bispectrum also has a noise-suppression property. For a stationary Gaussian process, the third-order cumulant vanishes, so Gaussian noise—white or colored—has zero expected bispectrum. A finite record still produces a noisy estimate, but Gaussian components are attenuated relative to non-Gaussian fault transients. This property is useful for spiral bevel gears because the measured signal is dominated by complex background noise and mesh harmonics rather than by a single clean fault frequency.
Local bispectrum as a compact CNN input
The bispectrum plane is highly redundant. For real signals, \(B(\omega_1,\omega_2)=B(\omega_2,\omega_1)\), and conjugate symmetries repeat information in other regions of the plane. A single quadrant therefore contains the principal nonredundant structure needed for classification, together with a limited number of redundant points. Cropping the top-left quadrant to \(128\times128\) pixels gives a compact image while avoiding the repetition across the full plane. In this context, “local” denotes a selected spatial region of the bispectrum, not a time-localized short-time bispectrum.
The crop has two practical effects. First, it reduces the input area by a factor of four relative to the full plane, which lowers the computational burden of CNN training. Second, it keeps the representation two-dimensional and phase-sensitive, unlike a one-dimensional raw vibration segment in which noise and fault impulses are mixed along a single axis. The trade-off is that the crop is fixed; if the relevant phase coupling were concentrated outside the selected quadrant, information would be lost. For a symmetric bispectrum of a real signal, however, the quadrant is a reasonable compact representation.
CNN architecture and training protocol
The CNN used in the experimental evaluation contained three convolutional layers, two activation layers, two pooling layers, and one fully connected layer. A convolutional layer computes
$$a_j^{l}=\sigma\left(\sum_{i\in M_j}a_i^{l-1}\ast w_{ij}^{l}+b_j^{l}\right)$$
where \(a_j^{l}\) is the output of neuron \(j\) in layer \(l\), \(M_j\) is the set of input feature maps, \(\ast\) denotes convolution, \(w_{ij}^{l}\) is the kernel weight, \(b_j^{l}\) is the bias, and \(\sigma\) is the activation function. Each kernel slides across the input with a fixed stride and produces one feature map. Pooling then reduces the spatial dimensions, and the fully connected layer maps the resulting features to four output classes through a softmax layer.
The input was a \(128\times128\) local-bispectrum image. The first convolutional layer used eight \(3\times3\) kernels with stride \(1\times1\), followed by \(2\times2\) pooling with stride \(2\times2\). The second convolutional layer used 16 \(3\times3\) kernels, followed by another \(2\times2\) pooling layer. The third convolutional layer used 32 \(3\times3\) kernels. The model was trained for 60 epochs with a learning rate of 0.0001 and a minibatch size of 50. Weight sharing in the convolutional layers keeps the number of trainable parameters much smaller than in a fully connected network of comparable width.
Four-class gear fault experiment
The experimental evidence comes from a four-class test on a spiral bevel gear reducer. Faulted gears were mounted on the input shaft, and an accelerometer on the same shaft measured vibration through a BK measurement system at a sampling frequency of 8,192 Hz. The input shaft ran at 1,200 r/min. Four conditions were tested: healthy, one-third tooth fracture, two-thirds tooth fracture, and root crack.
Each condition was divided into 500 segments of 1,024 samples, corresponding to 0.125 s per segment at 8,192 Hz. This produced 2,000 segments in total. A bispectrum was computed for each segment and cropped to a \(128\times128\) local image. From each class, 150 samples were randomly selected for training, giving 600 training samples, and the remaining 350 samples per class formed the test set of 1,400 samples.
Measured performance
After 60 epochs, a single training run reached 99.50% accuracy on the 1,400 test samples, with the loss approaching zero. Repeating the complete experiment 100 times gave an average accuracy of 99.56%. In one representative run, the local-bispectrum CNN classified 348, 348, 350, and 347 of the 350 test samples in the four classes, making seven errors in total.
| Method | Input representation | Average accuracy (%) | Training time (s) |
|---|---|---|---|
| Local bispectrum + CNN | \(128\times128\) cropped bispectrum quadrant | 99.56 | 15 |
| Full bispectrum + CNN | Complete bispectrum plane | 99.53 | 78 |
| Raw vibration + CNN | One-dimensional time signal | 92.65 | 18 |
| Local bispectrum + SVM | \(128\times128\) cropped bispectrum quadrant | 98.03 | 2,763 |
| Local bispectrum + BP | \(128\times128\) cropped bispectrum quadrant | 88.91 | 631 |
The full-bispectrum CNN reached 99.53%, only 0.03 percentage points below the local-bispectrum model, but required 78 s for training instead of 15 s. The local crop therefore reduced training time by about 81% while preserving essentially the same accuracy in this test. The raw-vibration CNN trained in 18 s but reached only 92.65%, showing that a fast one-dimensional representation did not retain the class information as effectively as the local bispectrum.
The local-bispectrum SVM reached 98.03% but required 2,763 s, about 184 times the training time of the local-bispectrum CNN. The SVM used an RBF kernel with \(\delta=3\) and penalty factor \(C=4\), selected by cross-validation. The BP network used 32 hidden nodes with ReLU activation and reached 88.91% after 631 s. Its shallow architecture limited its ability to separate the spatial patterns in the bispectrum images.
Why local bispectrum plus CNN is efficient
Three mechanisms explain the measured pattern. First, the local crop removes the redundancy created by bispectrum symmetry. The full plane repeats information across quadrants, so a CNN trained on the full plane spends computation on repeated patterns. The cropped input reduces the number of pixels and the scale of the convolution operations while retaining the phase-coupling structure used by the classifier.
Second, the bispectrum converts nonlinear fault signatures into a two-dimensional texture. Convolutional filters can learn local orientation and contrast patterns across that texture, and pooling provides a degree of translation tolerance. A raw one-dimensional waveform contains the same physical information in principle, but the network must learn noise rejection and fault separation directly from amplitude samples. With only 600 training samples, that is a harder learning problem.
Third, the CNN’s weight sharing and depth give it an advantage over shallow classifiers on image-like inputs. The SVM depends heavily on the quality of the fixed bispectrum features and constructs pairwise binary classifiers, which increases training cost. The BP network has a single hidden layer and limited nonlinear representation capacity. The CNN learns filters and classifiers jointly, so its training time is lower than the SVM’s and its accuracy is higher than the BP network’s on this dataset.
Scope, limitations, and deployment considerations
The evidence supports a specific conclusion: for this spiral bevel gear set, four discrete conditions, and a test speed of 1,200 r/min, a local-bispectrum CNN is an accurate and computationally efficient classifier. It does not establish that the same model transfers to other gear geometries, sensor locations, speed profiles, or load ranges. Early faults that do not yet produce measurable phase coupling may also be harder to detect than the fracture classes tested here.
The reported 15 s is model-training time, not inference latency. A deployed condition-monitoring system would need separate measurements of inference cost, memory use, and robustness on the target hardware. The bispectrum also assumes a stationary segment; strong speed fluctuations smear the spectrum and weaken the phase-coupling structure. Variable-speed applications may therefore require order tracking or angular resampling before segmentation.
The test set was balanced, with 350 samples per class, so overall accuracy is a reasonable summary of performance. Field data are rarely balanced, and the cost of a missed root crack differs from the cost of a false alarm. A practical system would need class-weighted metrics, confidence thresholds, and maintenance records for validation. The CNN is also a black-box classifier; the bispectrum input gives the model a physically meaningful basis, but the learned filters are not directly interpretable as specific fault mechanisms.
What the results support
The local bispectrum is a useful compromise between physical interpretability and computational efficiency for spiral bevel gear diagnosis. It retains the phase-sensitive, Gaussian-noise-resistant properties of the bispectrum while removing the redundancy that slows CNN training. On the four-class test, the local-bispectrum CNN achieved 99.56% average accuracy with 15 s of training, outperforming the full-bispectrum, raw-vibration, SVM, and BP alternatives in combined accuracy and training cost.
The broader engineering value is a fast offline classifier for gearbox test stands and maintenance diagnostics, where labeled fault examples are available and the operating condition is controlled. Before the method can be treated as a general diagnostic solution, it needs validation across speeds, loads, gear geometries, sensor placements, and incipient fault sizes—conditions that determine whether the phase-coupling signature remains stable outside the laboratory test matrix.