Double circular-arc spiral bevel gears are used where a compact final drive must transmit high torque quietly: the tooth flank is built from two circular arcs, one convex and one concave, so the pair contacts along a long, rolling line rather than at a concentrated point. The result is high load capacity, low noise and a contact pattern that tolerates misalignment. The same geometry is what makes incipient tooth-root cracks so hard to see in a vibration record. Contact line length and contact position change continuously through the mesh, so the mesh stiffness is a strongly nonlinear, time-varying quantity even at constant speed; when the speed itself is ramped, the excitation frequency sweeps with it and the crack signature is buried under a moving forest of mesh harmonics.
Why Variable Speed Erases the Crack Signature
Consider the local contact between a pinion and a wheel tooth. With a normal load Fn, an equivalent elastic modulus E′ and principal radii of curvature R1 and R2 at the contact point, Hertz theory gives the half-width of the contact ellipse as
$$a=\sqrt[3]{\frac{6F_n R_1R_2}{\pi E'(R_1+R_2)}}$$
and the local mesh stiffness follows from the elastic approach δ(t) of the contact zone:
$$k(t)=\frac{F_n}{\delta(t)}$$
Both radii of curvature vary along the path of contact, and for a double circular-arc profile they vary over a wider range than for an involute flank. The consequence is that k(t) is periodic but distinctly non-sinusoidal, containing strong higher harmonics of the mesh period. Inserting this stiffness into the single-degree-of-freedom model of the gear pair,
$$m\ddot{x}(t)+c\dot{x}(t)+k(t)x(t)=F(t)$$
with equivalent mass m, damping c, vibration response x(t) and external excitation F(t) from mesh impact and transmission error, shows that the stiffness variation acts as a parametric excitation. Under variable speed the mesh frequency itself moves, fm(t) = zω(t)/2π for a wheel of z teeth, so every harmonic of the mesh smears into a band and the spectrum loses the sharp comb that conventional demodulation relies on.
A tooth-root crack is a once-per-revolution event for a single damaged tooth: each time that tooth enters the mesh it produces a short impact, and the impact amplitude is modulated by the mesh stiffness. In the frequency domain the information therefore sits in sidebands spaced at shaft rate around the mesh harmonics, not at the mesh frequency itself. For the gear pair used in the validation below (26 pinion teeth, 500–3000 r/min) the pinion mesh frequency moves from roughly 217 Hz to 1300 Hz, sweeping across more than two and a half octaves in a single test run. A weak sideband family riding on that moving carrier is easily an order of magnitude below the mesh peaks, and any detector that thresholds raw spectral amplitude will lock onto the mesh instead of the crack.
Resonance-Based Decomposition of the Vibration Signal
The detection chain starts by separating the measured record into components with different resonance character rather than different nominal frequency ranges. The acceleration signal is modelled as
$$x(t)=x_h(t)+x_l(t)+n(t)$$
where xh is the high-Q, sustained-oscillation component dominated by mesh resonance, xl is the low-Q component that carries transient, impulse-like content, and n is broadband noise. A two-channel multi-rate filter bank with analysis filters H(ω), synthesis filters G(ω) and decimators Dh, Dl is used to build two wavelet dictionaries: a high-quality-factor basis tuned to sustained tones and a low-quality-factor basis tuned to short transients. Writing the signal as a sparse combination of dictionary atoms,
$$x(t)=D_h\alpha_h+D_l\alpha_l+n(t)$$
the decomposition is posed as a morphological component analysis problem,
$$U=\min_{\alpha_h,\alpha_l}\left\{\left\|x-D_h\alpha_h-D_l\alpha_l\right\|_2^2+\lambda_h\left\|\alpha_h\right\|_1+\lambda_l\left\|\alpha_l\right\|_1\right\}$$
in which the quadratic term forces fidelity to the measurement while the two ℓ1 penalties keep each coefficient set sparse in its own dictionary. The problem is solved with the split augmented Lagrangian shrinkage algorithm (SALSA), which alternates a shrinkage step on each coefficient block with a gradient step enforcing consistency with the data. Once the coefficients have converged, the components are reconstructed directly:
$$x_h(t)=D_h\alpha_h^{*},\qquad x_l(t)=D_l\alpha_l^{*}$$
Crack-induced impacts are broadband and short, so they preferentially populate the low-Q branch; concentrating on xl(t) removes most of the mesh-harmonic energy before any demodulation is attempted. The split is not a filter in the ordinary sense: because both dictionaries compete for the same signal, sustained mesh tones are absorbed by Dh even where their frequency overlaps the impulse band, which is the property that makes the approach viable when the carrier sweeps.
Minimum Entropy Deconvolution and Kurtosis-Guided Band Selection
Decomposition alone leaves the crack impacts weak relative to residual noise, because the transmission path from the tooth to the casing accelerometer acts as a low-pass filter that smears each impact over many samples. Minimum entropy deconvolution (MED) inverts that path. A finite-impulse-response inverse filter w(n) is sought such that the output
$$y(n)=w(n)*x_l(n)$$
has the smallest entropy — equivalently, the largest kurtosis — among all outputs consistent with the observed data. Impulsive fault trains are heavy-tailed and therefore low-entropy; noise and smeared resonance are high-entropy. Maximizing kurtosis of y consequently sharpens the periodic impacts and suppresses the ringing around them. One caveat applies in practice: kurtosis is also maximized by isolated outliers, so the filter length and the number of iterations must be held modest (30 iterations were used in the reported configuration) to avoid fitting a single spike.
The enhanced sequence is then used to choose the demodulation band objectively. A short-time Fourier transform gives the time–frequency matrix
$$Y(f,t)=\sum_{n=1}^{N}y(n)\,w(n-t)\,e^{-j2\pi fn}$$
and a kurtosis value is computed for each spectral band,
$$K(f)=\frac{E\left[|Y(f,t)|^4\right]}{\left(E\left[|Y(f,t)|^2\right]\right)^2}$$
Bands containing repetitive impacts stand out as high-kurtosis ridges in the kurtosis spectrum. The band [flow, fhigh] with the maximum K(f) is selected, and the enhanced signal is band-pass filtered with an M-order recursive filter,
$$y'(n)=\sum_{k=1}^{M}b(k)\,y(n-k+1)-\sum_{k=1}^{M}a(k)\,y'(n-k+1)$$
after which a Hilbert envelope isolates the modulation,
$$e(n)=\sqrt{y'(n)^2+H\left(y'(n)\right)^2}$$
and its discrete Fourier transform,
$$E(f)=\sum_{n=1}^{N}e(n)\,e^{-j2\pi fn}$$
exposes the shaft-order family produced by the cracked tooth. Candidate fault frequencies are those whose envelope amplitude exceeds a threshold T:
$$f_{\text{fault}}=\left\{f_i \mid \left|E(f_i)\right|>T\right\}$$
Kurtosis-guided band selection is what makes the demodulation robust across the speed ramp: instead of fixing a resonance band from a baseline measurement at one speed, the procedure re-selects the band for each record according to where the impulsive energy actually lies.
Order Normalization and RBF-Based Fault Mapping
A crack-related frequency is proportional to shaft speed, so it migrates during a ramp and cannot be fed directly to a classifier trained at a single speed. The extracted frequency is therefore normalized by the instantaneous speed r(t) measured by a tachometer or encoder,
$$f_{\text{norm}}=\frac{f_{\text{fault}}}{r(t)}$$
With r(t) expressed in revolutions per second, fnorm is the fault order — the harmonic index of shaft rotation — and is speed-invariant by construction. The order value, together with envelope amplitude and the kurtosis value of the selected band, forms the input vector pi of a radial basis function (RBF) network with H hidden neurons:
$$y_i=\sum_{j=1}^{H}w_j\,\varphi_j\left(\left\|p_i-c_j\right\|\right)$$
where cj are the hidden-layer centres, φj the Gaussian basis functions and wj the output weights. Because the mapping from input features to class is nonlinear only through the fixed-width kernels, the output weights follow from a linear least-squares problem,
$$\zeta=\min\sum_{i=1}^{N}\left\|y_i-f(p_i)\right\|^2$$
which makes training fast and free of local minima — an advantage when the training set has to be rebuilt for a new gearbox. Severity classes are mild, moderate and severe cracking, and the network output is the class label.
Test Stand, Specimens and Parameter Settings
The validation reported for this method used an automotive double circular-arc spiral bevel gear pair on a variable-speed stand. The driving train comprised an asynchronous motor, the gear pair and bearings, a braking load and an impeller; a single accelerometer was mounted on the gearbox casing close to the bevel pair and calibrated against a reference vibration signal before the test, with the deviation between the two traces remaining small enough to be neglected. Ambient conditions were 20 ± 5 °C and 50 ± 10 % relative humidity. Speed started at 500 r/min, rose to a maximum of 3000 r/min and was ramped at 500 r/min every 5 min.
The gears were made of 18CrNiMo7-6 low-carbon high-alloy steel, with 26 pinion teeth, 122 wheel teeth and a module of 12 mm — a ratio of about 4.7:1. The dataset contained 300 vibration records, 100 for each of the three severity classes, and four records of differing type were used as the principal detection test. Processing parameters were fixed as listed in Table 1.
| Parameter | Value |
|---|---|
| Quality factor, high-Q dictionary | 5 |
| Quality factor, low-Q dictionary | 2 |
| Sampling frequency | 10 000 Hz |
| SALSA iterations | 50 |
| MED iterations | 30 |
| Kurtosis-spectrum threshold | 3 |
| Envelope analysis band | 100–2000 Hz |
| FFT length | 4096 points |
| RBF learning rate | 0.01 |
| RBF training iterations | 1000 |
Two details of this configuration deserve comment. The 100–2000 Hz envelope band comfortably contains the pinion mesh frequency over the whole 500–3000 r/min range, so the demodulation window never loses the carrier even at top speed. And the low quality factor of 2 is deliberately aggressive: it produces short, poorly localized atoms that fit impulsive transients, at the cost of leaking some mesh energy into the impulse branch — a trade-off that the subsequent kurtosis selection is expected to correct.
Detection Results
In the reported comparison the proposed pipeline was set against an improved deep-forest planetary-gearbox diagnostic, a low-rank sparse decomposition method developed for milling-machine gearboxes, an AR-MED resonance-enhancement scheme for wind-turbine gearboxes, and an image-based crack detector (SCD-YOLO) originally built for road-surface inspection. The last of these operates on a different signal modality, so its inclusion is indicative rather than a like-for-like benchmark.
On the four test records, the proposed method reproduced the reference severity in every case, whereas the deep-forest method under-called the mild crack as moderate and the low-rank sparse method over-called the mild crack as severe and mislabelled one moderate case as severe (Table 2). With four samples the figure of 100 % agreement should be read as perfect agreement on a small set, not as an estimate of field accuracy.
| Record | Reference severity | Improved deep forest | Low-rank sparse decomposition | Proposed method |
|---|---|---|---|---|
| 1 | Mild | Moderate | Mild | Mild |
| 2 | Moderate | Moderate | Severe | Moderate |
| 3 | Severe | Severe | Severe | Severe |
| 4 | Moderate | Moderate | Moderate | Moderate |
Robustness to noise was assessed at signal-to-noise ratios of 20, 10 and 5 dB (Table 3). The proposed method held its detection stability above 99 % across the whole range, while the four comparison methods degraded as the ratio fell. The mechanism behind that behaviour is the order in which the processing steps act: MED restores the impulse train before any band decision is made, so the kurtosis spectrum is computed on a signal in which the impacts have already been raised above the noise floor. When band selection is performed first — as in conventional envelope analysis — a low-SNR record yields the wrong band and no amount of subsequent processing recovers the fault order.
| Signal-to-noise ratio | Proposed method | Comparison methods |
|---|---|---|
| 20 dB | > 99 % | Below proposed; degraded with lower SNR |
| 10 dB | > 99 % | Below proposed; degraded with lower SNR |
| 5 dB | > 99 % | Below proposed; degraded with lower SNR |
Limits of the Evidence and Practical Guidance
Three constraints frame the result. First, all 300 records come from one gear pair on one stand, so the classifier is trained and tested within a single machine identity; transfer to a different gearbox would require re-training or, at minimum, an order-domain normalization check against a healthy baseline. Second, the severity labels are class labels rather than measured crack lengths, which means agreement with the reference is agreement with an inspection verdict, not with a physical crack dimension. Third, the noise study perturbs measured records rather than reproducing a specific industrial acoustic environment, so the 99 % stability figure characterises sensitivity to added broadband noise, not to oil-film effects, lubricant starvation or sensor drift.
For deployment, the parameter set in Table 1 is a reasonable starting point for a 10 kHz acquisition chain, but the two values that most affect outcome are the quality factors and the MED iteration count. A quality factor of 2 with 30 MED iterations suits sharp, lightly damped impacts; a softer or more heavily damped transmission path calls for a longer inverse filter and a lower kurtosis threshold, otherwise the kurtosis spectrum selects a band dominated by a single outlier rather than by the periodic crack train. Because the method ultimately reports an order, not a frequency, the output remains interpretable across the whole speed range and can be tracked continuously during a ramp — which is the property that makes it usable for condition monitoring of bevel final drives whose operating speed is set by the vehicle or the load rather than by the analyst.