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Sparsity-based Algorithm for Detecting Faults in Rotating Machines

Wangpeng He, Yin Ding, Yanyang Zi, Ivan W. Selesnick

arXiv:1511.00067v1cs.SD

TL;DR

The paper addresses detecting periodic fault transients hidden in noisy rotating-machine vibration signals. It estimates periodic-group-sparse signals through a convex optimization formulation with an iterative solver, and reports effective extraction of bearing outer- and inner-race fault features, including compound faults.

  • Problem

    Periodic transient vibration features used for rotating-machine diagnosis are often buried in heavy background noise and irrelevant vibrations.

  • Method

    The paper estimates periodic-group-sparse signals using a non-convex penalty designed around fault periodicity while constraining the full objective to remain convex.

  • Results

    The processed motor-bearing signals reveal periodic impulses corresponding approximately to outer-race and inner-race characteristic frequencies, and the approach effectively detects and extracts useful bearing features.

  • Takeaways & Limitations

    Using machine-geometry frequency prior knowledge, the approach can separate compound bearing-fault features associated with different transient periods.

  • Takeaways & Limitations

    The signal model assumes periodic group sparsity with additive white Gaussian noise, and the MM implementation requires non-zero initialization to avoid zero-locking.

Abstract

from arXiv · show

This paper addresses the detection of periodic transients in vibration signals for detecting faults in rotating machines. For this purpose, we present a method to estimate periodic-group-sparse signals in noise. The method is based on the formulation of a convex optimization problem. A fast iterative algorithm is given for its solution. A simulated signal is formulated to verify the performance of the proposed approach for periodic feature extraction. The detection performance of comparative methods is compared with that of the proposed approach via RMSE values and receiver operating characteristic (ROC) curves. Finally, the proposed approach is applied to compound faults diagnosis of motor bearings. The non-stationary vibration data were acquired from a SpectraQuest's machinery fault simulator. The processed results show the proposed approach can effectively detect and extract the useful features of bearing outer race and inner race defect.

1 Introduction

The paper targets periodic transient detection in noisy vibration signals by modeling fault signatures as periodic group-sparse signals. It introduces a convex optimization approach that uses fault-period prior knowledge and can generalize non-periodic group-sparse denoising.

  • Rotating-machine faults in bearings and gearboxes are frequent causes of breakdown, making early detection important for preventing economic loss and personal casualties.
  • Periodic transient vibration signatures are often buried in heavy background noise and irrelevant vibrations, motivating specialized signal-processing methods.
  • The proposed model estimates a periodic group-sparse signal x from noisy observations y, assuming additive white Gaussian noise and grouped large-magnitude values.
  • The method uses fault characteristic frequency as prior knowledge, allowing groups to occur at periodic intervals rather than only contiguously.
  • The regularization term is designed to exploit periodic impulsive features and may separate compound faults with different transient periods.
  • The paper validates the approach through simulation and applies it to motor-bearing fault diagnosis after reviewing group-sparse denoising methods.

2 Review

The review develops overlapping group shrinkage using convex and non-convex penalties, then explains how majorization-minimization provides an iterative solution. Non-convex penalties can strengthen sparsity while suitable conditions preserve convexity of the full objective.

  • Convex and non-convex regularization: Although non-convex penalties can make optimization non-convex, designing them appropriately can preserve convexity of the total objective and avoid non-optimal local minima.
  • Overlapping Group Shrinkage: OGS estimates group-sparse signals in noise, while its non-convex regularized extension promotes sparsity more strongly and may extract periodic transient pulses.
  • Penalty conditions: The penalty function φ is required to be continuous, twice differentiable away from zero, increasing, and concave on the positive reals.
  • Penalty functions: The arctangent penalty is identified as the most strongly sparsity-inducing among the listed non-convex functions, while a = 0 yields the convex ℓ1-norm.
  • Majorization-minimization: Majorization-minimization converts the optimization into a sequence of simpler problems, but zero-locking can occur unless the algorithm uses a non-zero initialization.

3 OGS with binary weights

The section extends group-sparse denoising with binary weights that encode periodic structure, while constraining non-convex penalties so the overall objective remains convex. The resulting formulation supports a convergent MM algorithm and periodic fault recovery.

  • Binary-weighted grouping: A binary vector b partitions group positions into weighted and unweighted sets, with K = K0 + K1 and disjoint K0 and K1.The construction uses K0 for zero weights and K1 for nonzero weights.
  • Binary-weighted grouping: The binary-weighted grouping function θ measures the Euclidean norm of selected blocks of x after element-wise weighting.The block is defined through K-point subvectors Bn(x).
  • Binary-weighted grouping: The formulation contains ordinary OGS as a special case when all binary weights equal one, and scalar sparse denoising when K = K1 = 1.The group-sparse setting considered is N ≫ K ⩾ K1 > 0.
  • Convexity: Choosing the non-convexity parameter appropriately preserves strict convexity of the objective, avoiding non-optimal local minima in iterative optimization.The convexity condition depends on the nonzero binary weights and regularization parameter.
  • Algorithm: The MM procedure has an explicit minimizer and converges to the unique global minimizer when the penalty satisfies the convexity condition.Table 2 summarizes the explicit solution steps for OGS with binary weights.
  • Periodicity-induced OGS: For periodic transients, the binary pattern repeats N1 nonzero samples and N0 zero samples across M periods, with K0 = MN0 and K1 = MN1.Given fs and T, selecting N1 and M determines the pattern b.
  • Periodicity-induced OGS: POGS recovers periodic impulsive signatures by solving the binary-weighted problem with a periodic pattern, while conventional OGS remains available when the period is unknown.If OGS reveals a period, POGS may achieve better accuracy using that periodicity.

4 Simulation validation

The simulation tests periodic-group-sparse denoising on noisy periodic transients and compares POGS with OGS, wavelet denoising, and fast spectral kurtosis. POGS identifies transients cleanly and achieves almost perfect detection in ROC evaluation, while parameter choice and periodicity matching affect denoising quality.

  • Simulation setup: The simulated 1-second signal contains 50 faults at 80 Hz, with white Gaussian noise of standard deviation σ = 2.5 added after t = 0.36 seconds.The sampling rate is fs = 6400 Hz, and each transient is modeled using random sinusoidal components.
  • Proposed method: With the known period and N1 = 4, M = 4, POGS produces transients with an almost pure zero baseline, making them easily identifiable.The periodic binary pattern is computed from T = 1/80 seconds and fs = 6400 Hz, giving N0 = 76.
  • Parameter sensitivity: Using N1 = 2 yields slightly worse RMSE because this pattern does not match the simulated transients, representing an inappropriate-pattern worst case.The authors describe N1 = 2 as the lower limit of a realistic transient length.
  • Comparative denoising: Conventional OGS misses some faults and produces false detections, while wavelet denoising can recover large transients but imposes the chosen wavelet shape.OGS also produces false transients before t = 0.36 seconds and around 0.82 seconds.
  • Comparative denoising: Fast spectral kurtosis estimates an optimal carrier frequency of 1000 Hz, but the extracted repetitive transients remain surrounded by strong irrelevant noise.The corresponding transient envelope shows greater peak density after t = 0.36 seconds.
  • ROC evaluation: POGS achieves an almost perfect ROC detection result, with ROC curves nearly identical across the tested POGS parameter settings.In this example, OGS performs better than wavelet-based denoising; optimal λ varies approximately linearly with noise level and can be set as λ = rσ.

5 Experimental and engineering data validation

The experiments apply the proposed periodic group-sparse method to simulated and measured vibration data, including compound motor-bearing faults. The method extracts periodic transients and reveals outer- and inner-race characteristic frequencies despite strong noise.

  • 5.1 Example 2: Rolling bearing with defect on outer race: The outer-race bearing experiment uses a measured vibration signal from a locomotive bearing sampled at 12.8 kHz, with an estimated defect frequency of 57.8 Hz.The tested bearing has a slight scrape on its outer race and rotates at approximately 481 rpm.
  • 5.1 Example 2: Rolling bearing with defect on outer race: When transient duration is uncertain, POGS uses M = 4 and N1 = 2, with λ selected according to the estimated background noise level.Healthy data can provide the noise estimate; without healthy data, the paper estimates background deviation using median absolute deviation.
  • 5.1 Example 2: Rolling bearing with defect on outer race: The estimated background deviation is σ̂ = 0.1606, and the resulting POGS output reveals a clearly identifiable periodic phenomenon.The same noise-based parameter selection is also tested with M = 4 and N1 = 4.
  • 5.1 Example 2: Rolling bearing with defect on outer race: AR-MED promotes most impulses, but its relatively noisy baseline leaves the periodic regularity unclear compared with the proposed result.The AR filter order is selected by maximizing kurtosis, and the MED filter length is 50.
  • 5.2 Example 3: Motor bearing with multiple faults: The motor-bearing validation uses a Spectra Quest simulator containing intentional inner- and outer-race faults, making the experiment a compound-fault detection case.The measured signal, Fourier spectrum, and Hilbert envelope spectrum are shown before processing.
  • 5.2 Example 3: Motor bearing with multiple faults: POGS is run with separate group structures based on inner- and outer-race periods, while λ is chosen from healthy data to suppress healthy signals toward zero.The two structures are used to separate impulsive features associated with the two fault frequencies.
  • 5.2 Example 3: Motor bearing with multiple faults: The processed signals reveal outer-race transients near 73.2 Hz and inner-race transients near 117.8 Hz, with corresponding characteristic frequencies and harmonics visible in the spectra.The detected intervals are approximately 0.0133 seconds for the outer race and 0.0085 seconds for the inner race.
  • 5.2 Example 3: Motor bearing with multiple faults: Fast spectral kurtosis identifies a 2050 Hz optimal carrier, but its filtered envelope provides no further fault-related information beyond periodic transients linked to the motor speed.The envelope is consistent with the motor’s rotating speed of 23.89 Hz.

6 Conclusion

The conclusion presents periodic group sparsity as a fault-detection approach for rotating machinery. It emphasizes convexity-preserving non-convex penalties, prior-frequency-based feature extraction, compound-fault separation, and validation on simulated and experimental data.

  • 6 Conclusion: The proposed approach uses non-convex penalty functions to promote periodic group sparsity while constraining the objective function to remain convex.Its penalty models the periodicity of sparse groups for machinery fault feature extraction.
  • 6 Conclusion: The sparse-pulse period is selected from prior knowledge of machine geometry, enabling separate processing of fault features with different characteristic frequencies.The paper demonstrates this separation for outer- and inner-race bearing faults.
  • 6 Conclusion: Simulation and experimental results demonstrate the effectiveness of the proposed approach, which the paper reports as outperforming other methods.The validation includes compound bearing-fault data.

A Proof of Proposition 1

The proof establishes strict convexity by examining second derivatives of the objective components. It handles both nonzero and zero values of the scalar variable under the stated condition.

  • A Proof of Proposition 1: For v ≠ 0, positivity of γ + λφ′′(v; a) is sufficient to ensure p′′(v) > 0.The argument uses twice differentiability of φ(v; a) away from zero.
  • A Proof of Proposition 1: At v = 0, the proof invokes Lemma A and the condition p′(0−) < p′(0+) to establish strict convexity on R.The zero case complements the second-derivative argument for v ≠ 0.

B Proof of Proposition 2

The proof concludes that P is strictly convex when condition (19) is satisfied. The excerpt also references equations (10) and (18) without stating their full roles.

  • The proof references equation (10) in a relation involving k.
  • Equation (18) is introduced as being given by the preceding derivation.
  • P is strictly convex if condition (19) is satisfied.
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