Source-linked AI summary
A Harmonic-Regression ARMA model for rotating machinery dynamics identification and fault detection under varying conditions
Dimitrios M. Bourdalos, John S. Sakellariou, Spilios D. Fassois
TL;DR
Fault detection under varying operating conditions is complicated by changes in vibration responses and by the mixed cyclical and broadband spectra of rotating machinery. The paper introduces HR–ARMA, combining harmonic regression with ARMA dynamics in a Multiple Model framework. In single-stage gearbox experiments, it outperformed conventional ARMA and achieved at least 98% True Positive Rate at a 5% False Positive Rate.
Problem
Varying speeds and loads affect vibration responses, while purely rational ARMA models can restrictively represent signals with both cyclical lines and broadband dynamics.
Source: §1 · pp. 1–2Method
HR–ARMA combines harmonic regression for dominant deterministic cycles with ARMA for broadband dynamics, using a Multiple Model framework for varying operating conditions.
Source: §1 · p. 2, §2.1 · p. 3, §3 · p. 5Results
At least 98% True Positive Rate at a 5% False Positive Rate was achieved, and the method outperformed the conventional ARMA alternative in gearbox experiments.
Source: §6 · p. 12Takeaways & Limitations
The single-stage gearbox studies demonstrate that HR–ARMA can represent cyclical and broadband dynamics while using substantially fewer parameters than conventional ARMA.
Source: §6 · pp. 11–12, §5.4 · p. 11The model assumes Gaussian i.i.d. zero-mean residuals.
Source: §2.1 · p. 3
Abstract
from arXiv · showhide
Fault detection in rotating machinery under varying operating conditions remains a challenging problem. Existing approaches are mainly based either on non-parametric vibration signal representations, parametric deep learning models, or on parametric ARMA-type models. While deep learning approaches often lack transparency and require large training datasets, ARMA-type models rely on the assumption of a purely rational spectrum, which can be restrictive for rotating machinery vibration signals exhibiting mixed spectral characteristics. To address these limitations, a novel Harmonic Regression ARMA (HR-ARMA) model is presented for rotating machinery dynamics identification, which combines an explicit harmonic regression part for representing the dominant deterministic cyclical components with an ARMA part for the remaining broadband stochastic dynamics. Based on this model and within a Multiple Model framework, fault detection is achieved under varying operating conditions. The HR-ARMA modelling performance is validated through simulation and experimental studies on a single-stage gearbox, demonstrating accurate representation of both cyclical and broadband dynamics, improved parsimony and significantly higher fault detection performance compared with conventional ARMA models, achieving at least 98% True Positive Rate at 5% False Positive Rate in the experimental study.
1 Introduction
Rotating machinery fault detection must contend with operating-condition changes that affect vibration responses, and with signals combining cyclical spectral lines and broadband dynamics. The paper introduces HR–ARMA to represent both components explicitly and embeds it in a Multiple Model framework for varying conditions.
1 Introduction
- Variable rotational speeds and loads can significantly affect vibration responses and obscure fault-related effects.Most vibration-based fault-detection studies assume constant operating conditions.Source: §1 · p. 1
- Non-parametric methods depend on suitable representation selection and tuning, while deep-learning approaches typically require large datasets and can be difficult to interpret physically.Parametric statistical time-series models instead offer compactness and a transparent relationship between model parameters and underlying dynamics.Source: §1 · p. 2
- Purely rational ARMA spectra can be restrictive because rotating-machinery vibrations combine deterministic cyclical components with broadband stochastic dynamics.In a purely ARMA framework, cyclical spectral lines are only indirectly approximated.Source: §1 · p. 2
- HR–ARMA combines explicit harmonic regression for dominant cyclical components with an ARMA model for remaining broadband dynamics, within a Multiple Model fault-detection framework for varying conditions.The cyclical components include shaft and gear-meshing harmonics and modulation-induced sidebands.Source: §1 · p. 2
- The experimental study uses hundreds of single-stage gearbox experiments across four rotational speeds, five load levels, and three single-tooth crack levels, comparing HR–ARMA with conventional ARMA.The passage introduces comparative scatter plots and ROC curves but does not provide their outcomes.Source: §1 · p. 2
2 HR–ARMA model
The HR–ARMA model represents vibration signals as explicit deterministic cyclical components plus broadband stochastic dynamics. Its estimation initializes cyclical and ARMA parameters, then jointly optimizes the model parameters using residuals.
2.1 HR–ARMA model structure
- The model represents mixed-spectrum vibration as explicit harmonic regression for dominant deterministic cycles plus a rational ARMA component for remaining broadband dynamics.It is defined in the angular domain, so signals are angularly resampled to compensate for minor speed fluctuations that can smear deterministic spectral components.Source: §2.1 · p. 3
2.2 HR–ARMA model estimation
- HR–ARMA estimation minimizes a nonlinear least-squares criterion based on residuals that depend on the unknown parameter vector, using iterative trust-region-reflective optimization.The final parameter estimate follows joint optimization from an initialized parameter vector.Source: §2.2 · pp. 4–5
- Initialization detects cyclical orders by thresholding a ratio of differently smoothed periodograms, with the user-defined threshold typically in the range 8–15.The periodograms use rectangular moving-average windows of different lengths.Source: §2.2 · p. 4
- Given initial cyclical orders, ordinary least squares initializes harmonic coefficients, then an ARMA model fitted to the residual signal initializes the AR and MA parameters.The procedure then forms the full initial parameter vector for joint estimation.Source: §2.2 · pp. 4–5
3 Multiple HR–ARMA based fault detection under varying conditions
The method represents healthy dynamics across operating conditions with multiple HR–ARMA models, then assesses unknown signals against that baseline to detect faults.
3 Multiple HR–ARMA based fault detection under varying conditions
- The method has baseline-training and inspection phases within a Multiple Models framework for fault detection under varying operating conditions.The baseline consists of healthy HR–ARMA models estimated from vibration signals recorded under different operating conditions.Source: §3 · p. 5
- An unknown signal is passed through each baseline model, and its residuals are used to assess consistency with that model's healthy dynamics.The residual sequence is computed using the model's estimated parameter vector.Source: §3 · p. 5
- The consistency check uses the Pena–Rodriguez whiteness-test D statistic, with the minimum across baseline models defining the fault-detection metric.A user-defined threshold is then used for fault detection.Source: §3 · p. 5
4 Simulation Study
The simulation evaluates HR–ARMA on a gearbox signal with cyclical and broadband components, then compares its spectrum and parsimony with conventional ARMA.
4 Simulation Study
- The simulation combines shaft harmonics, modulated GMF harmonics, and transmission-path and structural vibrations represented by an ARMA component.The model uses a broadband ARMA component for transmission-path and structural vibration.Source: §4 · p. 6
- The simulated signal contains 14 deterministic cyclical components, including shaft and GMF harmonics with sidebands, plus stochastic dynamics represented by ARMA(4, 4).The simulation uses 80 input-shaft revolutions.Source: §4 · p. 6
- PRD initializes cyclical orders, BIC selects the true ARMA orders (4, 4), and nonlinear least squares converges to the true innovation-sequence RSS.The BIC minimum is attained at the true orders, and the final estimated RSS matches the simulation model's innovation sum of squares.Source: §4 · p. 6, Figure 1 · p. 7
- HR–ARMA(14, 4, 4) matched the true mixed spectrum, while ARMA(60, 60) missed deterministic spectral-line amplitudes and introduced artificial peaks and valleys.The HR–ARMA fit captured both the discrete line spectrum and continuous rational spectrum, including shaft orders and the first three GMF harmonics.Source: §4 · pp. 6–7, Figure 2 · p. 8
- 50 versus 120 parameters (58% fewer) were used by HR–ARMA(14, 4, 4) versus ARMA(60, 60); RSS/SSS was 15.43% versus 17.32%, respectively.RSS/SSS is RSS normalized by the series sum of squares.Source: §4 · p. 8
5 Experimental Study
The UNSW gearbox study evaluates HR–ARMA modeling and fault detection across varied speeds, loads, and crack sizes. HR–ARMA represents the signal’s cyclical and broadband components, and the experimental comparison reports better fault-detection rates than conventional ARMA.
5.1 The rotating machinery, the fault scenarios and the vibration signals
- The UNSW study uses a single-stage spur gearbox with four rotational speeds, five loads, three pinion-crack sizes, and 640 vibration signals.The cracks are small, medium, and large; signals span the considered health states and operating conditions.Source: §5.1 · p. 8
5.2 Experimental assessment of the HR-ARMA modeling
- For a healthy signal at 15 rev/s and 20 Nm, HR–ARMA(91, 80, 50) matched the periodogram-based PSD estimate across broadband dynamics and dominant spectral lines.The comparison includes lines around the first two gear-meshing-frequency harmonics.Source: §5.2 · p. 9, Figure 4 · p. 10
5.3 Experimental fault detection results
- The detector represents healthy gearbox dynamics with 20 HR–ARMA models; Figure 5 summarizes D-statistics and ROC curves for 620 inspection signals across four speeds and five loads.The inspection signals include healthy and faulty gearbox states.Source: §5.3 · p. 10, Figure 5 · p. 11
5.4 Comparison with the conventional ARMA model
- Conventional ARMA uses 42% more parameters than HR–ARMA yet captures sharp spectral lines less faithfully, with several modulation sidebands unrepresented.It captures the broadband continuous spectrum reasonably well, but some dominant cyclical components have amplitude mismatch.Source: §5.4 · p. 11
- At 5% FPR, conventional ARMA reaches 53% TPR for F1 and 73% for F2/F3, versus HR–ARMA’s 98% and 100%, respectively.The F1 case is the small crack; F2 and F3 are medium and large cracks.Source: §5.3 · p. 10, §5.4 · p. 11
6 Conclusions
HR–ARMA combines harmonic regression for deterministic cycles with ARMA for broadband dynamics, supporting Multiple Model fault detection under varying operating conditions.
6 Conclusions
- HR–ARMA explicitly represents shaft harmonics, gear-meshing frequencies, and sidebands, while ARMA represents remaining broadband dynamics.The broadband dynamics are associated with the transmission path and structural dynamics.Source: §6 · p. 11
- Figure 6 compares the conventional ARMA spectrum with a periodogram estimate and shows fault-detection statistics and ROC curves for 620 inspection signals across four speeds and five loads.Source: Figure 6 · p. 12
- At least 98% True Positive Rate at a 5% False Positive Rate was achieved, clearly surpassing the conventional ARMA alternative.The conventional ARMA alternative exhibited inadequate performance.Source: §6 · p. 12
- The model outperformed conventional ARMA in simulation and experimental studies on a single-stage gearbox while requiring substantially fewer estimated parameters.Source: §6 · p. 12
A Simulation Parameters
Table A1 lists configuration parameters for the simulation study.
A Simulation Parameters
- Table A1 presents the simulation study's configuration parameters.Source: Table A1 · p. 15