Generalised envelope spectrum-based signal-to-noise objectives: Formulation, optimisation and application for gear fault detection under time-varying speed conditions

TL;DR

This work introduces a Generalised Envelope Spectrum-based Signal-to-Noise (GES2N) objective for optimising FIR filter coefficients to enhance gear fault signatures in SES under time-varying speeds. By formulating $\psi$ as a ratio between weighted SES signal bands and noise bands, the authors derive multiple variants (including GES2N-Mean-Np, GES2N-Max-Np) that can reproduce and extend existing objectives like ICS2 and MSESHIRD. The approach uses gradient-based optimisation with SES as the diagnostic core and demonstrates superior fault-enhancement performance across three experimental gear datasets compared to CYCBD, ACYCBD, MOMEDA, and SES-based proxies. The results highlight the advantage of targeting specific SES bands and excluding the zero-order energy from the denominator to improve robustness under speed variation, offering practical gains for real-time gear fault detection. The work also provides code access and gradient derivations to facilitate adoption and extension.

Abstract

In vibration-based condition monitoring, optimal filter design improves fault detection by enhancing weak fault signatures within vibration signals. This process involves optimising a derived objective function from a defined objective. The objectives are often based on proxy health indicators to determine the filter's parameters. However, these indicators can be compromised by irrelevant extraneous signal components and fluctuating operational conditions, affecting the filter's efficacy. Fault detection primarily uses the fault component's prominence in the squared envelope spectrum, quantified by a squared envelope spectrum-based signal-to-noise ratio. New optimal filter objective functions are derived from the proposed generalised envelope spectrum-based signal-to-noise objective for machines operating under variable speed conditions. Instead of optimising proxy health indicators, the optimal filter coefficients of the formulation directly maximise the squared envelope spectrum-based signal-to-noise ratio over targeted frequency bands using standard gradient-based optimisers. Four derived objective functions from the proposed objective effectively outperform five prominent methods in tests on three experimental datasets.

Figures (15)

• Figure 1: An overview of the evaluation of the proposed Generalised Envelope Spectrum-Based Signal-to-Noise (GES2N) objective and associated objective function. Abbreviations: Finite Impulse Response (FIR); Squared Envelope Spectrum (SES); Velocity Synchronous Discrete Fourier Transform (VS-DFT). The natural logarithm is denoted $\ln$.
• Figure 2: The SES amplitudes $\boldsymbol{b}$ is presented against the corresponding cyclic orders $\boldsymbol{\alpha}$, with the amplitude corresponding to the cyclic order $\alpha[m]$ denoted $b[m]$. In (a), the cyclic order resolution $\Delta \alpha$ and the amplitudes of four targeted harmonics $k \cdot \alpha_{c}$ are shown, i.e., $N_h = 4$. In (b), four cyclic bands with a constant bandwidth $\Delta \alpha_{b}$ around the targeted harmonics $k \cdot \alpha_{c}$ are shown.
• Figure 3: Two examples of weighting matrices are shown for the Squared Envelope Spectrum (SES) in Figure \ref{['fig:Method:SES_Example']}(b). The SES has four bands $N_b = 4$. If the sum of the components in each band is calculated (e.g., like the MSESHIRD objective function), the cyclic order weighting matrix in Figure \ref{['fig:Method:WeightingMatrix']}(a) can be used. If the maximum of each band needs to be calculated, the cyclic order weighting matrix has the form shown in Figure \ref{['fig:Method:WeightingMatrix']}(b).
• Figure 4: The helical gearbox test bench in the Centre for Asset Integrity Management (C-AIM) laboratory is presented. (a) The components in the test bench are shown. (b) The sensors that were utilised are shown.
• Figure 5: The gear with localised damage is shown in \ref{['fig:Results:Exp:LGD:Gear:Before']} before the experiment was started and in \ref{['fig:Results:Exp:LGD:Gear:After']} after the experiment was completed.