Underdamped stochastic resonance with the bistable confining potential for weak fault diagnosis of rotating machinery

Stochastic resonance (SR) with a bistable confining potential, $U_{cb}(x)=U_0\exp(-x^2/L_0^2)+k|x|^q/q$, was proposed by Li and Shi (2018) as a method for extracting weak fault characteristic frequencies of rotating machinery from heavy noise, because its four parameters tune barrier height, well width, and wall steepness independently — an advantage over the classical quartic bistable potential. The method was formulated and validated only in the overdamped (first-order, inertia-free) regime. Characterize the output signal-to-noise ratio and weak-fault-diagnosis performance of the bistable confining potential SR in the underdamped (second-order, inertia-inclusive) regime, where intrawell resonances and damping alter SR behavior, and determine whether that advantage carries over to the underdamped regime as a higher output signal-to-noise ratio and fault-frequency prominence than underdamped classical bistable SR.

Background

Stochastic resonance (SR) is a nonlinear phenomenon in which noise, rather than masking a weak periodic signal, improves its detectability: a particle in a double-well potential driven jointly by noise and a weak sinusoidal force switches between wells most coherently when the noise-induced hopping rate matches the drive period, so the output signal-to-noise ratio (SNR) peaks at a nonzero noise intensity. In machinery fault diagnosis this is exploited by feeding the envelope of a measured vibration signal into such a bistable system: a repetitive impulse train at a fault characteristic frequency (the repetition rate of impacts produced, for example, by a spalled bearing race or a broken gear tooth) plays the role of the periodic drive, and SR transfers noise energy into that spectral line, lifting it above the background spectrum where direct envelope analysis fails.

The classical bistable SR (CBSR) uses the quartic potential $U_b(x)=-\frac{a}{2}x^2+\frac{b}{4}x^4$, whose barrier height $\Delta U=a^2/(4b)$ and well width cannot be tuned independently: adjusting $a$ or $b$ moves both at once, which limits how well the potential can be matched to a given signal. Two further limitations motivated the present problem. First, the CBSR output saturates: when the input is large, the output amplitude stabilizes at a low level instead of growing with the input. Second, essentially all fault-diagnosis SR systems, including CBSR, are formulated in the overdamped limit — the first-order Langevin equation $\frac{dx}{dt}=-\frac{dU}{dx}+A\sin(2\pi f_d t+\varphi)+\sqrt{2D}\xi(t)$, where $A$, $f_d$ are the drive amplitude and frequency, $D$ is the Gaussian white noise intensity, and $\xi(t)$ is zero-mean unit-variance noise — which is valid only when damping dominates inertia.

The underdamped regime instead obeys a second-order Langevin equation with inertia, $\frac{d^2x}{dt^2}=-\frac{dU}{dx}-\gamma\frac{dx}{dt}+A\sin(2\pi f_d t+\varphi)+\sqrt{2D}\xi(t)$ with damping coefficient $\gamma$. There, inertial motion introduces intrawell resonances — oscillations of the particle within a single well — whose interplay with noise-activated interwell hopping changes the SR response in ways absent from the overdamped picture (Alfonsi, Gammaitoni, Santucci and Bulsara, Physical Review E 62, 299 (2000)). Underdamped SR has been applied to fault diagnosis with several potentials: Lei, Qiao, Xu, Lin and Niu (Mechanical Systems and Signal Processing 94, 148 (2017)) matched stable states of an underdamped system to bearing signals; Zhang, Zhang, Zhang and Xiao (IEEE Access 6, 42431 (2018)) used an underdamped exponential bistable potential for bearing faults; and Li, Liu, Wang, He and Shan (Journal of Sound and Vibration 459, 114862 (2019)) and Li, Liu, He and Shan (IEEE Access 7, 141633 (2019)) developed underdamped SR with a multi-parameter shifted-quartic 'constrained' potential and with a periodic potential, respectively — the first carrying the same independent-parameter-tuning motivation but with a different potential.

The bistable confining potential itself was introduced in the SR physics literature by Heinsalu, Patriarca and Marchesoni (European Physical Journal B 69, 19 (2009)): $U_{cb}(x)=U_0\exp(-x^2/L_0^2)+k|x|^q/q$ with $q>1$, a Gaussian barrier of height set by $U_0$ and width by $L_0$, plus power-law confining walls whose steepness is set by $q$ and scale by $k$; in the overdamped regime they showed the confinement exponent $q$ sharpens the SR peak for hard walls ($q>2$) and suppresses it for soft walls ($q<2$). Li and Shi (Shock and Vibration, 2018) adopted this potential for weak fault characteristic extraction (the BCPSR method), tuning $q$, $L_0$, $U_0$, $k$ with an ant-colony parameter search (a swarm-intelligence optimization heuristic that explores the parameter space with many interacting candidate 'ants'), and demonstrated on simulated signals, a bearing inner-race fault, and planetary-gearbox missing- and broken-tooth experiments that BCPSR achieves a higher fault-frequency spectrum peak, recognition degree (the relative prominence of the fault-frequency spectral line in the output spectrum), and output SNR than CBSR. Their model, like Heinsalu's, was overdamped; their conclusion explicitly deferred the underdamped behavior of BCPSR to future work. What is not known is how the same confining potential behaves once inertia is included — the question stated here.

Problem Statement

For the underdamped (second-order, inertia-inclusive) Langevin system $\frac{d^2x}{dt^2}=-\frac{dU_{cb}}{dx}-\gamma\frac{dx}{dt}+A\sin(2\pi f_d t+\varphi)+\sqrt{2D}\,\xi(t),$ where $U_{cb}(x)=U_0\exp(-x^2/L_0^2)+k|x|^q/q$ with $q>1$ is the bistable confining potential, $\gamma$ is the damping coefficient, and $\xi(t)$ is zero-mean unit-variance Gaussian white noise of intensity $D$: (1) characterize the stochastic-resonance response of this underdamped BCPSR system — its output signal-to-noise ratio and its dependence on the potential parameters ($q$, $L_0$, $U_0$, $k$), damping, noise intensity, and driving frequency — including how inertia-driven intrawell dynamics interact with interwell hopping and what role the confinement exponent $q$ plays in the inertial regime, analytically and/or numerically; (2) determine whether the central advantage of the confining potential in the overdamped regime — a higher output signal-to-noise ratio and fault-frequency prominence than classical bistable SR, enabled by its independent adjustment of barrier height, well width, and wall steepness — carries over to the underdamped regime, with underdamped classical bistable SR (quartic potential) as the natural reference point, mirroring the source paper's own BCPSR-versus-CBSR comparisons; either a well-supported positive or a well-supported negative conclusion completely answers this open determination; and (3) build on these results to formulate a weak fault diagnosis method for rotating machinery based on underdamped BCPSR, including a parameter-matching or tuning scheme; demonstration on available vibration data (e.g., rolling-bearing and planetary-gearbox fault records) serves as illustrative validation of the formulated method rather than a mandatory completion requirement.

References

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