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Abstract

Stochastic resonance (SR) has proven to be effective in early fault signal detection. This paper proposes a multi-time-delayed feedback tri-stable stochastic resonance (MTFTSR) method to enhance signal detection capabilities. An improved tri-stable model incorporating a multi-time-delay structure is introduced, followed by the derivation of equivalent potential function and signal-to-noise ratio (SNR) expressions. And the MTFTSR method provides a nested approach for multi-parameter optimization to obtain the best output results. A comprehensive investigation of parameter influence on system performance leads to optimization for maximum system efficiency. Simulation and experimental results demonstrate superior SNR and output waveform clarity compared to envelope analysis and conventional SR methods. The findings highlight the method’s effectiveness in extracting bearing fault features and the method also has certain advantages in detecting gearbox fault signal.

Keywords

Stochastic resonance signal detection multi-time-delayed feedback tri-stable signal-to-noise ratio

Introduction

With the rapid development of equipment manufacturing industry, rotating machinery plays an important role in engineering field. Bearings^1–3 are important mechanical components, and their failures will affect normal operation of equipment, resulting in incalculable consequences. Therefore, it is necessary to carry out fault diagnosis of bearings, and extracting bearing fault features from noise is the key issue of bearing fault diagnosis.

In early fault diagnosis, noise is filtered or suppressed as a negative effect. There are many such processing methods, such as time-frequency analysis,^4,5 singular value decomposition,^6,7 empirical mode decomposition,^8,9 and Hilbert-Huang transform¹⁰. However, the early fault signal of the bearing is very weak, and the useful signal will be weakened while filtering out the noise, which will affect the signal detection result.

Therefore, SR method^11–14 is widely used in weak signal detection, which utilizes noise to enhance the strength of useful signals. This method was originally proposed by Benzi¹⁵ et al. As a nonlinear phenomenon, SR effectively uses the synergistic effect of noise, nonlinear environment and weak periodic signals to enhance the signal energy. At present, noise-induced SR has been extensively studied in the fields of chemistry, biology, quantum mechanics, lasers, etc.,^16–19 and fruitful results have been obtained in weak signal detection. Zhang et al.²⁰ consider the LSR phenomenon in a triple-well potential system driven by Gaussian chromatic noise and verify that the desired logic output can be obtained in a large noise region by adjusting the noise correlation time. Chi et al.²¹ propose a new monostable SR, which can accurately detect the fault characteristic frequencies of bearings and planetary gearboxes. Wang et al.²² adopt an unsaturated SR based on maximum cross-correlation kurtosis, which can effectively suppress high-level background noise. Jiang et al.²³ investigate the collective behavior of two coupled harmonic oscillators with a bipartite wave frequency and find that there is an optimal coupling strength that maximizes the output amplitude gain. The SR principle, often also referred to as noise enhancement theory, is a counter-intuitive dynamical phenomenon that amplifies weak coherent signals by introducing the right amount of noise. Three key elements are known to be required for the SR phenomenon to occur: noise, a nonlinear environment, and a weakly periodic signal. Addressing the following two important issues will help to apply SR principles to the design of fault detection algorithms. Firstly, the SR method does not eliminate the noise and thus also avoids the weakening of the fault signal; instead, its use of noise enhances the fault signal and makes it more convenient to be extracted. Secondly, in mechanical fault diagnosis, both weak periodic signals and noise have clear physical sources: the periodicity of weak fault signals comes from the repetitive rotation of the machinery, while the noise originates from the tiny vibration of other mechanical parts. The effectiveness of the SR system in extracting the signal features in fault diagnosis mainly depends on the complexity of the input signals as well as the selection of the system parameters. Usually, low-frequency signals below 1 Hz can be amplified by SR, while the characteristic frequencies of mechanical signals are usually tens to hundreds of Hz. Therefore, SR-based diagnostic methods need to first obtain the eigenfrequency of the faulty signal, then displace or rescale it to the applicable frequency range,²⁴ and use the scale-transformed eigenfrequency as the driving frequency. In addition, since the noise is a combination of tiny vibrations of individual mechanical parts and the sampling frequency of the experimental data is much higher than the actual eigenfrequency of the fault signal, we reasonably assume that the noise is Gaussian white noise. The selection of parameters will be discussed in detail later.

Compared with the classical stochastic resonance system, the underdamped stochastic resonance system has secondary filtering effect and the obtained system output is higher. Lai et al.²⁵ analyze a new parameter tuning method to deal with large parameters SR of Duffing oscillator, and verifies the feasibility in fault diagnosis. Cui et al.²⁶ apply a strategy based on EMD and cascaded adaptive second-order tri-stable SR for engineering signal processing. However, based on the actual situation, time delay is unavoidable. Time delay can be regarded as a transmission mechanism, providing input variables to delay output variables, that is, a single time delay.^27–29 The existence of multiple delay feedback items can make the historical information appear superimposed, thus enhancing the periodicity of the signal and obtaining a higher SNR.^30–32 Therefore, the MTFTSR method based on the second-order underdamped tri-stable potential function³³ is proposed for engineering signal processing. Firstly, it is a multi-parameter model, which can obtain more diverse potential function shapes, so as to achieve the best signal matching. Secondly, the influence of multiple time delay on SR is considered, and higher SNR is achieved through optimal parameter matching. This method takes advantage of multiple time delay to get better output waveform, and the characteristic frequency obtained is higher than the classical method, which is more effective in bearing fault diagnosis.

The paragraph arrangement of the following text is: in section 2, the model is introduced and the stationary probability density (SPD), the mean first passage time (MFPT), and the SNR are deduced, respectively, and the influence of the parameters is analyzed. In Section 3, the performance of MTFTSR method is evaluated by simulation analysis. In Section 4, practical bearings are used to verify the feasibility of the proposed method, and the conclusions are given in Section 5.

Muti-time-delayed feedback tri-stable SR system

System model

Since the noise and the weak periodic signal in the system are fixed, we often improve the performance of system by adjusting the potential function model. Due to the single structure of classical tri-stable model, the matching effect is negative for complex vibration signals. Therefore, a new tri-stable potential model based on the classical tri-stable stochastic resonance (CTSR) model³⁴ is introduced as follows $U (x) = a x^{2} + x^{2} {(b x^{2} - c)}^{2}$ (1)where $a, b, c$ are the system parameters, and $a > 0, b > 0, c > 0$ .

Next, consider a second-order underdamped stochastic resonance system with muti-time-delayed feedback based on tri-stable can be defined as $\frac{d^{2} x (t)}{dt} + λ \frac{dx (t)}{dt} = 2 (a + c^{2}) x (t) - 8 bc x^{3} (t) 6 b^{2} x^{5} (t) + \frac{ε}{L} \sum_{l = 1}^{L} x (t - l τ) + S (t) + η (t)$ (2)

In this formula, $S (t)$ represents a weak periodic signal and $S (t) = Acos Ω t$ , where $A$ and $Ω$ represent the amplitude and frequency of the signal. $η (t)$ denotes Gaussian white noise and $η (t) = \sqrt{2 D} δ (t)$ , $〈 η (t) 〉 = 0$ , $〈 η (t) η (t + τ) 〉 = 2 D δ (t)$ , where $D$ represents noise intensity and $δ (t)$ is Gaussian white noise with mean zero and variance one. $L$ , $ε,$ and $τ$ are the amount of muti-time-delayed feedback term, the feedback intensity, and the time delay, respectively.

Since the solution of the system is a non-Markov process,³⁵ small delay approximation method³⁶ can be used to convert the original equation into an equivalent equation. The formula can be sorted as $\frac{d^{2} x (t)}{dt} + λ \frac{dx (t)}{dt} = 2 (a + c^{2}) x (t) - 8 bc x^{3} (t) + 6 b^{2} x^{5} (t) ε x (t) + \frac{ε}{L} \sum_{l = 1}^{L} l τ [2 (a + c^{2}) x (t) - 8 bc x^{3} (t) + 6 b^{2} x^{5} (t) + ε x (t) + Acos Ω t] + \sqrt{2 D} δ (t) = (1 + \frac{ε}{L} \sum_{l = 1}^{L} l τ) {[2 (a + c^{2}) + ε] x (t) - 8 bc x^{3} (t) + 6 b^{2} x^{5} (t) + Acos Ω t} + \sqrt{2 D} δ (t)$ (3)

And the corresponding equivalent potential function is $U_{e} (x) = (1 + \frac{ε}{L} \sum_{l = 1}^{L} l τ) [(a + c^{2} + \frac{ε}{2}) x^{2} - 2 bc x^{4} + b x^{6}]$ (4)

Figure 1 gives the shape of potential function influenced by different parameters. As can be seen from Figure 1(a), with the increase of a, the depth of the potential well becomes smaller and the steepness of the potential wall gradually increases. Therefore, reducing the value of a in the small parameter range is conducive to the particle transition between potential Wells. In Figure 1(b), with the increase of b, the potential well becomes shallower so that the particle transition is easier, which is more conducive to the occurrence of SR phenomenon. In Figure 1(c), as c increases, resulting in larger energy required for particle transition and more difficult it is to transition between adjacent potential wells. In Figure 1(d) and (e), potential function is sensitive to small changes in $ε$ and $τ$ . The potential barrier is higher with the slight increase of $ε$ and $τ$ under the condition of small parameters, and more energy is required for the particle to transition between wells.

Figure 1.

U(x) versus $x$ under different parameters.

The change of potential well under the influence of different parameters is shown in Figure 2. The particle transitions under the synergistic effect of periodic signal and noise. As the particle transitions to a higher position, its energy gradually increases. In Figure 2(a), with the decrease of parameters a, b, c, the potential well tends to flatten gradually so that the transition between potential wells is easier to occur. In Figure 2(b), as $ε$ and $τ$ decrease in a certain range, the height of the potential barrier gradually decreases, and the energy required for transition is reduced, thus promoting the transition of particles between potential wells.

Figure 2.

The potential shape with different parameters.

The stationary probability density

To facilitate the following derivation, the above equation can be simplified as ${\begin{cases} \frac{dx}{dt} = y \\ \frac{dy}{dt} = (1 + \frac{ε}{L} \sum_{l = 1}^{L} l τ) {[2 (a + c^{2}) + ε] x (t) - 8 bc x^{3} (t) + 6 b^{2} x^{5} (t) + Acos Ω t} - λ y + \sqrt{2 D} δ (t) \end{cases}$ (5) $Let g (x) = (1 + \frac{ε}{L} \sum_{l = 1}^{L} l τ) [2 (a + c^{2}) + ε - 24 bc x^{2} + 30 b^{2} x^{4}]$ (6)

The linearized matrix is $(\begin{array}{l} 0 & 1 \\ g (x) & - λ \end{array})$

And its characteristic polynomial can be denoted as $f (q) = q^{2} + λ q - g (x)$ (7)

Let $f (q) = 0$ , so that $q$ can be obtained as $q = \frac{- λ \pm \sqrt{λ^{2} + 4 g (x)}}{2}$ (8)

Let $\frac{d x}{d t}, \frac{d y}{d t}$ , $A, D$ all be zero. Thus, the roots of the equivalent potential function are as follows $x_{0} = 0$ $x_{1}^{+} = \sqrt{\frac{4 c + \sqrt{4 c^{2} - 12 a - 6 ε}}{6 b}}, x_{1}^{-} = - \sqrt{\frac{4 c + \sqrt{4 c^{2} - 12 a - 6 ε}}{6 b}},$ $x_{2}^{+} = \sqrt{\frac{4 c - \sqrt{4 c^{2} - 12 a - 6 ε}}{6 b}}, x_{2}^{-} = - \sqrt{\frac{4 c + \sqrt{4 c^{2} - 12 a - 6 ε}}{6 b}} .$ Thus, ${\begin{cases} g (x_{0}) = (1 + \frac{ε}{L} \sum_{l = 1}^{L} l τ) [2 (a + c^{2}) + ε] \\ g (x_{1}^{+}) = g (x_{1}^{-}) = (1 + \frac{ε}{L} \sum_{l = 1}^{L} l τ) [\frac{8}{3} c^{2} - 8 a - 4 ε + \frac{8}{3} c \sqrt{4 c^{2} - 12 a - 6 ε}] \\ g (x_{2}^{+}) = g (x_{2}^{-}) = (1 + \frac{ε}{L} \sum_{l = 1}^{L} l τ) [\frac{8}{3} c^{2} - 8 a - 4 ε - \frac{8}{3} c \sqrt{4 c^{2} - 12 a - 6 ε}] \end{cases}$ (9)And the eigenvalues of the linearized matrix are ${\begin{cases} β_{1, 2} = \frac{- λ \pm \sqrt{λ^{2} + 4 g (x_{0})}}{2} = \frac{- λ \pm \sqrt{λ^{2} + (1 + \frac{ε}{L} \sum_{l = 1}^{L} l τ) [8 (a + c^{2}) + 4 ε]}}{2} \\ α_{1, 2} = α_{3, 4} = \frac{- λ \pm \sqrt{λ^{2} + 4 g (x_{1}^{+})}}{2} = \frac{- λ \pm \sqrt{λ^{2} + (1 + \frac{ε}{L} \sum_{l = 1}^{L} l τ) [\frac{32}{3} c^{2} - 32 a - 16 ε + \frac{32}{3} c \sqrt{4 c^{2} - 12 a - 6 ε}]}}{2} \\ α_{5, 6} = α_{7, 8} = \frac{- λ \pm \sqrt{λ^{2} + 4 g (x_{2}^{+})}}{2} = \frac{- λ \pm \sqrt{λ^{2} + (1 + \frac{ε}{L} \sum_{l = 1}^{L} l τ) [\frac{32}{3} c^{2} - 32 a - 16 ε - \frac{32}{3} c \sqrt{4 c^{2} - 12 a - 6 ε}]}}{2} \end{cases}$ (10)

According to the Kramer’s function,³⁷ let $P (x, y, t)$ denotes the SPD of the system at time $t$ , then the corresponding Fokker-Plank equation³⁸ is expressed as $\frac{\partial P (x, y, t)}{\partial t} = - \frac{\partial}{\partial x} [yP (x, y, t)] - \frac{\partial}{\partial y} [μ (x) P (x, y, t)] + D (\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}) P (x, y, t)$ (11)where $μ (x) = (1 + \frac{ε}{L} \sum_{l = 1}^{L} l τ) {[2 (a + c^{2}) + ε] x (t) - 8 bc x^{3} (t) + 6 b^{2} x^{5} (t) + Acos Ω t} - λ y$ (12)

Under the adiabatic approximation,³⁹ the SPD of the equation is $P_{st} (x, y, t) = \tilde{N} \exp [- \frac{{\tilde{U}}_{e} (x, y, t)}{D}]$ (13) $\tilde{N}$ is a normalization constant. According to the small parameters’ expansions, the generalized potential function $\tilde{U} (x, y, t)$ is denoted as ${\tilde{U}}_{e} (x, y, t) = \frac{y^{2}}{2} - (1 + \frac{ε}{L} \sum_{l = 1}^{L} l τ) [(a + c^{2}) x^{2} - 2 bc x^{4} + b^{2} x^{6} + \frac{ε x^{2}}{2} + Axcos Ω t] + λ xy$ (14)

SPD is an important physical quantity in nonlinear dynamical systems, which represents the probability of particles residing in potential wells. The influence of damping coefficient $λ$ , the feedback intensity $ε$ , and the time delay $τ$ on SPD are analyzed in Figure 3. The SPD is closely related to the position of its well. When the particle is in the middle of the well, the particle is in its most stable state. When the parameters $λ$ , $ε$ , $τ$ are changed, the potential well changed, leading to the change in the SPD. Moreover, it is clear that SPD is more sensitive to the changes of $ε$ and $τ$ , and the peaks on both sides of $P_{s t}$ increase with the increase of $ε$ and $τ$ .

Figure 3.

Three-dimensional diagram of $P_{s t}$ versus $x$ under different parameters.

Mean first passage time

After Taylor expansion of the probability transfer rate between the two wells, there is $R_{\pm} (t) = \frac{\sqrt{α_{1} α_{2}}}{2 π} \frac{\sqrt{α_{5} α_{6}}}{2 π} \sqrt{- \frac{β_{1}}{β_{2}}} \exp (\frac{- {\tilde{U}}_{e} (x_{0}, y_{0}, t) + {\tilde{U}}_{e} (x_{1}^{\pm}, y_{1}^{\pm}, t) + {\tilde{U}}_{e} (x_{2}^{\pm}, y_{2}^{\pm}, t)}{D})$ $= \frac{\sqrt{α_{1} α_{2}}}{2 π} \frac{\sqrt{α_{5} α_{6}}}{2 π} \sqrt{- \frac{β_{1}}{β_{2}}} \exp (\frac{1 + \frac{ε}{L} \sum_{l = 1}^{L} l τ}{- D} [(a + c^{2}) ({x_{1}^{\pm}}^{2} + {x_{2}^{\pm}}^{2}) - 2 bc ({x_{1}^{\pm}}^{4} + {x_{2}^{\pm}}^{4}) + b^{2} ({x_{1}^{\pm}}^{6} + {x_{2}^{\pm}}^{6}) + \frac{ε ({x_{1}^{\pm}}^{2} + {x_{2}^{\pm}}^{2})}{2} \pm A (x_{1}^{+} + x_{2}^{+}) \cos Ω t]) = \frac{\sqrt{α_{1} α_{2}}}{2 π} \frac{\sqrt{α_{5} α_{6}}}{2 π} \sqrt{- \frac{β_{1}}{β_{2}}} \exp (\frac{1 + \frac{ε}{L} \sum_{l = 1}^{L} l τ}{- D} [(a + c^{2}) ({x_{1}^{\pm}}^{2} + {x_{2}^{\pm}}^{2}) - 2 bc ({x_{1}^{\pm}}^{4} + {x_{2}^{\pm}}^{4}) + b^{2} ({x_{1}^{\pm}}^{6} + {x_{2}^{\pm}}^{6}) + \frac{ε ({x_{1}^{\pm}}^{2} + {x_{2}^{\pm}}^{2})}{2}]) [1 \mp \frac{A}{D} (x_{1}^{+} + x_{2}^{+}) \cos Ω t + \frac{1}{2} {(\frac{A}{D} (x_{1}^{+} + x_{2}^{+}) \cos Ω t)}^{2} \mp \dots]$ (15)where $R_{0} = \frac{\sqrt{α_{1} α_{2}}}{2 π} \frac{\sqrt{α_{5} α_{6}}}{2 π} \sqrt{- \frac{β_{1}}{β_{2}}} \exp (\frac{1 + \frac{ε}{L} \sum_{l = 1}^{L} l τ}{- D} [(a + c^{2}) ({x_{1}^{\pm}}^{2} + {x_{2}^{\pm}}^{2}) - 2 bc ({x_{1}^{\pm}}^{4} + {x_{2}^{\pm}}^{4}) + b^{2} ({x_{1}^{\pm}}^{6} + {x_{2}^{\pm}}^{6}) + \frac{ε ({x_{1}^{\pm}}^{2} + {x_{2}^{\pm}}^{2})}{2}])$ (16) $R_{1} α = \frac{A}{D} R_{0} (x_{1}^{+} {+ x}_{2}^{+})$ (17)Thus, the MFPT can be expressed as $T 12 (x^{+} \to x^{-}) = \frac{1}{R_{+}}$ (18)

The average time required for the system to transition from a steady state to another is expressed by MFPT, which is an important index to evaluate the instantaneous characteristics of the system. Figure 4 shows the three-dimensional diagram of MFPT changing with noise intensity D and different parameters in the ${x_{1}}^{+} \to {x_{1}}^{-}$ direction. MFPT decreases monotonously with the increase of D, which means that the increase of noise intensity within a certain range is conducive to the transition behavior of particles. In addition, with the increase of $ε$ and $τ$ , the value of MFPT also increases gradually, indicating that larger feedback intensity and time delay inhibit the transfer of particles, which is also verified in Figure 5(b). In Figure 5(a), the increase of parameters $a, b, c$ also has the same inhibitory effect.

Figure 4.

Three-dimensional diagram of $M F P T (x^{+} \to x^{-})$ versus $D$ under different parameters.

Figure 5.

$M F P T (x^{+} \to x^{-})$ versus $D$ for different parameters.

SNR of the MTFTSR system

The system output power is $S (ω) = S_{1} (ω) + S_{2} (ω)$ ${\begin{cases} S_{1} (ω) = \frac{π {x_{1}^{\pm}}^{2} {x_{2}^{\pm}}^{2} {(R_{1} α)}^{2}}{2 ({R_{0}}^{2} + {4 Ω}^{2})} [δ (ω - Ω) + δ (ω + Ω)] \\ S_{2} (ω) = [1 - \frac{{(R_{1} α)}^{2}}{2 ({R_{0}}^{2} + {4 Ω}^{2})}] \frac{2 {x_{1}^{\pm}}^{2} {x_{2}^{\pm}}^{2} R_{0}}{{R_{0}}^{2} + 4 Ω^{2}} \end{cases}$ (19)

According to the two-state model theory,⁴⁰ the SNR is expressed as $SNR = \frac{\int_{0}^{+ \infty} S_{1} (ω) d ω}{S_{2} (ω = Ω)} = \frac{π {R_{1}}^{2} α^{2} ({R_{0}}^{2} + {4 Ω}^{2})}{2 R_{0} (2 {R_{0}}^{2} + 8 Ω^{2} - {R_{1}}^{2} α^{2})}$ (20)

Based on the sensitivity of system to feedback intensity $ε$ and time delay $τ$ , the influence of $ε$ and $τ$ on SNR is studied. According to the expression of SNR, the three-dimensional diagram of SNR is investigated. In Figure 6(a), with the increase of $D$ and $ε$ , the SNR image presents a non-monotone structure. It means the system can cause SR phenomenon with the change of noise intensity and feedback intensity. Similar phenomenon can also be seen from Figure 6(b), indicating that the changes in noise intensity and time delay can also lead to SR. Figure 7 shows the change SNR with D at different values of parameters. From Figure 7(a), there is the traditional SR phenomenon. The peak of SNR increases gradually with the increase of parameters a, b, c, and the horizontal position of the peak shifts to the left. Figure 7(b) shows the change of SNR curve when the $λ$ , $ε$ , and $τ$ change, and the SNR increases with the decrease of $λ$ , $ε$ , and $τ$ . Therefore, the smaller $λ$ , $ε$ , and $τ$ are favorable to the occurrence of SR.

Figure 6.

Three-dimensional diagram of SNR versus $D$ under different parameters.

Figure 7.

SNR versus $D$ for different parameters.

Table 1.

Simulation results.

	A	b	c	$ε$	$τ$	$λ$	SNR_max
Optimum value	0.9803	0.1041	0.0218	0.7305	0.0703	0.2812	0.05
Correlation coefficient	0.07	0.03	0.04	0.43	0.29	0.14	0.05

Simulation

Sensitivity analysis

Based on the sensitivity of the stochastic resonance to the system parameters, proper tuning of the parameters can lead to an optimal level of the system and requires a global sensitivity analysis of the parameters. By means of Monte Carlo simulation method, the parameters are sampled randomly throughout the parameter space to evaluate the effect of each parameter on the system performance under different combinations as well as under interaction. Firstly, under the small parameter condition, the value ranges of five parameters (a, b, c, $λ$ , $ε$ , $τ$ ) are set within (0,1), and 100 sets of parameter sets are generated by randomly sampling within the parameter ranges. Then, the simulateMTFTSR function is called to calculate the system performance corresponding to each set of parameters. Finally, the correlation coefficients between each parameter and SNR are calculated and used to initially assess the degree of influence of each parameter on the system performance. The following are the results of simulating the optimal SNR that can be achieved for each parameter:

According to the simulation results, the main factors: $ε$ and $τ$ have great influence on SNR, which indicates that adjusting the values of $ε$ and $τ$ may be an effective way to improve the system performance. The secondary factors: the impact of $λ$ on SNR is relatively small. The unimportant factors: a, b, and c have little influence on SNR, so these parameters can be fixed as the optimal value first in the optimization process to simplify the model and optimization process Figure 8 and Table 1.

Figure 8.

Monte Carlo simulation process.

MTFTSR method for signal detection

The theoretical analysis of MTFTSR system has been carried out, but the system must be analyzed under the condition of small parameters because of the adiabatic approximation theory. In fact, the actual engineering signal usually has large parameters, which makes it impossible to obtain its analytical solution. Therefore, equation (5) can be solved by the Runge-Kutta method. ${\begin{cases} x_{1} = - U^{'} (x [n]) - λ y_{1} + S [n] + η [n] \\ x_{2} = - U^{'} (x [n] + \frac{y_{1} h}{2}) - λ y_{2} + S [n] + η [n] \\ x_{3} = - U^{'} (x [n] + \frac{y_{2} h}{2}) - λ y_{3} + S [n + 1] + η [n + 1] \\ x_{4} = - U^{'} (x [n] + y_{3} h) - λ y_{4} + S [n + 1] + η [n + 1] \\ x [n + 1] = x [n] + \frac{(y_{1} + 2 y_{2} + 2 y_{3} + y_{4}) h}{6} \\ y [n + 1] = y [n] + \frac{(x_{1} + 2 x_{2} + 2 x_{3} + x_{4}) h}{6} \end{cases}, \begin{array}{l} y_{1} = y [n] \\ y_{2} = y [n] + \frac{x_{1} h}{2} \\ y_{3} = y [n] + \frac{x_{2} h}{2} \\ y_{4} = y [n] {+ x}_{3} h \end{array}$ (21)

By selecting appropriate step size $h$ , muti-time-delayed feedback and input signal, the optimal system output can be obtained. Therefore, in order to keep generality and simplicity, the parameters a, b, c, $λ$ are fixed and the feedback intensity $ε$ , time delay $τ$ , and feedback term $L$ are adjusted to reach the optimal level of the system.

Based on the above results, a new nested strategy of weak enhancement in strong noise is carried out by MTFTSR method. Figure 9 shows the flowchart of the MTFTSR method, as follows:

1. Signal preprocessing: Firstly, the collected vibration signals are processed by bandpass filtering and envelope analysis and other common techniques.

2. Parameters initialization: According to the above simulation, parameters a, b and c with small impact factors are selected as their best values respectively, and then parameters with large impact factors are further optimized and adjusted. Set the system parameters $a = 0.9803, b = 0.1041, c = 0.0218$ , $λ = 0.2812$ . Limit the search scope of the following parameters under small parameter conditions and controlling the number of multiple time-delay feedback terms within 10 can effectively balance complexity and performance, as follows: $h \in (0, 1), ε \in (0, 1), τ \in (0, 1), L \in [1, 10]$ .

3. Output evaluation: The input signal of MTFTSR is the preprocessed target signal, and the system output is calculated by equation (22). SNR is usually chosen as the standard to measure the detection performance of weak signals. A higher SNR means better identification of useful information, which can be expressed as

SNR = 10 \log_{10} \frac{A_{d}}{\sum_{i = 1}^{N / 2} A_{i} - A_{d}}

(22)where

A_{d}

and

\sum_{i = 1}^{N / 2} A_{i} - A_{d}

represent the driving frequency and total noise power, respectively, N is the length of the weak signal.

4. Parameter optimization: If the SNR is the maximum and all parameters are within the search range, go to step 5, otherwise, return to parameters initialization.

5. Output results: Identify bearing fault characteristics and output the optimal power spectrum.

Figure 9.

Flowchart of the proposed MTFTSR method.

Among them, step 4 is the key step, we use adaptive particle swarm algorithm (APSO) to adjust the system parameters and simulate the signal optimal output results. APSO is an improved algorithm of particle swarm, which can adaptively update the weights and ensure the particles have faster convergence speed and global search ability. The specific design is as follows:

In the objective search space, there are a number of particles composed of a population, each particle is a potential solution, after substituting into the objective function to calculate its adaptive value, and then according to the size of the adaptive value to determine the advantages and disadvantages of the solution. The particles need to go through several iterations to obtain the optimal solution, and the position is updated once after each iteration. The position of the particle with the best fitness is taken as the optimal position of the current particle swarm, and then the step size and position of the particles are adjusted to calculate the fitness of the particles after updating, and the fitness of each particle is compared with the best position of all particles until the optimal solution is reached. The specific steps are as follows:

1) Initialize the step length and position of the particles, the size of the initial population is N0, the maximum number of iterations is MaxDT, the dimension of the search space is D, and the current position of each particle is temporarily set as the respective optimal position P_best.

2) Calculate the fitness of each function, store their optimal position and fitness, and select the position of the particle with the best fitness as the current best position Q_best, and then adjust the step size and position of the particle.

3) Calculate the particle’s updated fitness, then compare its fitness with the fitness corresponding to the previously experienced P_best, and take the best as the current P_best.

4) Compare the fitness of each particle with the Q_best of all particles, and take the optimal as the current Q_best.

5) If the maximum number of iterations or the optimal fitness is reached, the iteration is stopped and the optimal solution is output; if the above termination conditions are not reached, return to step 2).

Where the outer loop runs MaxDT times and the inner loop runs N0 times, the time complexity is O (MaxDT•N0), and the space complexity involves the storage of each particle, that is, O(N0•D). The iterative optimization is shown in Figure 10, at this point $ε$ = 0.7761, $τ = 0.0693$ , the value of SNR is further improved from the highest result of 0.05 in the simulation to 3.648, which indicates the effectiveness of the method.

Figure 10.

The iterative process of APSO algorithm.

Detection of the simulated signal

In the actual mechanical bearing fault diagnosis, considering that the bearing ball periodically rolls over the fault, a noisy series of unilateral attenuation impulses⁴¹ are selected as the analog signal for simulation analysis, namely $S (t) = \exp {- d {[t - n (t) T_{d}]}^{2}} \cdot Asin (2 π ft)$ (23)where $Asin (2 π f t)$ is the carrier signal, the signal amplitude is $A = 0.1$ , the carrier frequency is $f = 2 k H z, d = 300000$ represents the oscillation attenuation rate, $T_{d} = 0.01$ is the period of the fault signal, $n (t) = ⌊ \frac{t}{T_{d}} ⌋$ controls the periodicity of pulse signal and the sampling frequency of $f_{s} = 20 k Hz$ . In Figure 11(a), it is evident that the periodic pulse is drowned out by noise and almost all the oscillations used in the power spectrum are obscured. Then, the periodicity of pulse is revealed by envelope extraction method as shown in Figure 11(b). The relative pulse frequency of 100 Hz cannot be found in the power spectrum, and there are some interference frequencies, which may lead to misdiagnosis.

Figure 11.

Results of processing unilateral attenuated pulse signals by different methods: (a) noisy signal, (b) envelope signal, (c) CTSR method output, (d) TFSR method output, (e) MTFTSR method output.

In Figure 11(c), the same signal is analyzed by the CTSR method. The high-frequency jitter in the time domain waveform is significantly improved and the pulse is enhanced. However, the interference component can still be seen in the frequency domain diagram. Figure 11(d) shows the time-delayed feedback stochastic resonance (TFSR) method. The diagnostic performance of the TFSR method has been further improved, allowing for clearer identification of fault features, but the SNR output has not shown significant improvement compared to the CTSR method. Subsequently, MTFTSR method is used for analysis in Figure 11(e). The fault frequency can be clearly displayed, almost no interference characteristic frequency. Furthermore, our proposed method attains the higher output SNR = 3.4015. Therefore, MTFTSR method has certain advantages in extracting fault signals from simulation signals.

Experimental verification

CWRU external raceway detection

In this section, a group of defective bearing data is tested to verify the performance of MTFTSR method. Case Western Reserve University (CWRU) data center (Figure 12) provides the bearing data. The type of fault bearing used is 6205-2RS JEM SKF, and its detailed parameters are shown in Table 2. Electrical discharge machining is carried out on outer raceway. Among them, there are defects in the external and internal raceway with diameters ranging from 0.007 to 0.040 inches. The data of working conditions are shown in Table 3. After the fault bearing is installed in the motor, adjust the motor speed to 1720 to 1797 r/min, and record the vibration data tested by the motor. There are grooves with a width of 0.178 mm and a depth of 0.279 mm artificially processed in the outer ring. The sampling frequency and sample capacity of the sensor are 12 kHz and 4800 points, respectively.

Figure 12.

(a) Bearing test stand, (b) the cross-section of bearing.

Table 2.

Fault bearing parameters.

Ball number of inner and outer ring	Pitch diameter (inch)	Ball diameter (inch)
9	1.537	0.3126

Table 3.

Working condition data sheet.

Fault location	Fault diameter (inch)	Approximate motor speed (rpm)	Fault characteristic frequency (Hz)
External raceway	0.007	1750	107.36
Internal raceway	0.021	1797	162.18

Figure 13 is the analysis of defect signal of outer raceway. Figure 13(a) shows the time domain and the frequency domain diagrams of the original signal, and no fault pulse appears in the waveform. Figure 13(b) shows the result of the envelope analysis. Due to the modulation of the rotating frequency, the demodulation of the fault characteristic frequency is hindered, which affects the accurate judgment of the signal. Figure 13(c) is the output result of CTSR method. Most of the high-frequency noise is effectively processed. However, the high energy spike caused by non-fault pulses still exists, which makes the diagnosis results unconvincing. Hence, the classical method is prone to interference from low-frequency signals. The TFSR method is tested as shown in Figure 13(d), although the amplitude and the SNR output are higher than the other two methods, reaching 6.68 × 10⁻⁶ and 4.9092, respectively, there are still obvious interference spikes, which affects the accurate judgment of the fault signal. In Figure 13(e), the MTFTSR method is applied to analyze the signal. The pulse caused by the fault can be clearly identified in the time domain diagram, the high-frequency noise and low frequency are both suppressed, the fault peak exists independently, and the characteristic frequency obtained is higher. Therefore, compared with classical method, MTFTSR method has better filtering effect in detecting weak signals.

Figure 13.

Results of processing defect signals of CWRU outer raceway by different methods: (a) noisy signal, (b) envelope signal, (c) CTSR method output, (d) TFSR method output, (e) MTFTSR method output.

CWRU internal raceway detection

To verify the feasibility of MTFTSR method for the whole bearing fault detection, the internal raceway defect signal is analyzed. The fault pulse is submerged by noise and the fault pulse interval cannot be distinguished from Figure 14(a). The result of envelope analysis is shown in Figure 14(b), the frequency domain diagram shows that the fault can be pointed out by the pulse signal, but the chaotic waveform still exists. Figure 14(c) and (d) show the detection results of CTSR method and TFSR method, respectively. Firstly, for the CTSR method, the suppression effect of high-frequency noise is close to ideal. Nevertheless, there is still a high energy spike on the left side of the target signal caused by non-fault pulses. Therefore, it can be concluded that noise enhancement in low-frequency domain has relative priority. Secondly, Figure 14(d) shows the diagnostic effect of the TFSR method, and it can be found that its signal extraction of the faulty bearing is unexpectedly good, but the SNR output is only 0.2691, which is much smaller compared with the MTFTSR method. It concludes that the performance of the TFSR method in the detection process is not stable. For the MTFTSR method in Figure 14(e), the waveform is arranged neatly and have obvious periodicity in the time domain diagram. The suppression effect of high-frequency and low-frequency noises is ideal, and the fault peak exists alone in the frequency domain diagram. This result indicates the superiority of MTFTSR method in fault diagnosis.

Figure 14.

Results of processing defect signals of CWRU inner raceway by different methods: (a) noisy signal, (b) envelope signal, (c) CTSR method output, (d) TFSR method output, (e) MTFTSR method output.

FEMTO-ST bearing signal detection

In order to obtain more general comparison results, the second set of outer ring fault vibration signals comes from the FEMTO-ST Institute. This set of signal detection results show that the envelope signal still cannot detect the fault characteristics intact, and the detection results of CTSR and TFSR methods are not satisfactory, and there are interference peaks, which affect the judgment of correct results. Even though the amplitude and the SNR reach 2.544 × 10⁻⁶ and 1.8120, which are smaller than the amplitude of TFSR and SNR result 4.363 × 10⁻⁶ and 2.7832, only MTFTSR method can correctly distinguish the frequency to be detected, showing a powerful filtering effect Figure 15.

Figure 15.

Results of processing defect signals of FEMTO-ST outer ring fault by different methods: (a) noisy signal, (b) envelope signal, (c) CTSR method output, (d) TFSR method output, (e) MTFTSR method output.

Figure 16.

Results of processing defect signals of gearbox by different methods: (a) noisy signal, (b) envelope signal, (c) CTSR method output, (d) TFSR method output, (e) MTFTSR method output.

Gearbox signal detection

In order to test whether the MTFTSR method is limited to the fault detection of bearings, we try to extend the method to the fault signal detection of gearboxes, data from Southeast University. The limitations of traditional CTSR method and TFSR are obvious. There are obvious drawbacks in the fault detection process of gearbox, that is, the fault frequency that needs to be detected cannot be clearly identified, there are more or less interference signals, and their amplitude is higher than the fault frequency. From the perspective of amplitude, the fault can be accurately detected at the characteristic frequency of 30, and its amplitude is 8.125 × 10⁻⁷, which is much higher than that of 4.84 × 10⁻⁷ and 4.327 × 10⁻⁷ of the CTSR and TFSR methods. From the SNR point of view, the MTFTSR method reaches 2.3258, which is superior to the SNR of the CTSR and TFSR method. The experiment proves that the MTFTSR method mentioned in this paper can also be used for gearbox fault detection, and the effect is still better than the traditional method, and can achieve accurate signal detection under different working conditions Figure 16.

Comparison of simulation and experimental performance

Figure 17 shows the SNR when the CTSR, TFSR, and MTFTSR method proposed in this paper detect analog signals and four different defect signals respectively. It can be seen that the traditional CTSR method has certain disadvantages compared with the other two methods, both the previous amplitude level and the SNR level are in a low state. The output of TFSR method is not so stable. When detecting CWRU and FEMTO fault signals in the outer ring, SNR reaches 4.9092 and 2.7832, which is higher than the SNR values of 4.3642 and 1.8120 of MTFTSR method, but the difference is not much, and the output of MTFTSR method has been maintained at a high level. In terms of the overall amplitude level and the clarity of the detection effect, the performance of MTFTSR is still superior. Therefore, it also shows the superiority of the proposed method from the perspective of SNR output.

Figure 17.

Statistical chart of simulation and experimental performance results.

Conclusion

In this paper, an improved underdamped multi-delay feedback tri-stable SR system is studied and successfully applied to fault signal extraction of bearings and gearboxes. Through simulation and experiment, the superiority of the MTFTSR method in fault diagnosis is discussed.

Firstly, the SNR equation is derived by using the two-state model theory, and the effects of parameters on SPD, MFPT, and SNR are studied. Through parameter optimization and matching, higher resonant output signal can be obtained. It also reflects the sensitivity of the system to parameter variations. Then, a nested optimization algorithm is proposed. Through the two-layer optimization of the parameters, the SNR of the system is raised to the optimal level, and the numerical simulation is verified. Finally, the feasibility of the proposed method is verified by several sets of experimental data, and compared with the CTSR and the TFSR methods, the superiority of the proposed method in fault signal extraction is demonstrated. The analysis shows that the MTFTSR system has good bandpass filtering effect and can clearly highlight the fault signal characteristics.

In summary, the MTFTSR method obtains better results in fault diagnosis and has certain advantages over the envelope method and some classical methods, that is, better recognition, higher fault feature magnitude, and better performance. In addition, we expect that the MTFTSR method can also be widely used in other signal detection fields, such as underwater signal detection and sensor signal monitoring.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (No.62463016), the Key Project of the Gansu Natural Science Foundation (No.23JRRA882), and the Industrial Support and Guidance Project of Colleges and Universities of Gansu Province (No.2024CYZC-23).

ORCID iD

Jiangang Zhang

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An improved underdamped muti-time-delayed feedback tri-stable stochastic resonance and its research in bearing fault detection

Abstract

Keywords

Introduction

Muti-time-delayed feedback tri-stable SR system

System model

The stationary probability density

Mean first passage time

SNR of the MTFTSR system

Simulation

Sensitivity analysis

MTFTSR method for signal detection

Detection of the simulated signal

Experimental verification

CWRU external raceway detection

CWRU internal raceway detection

FEMTO-ST bearing signal detection

Gearbox signal detection

Comparison of simulation and experimental performance

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References