Stochastic resonance in a high-order time-delayed feedback tristable dynamic system and its application

Chaos, Solitons and Fractals 128 (2019) 155–166

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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Frontiers

Stochastic resonance in a high-order time-delayed feedback tristable dynamic system and its application Peiming Shi a,∗, Wenyue Zhang a, Dongying Han b, Mengdi Li a a b

School of Electrical Engineering, Yanshan University, Qinhuangdao, Hebei 066004, PR China School of Vehicles and Energy, Yanshan University, Qinhuangdao, Hebei 066004, PR China

a r t i c l e

i n f o

Article history: Received 16 March 2019 Revised 7 July 2019 Accepted 30 July 2019

Keywords: Stochastic resonance Tristable system The high-order term time-delayed feedback SNR Fault signature detection

a b s t r a c t A stochastic resonance (SR) tristable system based on a high-order time-delayed feedback is investigated and the feasibility of the system for weak fault signature extraction is discussed. The potential function, the mean first-passage time (MFPT) and the signal-to-noise ratio (SNR) are used to evaluate the model. Firstly, the potential function and stationary probability function (PDF) of the system are derived, and then the influence of the time delay parameters on the MFPT of the particles is analyzed. Secondly, the influences of time-delyed strength e and delyed length τ on the SR system from the perspective of the transition of the particles in the potential wells are discussed, and then the SNR and the effect of the parameters on the SNR are derived. In addition, the high-order time-delayed feedback tristable stochastic resonance (HTFTSR) system is used to deal with faulty bearing data and is compared with traditional tristable stochastic resonance (TSR). The result shows that the nonlinear system model can accurately identify the fault frequency and improve the energy of the characteristic signal under the appropriate system parameters. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Stochastic resonance (SR) is a common phenomenon that is induced by internal or external noise in many nonlinear systems in nature. The concept of SR was proposed by Benzi et al. in the beginning of 1980s [1], whose initial purpose was to introduce mathematical methods to analyze the phenomena of periodic climate change during the ice age. As a nonlinear phenomenon, SR is a unique detection method by enhancing signal energy with the cooperative effects between the noise and external forces. Up to now, SR has received considerable attention in Biology [2,3], neurology [4,5], meteorology [6], physics [7–15], laser and so on. For the past few years, the researchers have done a lot of studies on stochastic resonance and achieved fruitful results in the direction of weak signal detection [16–25]. Especially the dynamic behavior of time-delayed SR system has attracted great attention in the field of weak signal detection. When both the noise and delay time feedback are considered in the system, the interaction of noise and delay feedback will lead to a significant change in the dynamics of the system. Delayed-time feedback and the synergy of noise in nonlinear systems have become one of the frontier topics in the field of nonlinear systems currently.

∗

Corresponding author. E-mail address: [email protected] (P. Shi).

https://doi.org/10.1016/j.chaos.2019.07.048 0960-0779/© 2019 Elsevier Ltd. All rights reserved.

At present, most theories about stochastic resonance involve the traditional system, such as monostable systems and bistable systems. The researchers analyzed the nonlinear system in many ways such as MFPT and escape rate, SNR and the steady state probability, as well as approximate entropy [26–32]. The introduction of MFPT and the escape rate to describe the characteristics of escape. McNamara and Wiesenfeld [33] obtained the expression of SNR of SR by using the two-state model theory and adiabatic approximation theory. Li et al. [34] introduced the signal of the approximate entropy measure, obtained the optimal output of the bistable system and the result of the match with the original signal by using the adaptive condition. He et al. [35] studied the influence of multiplicative bisection noise on SR in over-damped fractional oscillator. Barbera and Spagnolo [36] studiedthe transient dynamics of interacting biological species extracted from two ecosystems. In the research field of SR system based on time-delay feedback, the response process is no longer Markov process because the system has time delay at this time. Some typical examples are as follows: Guillouzic et al. [37] used the method of small delay time approximation to transform non-Markov process with time delay into Markov process and obtained effective Fokker–Planck equation. Budini and Caceres [38] used the method of variable transformation. Frank [39,40] adopted the Nocikoc theory and the first-order wrestling theory to investigate the dynamics of nonlinear stochastic systems with time delay. In the traditional research

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of SR, there are many researches on the influence of delay feedback and many interesting achievements have been made. Masollar [41] found that the interaction of noise and delay feedback can help the system get resonance when studying single mode semiconductor laser with delay feedback. He used the probability distribution of dwell time to prove that this resonance phenomenon exists not only in the multi-attractor model he used, but also in other delay feedback systems. Ohira and Sato [42] studied the role of delay feedback in discrete binary models. Wu and Zhu [43] studied the interaction between time-delayed feedback and non-Gaussian noise. Li et al. [44] studied the properties of SPD and SNR for the bistable system under the combined effects of multiplicative noise, additive journaling white noise and linear delay feedback. In fact, the classical stochastic resonance model is a short memory system, and the physical model considering delay is closer to the actual system. However, few studies have focused on the delay in TSR system. At the same time, the stochastic resonance system with a high-order time-delayed feedback has not been studied. Therefore, it is necessary to study the principle of SR based on a high-order time-delayed feedback and its practical value. This paper proposes a SR system based on a high-order timedelayed feedback and discuss the feasibility of the system for weak fault signature extraction. In section second, derived from the potential function of the system and the stationary probability function. In Section 3, the influences of time delay length τ and strength e on the mean first-passage time are analyzed. The influences of e and τ on the stochastic resonance system from the perspective of the transition of the particles in the potential wells are discussed. In Section 4, the SNR and discuss the effect of the parameters b, c, d, e and τ on the SNR are derived. In Section 5, faulty bearing data is dealt with and it is compared with traditional TSR using the HTFTSR system. At last, Section 6 makes a summary.

(1)

√ Where 2Dξ (t ) is the Gaussian white noise item, in which D is the noise intensity and ξ (t) represents Gaussian white noise with zero mean and unit variance. The potential function U(x) can be defined as flows:

U (x ) =

b 2 c 4 d 6 x + x + x 2 4 6

(2)

Where parameters b, c, and d are real number.When a high order term time-delay feedback is introduced into system, the mathematicl model can be denoted as:

√ dx = − bx − cx3 − dx5 + e[x(t − τ )]5 + Acos(2π f t ) + 2D dt

ξ (t ) (3)

The THFTSR system potential function can be written as:

b 2 c 4 d 6 1 x + x + x − e[x(t − τ )]6 2 4 6 6

U (x ) =

(4)

From Eqs. (1) and (2), the Fokker–Planck equation is given by [45]

∂ p(x, t ) ∂ [A(x )P (x, t )] ∂ 2 [B(x )P (x, t )] =− + ∂t ∂x ∂ x2

(5)

Where A(x) is the conditional mean drift can be expressed as:

A (x ) =



+∞ −∞

h(x, xτ )P (xτ , t − τ |x, t )dxτ

And in Eq. (6) xτ = x(t − τ ), h(x, xτ ) = + exτ + Acos(2π f t ), P (xτ , t − τ |x, t ) is the zeroth-order approximate Markovian transition probability density that can be described as below:

1

 

P (xτ , t − τ |x ) = √ exp − 4π Dτ

xτ 6 − x6 − h (x )τ

(6)

− d x5

5

2 

4Dτ

(8)

h(x ) = −bx − cx3 − dx5 + ex5 + A cos (2π f t )

(9)

Where

The Markov Process can be obtained by simplified Eq. (6)



A ( x ) = 1 + 5 eτ x 4





−bx − cx3 − dx5 + ex5 + A cos(2π f t )

(10)

The equivalent Langevin Eq. (1) can be further obtained

     dx = 1 + 5eτ x4 −bx − cx3 − dx5 + e 1 + 5eτ x4 x5 dt √   + 1 + 5eτ x4 A cos(2π f t ) + 2D ξ (t )

(11)

Comparing Eq. (11) and Eq. (3), a coupling term 5eτ x4 (−bx − cx3 − dx5 + ex5 + A cos(2π f t )) is introduced into the HTFTSR. Delay and its feedback affect the stochastic resonance output. The generalized function can be written as flows:

Ue (x ) =

b 2 c 4 1 5 x + x + (d − e + 5eτ b)x6 + eτ cx4 2 4 6 8   1 10 5 + eτ (d − e )x + x + eτ x A cos(2π f t ) 2

(12)

The stationary probability density function (PDF) of the system can be defined as flows: Ue

The multistable SR model is a nonlinear system driven by periodic signal and Gaussian white noise. The Langevin equation can be written as:

ξ (t )

(7) −bx − cx3

Pst = ne− D

2. The tristable SR model driven with a high order-term time-delayed feedback

√ dx dU (x ) + = Acos(2π f t ) + 2D dt dt

B (x ) = D

(13)

Where n is normalization constant. The structural parameters affect the particle’s movement efficiency in the potential well, thereby affecting the stochastic resonance effect. Fig. 1 illustrates the variation of potential well width and depth with time-delayed strength e. It can be seen that the width and depth of well increase with the growth of e in the high order term time-delayed tristable system. Similarly, with the change of τ , the Fig. 2 shows the same trend as the Fig. 1. However, we can clearly observe that the influence of e on the potential function is more pronounced than τ at the same scale of change. The PDF describes the probability of the particles staying in the potential well. The influence of the research parameters on the probability density function is helpful to further analyze the motion state of the particles. Fig. 3 separately analyzes the effect of the growth of parameters e, τ and D on the steady-state probability density. It can be seen from Fig. 3(a) and (b) that both the increase of e and τ can increase the peak of pst , but the degree of influence of pst on them is different. When e and τ become larger, the potential well becomes deeper, and the probability that the particles stay in the potential well is greater. At this time particles are more stable. However, it can be seen from Fig. 3(c) that the influence of noise intensity on the pst of the particle is exactly the opposite, and the two peaks of pst decrease as D increases. This means that the smaller the D is, the greater the probability that the particles will stay in the potential well This means that the smaller the D is, the greater the probability that the particles will stay in the potential well. Particles will become more stable. 3. Mean first-passage time (MFPT) The noise induced escape problem has caused widespread concern in the field of stochastic dynamics. The mean first-passage

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Fig. 1. The potential function Ue (x) versus x for different e with b = 0.25, c = −0.73, d = 0.4, A = 0.2, and changes of Ue (x) at different time (a) t = π /2, (b) t = 3π /4, (c) t = π .

time is the average of the time required for the system to move from steady state to another steady state barrier. It is used to describe the transient characteristics of nonlinear systems and is also a fundamental measure of the escape problem of nonlinear stochastic systems. By the definition of mean first-passage time, MFPT of four different directions in the HTFTSR system can be obtained as flows:

MF P T (xS1 → xS2 ) = =



xS2 xS1



dx B(x )PSt (x )



x

=



xS2



xS1 = −xS3 =



xU2 x

xU1 = −xU2 =

2π 1

U x − U x e ( U1 ) e ( S2 )

MF P T (xS2 → xS3 ) = =



xS3 xS2



(15)

D dx B(x )PSt (x )

2π



x xU1

dyPSt (y ) 1

B(xS2 )|U  (xS2 )U  (xU2 )| 2

U x − U x e ( U2 ) e ( S2 )

exp

D

−



dyPSt (y )

B(xS2 )|U  (xS2 )U  (xU1 )| 2

exp



(14)

D dx B(x )PSt (x )

dx B(x )PSt (x )

xS3





+∞ x

dyPSt (y )

2π 1

)U  (xU2 )| 2 U x − U x e ( U2 ) e ( S3 )

B(xS3

)|U  (x

S3

D

(17)

xS2 = 0

1

xS1

xS2

Where Ue (x) is the time-delayed tristable potential function.

U x − U x e ( U1 ) e ( S1 )

MF P T (xS2 → xS1 ) =



exp

B(xS1 )|U  (xS1 )U  (xS2 )| 2

exp

=

dyPSt (y )

−∞

2π

MF P T (xS3 → xS2 ) =

(16)





 1 c − c 2 − 4b( d − e ) 2d − e 



 −1 c + c 2 − 4b( d − e ) 2 (d − e )

The dependence of MFPT on parameterτ and e is discussed separately as follows: The potential well of the system is asymmetric when t = π /2. And at this time, the probability of particles in each potential well is different. It can be seen that the potential well is the state in which the left side is deeper than the right and the middle well is the shallowest from Fig. 1. Thus, in Figs. 3 and 4, the effect of noise intensity on MFPT under different delay parameters is studied. First, at the same noise level, the MFPT increases as the parameter e increases. Form Figs. 3 and 4, it is easy to see that noise contributes to the particle escape rate when the particle traverses

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Fig. 2. The potential function Ue (x) versus x for different τ with b = 0.25, c = −0.73, d = 0.4, A = 0.2, e = 0.1 and changes of Ue (x) at different time (a) t = π /2, (b) t = 3π /4, (c) t = π .

from the first potential well to the second potential well. Therefore, the impact of noise D on the MFPT is closely related to the state of the system. Meanwhile, it also demonstrates the effect of noise on stochastic resonance system isn’t single. The MFPT versus the noise intensity D for different τ are discussed in Fig. 4. MFPT increase with τ increases. By observing Figs. 3 and 4, it can be seen that the effect of e on MFPT is greater than τ . However, the influence of the two parameters on the MFPT function is the same. The Fig. 4(a) shows a multi-peak phenomenon. From the influence of e and τ on the potential function in Figs. 1 and 2, it is easy to see that the barrier of the potential well increases with the increase of time delay, and the well width becomes narrower with the increase of time delay. When the delay parameters decrease, the height of the potential well decreases and the particles in the potential well are more likely to transit and the escape rate increases. Actually, there are in Fig. 3c and d, and in Fig. 4a, b, and d clear evidence of a non-monotonic behavior, which is a clear signature of the noise enhanced stability (NES) phenomenon. A significant sign of the NES effect is that the mean escape time exhibits nonmonotonic behavior and a peak value with a change in noise intensity. When e = 0.1, the change in T23 and T23 caused by additive white Gaussian noise changes from an attenuating structure to a multimodal structure. As the value of e increases, the NES phenomenon disappears. At the same time, the MFPT of the system also exhibits non-monotonic behavior when τ = 1.4. These phenomena all indicate that additive white noise can cause the NES effect of the system under appropriate delay strength. Under conditions with minimal noise, the particles will be trapped in the potential well (metastable); MFPT will become infinite at a low noise

level and will eventually return to exponential decay at a higher noise level. These phenomena are the same as those described in Ref. [46–48]. To summarize, a better output state can be achieved in the stochastic resonance system by adjusting the e and τ thus affecting the mean first-passage time of particles in the potential well. At the same time, the escape rate can be increased by increasing the intensity of additive noise in the presence of time delay. 4. SNR of high order term time-delay feedback TSR The SR phenomenon of the system on SNR is studied in this section. Based on previous research [48] the transition rates are shown that in the progressive state, the average correlation function of random variable is given by: < x(t )x(t + τ )>Average = f



=

1 −c − 8 (d − e ) 4



· e

 +

 · e



−R0 |τ |





1/ f



0

x(t )x(t + τ )dt

b d−e

R21 β 2

−1 −   2 2 R20 + (2π f )



+1+



R21 β 2 cos (2π f t )



2 R20 + (2π f )

2

 −1 c + c 2 − 4b( d − e ) 8 (d − e )

−R0 |τ |

R21 β 2

1−   2 2 R20 + (2π f )

+1+

R21 β 2 cos (2π f t )



2 R20 + (2π f )

2







 (18)

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159

Fig. 3. Stationary probability function versus x with fixed b = 0.25, c = −0.73, D = 0.4: (a) different delay intensity e; (b) different delay τ ; (c) different noise intensity D.

Fig. 3a. The MFPT versus the noise intensity D for different delay parameters with b = 0.15, c = −0.32, d = 0.8, A = 0.2, t = π /2, τ = 0.5.

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Fig. 4. The MFPT versus the noise intensity D for different delay parameters with b = 0.15, c = −0.32, d = 0.8, A = 0.2, t=π /2, e = 0.5.

The output power spectral density of a high order term timedelay feedback SR can be deduced as:

S (ω ) =



+∞ −∞

x(t )x(t + τ )Average e

− jωt

d τ = S1 ( ω ) + S2 ( ω )

(19)

Where S1 (ω) is the power special densities of the output signal and S2 (ω) is the output noise. It can be deduced as:



S1 ( ω ) =

1 −c − 8 (d − e ) 4



c+ b − d−e



c+ b − d−e

S2 ( ω ) = ·



8 (d − e )

2R0

(21)

R0 2 + ( 2π f )

2

Under approximate conditions, the probability transition rate between two potential traps can be expressed as:

R1 (t ) =

|U  (−x

1

)U  (−x 2π

Make x1 = 0;



x0 = −

x2 =

−1 c+ 2 (d − e )



)|





· exp −



φ (−x1 ) − φ (−x2 ) D

c 2 − 4b( d − e )

 −1 c − c 2 − 4b( d − e ) 2 (d − e )

−



2

1/2

= ·e



−







π

b 2

 ) eτ (g−k )+ 4c (g2 −k2 )+ d−e+5 (g3 −k3 )+ 5e8τ c (g4 −k4 )+ eτ (d−e (g5 −k5 ) /D 6 2

(23)



 −2 g= c + c 2 − 4b( d − e ) , k = 2 (d − e )



c b − . d−e d−e

Fig. 5 depicts the relationship between transformed noise intensity and SNR. The high-order delayed feedback stochastic resonance has similar characteristics as the classical stochastic resonance: The SNR increases with increasing noise, reaches a maximum value, and then starts to decrease. In order to clearly describe the energy distribution of the system output signal, the SNR function can be written as follows: ∞

SNR =

(22) =

0 s1 (ω )dω s2 ( ω = 2π f )





−C/8(d − e ) − 1/4 b/d − e − c +





C/8(d − e ) + 1/4 b/d − e − c +

·

 −1 c + c 2 − 4b( d − e ) 2 (d − e )

 

b + 3cg + 5(d − e )g2 · b + 3ck + 5(d − e )k2



(20) c 2 − 4b( d − e )



Here

8 (d − e )

π R21 β 2 ·   δ ( ω − 2π f ) 2 2 2 R0 + ( 2π f ) 

 1 c − 8 (d − e ) 4

c 2 − 4b( d − e )



R0 = 2 · R1 (t )|A cos(2π f t )=0

   π R0 A2 −C/(d − e ) − 2 b/d − e 4D2







c 2 − 4b( d − e ) /8 ( d − e )



c 2 − 4b( d − e ) /8 ( d − e )

(24)

SNR and 3D contour as the drawing function of system parameters b and noise intensity D in Fig. 6. After simulation, when the noise intensity D is a fixed value, SNR increases with the decrease of b within a certain range. Subsequently, the relationship between

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161

In this part, a set of faults bearing data provided are used by CWRU and a set of faults bearing data collected in the actual project analysis to verify the HTFSTR method in fault diagnosis [49]. At the same time, this method is compared with the traditional multistable processing method. We have previously analyzed HTFTSR systems theoretically, but the former must satisfy the adiabatic approximation theory and be limited to small parameter limits. For discrete HTFTSR systems, the output can be obtained using the RK4 equation.

Fig. 5. SNR versus noise intensity D with b = 0.35, c = −0.12, d = 0.9, e = 0.2, τ = 0.2 and A = 0.2.

system parameters c, d and SNR are described through Figs. 7 and 8. It is not difficult to find that the two graphs have the same monotonicity, SNR increases with the increase of b and c. Fig. 9 depicts the SNR and 3D counter as the drawing function of system D and the e. Stochastic resonance phenomena occur when the signal-to-noise ratio curve has obvious peak value. The SNR changes in the 0.2 ≤ e ≤ 0.25 are discussed and found that the stochastic resonance effect is most pronounced when 0.2 ≤ e ≤ 0.21. With the increase of e, SNR decreases. Fig. 10 is a three-dimensional graph of SNR as a function of D for different value of τ . However, the SNR can be achieved a very good value when 0.2 ≤ τ ≤ 0.22. Comparing Fig. 9 with Fig. 10, we can draw a conclusion that adjusting the time-delayde strength τ is more sensitive that of the time-delayed e in controlling stochastic resonance. Analyzing the effect of each parameter on SNR, No matter what the other system parameters are, the corresponding noise intensity D is always the same when the peak value of the signal-to-noise ratio curve appears. This shows that in the process of parameter adjustment, we can first determine the optimal noise D, and then adjust other parameters to reach the resonance state of the system with high signal-to-noise ratio. 5. Stochastic resonance and engineering application a) 5.1 Analysis of rolling bearing faults data from CWRU

⎧ k = h{−U  (x[n] ) + ex[n − round (τ × f s )] + S[n] + N[n]} ⎪ ⎪ 1 ⎪ ⎪ k = h{−U  (x[n] + k1 /2) ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ + e(x[n − round (τ × f s )] + k1 /2) + S[n] + N[n]} ⎪ ⎪ ⎪ ⎨k3 = h{−U  (x[n] + k2 /2) ⎪ + e(x[n − round (τ × f s )] + k2 /2) + S[n + 1] + N[n + 1]} ⎪ ⎪ ⎪ ⎪ ⎪ k = h{−U  (x[n] + k3 ) 4 ⎪ ⎪ ⎪ ⎪ ⎪ + e(x[n − round (τ × f s )] + k3 ) + S[n + 1] + N[n + 1]} ⎪ ⎪ ⎩ x[n + 1] = x[n] + (k1 + 2k2 + 2k3 + k4 )/6 (25) In this paper, the fault data comes from the Case Western Reserve University (CWRU) Bearing Data Center website. The experimental setup is shown in Fig. 11. The bearings used in this test are deep groove ball bearings of the type 6205–2RS JEM SKF with the speed of 1730 r/min, and the sampling frequency is 12 kHz. The details of the geometry of this type of bearing are provided in Table 1 and the characteristic frequencies are shown in Table 2. Taking the inner raceway faults as an example, the theoretical value of the fault frequency of the inner ring is 156.14 Hz. Due to a fault in the frequency of large projects, small parameter does not meet the conditions, so a twice sampling frequency transformation method is adopted in the algorithm for data processing. The twice sampling frequency is 6 Hz. Firstly, the theoretically fault frequency is 156.14 Hz. Inner race fault signal frequency spectrum shown in Fig. 12, it can be seen significant impact periodicity in the time domain waveform. From the Fig. 12(b), it can be observed that the high amplitude signals are mainly concentrated between 0 and 2 kHz. Because of the serious interference of background noise, the fault characteristic frequency can not be obtained in the waveform. Therefore, the proposed method is applied to fault feature detection. In this part, we analyze and compare the possibility of the delay feedback tristable state (TFTSR) system and HTFTSR system in

Fig. 6. (a) SNR. (b) The contour of SNR as a function of the noise intensity D and the system parameter b with c = −0.15, d = 0.9, e = 0.2, τ =0.2, A = 0.2.

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Fig. 7. (a) SNR (b) The contour of SNR as a function of the noise intensity D and the system parameter c with b = 0.3, d = 0.9, e = 0.2, τ = 0.2, A = 0.2.

Fig. 8. (a) SNR (b) The contour of SNR as a function of the noise intensity D and the system parameter d with b = 0.3, c = −0.15, e = 0.2, τ =0.2, A = 0.2.

Fig. 9. (a) SNR (b) The contour of SNR as a function of the noise intensity D and time delay strength e with b = 0.3, c = −0.15, d = 0.9 τ = 0.2, A = 0.2.

Table 1 The main parameters of the rolling bearings. Inner diameter/mm

Outer diameter/mm

Pitch diameter/mm

Ball diameter/mm

Ball number

contact angle/(°)

25.001

51.999

39.040

7.940

10.000

0

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163

Fig. 10. (a) SNR (b) The contour of SNR as a function of the noise intensity D and time delay τ with b = 0.3, c = −0.15, d = 0.9, e = 0.2, A = 0.2.

Fig. 11. Experiment platform. Table 2 Rolling bearing fault feature frequency. Bearing element

Inner ring

Outer ring

The retainer

Rolling body

Failure frequency

5.4152 fr

3.5848 fr

0.39828 fr

4.7135 fr Fig. 13. Comparison of the waveform and spectra of TFTSR and HTFTSR.

is also increased. However, there are still some intermediate frequency interference, so the characteristic frequency is not obvious. By observing Fig. 13(b) and (d), the frequency of the inner fault is 156 Hz, which is close to the theoretical value of 156.14 Hz. By comparing the spectra of TSR with HTFTSR can be seen that both can detect weak fault signals, but the HTFTSR filtering effect is more obvious. The amplitude at the fault frequency of the HTFTSR system is higher than the TFTSR system, increasing to 0.4036 from 0.2811. At the same time, the signal-to-noise ratio of this processing result is analyzed, the SNR increases from 0.9723 dB to 2.1840 dB. In this section, the fault bearing data of CWRU are used to verify the effectiveness of HTFSTR method in fault diagnosis. The result shows that the HTFSTR method proposed in this paper has better filtering effect than the TFTSR method. a) 5.2 Fault characteristic analysis of bearing outer ring Fig. 12. Waveform (a) and spectra (b) of a bearing inner ring fault signal.

the actual fault feature extraction with b = 0.1, c = −0.2, d = 0.15, e = 0.08, τ = 0.05. As shown in Fig. 13, the original signal is more susceptible to noise. The signal processed by the TSR method is shown in Fig. 13(b), which can be observed that the pulse profile can already be seen. Most of the high frequency interference is filtered and the amplitude of the characteristic frequency

The gear fault data used in this section are derived from the rotating machinery fault diagnosis experimental platform and the test platform is shown in Fig. 14. the failure data of the middle speed shaft outer ring (bearing model = ER-16 K) are used. The measuring point is the middle speed front bearing (near the end of the motor) and the radial load. The working condition 29 data is adopted and set the sampling parameters as follows: the sampling frequency was 12 kHz, the sampling time was 6 s and the

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Fig. 14. Test platform.

Fig. 17. The SNR of TSR and HTFTSR system.

Fig. 15. Waveform (a) and spectra(b) of the bearing outer ring fault signal.

The processing result is shown in Fig. 16. As can be seen from Fig. 16, the characteristic frequency of the fault signal processed by stochastic resonance system is 41.02 Hz, which is very close to the theoretical value of 41.435 Hz. Compared with Fig. 16(b) and (d), it can be seen that the characteristic signal has been clearly displayed, but the result processed by HTFTSR system is more prominent than that processed by TFTSR system, and the fault frequency amplitude increases from 0.4979 to 0.912. This shows that HTFTSR has a more obvious effect on bearing fault feature extraction. The SNR also increases from 0.5252 dB to 1.6248 dB. This shows that HTFTSR has a more obvious effect on bearing fault feature extraction. 5.3. Signal-to-noise ratio To quantitatively discuss the superiority of the system, faulty bearing data from CWRU are used as input. At the same time, noise is used to drive the system and analyze the change of signal to noise ratio of TSR and HTFTSR system. SNR is an important index for measuring the effect of SR, which is defined as [18]



SNR = 10log10

Fig. 16. Comparison of the waveform and spectra of TFTSR and HTFTSR.

maximum speed of the motor was 30 0 0 rpm. The speed ratio and load ratio were both 80%. The motor speed was calculated 2400 rpm and the calculated speed shaft outer ring fault characteristic frequency theory value was 41.435 Hz [18]. Fig. 15 is the time domain diagram and frequency domain diagram of bearing outer ring fault signal. A large amount of noise in the original signal can be observed, so that failure information can not be obtained. The TSR system is used and TFTSR system is used to analyze the fault data. the processing result is shown in Fig. 16. As can be seen from Fig. 16, the characteristic.

Ad An



(26)

where Ad is the amplitude value corresponding to the forcing frequency, and An is the sum of all the amplitude values except Ad in the amplitude spectrum. We know that stochastic resonance is a negative feedback system. For stochastic resonance without delay, the negative feedback term of the system is the potential trap force. The delay stochastic resonance introduces an additional long memory feedback term into the system. Under appropriate conditions, the feedback term can superimpose the historical information onto the current output, thereby improving the detection result of the periodic weak signal. Fig. 17 shows the signal to noise ratio with D in different systems. It can be found that it has the same trend as our previously theoretical derived SNR, but the SNR of the HTFTSR system is significantly higher than that of the TFTSR. It can be observed that in HTFTSR system, SNR reaches a maximum value of 2.0738 when D = 0.3. Similarly, SNR maximum value is 0.9314 when D = 0.15 in TFTSR system. It can be found that, in the HTFTSR system, the SNR slowly varies from 2.001 to 2.006 at D = 0.05–0.6, indicating that the HTFTSR system has considerable processing power for this range of noise intensities. By comparing the signal-to-noise ratio

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of actual signals, it is shown that HTFTSR is more capable of transforming noise energy into weak signal energy than TFTSR, so that the amplitude of fault signal can be improved. Therefore, compared with the traditional multi-stochastic resonance system, the multi-stochastic resonance system based on high-order delay feedback proposed by presented in this paper can extract the faint fault characteristic signals with strong noise effectively. 6. Conclusion In this paper, a stochastic resonance system based on highorder delay feedback is proposed and discussed the feasibility of the system for weak fault signature extraction. The main conclusion can be concluded as follows: 1. The effective potential function and steady-state probability density function of the system are deduced and the influences of time delay strength e and depth with time delay τ are discussed. 2. According to the relationship between parameters e, τ , D and MFPT, it can be found that the delay intensity e and τ plays a significant role in controlling MFPT. The influence of e and τ on the stochastic resonance system from the perspective of the transition of the particles in the potential wells are discussed. 3. The SNR and the effect of the parameters b, c, d, e and τ on the SNR are discussed. It can be found that adjusting the time delay strength e is more sensitive that of the time delay τ in controlling SR. 4. The HTFTSR system are used to deal with faulty bearing data. The advantages of the HTFTSR system for extracting weak fault signature signals are clarified by comparing with the results of conventional TSR system. The results show that this method can effectively detect frequency weak signals submerged in noise. The test results are clear and accurate. Declaration of competing interest The authors declare that they have no conflict of interest. Acknowledgments The studies were funded by the National Natural Science Foundation of China [Grant numbers 51475407 and 51875500], Hebei Provincial Natural Science Foundation of China (Grant number E2019203146), and Hebei Province Graduate Innovation Funding Project [Grant number 20190 0 0629]. References [1] Benzi R, Sutera A, Vulpiani A. The mechanism of stochastic resonance. J Phys A: Math Gen 1981;14:453–7. [2] Douglass JK, Wilkens L, Pantazelou E, Moss F. Noise enhancement of information transfer in crayfish mechancreceptors by stochastic resonance. Nature 1993;365:337–40. [3] Levin JE, Miller JP. Broadband neural encoding in the cricket cercal sensory system enhanced by stochastic resonance. Nature 1996;380:165–8. [4] Stacey WC, Durand DM. Stochastic resonance improves signal detection in hippocampal CAI neurons. J Neuropysicol 20 0 0;83:1394–402. [5] Applebaum D. Extending stochastic resonance for neuron models to general Lévy noise. IEEE T Neural Netw 2009;20:1993–5. [6] Sharma A, Kohar V, Shrimali MD, Sinha S. Realizing logic gates with time-delayed synthetic genetic networks. Nonlinear Dyn 2013;76:431–9. [7] Jin YF. Noise-induced dynamics in a delayed bistable system with correlated noises. Physica A 2012;391:1928–33. [8] Xu PF, Jin YF. Mean first-passage time in a delayed tristable system driven by correlated multiplicative and additive white noises. Chaos Soliton Fract 2018;112:75–82. [9] Shi PM, Xia HF, Han DY, Fu RR, Yuan DZ. Stochastic resonance in a time polo-delayed asymmetry bistable system driven by multiplicative white noise and additive color noise. Chaos Soliton Fract 2018;108:8–14.

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