Fractional Order Derivative and Time-Delay Feedback Enabled Stochastic Resonance for Bearing Fault Diagnosis

Te benefts of noise can be found in nonlinear systems where a type of resonances can inject the noise into systems to enhance weak signals of interest, including stochastic resonance, vibrational resonance, and chaotic resonance. Such benefts of noise can be improved further by adding some items into the nonlinear systems. Considering the time-dependent memory of fractional-order derivative and time-delay feedback which makes the nonlinear systems take advantage of their historical information and makes the output of nonlinear systems afect the input by feedback control, therefore, we attempt to design the model of stochastic resonance (SR) enhanced by both fractional-order derivative andtime-delay feedback. Among them,fractional-order derivative andtime delay would reinforcethe memory of nonlinear systems for historical information and feedback would use the output of systems to control the systems precisely. Terefore,


Introduction
Stochastic resonance (SR) is a kind of nonlinear phenomena where noise is injected into nonlinear systems to enhance their outputs [1].Terefore, SR lets us discover the benefts of noise [2].With the development of research, it is found that there are lots of the benefts of noise in real world in addition to SR [3], for example chaotic resonance [4][5][6] and vibrational resonance [7].Such a behavior has attracted sustaining attention to apply it into various felds such as image enhancement [8], information transmission [9], and fault detection and fault diagnosis of rotating machinery [10][11][12][13], especially the development and future directions of SR in machine fault diagnosis have been summarized and proposed by [10,14].
Among them, SR-based fault detection and fault diagnosis has become one of the most widely studied and applied directions.Tey can be categorized into novel potentials [15], novel nonlinear systems [16], novel behaviors [17], novel methodologies [18], and others.Because this paper focuses on improving novel nonlinear systems, we would review the development of novel nonlinear systems, which mainly contains two aspects.
On the one hand, some scholars attempt to add the fractional-order derivative into nonlinear systems to induce more abundant dynamics for better weak signal detection than those without fractional-order derivative.For example, Zheng et al. applied genetic algorithms to optimize the multi parameters of fractional-order SR for weak signal detection [19].Yang et al. investigated stochastic P-bifurcation and SR in fractional-order bistable nonlinear systems, indicating that the fractional-order SR can enhance the weak signal better than integral-order ones [20].Zeng et al. designed a new indicator, namely, weighted correctional signal-to-noise ratio, to tune the parameters of fractional-order SR for mechanical fault diagnosis [21].Qiao et al. employed the fractional-order derivative to improve the second-order SR for enhancing weak fault signature of machinery [22].Guo et al. studied the efect of random mass and signal-modulated noise on SR in fractional-order harmonic oscillators [23].Yu et al. explored the SR in two coupled fractional harmonic oscillators induced by a dichotomous fuctuating mass, indicating that coupling strength and the fractional order both beneft to weak signal detection [24].Zhong et al. studied the collective SR in globally coupled fractional-order harmonic oscillators induced by multiplicative noise [25].Te above researches indicated that the fractional-order derivative is able to improve SR for weak signal detection than integral-order ones.
On the other hand, time delay can utilize the historical information of nonlinear systems to enhance weak signal detection, whereas feedback could achieve the precise control of nonlinear systems for improving the output.Terefore, some scholars pay more attention to studying the SR with time delay and feedback.For example, Wu and Zhu investigated the SR in a bistable system with time-delayed feedback driven by non-Gaussian noise by using two indicators including quasi-steady-state probability distribution function and signal-to-noise ratio (SNR) [26].Lu [35].In sum, both time delay and feedback can improve the weak signal detection of SR by precise control and information memory.
Te above summary of literature has indicated that both fractional-order derivative and time-delayed feedback can improve the weak signal detection of SR from historical information memory, precise control, and so on.Up to now, however, the SR with both fractional-order derivative and time-delayed feedback has not been studied and applied to mechanical fault diagnosis yet.Terefore, this paper attempts to fuse their advantages to improve the weak signal detection of SR further, thereby enhancing the weak early fault signature of the machinery.Inspired by such an idea, the remainder of this paper is organized as follows.Section 2 designs a SR model enhanced by fractional-order derivative and time-delay feedback and provides its mathematical expression.In Section 3, we apply the SR enhanced by fractional-order derivative and time-delay feedback to bearing fault diagnosis and a bearing fault experiment is performed to demonstrate its efectiveness and feasibility.Even the comparison between those with or without fractional-order derivative and time-delay feedback is made.Finally, conclusions are drawn in Section 4.

An SR Model Enhanced by Fractional-Order Derivative and Time-Delay Feedback
Te existing nonlinear systems of SR are of two kinds: overdamped and underdamped ones [36].Overdamped SR characterizes the low-pass fltering properties [37,38], while underdamped SR characterizes the band-pass fltering properties which is more suitable to enhance weak signals under strong background noise than overdamped ones [39,40].Hence, we pay attention to underdamped SR and it can be described as follows: where c is the damping factor and c > 0, A and ω 0 are the amplitude and angular frequency of the periodic signal.x and t are moving trajectory of Brownian particles in U(x) as time varies and time, respectively.ε(t) is the Gaussian white noise.U(x) is the harmonic-Gaussian double-well potential given by equation ( 2) and has richer dynamics than classical bistable potential.Terefore, it can be used to design the SR model enhanced by fractional-order derivative and timedelay feedback.
where k, α, and β are the adjusting parameters.Te harmonic-Gaussian double-well potential has two stable states and one unstable state located at  Shock and Vibration Furthermore, the Grunwald-Letnikov fractional-order derivative [41,42] is added into equation ( 1) as follows: where ϑ is the fractional order and ϑ ∈ (0, 2] [43].We rewrite equation ( 3) for numerical solution easily as follows: According to the defnition of the Grunwald-Letnikov fractional-order derivative, equation ( 4) can be transformed into the following discrete expression: where and l � 1, 2, ..., N in which N is the length or the number of sampling points of the signal F(t) � A cos ω 0 t + ε(t).Te variable h is the integral step.x(l), y(l), and F(l) are the corresponding discrete expressions of x(t), y(t), and F(t), respectively.It can be seen from equation ( 5) that adding the fractional-order derivative into the underdamped SR can make the current value of output x(t) highly depend on historical values of the past.Due to the above reason, the fractional-order derivative can enhance the weak signal detection of SR.Moreover, such a property is consistent with a mechanical signal which has high dependence between each value.Furthermore, we consider the memory of time delay to historical information and the precise control of feedback to the output of SR [44].Te time-delay feedback item is added into equation ( 4) to obtain the following equation: Shock and Vibration 3 where θ > 0 is the feedback strength and τ is the time delay.Equation ( 6) can be discretized to solve it numerically as follows: x where f s is the sampling frequency and the notation ⌊•⌋ stands for round down.In equation ( 7), when − ⌊τ/h⌋ < 0, x(l − ⌊τf s ⌋) � x(0) � 0. According to the conclusions in [45,46], the small delay is acceptable when τ < 1.
Terefore, a SR model enhanced by fractional-order derivative and time-delay feedback is built in equation ( 6) and its numerical solution expression can be given by equation (7).In the model, there are eight tuning parameters to control the SR for enhancing weak signal detection, including fractional order ϑ ∈ (0, 2], damping factor c > 0, harmonic-Gaussian double-well potential parameters including k, α, and β with the condition k < 2αβ, feedback strength θ, time delay 0 < τ < 1, and integral step 0 < h < 1. Obviously, it is impossible for us to set up these tuning parameters artifcially.For this purpose, we would use the optimization algorithms to tune these parameters automatically in this proposed method for enhancing weak fault signature of machinery.

Fractional-Order Derivative and Time-Delay Feedback Enabled Stochastic Resonance for Bearing Fault Diagnosis
In this section, we would propose an adaptive SR method based on the model built in Section 2 to enhance weak fault signature for identifcation.Moreover, an experiment on bearing faults was performed to verify the feasibility and efectiveness of the proposed method.
3.1.Te Proposed Method.Te proposed fractional-order derivative and time-delay feedback enabled stochastic resonance method for bearing fault diagnosis is shown in Figure 2 and the detailed processes are given as follows: (1) Parameter initialization: Te proposed method is based on the SR model built in equation ( 6) and, therefore, it has eight tuning parameters including fractional order ϑ, damping factor c, potential parameters k, α and β, feedback strength θ, time delay τ, and integral step h.According to their defnitions and physical meanings, they are initialized as ϑ ∈ (0, 2], c ∈ (0, +∞], k ∈ (0, +∞], α ∈ (0, +∞], β ∈ (0, +∞], θ ∈ (0, +∞], τ ∈ (0, 1), and h ∈ (0, 1), respectively.(2) Output solution: Te vibration signal F(t) of bearings with the sampling frequency f s and the data length N is fed into equation (6).Ten, the output x(l) can be calculated by using equation (7) where In the solving process, the initialization values of these tuning parameters including ϑ, c, k, α, β, θ, τ, and h would be substituted into equation (7) for solving x(l).(3) Parameter optimization: According to equation (8), we can calculate the local signal-to-noise ratio (LSNR) of the output x(l) as an objective function of the genetic algorithms.For fault diagnosis of machines, the intensity of fault signature in the local frequency band would be cared instead of the whole frequency band.Terefore, LSNR indicator has more meaning to evaluate weak fault signature enhanced by SR.Furthermore, the solving issue can be transformed into the following optimal issue in equation (9).
where A(i) denotes the amplitude at i-th spectral line of the frequency spectrum of the output signal x(l) and M is the tuning range for selecting a local frequency band, and here M � 50.f c denotes the theoretical value of the fault characteristic frequency of bearings, which can be calculated by virtue of structural parameters and operating speed of bearings.
(4) Optimal output calculation and fault identifcation.Te optimal parameter set for maximizing the LSNR is substituted into equation (7) for solving x opt (l).
Ten, common spectral analysis such as frequency spectrum and envelope spectrum is performed on the x opt (l) to recognize the weak fault signature of roller bearings, which especially pays close attention to the spectral peaks at the fault characteristic frequencies of the bearings.

Experimental Verifcation.
In this subsection, the proposed fractional-order derivative and time-delay feedback enabled stochastic resonance method is applied to bearing fault diagnosis for verifying its feasibility and efectiveness.Due to contact fatigue, uneven lubrication, misalignment, and so on, it is inevitable for roller bearings to occur wears and other types of defects [47][48][49].In early stage of defects, the changes of vibration and oil temperature of bearings are too weak to be visible to the naked eye [50].In serious stage of defects, the changes of vibration and oil temperature of bearings are very clear to be observed by ears and fngers [51].Terefore, it is a challenge for us to detect the early defects of bearings by using advanced methods and technologies.For this purpose, we would apply the proposed method to early fault diagnosis of bearings.Four Rexnord ZA-2115 double row bearing run-tofailure experiments under the rotating speed 2000 rpm and radial load 6000 lbs were performed to acquire the bearing failure data by using PCB 353B33 accelerometers and a data acquisition card.Te bearing experimental rig is shown in Figure 3(a) and the corresponding sensor placement is illustrated in Figure 3(b).Tis experimental rig is composed of four tested bearings, an AC motor, and rub belts.In the bearing run-to-failure experiment, the sampling frequency is 20 kHz and the sampling time is 1.024 seconds.Te experimental parameters of tested bearing are shown in Table 1.In the process of experiment, an inner race defect occurs on the tested bearing 3, as shown in Figure 3(c).According to the equation that f inner � r speed /60 * 1/2 * n(1 + d/D cos ψ), where f inner is inner race fault characteristic frequency, r speed is the rotating speed, n is the number of rollers, d is the roller diameter, D is the pitch diameter, and ψ is the contact angle, and the theoretical inner race fault characteristic frequency f inner � 296.93 Hz.
Te time-domain signal collected from a bearing usually changes when a damage occurs in a bearing.Both its amplitude and distribution may be diferent from those of the time-domain signal of a normal bearing.Root mean square (RMS) refects the vibration amplitude and energy in time domain [53].Figure 4(a) depicts the RMS of the tested bearing 3. It can be found from Figure 4(a) that the RMS starts to be stable and then becomes larger slowly and fnally degenerates rapidly.To the end, a serious defect occurred on the tested bearing 3. Terefore, we choose the vibration signal at the early stage of defects to perform the spectral analysis, as shown in Figure 4(b).From the raw signal, we can see weak impacts but they are submerged by strong noise.In Figure 4(c), there are clear spectrum peaks at 232.5 Hz, 493.8 Hz, etc., but we cannot see the spectral peaks at the inner race fault characteristic frequency and its harmonics.Meanwhile, we also cannot see the corresponding spectral peaks from the zoomed envelope spectrum in Figure 4(d).
Te raw signal in Figure 4(b) is fed into the proposed method and the underdamped SR without both fractional derivative and time-delay feedback to enhance weak fault signature, respectively.Te enhanced results are plotted in Figure 5.It can be found that there are clear repetitive impacts in the time-domain signal as shown in Figure 5(a) and we can see the eye-catching spectral peaks at the inner race fault characteristic frequency and its harmonics from the corresponding frequency spectrum in Figure 5(b) and even side frequency bands with the interval rotating frequency.Te above information tells us the fact that an inner race wear has happened in the tested bearing 3. Te diagnosis results are consistent with the experimental results.Meanwhile, Figures 5(c) and 5(d) show the enhanced signal and its zoomed frequency spectrum by using the underdamped SR without both fractional derivative and timedelay feedback, where the objective function is usually SNR instead of LSNR.It can be noticed that there exists a eyecatching spectral peak at the inner race fault characteristic frequency, but we cannot see other any diagnostic information, that is, because the SNR indicator would be calculated by the ratio between the energy at the inner race fault characteristic frequency and the energy at the whole frequency band in addition to fault signature, resulting in energy concentration and ignoring other diagnostic information.Moreover, the spectral peak at the inner race fault Shock and Vibration

Initialize eight tuning parameters
Input the vibration signal into the model in Eq. ( 6) Solve the output x (l) with l=1, 2, ..., N numerically of Eq. ( 6) by using Eq. ( 7) Calculate the LSNR of the output x (l) as the objective function of the genetic algorithm

Maximum of LSNR?
Output the optimal parameter set and substitute it into Eq.( 7) to solve the optimal output x opt (l) Perform the spectral analysis on the optimal output 6 Shock and Vibration  Shock and Vibration 7 characteristic frequency using the proposed method is higher than that one using underdamped SR without both fractional derivative and time-delay feedback.Te above comparison demonstrates that the proposed method has richer diagnostic information and higher spectral peak at the inner race fault characteristic frequency than underdamped SR without both fractional derivative and time-delay feedback.
For comparison, the advanced infogram method [54][55][56] is applied to process the raw vibration in Figure 4(b), and the infogram for selecting the most informative frequency band is shown in Figure 6.It can be seen that the SE   infogram in Figure 6(a) and SES infogram in Figure 6(b) select diferent fltered frequency band and its parameters.Terefore, we apply two fltered frequency bands to flter the raw signal and the results are depicted in Figure 7.It can be seen that the SE infogram flters out the cyclostationary information as shown in Figure 7(a), but we cannot see the peaks at the inner race fault characteristic frequency and its harmonics from Figure 7(b).On the contrary, the SES infogram flters out the impulsivity information as shown in Figure 7(c), but we cannot also see the peaks at the inner race fault characteristic frequency and its harmonics from Figure 7(d).Te comparison with other non-SR methods demonstrates the feasibility and superiority of the proposed method further.

Conclusions
Stochastic resonance (SR) has become a hot signal processing method and has been widely used in mechanical fault diagnosis, such as fractional-order SR and time-delayed feedback SR.Here, fractional-order SR can utilize the historical information to enhance weak fault signature, and time delay and feedback can improve the memory of SR and tune it precisely.Terefore, we attempt to fuse fractional-order derivative and time-delayed feedback to develop the better SR method.Inspired by the idea, fractional-order derivative and time-delay feedback enabled SR method for bearing fault diagnosis is proposed in this paper.Te comparison with the advanced infogram and the SR without fractional-order derivative and timedelay feedback is made.Te results indicate that the proposed method has a little superiority in enhancing weak fault signature and richer diagnostic information in the enhanced results than the SR without fractional-order derivative and time-delay feedback.In future work, we would pay more attention to design SR-based flters to take the place of flters in infogram for exploring the SR-based infogram.
To keep the bistability of harmonic-Gaussian double-well potential, 2αβ/k > 0 and further k < 2αβ.Te potential is plotted in Figure1under diferent tuning parameters.It can be found that when α and β are kept unchanged, tuning k can change the depth and width of double wells, as shown in Figure1(a), where k < 2αβ.When k starts to become larger and k > 2αβ, the bistability of harmonic-Gaussian double-well potential loses becomes a monostable potential, as shown in Figure 1(b).Terefore, the following work would be performed under the condition of k > 2αβ.

Figure 3 :
Figure 3: Bearing test rigs and sensor placement illustration: (a) bearing test rigs, (b) sensor placement illustration, and (c) the tested bearing with an inner race wear [52].

Figure 2 :
Figure 2: Te fowchart of the proposed method.

Figure 4 :
Figure 4: Te RMS and raw signal of the tested bearing 3: (a) the RMS, (b) raw signal, (c) frequency spectrum, and (d) zoomed envelope spectrum.

Figure 5 :
Figure5: Te enhanced result and its spectrum using the proposed method and underdamped SR without both fractional derivative and time-delay feedback: (a) time-domain signal and (b) its zoomed frequency spectrum using the proposed method and (c) time-domain signal and (d) its zoomed frequency spectrum using underdamped SR without both fractional derivative and time-delay feedback.

Figure 7 :
Figure 7: Te detected results using infogram: (a) the fltered signal and (b) its amplitude spectrum using SE infogram and (c) the fltered signal and (d) its amplitude spectrum using SES infogram.
[27]l.proposed a time-delayed feedback SR method for mechanical fault diagnosis[27].Shiet al. presented a timedelayed tristable SR method for mechanical fault diagnosis, suggesting that time delay and feedback afect the noise enhanced stability [28].Meanwhile, Shi et al. designed a high-order time-delayed feedback tristable SR method for enhancing weak fault signature [29].Wadop Ngouongo et al. reported the SR with memory efects in a deformation potential [30].Wang et al. studied the efect of fractional damping and time-delayed feedback on SR [31].Li et al. studied the time-delayed feedback monostable SR in bearing fault diagnosis by fusing minimum entropy deconvolution [32].Wang et al. explored the infuences of time-delayed feedback on logical SR and the results indicated that the delay time can enhance the output SNR [33].Yang et al. indicated that the time delay can control primary resonance and SR, and increasing feedback intensity can suppress the vibrations [34].Liu et al. proposed a controlled SR method with time-delayed feedback to enhance weak fault signature of machinery

Table 1 :
Experimental parameters of tested bearings.