Adaptive noise cancellation based on beehive pattern evolutionary digital filter

Mechanical Systems and Signal Processing 42 (2014) 225–235

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Adaptive noise cancellation based on beehive pattern evolutionary digital filter Xiaojun Zhou, Yimin Shao n State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing 400044, China

a r t i c l e i n f o

abstract

Article history: Received 3 October 2011 Received in revised form 4 June 2013 Accepted 17 August 2013 Available online 14 September 2013

Evolutionary digital filtering (EDF) exhibits the advantage of avoiding the local optimum problem by using cloning and mating searching rules in an adaptive noise cancellation system. However, convergence performance is restricted by the large population of individuals and the low level of information communication among them. The special beehive structure enables the individuals on neighbour beehive nodes to communicate with each other and thus enhance the information spread and random search ability of the algorithm. By introducing the beehive pattern evolutionary rules into the original EDF, this paper proposes an improved beehive pattern evolutionary digital filter (BP-EDF) to overcome the defects of the original EDF. In the proposed algorithm, a new evolutionary rule which combines competing cloning, complete cloning and assistance mating methods is constructed to enable the individuals distributed on the beehive to communicate with their neighbours. Simulation results are used to demonstrate the improved performance of the proposed algorithm in terms of convergence speed to the global optimum compared with the original methods. Experimental results also verify the effectiveness of the proposed algorithm in extracting feature signals that are contaminated by significant amounts of noise during the fault diagnosis task. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Beehive pattern structure Evolutionary filter Convergence speed Fault feature signal

1. Introduction The faulty feature signals from structures in a mechanical transmission system are usually masked by the significant background noise. A successful fault diagnosis may be achieved based on the improvement of the signal to noise ratio (SNR) of the diagnostic signal [1,2]. The de-noising can be achieved using an adaptive noise cancellation filter (ANC), which is a special case of optimal filtering when some information about the reference noise signal is available [3–6]. The object is to identify or design an appropriate filter to optimally estimate the noise according to the reference signal and subtract it from the primary noisy signal [7]. It is noted that the reference signal does not have to be identical to the corresponding part of the primary signal, just related to it by a linear transfer function. The ANC procedure adaptively finds the transfer function, and can thus subtract the modified reference signal from the primary signal, leaving the desired component [8]. The reference noise signal and the noise contained in the primary noisy signal have an unknown linear dynamic relationship. The least mean square (LMS) and its modified algorithms are the most popular adaptive search algorithms used [9,10]. However, the LMS will converge to a local optimum in solving a multiple-peak problem since it is based on the gradient search algorithm. To overcome the shortcoming of local optimum in the multiple-peak problem, Abe et al. [11] proposed the

n

Corresponding author. Tel.: þ86 2365112520. E-mail address: [email protected] (Y. Shao).

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EDF algorithm. It is similar in concept to the genetic algorithm (GA) utilising the evolutionary strategies of living creatures [12], but is different concerning the genetic operator and the representation of individuals. The EDF algorithm is a population-based and robust optimisation method, with random search capability to get the evolutionary relationship between generations and reach the global optimisation using cloning and mating rules. It has been shown that the EDF can find the global minimum in a multiple-peak surface and exhibits smaller adaptation noise than other algorithms, such as the least mean square based adaptive digital filter (LMS-ADF) and the simple genetic algorithm based adaptive digital filter (SGA-ADF) [13]. The characteristics of EDF and its enhanced algorithm are further studied in [14–18]. Therefore, the EDF algorithm is not susceptible to local optimum which results from a multiple-peak surface, and has good de-noising performance in improving the SNR. However, the convergence speed and time consumption of the EDF are restricted by complex multiplications. These defects make the EDF-based algorithms not suitable for the on-line condition monitoring. To improve the convergence speed and noise cancellation ability of the EDF algorithm, an improved Beehive Pattern Evolutionary Digital Filter (BP-EDF) is proposed in this paper. The beehive pattern model is established and a new evolutionary rule combining the competing cloning, complete cloning and assistance mating is constructed to realise information communication among the generations and individuals. The optimum faulty feature signal can be subsequently extracted from the noisy signal by cancelling the global optimum noise using the proposed BP-EDF. The de-noising performance and convergence speed of the proposed BP-EDF algorithm are verified through simulation and experimental signals. This paper is organised as follows: in Section 2 the theoretical fundamentals of the new evolutionary rule and the proposed BP-EDF algorithm are presented. In Section 3 the de-noising performance and convergence speed of the proposed algorithm are verified using simulated signals. Section 4 presents the experimental verified. Section 5 provides the conclusions. 2. Adaptive noise cancellation based on the beehive pattern evolutionary digital filter 2.1. Review of the classic evolutionary rules The rules used in the original EDF are defined as the “original evolutionary rules”. The digital filter coefficients are updated as individual features according to the cloning and mating method employed in the rules [11]. Fig. 1 shows the original evolutionary rules. There are four steps in the rules: 1) Population creation. Suppose there are 2n individuals in the jth generation, they are sorted according to the fitness value in a descending order. 2) The cloning method. The first n individuals use the cloning method to get the offspring. The offspring are similar in characteristic to their parents. The kth parent and its offspring form a new family with (n þ1) individuals, and the

offspring m

Sorted according to the fitness

Sorted according to the fitness

parents 2n

cloning

mating

jth generation

(j+1)th generation

Fig. 1. Illustration of the original evolutionary rules.

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individual with the maximum fitness in this family is selected as the kth individual in the (jþ1)th generation. The coefficients are scattered in a narrow area, thus, the cloning method corresponds to a local search. 3) The mating method. The other n individuals use the mating method to get the offspring. Two parents in this group are randomly selected to create an offspring. The parent with higher fitness value and the offspring in each family survive into the (jþ1)th generation, while the parents with lower fitness die out. The mating method keeps various features of individuals, so it corresponds to the global search. 4) Regeneration. The survived 2n individuals form the (jþ1)th generation, and steps (1–3) are repeated until the requirements are satisfied. According to the rules, half of the population attend the cloning method which is much more complicated, thus causing the low convergence speed problem of the classic EDF. In the mating method, information is only communicated between two individuals in each family, which results in the information in one generation not being communicated completely. 2.2. Rules of the beehive pattern evolutionary method The rules proposed in this work are defined as “beehive pattern evolutionary rules”. In the new rules, the inner digital filter coefficients as updated according to the competing cloning, complete cloning and assistance mating methods. Fig. 2 shows a sketch diagram of the beehive pattern evolutionary rules. There are five steps in the rules: 1) Population creation. The beehive with Ncell layers is constructed by beehive cells which have six nodes. The individuals are cell distributed on the nodes, thus the number of the individuals can be expressed as ∑N i ¼ 1 ð12i6Þ. According to the special structure of the beehive, each individual belongs to two or three beehive cells simultaneously and it can have three modes to communicate information and create offspring. 2) The competing cloning method. As shown in Fig. 2, the innermost layer is defined as the competing cloning layer. The individuals on the nodes in this layer take the competing cloning method to create offspring and select the optimum individuals surviving to the next generation. This process aims to search out the optimum in different local areas. In the jth generation, the individual on the nth node of this layer is a parent, and it creates m offspring which are creations of small fluctuations around the parent. The (m þ1) candidates form a family and the individual with the highest fitness value survives from this family into the (jþ1)th generation. Thus, this operation corresponds to the local search and it will lead the population to evolve towards the optimum individual with optimum digital filter coefficient vectors. 3) The assistance mating method. The outermost layer is defined as the assistance mating layer. The individuals on the nodes in this layer take the assistance mating method to create offspring and select the individuals surviving to the next generation. This method aims to increase the random search performance with full information communication among the individuals. The assistance mating method can be divided into two cases: i. If the individual has two neighbours in the unit beehive cell area, the offspring are created by the population in the beehive cell to which it belongs. That is, 6 individuals create one offspring. ii. If the individual have three neighbours in the unit beehive cell area, the offspring are created by the population in the unit beehive cell area. That is, 4 individuals create one offspring. The offspring can combine the information from the parents with different population scales. The offspring all survive to form the next generation while the parents die out. Thus, this operation corresponds to the global search, and it enhances communication and transmission of information carried by the population. This operation also keeps the various features of individuals.

Beehive with 3 layers

Assistance mating layer

Offspring with m individuals

Parents with n individuals The jth generation

Assistance mating Competed cloning

Sorted according to the fitness value

Complete cloning layer

Complete cloning

Sorted according to the fitness value

Competed cloning layer The (j+1) generation

Fig. 2. Illustration of the proposed beehive pattern evolutionary rules.

Beehive cell

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4) The complete cloning method. The other beehive layers are defined as the complete cloning layer. The individuals on these nodes in these layers use the complete cloning method to create offspring of the next generation. This process aims to let the population convergence to the optimum steadily. The offspring in the (jþ1)th generation will inherit from the jth generation parents directly on the complete cloning layer. That is, they will carry the same information as the parents. 5) Regeneration. The surviving individuals form the (jþ1)th generation, and steps (1–4) repeat until the requirements are satisfied. In the beehive pattern evolutionary rules, the beehive structure enables individuals to belong to several beehive cells simultaneously, and the individuals on the neighbour nodes can communicate with each other. The spread of information and the random search ability of the algorithm can be enhanced. It overcomes the local optimum limitation and converges to the global optimum individual much more quickly.

2.3. Adaptive noise cancellation based on the BP-EDF filter Fig. 3 shows a diagram of the proposed adaptive noise cancellation algorithm based on the BP-EDF. The parameters of the inner digital filter evolve to the optimum according to the beehive pattern evolutionary rules. The details of the proposed BP-EDF algorithm in Fig. 3 can be expressed as follows: 1) construction of the digital adaptive filter coefficients The input reference signal X(k) of the inner filter is injected commonly to the ith inner digital filter Fi, and the output

Reference Signal X(k)

Primary Signal S(k)

Construct the adaptive filter coefficients as individual feature vector on the beehive nodes W(k) Calculate the residual signal and the fitness value of the individuals Sorted the individual according to the fitness

Positionofthe individualinthe beehive

competed cloning method

complete cloning method

assistance mating method

Reform the (j+1)th generation beehive Calculate the indicators of the output signal from the beehive nodes

No

| e - |< Yes Optimum output signal of the BP-EDF algorithm

Fig. 3. The diagram of the proposed BP-EDF algorithm.

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signal yi(k) is given by [13] N

M

n¼1

m¼0

yi ðkÞ ¼ ∑ an;i ðkÞyi ðknÞ þ ∑ bm;i ðkÞXðkmÞ;

i ¼ 1; 2; :::; Q

ð1Þ

where Q is the number of the individuals, M and N are the order of the moving average part and the auto-regressive part, respectively. The filter coefficients of each inner digital filter Fi at time k form the feature vector Wi ðkÞ as [13] Wi ðkÞ ¼ ½a1;i ðkÞ; a2;i ðkÞ; ⋯; aN;i ðkÞ; b0;i ðkÞ; b1;i ðkÞ; ⋯; bM;i ðkÞT

ð2Þ

2) The fitness function The fitness of the output signal for filter Fi can be calculated as [13] ei ¼

1 L ∑ ðSðkÞyi ðkÞÞ2 Lk¼1

ð3Þ

where L is the data sample of the input primary signal S(k). 3) The competing cloning, complete cloning and assistance mating methods The individuals are distributed on the beehive nodes and sorted according to the fitness value in descending order from the innermost to the outermost layer of the beehive. In the competing cloning method, the parent individuals live on the competing cloning layer. The competing cloning method updates the inner digital filter coefficients as individual features according to Wc;i;j ¼ Wp;i;j þr U nl

ð4Þ

where i¼1,2… Nap, l¼1,2… Nac, Nap and Nac are the number of the parents and the offspring in the competing cloning method. r denotes the cloning fluctuation, nl is a Gaussian random variable vector with zero mean and unit variance. Wp,i,j and Wc,i,j are the inner filter coefficient feature vectors of the parents and the offspring, respectively. Finally, the individual with the highest fitness value survives to the (jþ1)th generation from the family which is formed by the parent and the Nij offspring. In the complete cloning method, the parent individuals live on the complete cloning layer, the offspring inherit the coefficient vectors completely from the parents. The complete cloning method updates the inner digital filter coefficients as individual features according to Wi;j þ 1 ¼ Wi;j

ð5Þ

where Wj and Wj þ 1 are the jth and (jþ1)th generation individuals. i¼1,2… Nbp, Nbp is the number of the population in the complete cloning method. In the assistance mating method, the parent individuals live on the assistance mating layer. The assistance mating method updates the inner digital filter coefficients as individual feature according to Wi;j þ 1 ¼ q U nl þ ∑pm ¼ 1 U m Wi;j;m

ð6Þ

where i¼ 1,2…Nsp, Nsp is the number of the individuals in the assistance mating method. q denotes the mating fluctuation. nl is the a Gaussian random variable vector with zero mean and unit variance. p is the number of the neighbours in the unit area of the ith individual, with p¼ 6 for case (i) of the assistance mating method and p¼4 for case (ii). Um are the weight parameters. 4) The output of BP-EDE The output zj(k) from the filter with the jth generation coefficient vectors is given by [13] zj ðkÞ ¼ ymaxðei ðkÞÞ ðkÞ

ð7Þ

where zj(k) is the output signal of the inner filters that has the maximum fitness. 5) Convergence judgement For an output with neighbouring generation fitness as ej and ej þ 1, it is defined that if a ξ exists to make 8 ε 4 0 & jΔeξj o ε; where Δe ¼ ej þ 1 ej

ð8Þ

then it is determined that the fitness rate converges to ξ and fitness converges to e. If the fitness value reaches a stable convergence, the corresponding signal is outputted and taken as the feature estimation signal of the optimum de-noising; or, return to step 1 and enter the next evolutionary iteration using the generated new population until convergence is reached. 3. Simulation validation and performance study of the proposed algorithm 3.1. Simulation validation The nonlinear bearing vibration signal is selected as the feature signal D(k), and the simulated signal can be expressed as [19] DðtÞ ¼ eαt  sin 2πf c nT

ð9Þ

where t¼mod(nT,1/fm), α is the exponential frequency, fm is the modulation frequency, fc is the carrier frequency, T is the sampling time interval. Fig. 4 shows a simulated impulse feature signal consisting of 5 impulses with time interval T¼1/10,000 s

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15

Amplitude (A/g)

10 5 0 -5 -10 -15

0

0.05

0.1

0.15

0.2

0.25

0.3

Time (t/s) Fig. 4. Waveform of the bearing feature signal.

15

Amplitude (A/g)

10 5 0 -5 -10 -15

0

0.05

0.1

0.15

0.2

0.25

0.3

0.25

0.3

Time (t/s) Fig. 5. Waveform of the noise signal.

15

Amplitude (A/g)

10 5 0 -5 -10 -15

0

0.05

0.1

0.15

0.2

Time (t/s) Fig. 6. Waveform of the primary signal.

for α¼800, fc ¼ 300 Hz, fm ¼15 Hz. The total data is 16,384 sample points. The amplitude unit g is an acceleration unit standing for 9.8 m/s2. The cosine signal with frequency 17 Hz and amplitude 1 g is created as the noise signal Y(k) representing the vibration signal from other machine components. The Gaussian random noise signal with zero mean, unit variance and energy power equal to ten works as the noise signal Z(k). Fig. 5 shows the total noise signal (Y(k)þZ(k)). The primary signal S(k)¼D(k) þμ  (Y(k)þZ(k)) is obtained by combining the feature signal D(k) and the noise signal (Y (k)þZ(k)) with a  10 dB SNR, as shown in Fig. 6. Obviously, the significant noise weakens the periodic characteristic of the simulated feature signal. Fig. 7 shows the reference signal. The reference signal X(k) is related to the noise signal (Y(k)þZ(k)) but has different amplitude and phase information. Fig. 8 shows the comparison of the output signal in the time domain between the proposed BP-EDF algorithm, the original EDF algorithm [11] and the adaptive noise cancellation based on the complete cloning method. The beehive is three layers with Ncell ¼ 3. The parameters used in the calculations are 54 individuals in each generation and 10 offspring from one parent. The evolutionary generation is 100. The cloning fluctuation and the mating fluctuation are r ¼q ¼0.09 [11].

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15

Amplitude (A/g)

10 5 0 -5 -10 -15

0

0.05

0.1

0.15

0.2

0.25

0.3

Time (t/s) Fig. 7. Waveform of the reference signal.

30

Classic method Cloning method Proposed method

Amplitude (A/g)

20 10 0 -10 -20 -30

0

0.05

0.1

0.15

0.2

0.25

0.3

Time (t/s)

8

8

7

7

Kurtosis

Kurtosis

Fig. 8. Comparison of the output signals among the algorithms.

6 5 Classic method Cloning method Proposed method

4 3 0

20

40

60

80

Generation

100

6 5 Classic method Cloning method Proposed method

4 3 0

20

40

60

80

100

120

140

160

Time (t/s)

Fig. 9. Comparison of kurtosis with (a) generation and (b) time consumption among the algorithms.

Comprising Figs. 8 and 6, it can be seen that the periodic feature of the output signal using the BP-EDF algorithm is as noticeable as in the original signal and the amplitudes agree well, which supports the effective de-noising performance of the BP-EDF algorithm. It can also be seen that the feature signals are all well recovered using the three different algorithms. This suggests that the evolutionary rules in the original EDF and the modified rules in the proposed algorithm can both reach the global optimal estimation of the inner filter coefficients.

3.2. Convergence speed comparison To compare the performance of different algorithms, the indicator Kurtosis is chosen for the evaluation. Kurtosis is sensitive to the shape of the signal, and is thus well suited to the impulsive nature of the stimulating forces generated by component damage. The expression for Kurtosis is given by [20] Kurtosis ¼ ð1=LÞ∑Lk ¼ 1

ðxðkÞxÞ4 ½ð1=LÞ∑Lk ¼ 1 ðxðkÞxÞ2 2

ð10Þ

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where N is the number of data points in the signal, x(k) is the amplitude of the signal at the kth point, and x is the mean value of the signal. For a mechanical transmission system working under normal conditions, the vibration signal follows the Gaussian distribution and the kurtosis value equals to 3. A higher kurtosis value indicates the appearance of impulse and local fault in the system. Fig. 9 shows the evolution of kurtosis value with generation and time consumption in searching for the optimum output. In Fig. 9 (a), it can be seen that the kurtosis of the output signals using the three different algorithms all converge to the value 7.5, which is about 2.5 times the kurtosis value 3 of the primary input signal. These indicate that the optimum estimations are obtained. The failure impulse signals are much clearer and the noise is effectively cancelled with the application of the de-noising algorithms. It is also noted that the kurtosis value of the proposed algorithm always keeps ahead in each generation and converges earlier than the other two algorithms. This suggests that the proposed algorithm exhibits a higher convergence rate than the classic algorithm. Fig. 9(b) shows the time consumption for the three algorithms. It can be clearly seen that the proposed algorithm takes the least time to achieve the optimum output. The time cost of the proposed algorithm is about 1/3 of the classic evolutionary algorithm and 1/4 of the cloning method. Table 1 shows the comparison of number of multiplications per iteration among the three algorithms and the multiplication cost. It can be seen that the proposed algorithm takes the least number of multiplications among the three algorithms in the noise cancellation process. 3.3. Relationship between convergence speed and beehive layer The beehive layer represents the number of individuals participating in the evolution. To evaluate the effect of beehive layer on the de-noising and convergence performance of the BP-EDF, beehives with three different layers as 3, 4 and 5 are selected and tested. The 3 layer beehive is the simplest construct applicable for the proposed algorithm. Fig. 10 shows the comparison of evolution of kurtosis with the generation and time consumption for different beehive layers. It can be seen in Fig. 10(a) that the output signals from the BP-EDF with different beehive layers all converge to the optimum estimation steadily. The impulse feature signals are extracted and the related noise signals are cancelled. In the evolution progress, the kurtosis of the algorithm with a 5 layered beehive keeps ahead in each generation. This indicates that the de-noising performance of the BP-EDF is largely related to the individuals participating in the assistance mating. The search capability increases with the number of individuals. However, it is seen in Fig. 10(b) that the time consumption for the convergence increases with beehive layers. This is because, as the number of individuals increases, the calculation steps increase in each generation. 4. Experimental verification In order to evaluate the effectiveness of the proposed algorithm, vibration signals collected from an experimental rolling element bearing were used. The gearbox transmission system and its diagram are shown in Fig. 11. It comprises a 3-phase electrical induction motor, a two stage tested helical gearbox and the load device. The parameters of the gearbox transmission system are listed in Table 2. A tested ball bearing was installed on the output shaft close to the load device end, as shown in Fig. 11(b). A fault was introduced on the inner race of the tested bearing, as shown in Fig. 11(c). The motor and Table 1 Comparison of number of multiplications per iteration. Number of multiplications for the adaptive process

Multiplications in the simulation

Proposed Classic Cloning

[Nap(Nac þ 1) þNbp þ 2Nsp](N þ Mþ 1)L [Nap(Nac þ 1) þ2Nsp](N þM þ 1)L Nap(Nac þ1)(Nþ M þ1)L

[6  (10þ1) þ 18þ 2  30]  (3 þ3 þ1)  16,384 ¼16,515,072 [27  (10þ 1) þ2  27]  (3 þ3þ 1)  16,384 ¼40,255,488 54  (10þ 1)  (3 þ3 þ1)  16,384 ¼68,124,672

8

8

7

7

Kurtosis

Kurtosis

Algorithm

6 5 3 layer 4 layer 5 layer

4 3

0

20

40

60

Generation

80

100

6 5 3 layer 4 layer 5 layer

4 3

0

20

40

60

80

100

120

Time (t/s)

Fig. 10. Comparison of evolution of kurtosis with (a) generation and (b) time consumption for different beehive layers.

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Fig. 11. (a) Experimental system, (b) diagram of the system and (c) fault on the inner race of the experimental bearing.

Table 2 Parameters of the transmission system. Parameter

Value

Number of elements NR Ball diameter d (in.) Pitch diameter B (in.) Contact angle θ (deg) Teeth number Input motor speed (rpm) Sampling frequency fs (Hz) Sampling data length L Sampling time t (s)

8 0.3125 1.319 20 16/48/24/40 3,000 20,000/3 133,328 2

gearbox were housed in the same foundation and thus it was possible to simulate the noise conditions under different loads and speeds. The vibration signals were collected and used to test the diagnosis performance of the proposed method. An accelerometer was mounted on the top of the gearbox to collect vibration signals from the faulty bearing and this signal was used as the primary input signal to the proposed algorithm. Another accelerometer was mounted close to the motor to collect vibration signals and these signals were used as the reference input signals [3,8]. The datasets analysed in the following part were collected with the input motor speed at 3000 rpm. The characteristic frequency of the fault on the bearing inner race can be calculated by the following expression:   f d f i ¼ r NR 1 þ cos θ ð11Þ B 2 where fr is the bearing rotational frequency, NR is the number of elements, θ is the contact angle and d and B are the ball and pitch diameters, respectively. For the parameters of the tested bearing employed, the faulty characteristic frequency thus has the value of fi ¼49.48 Hz with fr ¼50 Hz. Fig. 12 shows the vibration signals of the two accelerators measured from the gearbox house and the motor. The signals are taken as the primary input signal S(k) and the reference input signal X(k), respectively. Several impulses can be observed from the vibration signal collected from the system with a faulty bearing; however, they are masked by noise and not evident enough to detect the existence of the fault. Then the proposed method is applied to processing the vibration signal. Fig. 13 shows the de-noised signal using the proposed algorithm for a three layer beehive in both time and frequency domain. The parameters used in the calculation are identical as that in the simulation signal shown in Fig. 8. The de-noised signal in Fig. 13(a) reveals clearer impulses compared with the original signal shown in Fig. 12(a). The time interval is about 0.0202 s corresponding to the frequency of 49.38 Hz. This is very close to the calculated faulty feature frequency of 49.48 Hz and also the rotating frequency of 50 Hz. However, it can be seen from Fig. 13(b) that no meshing frequency of the input shaft (fm ¼800 Hz) and its harmonics are observed in the spectrum of the de-noised signal. Thus, it is determined that this faulty feature frequency is not arising from the input shaft gear, but due to the faulty bearing.

X. Zhou, Y. Shao / Mechanical Systems and Signal Processing 42 (2014) 225–235

0.04

0.04

0.02

0.02

Amplitude (A/g)

Amplitude (A/g)

234

0 -0.02

0 -0.02 -0.04

-0.04 0

0.1

0.2

0.3

Time (t/s)

0.4

0.5

0

0.1

0.2

0.3

0.4

0.5

Time (t/s)

Fig. 12. Vibration signals of experimental rolling element bearing (a) primary input signal and (b) reference input signal.

Fig. 13. Output signal using the proposed algorithm (a) times series and (b) spectrum.

Fig. 14. Demodulation spectrum of the output signal (a) original noisy signal and (b) de-noised signal.

Fig. 14 shows the envelope demodulation spectrums of the original noisy signal and the de-noised signal. Although the envelope spectrum in Fig. 14(a) reflects the fault frequency (fi) and its harmonics of the bearing inner race, the SNR is much lower than that of Fig. 14(b). In Fig. 14(b), the fault frequency of the bearing inner race (fi) and its harmonics are more evident. It suggests that the proposed method is able to more clearly extract fault characteristics. Fig. 15 shows the comparison of the evolution of kurtosis with generation and time consumption among the three algorithms. It can be seen from Fig. 15(a) that the kurtosis value converges to 8 for the proposed algorithm, while the other two algorithms converge to about 7. This supports the better de-noising performance of the proposed algorithm. Fig. 15(b) shows that the proposed algorithm exhibits the least time to reach the convergence and thus to get the optimum estimation output in the three algorithms. The results from experimental datasets also show that the proposed BP-EDF can successfully discover the faulty characteristics from signals with significant noise and exhibit good performance in noise cancellation and convergence speed. 5. Conclusions In this study, an improved adaptive noise cancellation algorithm based on the beehive pattern evolutionary digital filter (BP-EDF) has been proposed to overcome the shortcomings of low convergence speed of the original evolutionary digital filter (EDF). A new evolutionary rule has been constructed by combining the competing cloning, complete cloning and assistance

10

10

9

9

8

8 Kurtosis

Kurtosis

X. Zhou, Y. Shao / Mechanical Systems and Signal Processing 42 (2014) 225–235

7 6

235

7 6 5

5 Classic method Cloning method Proposed method

4 3 0

20

40

60

Generation

80

100

Classic method Cloning method Proposed method

4 3 0

500

1000

1500

Time (t/s)

Fig. 15. Comparison of evolution of kurtosis with (a) generation and (b) time consumption among the algorithms.

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