Separation Method of Impulsive Fault Component for Gasoline Engine Based on Acoustic Signal Analysis

(is paper presents a developed dislocation superimposed method (DSM) for automatically extracting the component of impulsive signals from abnormal noise signals of an engine at a single speed range on the basis of the initial DSM. (is method consists of three steps: using a correlation analysis to select an appropriate starting superposition point, superimposing abnormal sound signals to improve the signal-to-noise ratio, and intercepting superimposed signals to separate the fault component. Experimental results show that the developed DSM can effectively extract the fault characteristics of cylinder knocking and connecting rod bearing knocking. (e developed approach can be applied to separate the fault characteristics of other types of rotating machines.


Introduction
An automobile engine is a core component of a car powered by fuel, and its structure is complicated.e failure of a part not only affects the machine operation but may also cause considerable economic losses or even catastrophic consequences.Common engine failures include wear, distortion, cavitation, and corrosion, and different faults can be detected in various approaches, such as tail gas, power and speed, oil, and vibration and noise analytical methods [1][2][3][4][5].
In comparison with other detection methods, the use of acoustic signals for extracting fault components has the advantage of detection without disassembling devices.e abnormal noise and vibration signals of an engine often contain considerable dynamic information about the engine state, including fault components, acoustic signals of other components working properly, and background noise [6].
e fault component of some faults often has quasiperiodicity.By processing abnormal sound signals, the extracted fault component can be used as an effective basis for analysing the cause of engine fault.As the fault component of abnormal signals has a low signal-to-noise ratio, the extraction of fault components is often the most important and difficult problem, which directly affects the accuracy and reliability of diagnosis [7].Methods for processing abnormal noise signals mainly include fast Fourier transform (FFT), wavelet transform, and empirical mode decomposition (EMD) [5,8,9].Classical signal processing methods, such as FFT and short-time Fourier transform, cannot analyse strong nonlinear and nonstationary signals.
Wavelet analysis can have good localization characteristics in the time and frequency domains; thus, it is often used to extract time-frequency characteristics [10].Figlus [11] used a continuous wavelet transform (CWT) to process vibration signals qualitatively, and they quantitatively identified the damage degree of the timing chain tensioner.Figlus [5,12] used the discrete wavelet transform (DWT) and wavelet packet decomposition to detect the valve clearance of a diesel engine automatically.Albarbar [13] proved that CWT is sensitive to the engine speed and load change by using it to manage diesel engine acoustic signals.Shirazi and Mahjoob [14] used the DWT transfer vibration signals of a typical four-cylinder engine from the time domain to the timefrequency domain for fault detection.Although the wavelet transform has the advantage of multiresolution analysis of signals, appropriate wavelet basis functions are difficult to be selected before processing signals [15].To solve this problem, Huang et al. [9] proposed the EMD, which can efficiently decompose nonlinear and nonstationary signals without any set of basis functions.Vernekar et al. [16] presented an engine gearbox fault diagnosis on the basis of EMD and naive Bayes algorithm to manage vibration signals.ey classified healthy and different simulated faulty conditions of gear and bearing, and results showed that the classification accuracy of their method can reach 98.88%.Zhao et al. [17] combined the EMD and autoregressive model to diagnose the faults of crankshaft bearing and connecting rod bearing and obtained the fault spectrum.To overcome the noise effects, Xu et al. [18] proposed a new method for engine vibration diagnosis on the basis of an EMD adaptive threshold filter and presented the correlation dimension.Experimental results indicated that the proposed method can effectively remove the noise of engine vibration signal and improve the accuracy and efficiency of EMD.Ye and Shao [19] introduced a new inspection method based on EMD and envelope spectrum analysis to analyse vibration signals.Results showed that the proposed method has excellent performance in quality inspection in the presence of abnormal clearance between the engine crankshaft and connecting rod.Tan et al. [20] used the EMD method to remove diesel engine vibration signals.Test results indicated that the true vibration signal from the diesel engine could be obtained by reconstructing the remaining IMF components.Although EMD has been successfully applied in the field of fault diagnosis, it has limitations, such as the frequent occurrence of mode mixing.To overcome this problem, Wu and Huang [21] proposed ensemble empirical mode decomposition (EEMD) in 2009.EEMD adds finite amplitude Gauss white noise to the signal when the signal is decomposed, thereby eliminating mode mixing to a certain extent.Wang et al. [6] presented a new adaptive wavelet packet thresholding function based on adaptive wavelet threshold denoising, EEMD, and correlation dimension for vibration signal denoising.is new method can effectively extract the impact signal features induced by vibrations.Zhang et al. [22] used EEMD and other methods to process engine abnormal sound and the frequency-dependent contributions of different engine parts to different test points under different speeds.
In recent years, numerous scientists have proposed processing methods for nonstationary signals.Smith [23] introduced the local mean decomposition (LMD) method in 2005.Frei and Osorio [24] presented the intrinsic time-scale decomposition (ITD) method in 2007.However, similar to the EMD, the LMD and ITD have the disadvantage of mode mixing.In 2015, Dayong et al. [25] proposed a dislocation superimposed method (DSM) on the basis of the random decrement technique.In comparison with other methods, the DSM is simple and will not change the concerned component in the mixed signal.e fault component is a quasi-periodic signal; thus, it requires people to select the starting point and superposition length when DSM is used to process abnormal acoustic signals, which greatly reduce the efficiency of processing experimental data.
is paper presents a developed DSM for automatically extracting the impulsive fault component from the abnormal noise signal of an engine.

DSM Review
e mathematical model of DSM is defined by using the following equation: where A(n) is the original signal, k denotes the superposition number (k � 1, 2, 3, . ..), and T represents a superposition step.
Figure 1 presents a simple example of the DSM, where A 1 and A 2 are two periodic signals with different periods and amplitudes, in which A 1 indicates the target signal and its period is T. A 3 refers to a synthetic signal of A 1 and A 2 .To extract A 1 from the mixed signal A 3 , A 3 is moved to the right for T and then added with A 3 to obtain A 4 .e period of A 1 does not change, but the amplitude is doubled during superposition, whereas A 2 is destroyed.Repeating the previous process can increase the amplitude of the target signal.Dividing the superimposed signal by the corresponding superposition number can obtain the target signal.
A correlation analysis is performed between separated signal A 5 and target signal A 1 to calculate their similarity.
e correlation coefficient can be calculated by equation ( 2), where v is the length of signals A 1 and A 5 .e larger the |ρ|, the stronger the correlation between two signals.e DSM can extract periodic signals from the mixed signal.As the superposition number increases, the extraction result becomes further accurate: (2)

Introduction of Improved DSM
A slight change in the engine speed is observed.us, the impulsive fault components generated by certain parts are quasi-periodic signals, as shown in Figure 2, which could not be extracted by moving the fault component cycle.erefore, artificially selecting a suitable starting point is necessary at the front end of each dislocation superimposed interception signal containing fault components.To overcome the shortcomings of human participation in processing data, this paper presents a developed DSM for automatically extracting impulsive fault components from the abnormal noise signal of an engine.is new method consists of three steps: selection of starting points, superimposition of interception signal, and separation of fault components.Assume that the engine produces an impulsive fault every q revolutions of the crankshaft.e angle of q revolutions is divided into H parts. en, an area of length w (W area) is intercepted at any phase, where w represents the length of the initial area containing the main energy of the fault component.Waveforms of the W area in abnormal sound at di erent times are similar.erefore, W 1 and W 2 are intercepted at the same phase of two consecutive q revolutions for correlation analysis.ρ(W 1 , W 2 ) and M 1 (which is a threshold for preliminarily judging whether the phase position can be used as the starting point) are compared.If the W 1 and W 2 areas include the main energy area of two fault components, as shown in Figure 3(a), where ρ(W 1 , W 2 ) is greater than M 1 , then the starting point is preliminarily selected at the phase.If the W 1 and W 2 areas contain background noise or few fault components, as shown in Figure 3(b), where ρ(W 1 , W 2 ) is less than M 1 , then the starting point should be reveri ed at the next phase.

Selection of the
A positional deviation may exist between the fault components intercepted at the same phase of di erent revolutions due to slight changes in the engine speed during operation.Figure 4 shows that ρ(W 1 , W 2 ) calculated at θ i is greater than M 1 ; however, a large positional deviation is observed between the fault components.e fault component will be destroyed in the superimposition process.
erefore, the starting point could not be selected at phase θ i and must be reveri ed at θ i+1 .When the phase is constantly approaching the position where the fault component exists, the positional deviation between the fault components is gradually reduced.If the deviation can be regarded as a natural error, then the starting point can be determined at the phase.
To determine whether the positional deviation between the fault features is negligible, some W areas are superimposed, and the correlation coe cient between the superimposed W areas is calculated under the same number of superposition times.If the correlation increases with the number of superpositions, then the positional deviation can be ignored.e steps of the judgment method are as follows: Step 1. Divide consecutive 2m W areas at the same phase of di erent revolutions into two groups in accordance with the parity of serial numbers and guarantee that samples 2m are enough.Superimpose W areas in each group by using the following equation: where D Wuk is the superimposed W areas; u denotes the group number, u 1, 2; and k indicates the number of superpositions, k 1, 2, 3, . .., m − 1.
Step 2. Calculate the correlation coe cient ρ(D W1k , D W2k ) between superimposed signals D W1k and D W2k under the same number of superpositions, as shown in Figure 5.If ρ(D W1k , D W2k ) increases with the number of superpositions until it is greater than threshold M 2 (M 2 is for assessing whether the positional deviation is negligible), then the positional deviation can be ignored, and the selected starting point is valid.Otherwise, the starting point is preliminarily assessed again at the next phase.ρ(D W1k , D W2k ) is not strictly monotonous with the number of superpositions due to the in uence of background noise.Hence, ρ(D W1k , D W2k ) is indirectly compared under the number of superpositions by using the following equation: A phenomenon that the fault component is destroyed when ρ(W 1 , W 2 ) is greater than M 1 also exists, as shown in Figure 6.To solve this problem, H parts of the angle of q revolutions must be su cient to satisfy d i < w/2, where d i is the length of the abnormal signal between phases θ i and θ i+1 .
e reason is that ρ(W 1 , W 2 ) must be recalculated at phase θ i+1 and compared with M 1 when ρ(W 1 , W 2 ) at phase θ i is less than M 1 .If d i ≥ w, then the main energy area of the fault components will be destroyed or skipped, as shown in Figures 7(a In addition, if the rst ρ(W 1 , W 2 ) is greater than M 1 , then the result is discarded and ρ(W 1 , W 2 ) is recalculated at the next phase until it is less than M 1 and then greater than M 1 again.
Considering the di erent lengths of the main energy areas of di erent fault characteristics, w should be set to a  Sampling points Amplitude  in other studies, as shown in Table 1.e length of the main energy area is determined through experience: Equation ( 5) is an empirical formula, which can effectively separate impulsive fault components, including cylinder knocking and connecting rod bearing knocking; however, its versatility needs further verification.
When slight changes in engine speed are not considered, d i is calculated using the following equation: where d i is the length of the abnormal signal between phases θ i and θ i+1 , q denotes the number of revolutions of the crankshaft, h represents the sampling frequency, ω indicates the engine rotating speed, and H refers to the number of parts into which the angle of q revolutions is divided.erefore, H should satisfy the following equation: e encoder can generate a certain number of pulse signals, and the angle between any adjacent pulse signals is equal.e encoder is used to divide the angle in the experiment.Figure 8 and equation (8) show that M 3 is a threshold for assessing the rising edge of the pulse signal.Two consecutive discrete points g(n) and g(n + 1) of the pulse signal are compared with threshold M 3 .When g(n) is smaller and g(n + 1) is larger than M 3 , a rising edge is considered to exist between points: e W areas are obtained as follows: simultaneous data acquisition of the abnormal sound signal f(n) and pulse signal g(n) is conducted under the same sampling frequency and one-to-one correspondence between signal data points.In Figure 9 and equation (9), f(n r + 1) corresponding to g(n r + 1) is selected as the starting point, and w points are intercepted backward as W areas: W j � fn (j−1)qN+r + 1, n (j−1)qN+r + 2, n (j−1)qN+r + 3, . . ., n (j−1)qN+r + w, (9) where j is the serial number of the W areas, j � 1, 2, 3, . ..; r denotes the serial number of the rising edge, r � 1, 2, 3, . ..; q represents the number of revolutions of the crankshaft; and N indicates the number of pulses triggered by the encoder at one revolution.

Superposition of Abnormal Sound Signals.
e abnormal sound signal is divided into different segments I j at valid phases.Different sections I j are superimposed to improve the signal-to-noise ratio.
I j is obtained as follows: in Figure 10 and equation (10), f(n (j−1)qN+r + 1) corresponding to g(n (j−1)qN+r + 1), which is used to assess the ((j − 1)qN + r)th rising edge, is selected as the first point of I j , and f(n jqN+r ) corresponding to g(n jqN+r ), which is used to judge the (jqN + r)th rising edge, is set as the last point of I j :

Shock and Vibration
where j is the serial number of the I j areas, j 1, 2, 3, . ..; r denotes the serial number of the rising edge, r 1, 2, 3, . ..; q represents the number of revolutions of the crankshaft; and N indicates the number of pulses triggered by using the encoder at one revolution.Superimposition is conducted as follows: consecutive 2m segments I j are divided into two groups in accordance with the parity of serial numbers.W areas are superimposed using equation (11) in each group.e superimposed length is the minimum length of 2m segments: where D Iuk is the superimposed segment signals; u indicates the group number, u 1, 2; and k denotes the number of superpositions when ρ(D W1k , D W2k ) is greater than M 2 .

Separation of Fault
Components. e superimposed segments are composed of the noise-reduced fault component and superimposed background noise.e fault component is located at the front end of the superimposed segment signal.erefore, the superimposed segments must be divided to separate the fault features.
Figures 11(a) and 11(b) show the separation method.e correlation coe cient ρ(P 1k (l), P 2k (l)) between the areas of length p (P areas) of the front end of the superimposed segments D Iuk is calculated.ρ(P 1k (l), P 2k (l)) is compared with M 4 (which is the threshold for assessing fault characteristics).If ρ(P 1k (l), P 2k (l)) is greater than threshold M 4 , then the P area can be regarded as a fault component.If ρ(P 1k (l), P 2k (l)) is smaller than M 4 , then the length of the P area is modi ed to 90% of the original length and then rounded down.ρ(P 1k (l), P 2k (l)) is recalculated and compared with M 4 .
e selection of starting point is a process of xed length analysis that fails to determine the amount of fault components in the W area. e fault component can be qualitatively separated by comparing ρ(P 1k (l), P 2k (l)) with M 4 .M 4 should be smaller than ρ(D W1k , D W2k ) to ensure that the length of the separated fault component is greater than w.In this study, M 4 is set to ρ(D W1k , D W2k )−0.05, and the initial value of p is set to 0.9 × L, where L is the length of the superimposed segments.e P area is calculated using the following equations: where P uk (l) is the area of length p of the front end of the superimposed segments D Iuk ; R 2 (n) represents a rectangular window function with a length of 0.9 l × L; l denotes the number of times that the length of the P area changes, l 1, 2, 3, . ..; u indicates the group number, u 1, 2; k is the number Rising edge Figure 8: Judgment method of the rising edge.When g(n) is smaller and g(n + 1) is larger than M 3 , a rising edge is considered to exist between points.
g (n r ) Sampling points f (n qN+r + 1) Amplitude f (n qN+r + w) Figure 9: Acquisition principle of W areas. f(n r + 1) corresponding to g(n r + 1) is selected as the starting point, and w points are intercepted backward as W areas.
6 Shock and Vibration of superpositions when ρ(D W1k , D W2k ) is greater than M 2 ; and L refers to the length of the superimposed segments.⌊.⌋ is a round-down function.e developed DSM encompasses all of the above steps, and the ow chart of Figure 12 is drawn for a better understanding of the method.

Experiment Condition
Figure 13 shows the physical diagrams of the experimental bench.e data acquisition system consists of acoustic sensors, encoders, data acquisition cards, and computers.
e acoustic sensor with a frequency ranging from 20 Hz to 20 kHz is placed above the cylinder to receive the abnormal sound signals generated by the engine.e encoder model is ZSP3806GC, which triggers 100 electrical pulses per revolution.
e encoder can synchronously revolve with the crankshaft by xing it on the front end of the crankshaft through a coupling.
e data acquisition card model is USB-6341 produced by the National Instruments Company.e sampling frequency is 44100 Hz.Data acquisition and processing are conducted using MATLAB software.e engine of the experimental stand was disassembled from a car, whose parameters are shown in Table 2.

Experimental Data Processing
Figure 14 presents the abnormal signal caused by the fault of cylinder knocking and pulse signal triggered by the encoder under high speed (1800 r/min).We can observe the considerable di erence from the normal and faulty acoustic signals.
e improved DSM is used to process the abnormal sound signal.Firstly, the angle of the two revolutions of the crankshaft is divided into 100 parts, that is, dividing 200 rising edges into 100 parts, which satis es the requirements of equation ( 7) (H should be greater than 49), to ensure that each intercepted segment contains a fault component.
reshold M 1 is set to 0.3.Secondly, ρ(W 1 , W 2 ) is calculated at di erent rising edges.Table 3 shows the ρ(W 1 , W 2 ) values calculated from the rst to the ninth rising edges.ρ(W 1 , W 2 ) calculated at the ninth rising edge is 0.5012, which is greater than M 1 .
erefore, the starting point is preliminarily Sampling points g (n r + 1) g (n qN+r + 1) g (n 2 qN+r + 1) g (n 3 qN+r + 1) Figure 10: Acquisition principle of I j .f(n (j−1)qN+r + 1) corresponding to g(n (j−1)qN+r + 1), which is used to assess the ((j − 1)qN + r)th rising edge, is selected as the rst point of I j , and f(n jqN+r ) corresponding to g(n jqN+r ), which is used to judge the (jqN + r)th rising edge, is set as the last point of I j .
Calculate ρ(P 1k (l), P 2k (l)) Calculate ρ(P 1k (l), P 2k (l)) P 1k (l) and P 2k (l) are the separate fault features p = L × 0.9 p = p × 0.9 Shock and Vibration selected at the position of the rising edge.irdly, the 22 W areas acquired at the ninth rising edge are divided into two groups.e odd-and even-numbered W areas are the rst and second groups, respectively.reshold M 2 is set to 0.8. Figure 15 shows ρ(D W1k , D W2k ) under di erent superposition numbers (ρ(D W10 , D W20 ) represents ρ(W 1 , W 2 )).ρ(D W14 , D W24 ) is greater than M 2 , and ρ(D W1k , D W2k ) increases with the number of superpositions under the superimposing condition of 1-4 times.Hence, the starting point selected at the ninth rising edge is valid.Equation ( 10) is used to intercept the abnormal signal per two revolutions of the crankshaft at the ninth rising edge.Ten intercepted sections are divided into two groups on the basis of the parity of serial numbers, and Figures 16(a Recalculate ρ(P 1k (l), P 2k (l)) after the length of the P area modified to 90% of the original length and then round down P 1k (l) and P 2k (l) can be considered as fault components Set the threshold M 2 , and calculate ρ(D W1k ,D W2k ) Set the threshold M 4 , and calculate ρ(P 1k (l), P 2k (l)) q revolutions are divided into H parts ) monotonically increasing and can exceed M 2 ρ(P 1k (l), P 2k (l)) > M 4 8 Shock and Vibration between the P areas of two superimposed signals is calculated.ρ(P 14 (15), P 24 (15)) is 0.7860 after p changed 15 times, which is greater than M 4 .us, P u4 (15) can be regarded as fault components, as shown in Figure 18.Although the front end of the isolated faulty component contains a minimal background noise, most of the noise has been removed.e result proves that the developed DSM can automatically extract the fault features from the abnormal sound.If traditional DSM is used in this chapter, we need to slowly select the appropriate starting point, which greatly reduces the computational e ciency.

Other Applications of the Developed Method
To verify the applicability of the developed method, the abnormal noise signal of the connecting rod bearing knocking is collected under a speed of 1100 r/min, as shown in Figure 19.Data acquisition and processing methods are the same as before, and the sampling rate is 44100 Hz.
e angle of one revolution of the crankshaft is divided into 100 parts, which satis es the requirements of equation ( 7) (H should be greater than 45).reshold M 1 is set to 0.3.Table 4 shows ρ(W 1 , W 2 ) calculated from the rst to the 23rd rising edges, and ρ(W 1 , W 2 ) calculated at the 23rd rising edge is greater than M 1 .Hence, the starting point is preliminarily selected at the position of the 23rd rising edge.
Figure 20 shows ρ(D W1k , D W2k ) under 1-10 numbers of superposition.reshold M 2 is set to 0.8.ρ(D W110 , D W210 ) is 0.8016, which is greater than M 2 , and ρ(D W1k , D W2k ) increases with the number of superpositions under the superimposing condition of 1-10 times.us, the starting point selected at the 23rd rising edge is valid.
e abnormal signal per revolution of the crankshaft is intercepted at the 23rd rising edge.e 22 intercepted sections are divided into two groups and superimposed.Figure 21 shows the result.
From the above two examples, we nd that the improved method can automatically extract the component of impulsive signal from the abnormal noise signal of an engine.However, the new method is currently only e ective at a single speed range and cannot process signals at di erent speed range.e reason is that the same sampling rate and varying speed result in a large di erence in length between di erent fault features in W regions so that the correct starting point cannot be selected.

Precision Analysis
Damaged parts are arti cially used to create a fault component, which is regarded as an actual fault component, with high signal-to-noise ratio in a quiet environment.e separated fault component is compared with the actual fault component for accuracy analysis.Figure 23 shows the areas of remarkable characteristics with a length of 300 of the actual and separated fault components of cylinder knocking.
is area is used for analysis to avoid interference from background noise at the rear end of the arti cially manufactured fault component.Separated fault components 1 and 2 are the segments of P 14 (15) and P 24 (15), respectively.e correlation coe cient between separated fault components 1 and 2 and the actual fault component are 0.7346 and 0.6655, respectively.Figure 24 presents the actual and separated fault components of connecting rod bearing knocking after normalization.e correlation coe cient between separated fault components 1 and 2 and the actual fault component are 0.8452 and 0.8685, respectively.
On the basis of the analysis result, the developed method can e ectively extract fault characteristics from an abnormal sound signal, and we can change the value of M 2 to increase the number of superimpositions for improving the accuracy of separated fault components.
To classify the damage, a database contained di erent fault characteristics should be built.e fault waveform is separated from complex noise by the developed DSM when engine fails.Next, the fault types can be determined by comparing the separated fault features with those in the database or classi cation methods such as neural networks.

Conclusion
In this study, a developed DSM is used to extract the characteristics of cylinder knocking and connecting rod            Shock and Vibration bearing knocking automatically from the abnormal sound signal of the engine.e main conclusions are as follows.e traditional DSM is a theoretical algorithm for ideal uniform velocity conditions, and it cannot analyse actual abnormal noise signals independently.
e developed method can automatically select appropriate starting points to superimpose the abnormal sound signals and nally separate the fault characteristics.Although the front end of the separated fault component contains minimal noise, most of the noise has been removed.Generally, the new method greatly improves the practicability, has a good application prospect, and supplements the methods of extracting fault features in the time domain.
Data Availability e data used to support the ndings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no con icts of interest regarding the publication of this paper.
) and 7(b).If w/2 ≤ d i < w, then the main energy area is destroyed, as shown in Figure7(c).If d i < w/2, then the main energy area is hardly destroyed.In this study, d i < w/5.

Figure 1 :
Figure 1: Graphical representation of DSM.A 1 indicates the target signal, A 2 is an interference signal, A 3 represents a mixed signal of A 1 and A 2 , A 4 refers to a superimposed signal after A 3 and A 3 moved to the right for T, and A 5 denotes a separated target signal.

Figure 4 :
Figure 4: Relationship between fault features acquired at di erent phases.(a) Intercepting abnormal signals at di erent phases.(b)Intercepted results at θ i and θ i+1 , where f(n) is the abnormal sound signal generated during engine operation, d i denotes the length of the abnormal signal between phases θ i and θ i+1 , B 1 represents the fault component intercepted at θ i , B 2 refers to the fault component intercepted at the next θ i , B 3 indicates the fault component intercepted at θ i+1 , and B 4 signi es the fault component intercepted at the next θ i+1 .In comparison with θ i+1 , θ i is far from the position where the fault component exists, and a large positional deviation may occur between B 1 and B 2 acquired at θ i in di erent q revolutions due to slight changes in the engine speed.θ i+1 is close to the position where the fault component exists.If the deviation between B 3 and B 4 is small enough and can be regarded as a natural error, then the starting point can be selected at the position of θ i+1 .

Figure 5 :Figure 3 :Figure 6 :
Figure 5: Relationship between fault features at di erent phases.If ρ(D W1k , D W2k ) increases with the number of superpositions until it is greater than threshold M 2 , then the positional deviation can be ignored.ρ(D W1k , D W2k ) is not strictly monotonous with the number of superpositions due to the in uence of background noise, such as ρ(D W13 , D W23 ) is less than ρ(D W12 , D W22 ) and ρ(D W16 , D W26 ) is less than ρ(D W15 , D W25 ).Hence, ρ(D W1k , D W2k ) is indirectly compared under the number of superpositions by using equation (4).

Figure 7 :
Figure 7: Influence of the relationship between w and d i on fault components.(a) d i > w. (b)d i � w.(c) w/2 ≤ d i < w.(d) d i < w/2.If d i ≥ w,then the main energy area of the fault components will be destroyed or skipped, as shown in Figures7(a) and 7(b).If w/2 ≤ d i < w, then the main energy area is also destroyed, as shown in Figure7(c).If d i < w/2, then the main energy area is hardly destroyed.

Figure 11 :
Figure 11: Method for separating fault components.(a) e principle of fault component separation.(b) e ow chart for fault component separation.p is decreased to improve the correlation between P areas.When ρ(P 1k (l), P 1k (l)) is greater than threshold M 4 , the P areas can be regarded as fault components.
) and16(b)  show the sections.en, the intercepted sections are superimposed four times in each group; the superimposed length is 2739.Figure17presents the result.Finally, in order to separate fault components, the initial value of p is set to ⌊0.9 × 2739⌋ 2465 and threshold M 4 is set to ρ(D W14 , D W24 ) − 0.05 0.7833.e correlation coe cient Engine Set the threshold M 1 and calculate ρ(W 1 , W 2 ) at the rising edge of the first part ρ(W 1 , W 2 ) at the next part

Figure 12 :
Figure 12: e ow chart of the developed DSM.

Figure 14 :
Figure 14: Abnormal noise signal of cylinder knocking and pulse signal under 1800 r/min speed.

Figure 19 :
Figure 19: Abnormal noise signal of connecting rod bearing knocking and pulse signal under 1100 r/min speed.

Figure 23 :
Figure 23: Separated and actual fault components of cylinder knocking.

Figure 24 :
Figure 24: Separated and actual fault components of connecting rod bearing knocking.

Table 1 :
Length of the main energy areas of engine fault components and mechanical faults found in other studies.

Table 3 :
ρ(W 1 , W 2 ) calculated from the rst to the ninth rising edges.

Table 4 :
ρ(W 1 , W 2 ) calculated from the rst to the ninth rising edges.