# Nonlinear Vibration Feature Recognition Method for Reciprocating Compressor Cylinder Based on VMD-Multifractal Spectrum

Authors: Fengxia Lyu; Caiqian Xie; Fengfeng Bie; Xinting Miao; Yifan Wu; Ying Zhang
Journal: Shock and Vibration
 (2023)

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## Abstract

The failure probability of piston ring abrasion occupies the forefront in reciprocating compressor cylinder assembly. Due to the reciprocating friction between the piston ring and cylinder liner, the local vibration signal contains typical nonstationary characteristics with a seriously overlapped frequency band of impact signals, while the different featured signals are coupled with each other. Therefore, the identification of cylinder friction and abrasion fault modes has always been a hot and difficult problem in fault diagnosis of reciprocating compressors. In order to monitor the friction and abrasion between low linear speed nonmetallic piston ring and cylinder liner of reciprocating compressor in refinery and provide a theoretical basis for preventive maintenance, an analysis method based on VMD-multifractal spectrum was presented in this paper. First, the feasibility of the proposed method was proven based on the cylinder vibration experiment of a reciprocating compressor in the laboratory. Then, an in-service reciprocating compressor in a refinery was taken as the research object, from which the cylinder friction vibration signals were processed by using variational mode decomposition to obtain a band-limited intrinsic mode function (BLIMF). Multifractal detrended fluctuation analysis is employed in the final abrasion pattern identification. The results show that the proposed method based on VMD-multifractal spectrum analysis can effectively obtain the abrasion state of a piston ring, which can avoid blind overhaul or provide the basis for preventive maintenance and a practicable engineering route for the study of the piston ring abrasion degree.


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## Body

## 1. Introduction

As an important fluid conveying machine in the petrochemical industry, the common faults of the reciprocating compressor include valve failure, packing leakage, piston parts wear, big and small end bearing shells, crosshead pin fractures, etc., among which unplanned downtime caused by piston ring wear and fracture accounts for 10% of the total faults. Therefore, it is very important to study the wear monitoring of piston rings.Comparing with traditional wear monitoring methods, including the cylinder pressure monitoring method and the oil monitoring method, the analysis method for vibration monitoring is the most practicable [1]. Compared with acoustic and thermal imaging, vibration monitoring also has some advantages. Vibration analysis, usually an online technique, has a relatively high efficiency in the data acquisition process, which can find problems at an early stage of failure. Although acoustic-based fault diagnosis is easier to install, the selection of the installation location is particularly crucial, and the selection of the measuring point location relies on experience, while the multichannel acquisition and manual or machine screening are additionally needed. Thermal imaging technology can quickly locate equipment components with large temperature changes when a fault occurs, usually for transformers, motors, etc. However, the precise fault diagnosis on the typical moving parts of the machinery system takes more time, which is not ideal as required. The frictional vibration of the piston ring and cylinder liner contains a lot of information reflecting the tribological characteristics and friction state of the system [2]. Some researchers have developed vibration-based wear monitoring for life prediction [3]. However, the vibration signals collected directly are mixed with other mode signals of a reciprocating compressor, which are coupled with each other. The actual operating status of the reciprocating compressor could not be directly determined from the original friction vibration signals since the signal has nonstationary and non-linear characteristics [4]. The method of vibration analysis focuses on the frequency band energy and time-frequency domain characteristics of the vibration signal collected from the assembly surface. As for the typical study achievements, the wavelet Kullback–Leibler distance method [5], Hilbert’s method of empirical mode decomposition [6], the local wave time-frequency method [7], etc., are presented gradually. However, for the nonlinear and nonstationary characteristics of the vibration, there are still some limitations in the classification of mode aliasing, preset wavelet analysis basis functions, or the number of intrinsic mode functions. Variational mode decomposition (VMD) was proposed as a new signal processing method by Dragomiretskiy and Zosso in 2014 [8], the core idea of which is the maxima problem with variational constraints. The band-limited intrinsic mode function (BLIMF) is obtained, which is different from EMD [9] and LMD [10]. The EMD and LMD methods used a cyclic sieving and stripping signal processing method. Therefore, the VMD method overcomes the mode aliasing problem and shows strong robustness in denoising [11–14]. In recent years, many scholars have applied it to signal analysis in related fields [15–17]. In view of the typical features of the nonlinear vibration of the reciprocating compressor cylinder block, including the impact of the air valve, the pulsation of the airflow, the friction vibration of the piston assembly, the mutual excitation and coupling between the moving parts, and so on [18], the basic VMD method could be applicable by presetting the scale K and constructing the variational framework, where the signal is decomposed into the variational model and the frequency center and bandwidth of the BLIMF components are determined by searching the optimal solution iteratively. This method could satisfy the condition of adaptive signal decomposition, and separates BLIMF components effectively by frequency domain segmentation, highlighting the local scale characteristics of data. Due to the periodicity and long-term unpredictability of the vibration signals of the reciprocating machine, the details of other reciprocating periods could be ignored when merely considering the discrete points of the vibration sequence. For the prediction of this kind of signal, it is necessary to describe the wave characteristics and invariable structure of a certain component of the signal at different time scales. Multifractal detrended fluctuation analysis [19] (MFDFA) is based on detrended fluctuation analysis proposed by Kantelhardt in 2002. The unsteady time series analysis method of DFA is a rapidly emerging nonlinear signal processing theory in recent years. It could divide a complex fractal into many small regions with different degrees of singularity, which is suitable for the self-similarity of complex systems analysis. Multifractal spectrum can describe the singularity of non-linear vibration signals well [20–22].Aiming at the nonlinear vibration characteristic involved in the piston assembly at various wear statuses of the piston ring and cylinder liner, a method based on variational mode decomposition (VMD) and multifractal spectrum is presented through an experimental and engineering case study. This paper is organized in the following manner: Section1 is the basic theory and research ideas; Section 2 is an experimental demonstration. The characteristic values of wear are obtained by processing the experimental signals with the proposed method. Section 3 is engineering verification, in which the proposed method is basically verified in an engineering application. The last part is the conclusion.


## 2. Description of Theoretical Background

### 2.1. Variational Mode Decomposition

In the VMD method, the intrinsic modal function (IMF) is expressed as follows:(1)ukt=Aktcosϕkt,where ukt is the IMF component from the vibration signal decomposition. Akt is the instantaneous amplitude. ϕkt is an instantaneous phase. ωt is the derivative of ϕkt, which is the instantaneous frequency decomposition process and the solution process for variational problems.If each modal component has the limited bandwidth of different center frequencies, the center frequency and bandwidth are continuously updated iteratively during the decomposition process, and the input signalF is decomposed into K discrete subsignals to minimize the estimated sum of the bandwidths of each subsignal. The variational process is specifically constructed as follows: an analytic signal of uk is first obtained from the unilateral spectrum by Hilbert transformation, and then a central frequency index is added to obtain the corresponding base band. The square of the demodulation signal gradient L2 is calculated, and the bandwidth of each subsignal is estimated. The constrained variational problem becomes(2)minuk,ωk∑k∂tδt+jπt∗ukte−jwkt22S.t.∑kuk=f,where uk=u1,⋯,uk represents K IMF components obtained by decomposition and ωk=ω1,⋯,ωk is the center frequency of each component obtained by decomposition.Equilibrium constraint parameterα and Lagrange multiplication operator λt are used to solve the constructed constraint variational problem, so that the variational problem is no longer limited by constraint conditions, and the augmented Lagrange is introduced L as follows:(3)Luk,ωk,λ=α∑k∂tδt+jπt∗ukte−jwkt22+ft−∑kukt22+λt,ft−∑kukt.The “saddle point” problem is solved by using the multiplier alternating direction algorithm. The components and frequencies are updated constantly. Finally, the “saddle point” of the unconstrained model is obtained, which is the optimal solution to the original problem. All component solutions are(4)u^kn+1w=f^ω−∑i≠ku^iω+λ^ω/21+2αω−ωk2,where u^kn+1w,f^ω,andλ^ω, respectively, represent the Fourier transforms of μkn+1ω,fω,λω. Meanwhile, the updated modal center frequency is expressed as follows:(5)ωkn+1=∫0∞ωμkω2dω∫0∞μkω2dω.VMD algorithm flow is as follows:(1)
Initialize:uk1,ωk1,λ1, N.(2)
N = N + 1, execute the loop program.(3)
FromK = 1: the number of preset modes K, update uk,ωk.(4)
Updateλn+1=λn+τf−∑ku^kn+1w.(5)
Repeat steps 3 and 4 until the iteration termination condition is satisfied as follows:(6)∑kukn+1−ukn22ukn22<ε.


### 2.2. Theory and Algorithm of Multifractal Detrended Wave Analysis

The multifractal detrended wave analysis method can accurately calculate the multifractal spectrum of vibration signals. The feature of multifractal spectrum is more sensitive to the change of vibration signal, MFDFA algorithm step [19] is as follows:(1)
For the time seriesxii=1,2…,n, calculate the deviation sequence Yi.(7)Yi=∑k=1ixi−x¯i=1,2…,n,x¯ is the average value of xi.(2)
DivideYi into a Ns subinterval whose length is equal to s. Also, the subinterval is continuous and nonoverlapping.(8)Ns=intNs.WhenN cannot divide s exactly, to ensure the integrity of information, the data are reversely repeated in this segmentation process to obtain 2 Ns subintervals.(3)
The mean square error was calculated, and thek-order multiform fitting was carried out by the following least squares method:When the interval ist=1,2,3,…,Ns, the mean square error is F2s,t.(9)F2s,t=1s∑i=1sYt−1s+i−yti2.When the interval ist=Ns+1,Ns+2,…,2Ns, the mean square error is F2s,t.(10)F2s,t=1s∑i=1sYN−t−1s+i−yti2.(4)
Calculate theq-order wave function Fqs.(11)Fqs=12N∑t=12NsF2s,tq/21/q.Whenq > 0, it reflects the big fluctuation trend of the time series; otherwise, it represents the small fluctuation trend of the time series.(5)
Calculate the power relation between the logarithmic function ofFqs to order and the time scale s.Fqs∼shq. hq is the generalized Hurst index.(6)
Analyze the logarithmic function ofFqs with respect to scale to obtain the wave function τq.(12)τq=qhq−1.(7)
Calculate the global singularity of the multifractal and the dimension of the multifractal set as follows:(13)α=dτqdq=hq+qhq˙,fα=qα−τq=qα−hq+1.


### 2.3. Diagnostic Model and Research Ideas

This paper constructs a fault diagnosis model based on VMD-MFDFA for the fault diagnosis of a reciprocating compressor cylinder block. The block diagram of the overall review of the research is shown in Figure1.Figure 1
Research flow block diagram.The specific steps are as follows:(1)
Set up a test platform to collect nonlinear vibration signals of the piston ring under normal and wear conditions.(2)
Variational mode decomposition (VMD) was used to process the signal. After the decomposition, the vibration signal of the air valve was removed and the signal was reconstructed.(3)
Multifractal processing was performed on each group of signals. Extract the characteristic value of wear, such as the Hurst index.(4)
Carry out an engineering application to verify the feasibility of the method.The technical route is shown in Figure2.Figure 2
Technical route.


## 2.1. Variational Mode Decomposition

In the VMD method, the intrinsic modal function (IMF) is expressed as follows:(1)ukt=Aktcosϕkt,where ukt is the IMF component from the vibration signal decomposition. Akt is the instantaneous amplitude. ϕkt is an instantaneous phase. ωt is the derivative of ϕkt, which is the instantaneous frequency decomposition process and the solution process for variational problems.If each modal component has the limited bandwidth of different center frequencies, the center frequency and bandwidth are continuously updated iteratively during the decomposition process, and the input signalF is decomposed into K discrete subsignals to minimize the estimated sum of the bandwidths of each subsignal. The variational process is specifically constructed as follows: an analytic signal of uk is first obtained from the unilateral spectrum by Hilbert transformation, and then a central frequency index is added to obtain the corresponding base band. The square of the demodulation signal gradient L2 is calculated, and the bandwidth of each subsignal is estimated. The constrained variational problem becomes(2)minuk,ωk∑k∂tδt+jπt∗ukte−jwkt22S.t.∑kuk=f,where uk=u1,⋯,uk represents K IMF components obtained by decomposition and ωk=ω1,⋯,ωk is the center frequency of each component obtained by decomposition.Equilibrium constraint parameterα and Lagrange multiplication operator λt are used to solve the constructed constraint variational problem, so that the variational problem is no longer limited by constraint conditions, and the augmented Lagrange is introduced L as follows:(3)Luk,ωk,λ=α∑k∂tδt+jπt∗ukte−jwkt22+ft−∑kukt22+λt,ft−∑kukt.The “saddle point” problem is solved by using the multiplier alternating direction algorithm. The components and frequencies are updated constantly. Finally, the “saddle point” of the unconstrained model is obtained, which is the optimal solution to the original problem. All component solutions are(4)u^kn+1w=f^ω−∑i≠ku^iω+λ^ω/21+2αω−ωk2,where u^kn+1w,f^ω,andλ^ω, respectively, represent the Fourier transforms of μkn+1ω,fω,λω. Meanwhile, the updated modal center frequency is expressed as follows:(5)ωkn+1=∫0∞ωμkω2dω∫0∞μkω2dω.VMD algorithm flow is as follows:(1)
Initialize:uk1,ωk1,λ1, N.(2)
N = N + 1, execute the loop program.(3)
FromK = 1: the number of preset modes K, update uk,ωk.(4)
Updateλn+1=λn+τf−∑ku^kn+1w.(5)
Repeat steps 3 and 4 until the iteration termination condition is satisfied as follows:(6)∑kukn+1−ukn22ukn22<ε.


## 2.2. Theory and Algorithm of Multifractal Detrended Wave Analysis

The multifractal detrended wave analysis method can accurately calculate the multifractal spectrum of vibration signals. The feature of multifractal spectrum is more sensitive to the change of vibration signal, MFDFA algorithm step [19] is as follows:(1)
For the time seriesxii=1,2…,n, calculate the deviation sequence Yi.(7)Yi=∑k=1ixi−x¯i=1,2…,n,x¯ is the average value of xi.(2)
DivideYi into a Ns subinterval whose length is equal to s. Also, the subinterval is continuous and nonoverlapping.(8)Ns=intNs.WhenN cannot divide s exactly, to ensure the integrity of information, the data are reversely repeated in this segmentation process to obtain 2 Ns subintervals.(3)
The mean square error was calculated, and thek-order multiform fitting was carried out by the following least squares method:When the interval ist=1,2,3,…,Ns, the mean square error is F2s,t.(9)F2s,t=1s∑i=1sYt−1s+i−yti2.When the interval ist=Ns+1,Ns+2,…,2Ns, the mean square error is F2s,t.(10)F2s,t=1s∑i=1sYN−t−1s+i−yti2.(4)
Calculate theq-order wave function Fqs.(11)Fqs=12N∑t=12NsF2s,tq/21/q.Whenq > 0, it reflects the big fluctuation trend of the time series; otherwise, it represents the small fluctuation trend of the time series.(5)
Calculate the power relation between the logarithmic function ofFqs to order and the time scale s.Fqs∼shq. hq is the generalized Hurst index.(6)
Analyze the logarithmic function ofFqs with respect to scale to obtain the wave function τq.(12)τq=qhq−1.(7)
Calculate the global singularity of the multifractal and the dimension of the multifractal set as follows:(13)α=dτqdq=hq+qhq˙,fα=qα−τq=qα−hq+1.


## 2.3. Diagnostic Model and Research Ideas

This paper constructs a fault diagnosis model based on VMD-MFDFA for the fault diagnosis of a reciprocating compressor cylinder block. The block diagram of the overall review of the research is shown in Figure1.Figure 1
Research flow block diagram.The specific steps are as follows:(1)
Set up a test platform to collect nonlinear vibration signals of the piston ring under normal and wear conditions.(2)
Variational mode decomposition (VMD) was used to process the signal. After the decomposition, the vibration signal of the air valve was removed and the signal was reconstructed.(3)
Multifractal processing was performed on each group of signals. Extract the characteristic value of wear, such as the Hurst index.(4)
Carry out an engineering application to verify the feasibility of the method.The technical route is shown in Figure2.Figure 2
Technical route.


## 3. Experimental Demonstration

### 3.1. Experimental Platform

The experiment was conducted on a SpectraQuest mechanical fault simulation test bench, as shown in Figure3. The compressor was a vertical, single cylinder compressor driven by a belt.Figure 3
Experimental bench and test system.The structure of the test rig is shown in Figure4. The structural dimensions of the equipment are listed in Table 1.Figure 4
The structure of the test rig.Table 1
Experimental system and structure size parameter table.
Inner diameter of cylinder block5.08 cmCompressor flow0.07 m3/sOperation period0.078 sPiston ring radial early thickness3 mmLubrication formOil freeAcceleration sensorDH1A110ESampling frequency12800 HzStroke3.81 cmOutlet pressure0.8 MPaPiston ring materialPTFEPiston ring radial end thickness2.8 mmMotor speed1420 r/minSensitivity5.088 mV/(m/s2)Sampling time1.28 s


### 3.2. Data Collection

In order to accurately monitor the operation of the piston ring, the outside of the cylinder block where the piston reciprocates was selected as the vibration collection point [23]. The cylinder vibration data in the early fault and late period of compressor operation were collected, as shown in Figures 5(a) and 5(b).Figure 5
Time-frequency diagram of vibration signals. (a) Early stage. (b) Late stage.
(a)(b)Figure5 shows the vibration time domain waveform of the compressor cylinder at the early and late stages. The overall signal is nonlinear and nonstationary. The vibration signal of the cylinder block includes the vibration and impact signal brought by the airflow when the valve is on and off. With the increasing wear of the piston assembly, the vibration amplitude of the late stage is lower than that of the early stage, but the waveform becomes more disorderly and irregular. Figure 6 shows the time-frequency spectrum of cylinder vibration. Since the piston ring of the compressor is self-sealing, the piston ring can be close to the sides of the cylinder liner and ring groove by means of the pressure difference on both sides (the upper and lower sides and the left and right sides), so the switch function of the air valve is still needed in the measurement process. The friction vibration signal of the piston component is drowned in other vibration signals, so the nonlinear vibration of the cylinder block cannot be studied only by analyzing this signal.Figure 6
Time domain waveform of a vibration signal.


### 3.3. VMD

Variational mode decomposition was used to decompose the signal. The optimal component was confirmed by the central frequency observation method and the spectrum map. The solution parameter isK = 2. The bandwidth limit α = 1000. Decompose it into two modal components. The early and end-stage VMD are shown in Figures 7–10.Figure 7
Two groups of BLIMF components of early VMD.Figure 8
Two groups of BLIMF components of end-stage VMD.Figure 9
Fourier transforms of two early BLIMF components.Figure 10
Fourier transforms of two BLIMF components in the terminal stage.According to the time-frequency image of the nonlinear vibration of a cylinder block shown in Figure6, the frequency range of the gas impact caused by the switch of the air valve is about 1600 Hz to 1800 Hz. In addition, the friction vibration between the piston ring support ring and cylinder block at the cylinder block side is considered to be characterized by low frequency and amplitude. Therefore, BLIMF2 was selected as an effective signal to remove the influence of the gas valve and carry out reconstruction. The reconstructed signal is shown in Figures 11(a) and 11(b).Figure 11
Reconstructed signal diagram. (a) An early reconstruction signal. (b) A late reconstruction signal.
(a)(b)


### 3.4. Multifractal Detrended Wave Analysis of Cylinder Block Nonlinear Vibration

A multifractal analysis algorithm was used to analyze the reconstructed signal. The signal had sixteen reciprocating operation cycles. By setting different scalesS and different orders q, the signals were grouped and the overall wave function was calculated. The scales S were 16, 32, 64, 128, 256, 512, and 1024. When q was 2, local fluctuation and global fluctuation of early data were calculated respectively.In Figure12, the blue line represents local fluctuations. It could be observed that, in the same order, fluctuations at the small scale are more obvious than those at the large scale. Since there are more data at the large scale, there is less difference in fluctuations after averaging with each other. The red line represents global fluctuations at different scales, with smaller global fluctuations at smaller scales. By setting q differently, the fluctuation of the time series could be distinctively reflected.Figure 12
Mean square error at different scales.Takingq in the following order: −5, −3, −1, 1, 3, and 5. The nonlinear vibration of the cylinder block at the early and late phases of operation is analyzed. The scale S of 16, 32, 64, 128, 256, 512, and 1024 were taken as the study objects, respectively. It could be found that different order is obviously different from the q-order wave function through Figures 13(a) and 13(b). The reason is that the small scale does not span a reciprocating motion period, while at the large scale, the wave functions of all orders tend to keep the same.Figure 13
Logarithmic regression line of wave function and scale. (a) Early function. (b) Late function.
(a)(b)The Hurst exponent defines the growth rate of fluctuation of a nonlinear vibration signal at different sample scales, which is the slopeHq of the regression line. If Hq > 1, it indicates that the time series has a long-term dependence. When Hq is between 0.5 and 1, it indicates that the time series has a long correlation structure and the vibration trend is more stable. Hq is between 0 and 0.5, indicating that the time series has an inverse correlation structure. Based on Figures 13 and 14, multifractal characteristics could be observed in the nonlinear vibration of a cylinder block. Combined with Table 2, the Hurst index of the early phase is less than the last phase, which indicates that the piston ring, the supporting ring, and the cylinder block in the case of normal oil injection, with the thinning of the thickness, the constraint between each other becomes smaller, the working face is much smoother, and the vibration of mutual excitation is reduced, while the frictional vibration goes stable.Figure 14
Hurst exponents at the early and late phases.Table 2
Hurst exponent of the nonlinear vibration signal of the cylinder block.
Hurst exponentsq = −5q = −3q = −1q = 1q = 3q = 5Early phase1.0740.9740.7860.5740.4410.380Late phase1.1741.1060.9980.8800.7940.744Combined with Figure15 and Table 3, for the order q > 0, the scaling exponent of the late phase is larger than the early phase. For the order q < 0, the late value is smaller than the early value.Figure 15
Early and late scaling exponent.Table 3
Scale index of the nonlinear vibration signal of the cylinder block.
Index of the scalesq = −5q = −3q = −1q = 1q = 3q = 5Early phase6.3733.9221.7860.4250.3250.902Late phase6.8734.3201.9980.1191.3822.722Table4 illustrates the multifractal spectrum parameters of nonlinear vibration signals of the cylinder block, in which αmax is the maximum value and αmin is the minimum value of the singular index. δα=αmax−αmin. Fαmax is the maximum value of the fractal dimension function of the singular exponent of the multifractal set. Fαmin is the minimum value of the fractal dimension function of the singular exponent of the multifractal set. δF=Fαmax−Fαmin. αfmax is the singular index after the fractal function reaches the maximum value. The width of the multifractal spectrum is 1.005 at the initial stage of the nonlinear vibration of a cylinder block and 0.406 at the end. δα shows a gradually smaller trend. From the physical significance of δα in the late stage of piston ring operation, the amplitude distribution range of the vibration signal gets smaller. In the early stage of compressor operation, the vibration fluctuation between the piston assembly and cylinder block is relatively violent, the signal energy is strong, and the multifractal characteristics are stronger. The early value αmin is less than the late value, which reflects that the signal changes rapidly and the vibration signal fluctuates greatly. The early stage δF is 0.898 and the late stage δF is 0.691, which shows a decreasing trend of δF, i.e., the larger the δα, the larger the δF, the smaller the αfmax, the stronger the multifractal properties.Table 4
Dimensions of multifractal global singular values and multifractal sets.
Spectrum parametersαmaxαminδαFαmaxFαminδFαfmaxEarly phase1.2500.2441.00510.10170.8980.670Late phase1.1790.7720.40610.30890.6910.947From the fractal spectrum shape, the early parabola has a wide distribution of zeros, and the extreme point is closer to theY-axis. Compared with the early working conditions, the piston ring and support ring components will wear to different degrees, and the extreme point will shift to the right end.In conclusion, MFDFA can fully reveal the multifractal characteristics of time series in nonlinear vibration and the multifractal spectrum. Figure16 can describe the dynamic behavior of a time series. The shape, spectrum width, extreme point, and other characteristic parameters of a multifractal spectrum are sensitive to changes in system state and can be used as characteristic parameters to characterize the system state of a reciprocating compressor cylinder block.Figure 16
Multifractal spectrum.


## 3.1. Experimental Platform

The experiment was conducted on a SpectraQuest mechanical fault simulation test bench, as shown in Figure3. The compressor was a vertical, single cylinder compressor driven by a belt.Figure 3
Experimental bench and test system.The structure of the test rig is shown in Figure4. The structural dimensions of the equipment are listed in Table 1.Figure 4
The structure of the test rig.Table 1
Experimental system and structure size parameter table.
Inner diameter of cylinder block5.08 cmCompressor flow0.07 m3/sOperation period0.078 sPiston ring radial early thickness3 mmLubrication formOil freeAcceleration sensorDH1A110ESampling frequency12800 HzStroke3.81 cmOutlet pressure0.8 MPaPiston ring materialPTFEPiston ring radial end thickness2.8 mmMotor speed1420 r/minSensitivity5.088 mV/(m/s2)Sampling time1.28 s


## 3.2. Data Collection

In order to accurately monitor the operation of the piston ring, the outside of the cylinder block where the piston reciprocates was selected as the vibration collection point [23]. The cylinder vibration data in the early fault and late period of compressor operation were collected, as shown in Figures 5(a) and 5(b).Figure 5
Time-frequency diagram of vibration signals. (a) Early stage. (b) Late stage.
(a)(b)Figure5 shows the vibration time domain waveform of the compressor cylinder at the early and late stages. The overall signal is nonlinear and nonstationary. The vibration signal of the cylinder block includes the vibration and impact signal brought by the airflow when the valve is on and off. With the increasing wear of the piston assembly, the vibration amplitude of the late stage is lower than that of the early stage, but the waveform becomes more disorderly and irregular. Figure 6 shows the time-frequency spectrum of cylinder vibration. Since the piston ring of the compressor is self-sealing, the piston ring can be close to the sides of the cylinder liner and ring groove by means of the pressure difference on both sides (the upper and lower sides and the left and right sides), so the switch function of the air valve is still needed in the measurement process. The friction vibration signal of the piston component is drowned in other vibration signals, so the nonlinear vibration of the cylinder block cannot be studied only by analyzing this signal.Figure 6
Time domain waveform of a vibration signal.


## 3.3. VMD

Variational mode decomposition was used to decompose the signal. The optimal component was confirmed by the central frequency observation method and the spectrum map. The solution parameter isK = 2. The bandwidth limit α = 1000. Decompose it into two modal components. The early and end-stage VMD are shown in Figures 7–10.Figure 7
Two groups of BLIMF components of early VMD.Figure 8
Two groups of BLIMF components of end-stage VMD.Figure 9
Fourier transforms of two early BLIMF components.Figure 10
Fourier transforms of two BLIMF components in the terminal stage.According to the time-frequency image of the nonlinear vibration of a cylinder block shown in Figure6, the frequency range of the gas impact caused by the switch of the air valve is about 1600 Hz to 1800 Hz. In addition, the friction vibration between the piston ring support ring and cylinder block at the cylinder block side is considered to be characterized by low frequency and amplitude. Therefore, BLIMF2 was selected as an effective signal to remove the influence of the gas valve and carry out reconstruction. The reconstructed signal is shown in Figures 11(a) and 11(b).Figure 11
Reconstructed signal diagram. (a) An early reconstruction signal. (b) A late reconstruction signal.
(a)(b)


## 3.4. Multifractal Detrended Wave Analysis of Cylinder Block Nonlinear Vibration

A multifractal analysis algorithm was used to analyze the reconstructed signal. The signal had sixteen reciprocating operation cycles. By setting different scalesS and different orders q, the signals were grouped and the overall wave function was calculated. The scales S were 16, 32, 64, 128, 256, 512, and 1024. When q was 2, local fluctuation and global fluctuation of early data were calculated respectively.In Figure12, the blue line represents local fluctuations. It could be observed that, in the same order, fluctuations at the small scale are more obvious than those at the large scale. Since there are more data at the large scale, there is less difference in fluctuations after averaging with each other. The red line represents global fluctuations at different scales, with smaller global fluctuations at smaller scales. By setting q differently, the fluctuation of the time series could be distinctively reflected.Figure 12
Mean square error at different scales.Takingq in the following order: −5, −3, −1, 1, 3, and 5. The nonlinear vibration of the cylinder block at the early and late phases of operation is analyzed. The scale S of 16, 32, 64, 128, 256, 512, and 1024 were taken as the study objects, respectively. It could be found that different order is obviously different from the q-order wave function through Figures 13(a) and 13(b). The reason is that the small scale does not span a reciprocating motion period, while at the large scale, the wave functions of all orders tend to keep the same.Figure 13
Logarithmic regression line of wave function and scale. (a) Early function. (b) Late function.
(a)(b)The Hurst exponent defines the growth rate of fluctuation of a nonlinear vibration signal at different sample scales, which is the slopeHq of the regression line. If Hq > 1, it indicates that the time series has a long-term dependence. When Hq is between 0.5 and 1, it indicates that the time series has a long correlation structure and the vibration trend is more stable. Hq is between 0 and 0.5, indicating that the time series has an inverse correlation structure. Based on Figures 13 and 14, multifractal characteristics could be observed in the nonlinear vibration of a cylinder block. Combined with Table 2, the Hurst index of the early phase is less than the last phase, which indicates that the piston ring, the supporting ring, and the cylinder block in the case of normal oil injection, with the thinning of the thickness, the constraint between each other becomes smaller, the working face is much smoother, and the vibration of mutual excitation is reduced, while the frictional vibration goes stable.Figure 14
Hurst exponents at the early and late phases.Table 2
Hurst exponent of the nonlinear vibration signal of the cylinder block.
Hurst exponentsq = −5q = −3q = −1q = 1q = 3q = 5Early phase1.0740.9740.7860.5740.4410.380Late phase1.1741.1060.9980.8800.7940.744Combined with Figure15 and Table 3, for the order q > 0, the scaling exponent of the late phase is larger than the early phase. For the order q < 0, the late value is smaller than the early value.Figure 15
Early and late scaling exponent.Table 3
Scale index of the nonlinear vibration signal of the cylinder block.
Index of the scalesq = −5q = −3q = −1q = 1q = 3q = 5Early phase6.3733.9221.7860.4250.3250.902Late phase6.8734.3201.9980.1191.3822.722Table4 illustrates the multifractal spectrum parameters of nonlinear vibration signals of the cylinder block, in which αmax is the maximum value and αmin is the minimum value of the singular index. δα=αmax−αmin. Fαmax is the maximum value of the fractal dimension function of the singular exponent of the multifractal set. Fαmin is the minimum value of the fractal dimension function of the singular exponent of the multifractal set. δF=Fαmax−Fαmin. αfmax is the singular index after the fractal function reaches the maximum value. The width of the multifractal spectrum is 1.005 at the initial stage of the nonlinear vibration of a cylinder block and 0.406 at the end. δα shows a gradually smaller trend. From the physical significance of δα in the late stage of piston ring operation, the amplitude distribution range of the vibration signal gets smaller. In the early stage of compressor operation, the vibration fluctuation between the piston assembly and cylinder block is relatively violent, the signal energy is strong, and the multifractal characteristics are stronger. The early value αmin is less than the late value, which reflects that the signal changes rapidly and the vibration signal fluctuates greatly. The early stage δF is 0.898 and the late stage δF is 0.691, which shows a decreasing trend of δF, i.e., the larger the δα, the larger the δF, the smaller the αfmax, the stronger the multifractal properties.Table 4
Dimensions of multifractal global singular values and multifractal sets.
Spectrum parametersαmaxαminδαFαmaxFαminδFαfmaxEarly phase1.2500.2441.00510.10170.8980.670Late phase1.1790.7720.40610.30890.6910.947From the fractal spectrum shape, the early parabola has a wide distribution of zeros, and the extreme point is closer to theY-axis. Compared with the early working conditions, the piston ring and support ring components will wear to different degrees, and the extreme point will shift to the right end.In conclusion, MFDFA can fully reveal the multifractal characteristics of time series in nonlinear vibration and the multifractal spectrum. Figure16 can describe the dynamic behavior of a time series. The shape, spectrum width, extreme point, and other characteristic parameters of a multifractal spectrum are sensitive to changes in system state and can be used as characteristic parameters to characterize the system state of a reciprocating compressor cylinder block.Figure 16
Multifractal spectrum.


## 4. Engineering Verification

### 4.1. Research Object

The test data came from the reciprocating compressor in service in a refinery. The engineering test is shown in Figure17. The unit is a symmetrical balance type with four columns and three-stage compression. The unit innovatively adopts a new hydrogen and cycle hydrogen combined compressor, in which three cylinders are used for three-stage compression of new hydrogen and another cylinder is used to boost the pressure of cycle hydrogen. The system and construction parameters are shown in Tables 5 and 6, where the test data collected from the cycling cylinder is set as the object.Figure 17
Real-time vibration data acquisition of a reciprocating compressor.Table 5
Dimensions and parameters of a refinery recovery machine.
Unit model4HHE-VG-3-1Compressor flow26928 Nm3/sOutlet pressure17 MPaMediumCirculating hydrogenOuter dead center clearance28.48 mmLubrication formOil lubricationLubricating oil name:SHELL S1 B460Stroke323.85 mmInlet pressure14 MPaPiston ring materialCarbon-PTFEPiston ring interface mode45° oblique incisionInner dead center clearance1.57 mmCylinder size171.4 mmMotor speed330 r/minTable 6
Signal acquisition system.
Instrument nameBH550Analysis frequency100–20000 HzAnalysis frequency10000 HzPlatformWindowsCEFrequency resolution1/64 HzAnalysis line number1600


### 4.2. Data Collection

The cylinder vibration data in the early and late period of compressor operation were collected with a sampling frequency of 25600 Hz and 4096 points, as shown in Figures18(a) and 18(b).Figure 18
Time domain waveform of a vibration signal. (a) Early stage. (b) Late stage.
(a)(b)


### 4.3. VMD

The number of modes was confirmed by comparing the center frequencies of differentK values. Bandwidth limit α = 2000. The following table shows the processing of end-stage data.As can be observed from Table7, when the number of modes is set to 7, components with similar center frequencies of 5671 Hz and 6518 Hz appear, and thus the K value is finally set at 6. Through VMD, the nonlinear vibration signal of a cylinder block can be effectively decomposed from a high-frequency to a low-frequency mode, and the phenomenon of mode aliasing can be restrained to a greater extent. In addition, the vibration of air valve excitation is effectively distinguished. The VMD of end-stage signals is shown in Figures 19 and 20.Table 7
Center frequencies under differentK values.
KCenter frequencyK = 223555867K = 3233452946798K = 41890444059058024K = 517493951556965038782K = 6163733184991597178179697K = 71603298244095671651880739966Figure 19
BLIMF components obtained by VMD.Figure 20
Spectrum of the BLIMF components.From the time-frequency image of the nonlinear vibration of a cylinder block in Figure21, the switching frequency segment of the air valve is about 3750 Hz to 7500 Hz. Therefore, BLIMF1 was selected as an effective signal to remove the influence of the gas valve and carry out reconstruction. The reconstructed signal is shown in Figures 22(a) and 22(b).Figure 21
Time-frequency diagram of the cylinder vibration signal.Figure 22
Reconstructed signal diagram. (a) An early reconstruction signal. (b) A late reconstruction signal.
(a)(b)


### 4.4. Multifractal Detrended Wave Analysis of Cylinder Block Nonlinear Vibration

Similarly, the scales S were 16, 32, 64, 128, 256, 512, and 1024. Whenq was 2, the local fluctuation and the overall fluctuation of the final data were calculated, respectively, as shown in Figure 23.Figure 23
Wave at different scales.Taking the orderq as −5, −3, −1, 1, 3, and 5 individually, the logarithmic relationship between scale and Fqs was calculated, as shown in Figures 24(a) and 24(b).Figure 24
Logarithmic regression line of wave function and scale. (a) Early function. (b) Late function.
(a)(b)On a small scale, different orders has a great difference in the q-order wave function, which is more obvious in the early stage of piston ring operation. While on a relatively large scale, all order wave functions also tend to be consistent.According to Figure25, when the order q > 0, the late value is larger than the early value. Also, when the order is q < 0, the late value is also basically smaller than the early value.Figure 25
Early and late scaling exponents.It can be seen from Figure26 that the nonlinear vibration of the cylinder block contains multifractal characteristics, while the Hurst index in the early stage is basically smaller than in the late stage, which is consistent with the experimental study. It can be considered that the Hurst index will increase at different orders at the end with the aggravation of piston ring wear.Figure 26
Hurst exponent of cylinder block nonlinear vibration.Table8 shows the multifractal spectrum parameters of the nonlinear vibration signals of the cylinder block, and the characteristics of the spectrum parameters are similar to the results of the experimental platform. According to the fractal spectrum (Figure 27), the early parabolic extreme point is closer to the Y-axis, and the extreme point also transforms to the right end of the parabola compared with the early conditions, and the zeros distance becomes narrower. Figure 28 shows the comparison between the old piston ring and the new piston ring during unit maintenance. The radial thickness of the new piston ring is 12.8 mm, while the radial thickness of the old piston ring is uneven, with an average of 9.12 mm and a local value of 6.29 mm. To sum up, their multifractal spectrum contains the abovementioned corresponding characteristics as the piston ring and support ring components in different wearing statuses.Table 8
Dimensions of multifractal global singular values and multifractal sets.
Spectrum parametersαmaxαminδαFαmaxFαminδFαfmaxEarly phase0.6280.1690.79810.03780.9620.168Late phase0.5190.0740.59310.49470.5050.204Figure 27
Multifractal spectrum.Figure 28
Comparison of early and late piston ring thicknesses.Experiments and engineering have shown that the Hurst index of the early phase is less than the late phase. The shape, spectrum widthδα, extreme point αfmax, and other characteristic parameters of a multifractal spectrum are sensitive to changes in system state and can be used as characteristic parameters to characterize the system state of a reciprocating compressor cylinder block. As the wear and tear intensifies, δα,δF becomes smaller and αfmax becomes larger.


## 4.1. Research Object

The test data came from the reciprocating compressor in service in a refinery. The engineering test is shown in Figure17. The unit is a symmetrical balance type with four columns and three-stage compression. The unit innovatively adopts a new hydrogen and cycle hydrogen combined compressor, in which three cylinders are used for three-stage compression of new hydrogen and another cylinder is used to boost the pressure of cycle hydrogen. The system and construction parameters are shown in Tables 5 and 6, where the test data collected from the cycling cylinder is set as the object.Figure 17
Real-time vibration data acquisition of a reciprocating compressor.Table 5
Dimensions and parameters of a refinery recovery machine.
Unit model4HHE-VG-3-1Compressor flow26928 Nm3/sOutlet pressure17 MPaMediumCirculating hydrogenOuter dead center clearance28.48 mmLubrication formOil lubricationLubricating oil name:SHELL S1 B460Stroke323.85 mmInlet pressure14 MPaPiston ring materialCarbon-PTFEPiston ring interface mode45° oblique incisionInner dead center clearance1.57 mmCylinder size171.4 mmMotor speed330 r/minTable 6
Signal acquisition system.
Instrument nameBH550Analysis frequency100–20000 HzAnalysis frequency10000 HzPlatformWindowsCEFrequency resolution1/64 HzAnalysis line number1600


## 4.2. Data Collection

The cylinder vibration data in the early and late period of compressor operation were collected with a sampling frequency of 25600 Hz and 4096 points, as shown in Figures18(a) and 18(b).Figure 18
Time domain waveform of a vibration signal. (a) Early stage. (b) Late stage.
(a)(b)


## 4.3. VMD

The number of modes was confirmed by comparing the center frequencies of differentK values. Bandwidth limit α = 2000. The following table shows the processing of end-stage data.As can be observed from Table7, when the number of modes is set to 7, components with similar center frequencies of 5671 Hz and 6518 Hz appear, and thus the K value is finally set at 6. Through VMD, the nonlinear vibration signal of a cylinder block can be effectively decomposed from a high-frequency to a low-frequency mode, and the phenomenon of mode aliasing can be restrained to a greater extent. In addition, the vibration of air valve excitation is effectively distinguished. The VMD of end-stage signals is shown in Figures 19 and 20.Table 7
Center frequencies under differentK values.
KCenter frequencyK = 223555867K = 3233452946798K = 41890444059058024K = 517493951556965038782K = 6163733184991597178179697K = 71603298244095671651880739966Figure 19
BLIMF components obtained by VMD.Figure 20
Spectrum of the BLIMF components.From the time-frequency image of the nonlinear vibration of a cylinder block in Figure21, the switching frequency segment of the air valve is about 3750 Hz to 7500 Hz. Therefore, BLIMF1 was selected as an effective signal to remove the influence of the gas valve and carry out reconstruction. The reconstructed signal is shown in Figures 22(a) and 22(b).Figure 21
Time-frequency diagram of the cylinder vibration signal.Figure 22
Reconstructed signal diagram. (a) An early reconstruction signal. (b) A late reconstruction signal.
(a)(b)


## 4.4. Multifractal Detrended Wave Analysis of Cylinder Block Nonlinear Vibration

Similarly, the scales S were 16, 32, 64, 128, 256, 512, and 1024. Whenq was 2, the local fluctuation and the overall fluctuation of the final data were calculated, respectively, as shown in Figure 23.Figure 23
Wave at different scales.Taking the orderq as −5, −3, −1, 1, 3, and 5 individually, the logarithmic relationship between scale and Fqs was calculated, as shown in Figures 24(a) and 24(b).Figure 24
Logarithmic regression line of wave function and scale. (a) Early function. (b) Late function.
(a)(b)On a small scale, different orders has a great difference in the q-order wave function, which is more obvious in the early stage of piston ring operation. While on a relatively large scale, all order wave functions also tend to be consistent.According to Figure25, when the order q > 0, the late value is larger than the early value. Also, when the order is q < 0, the late value is also basically smaller than the early value.Figure 25
Early and late scaling exponents.It can be seen from Figure26 that the nonlinear vibration of the cylinder block contains multifractal characteristics, while the Hurst index in the early stage is basically smaller than in the late stage, which is consistent with the experimental study. It can be considered that the Hurst index will increase at different orders at the end with the aggravation of piston ring wear.Figure 26
Hurst exponent of cylinder block nonlinear vibration.Table8 shows the multifractal spectrum parameters of the nonlinear vibration signals of the cylinder block, and the characteristics of the spectrum parameters are similar to the results of the experimental platform. According to the fractal spectrum (Figure 27), the early parabolic extreme point is closer to the Y-axis, and the extreme point also transforms to the right end of the parabola compared with the early conditions, and the zeros distance becomes narrower. Figure 28 shows the comparison between the old piston ring and the new piston ring during unit maintenance. The radial thickness of the new piston ring is 12.8 mm, while the radial thickness of the old piston ring is uneven, with an average of 9.12 mm and a local value of 6.29 mm. To sum up, their multifractal spectrum contains the abovementioned corresponding characteristics as the piston ring and support ring components in different wearing statuses.Table 8
Dimensions of multifractal global singular values and multifractal sets.
Spectrum parametersαmaxαminδαFαmaxFαminδFαfmaxEarly phase0.6280.1690.79810.03780.9620.168Late phase0.5190.0740.59310.49470.5050.204Figure 27
Multifractal spectrum.Figure 28
Comparison of early and late piston ring thicknesses.Experiments and engineering have shown that the Hurst index of the early phase is less than the late phase. The shape, spectrum widthδα, extreme point αfmax, and other characteristic parameters of a multifractal spectrum are sensitive to changes in system state and can be used as characteristic parameters to characterize the system state of a reciprocating compressor cylinder block. As the wear and tear intensifies, δα,δF becomes smaller and αfmax becomes larger.


## 5. Conclusion

In this paper, the compressor platform was built, and the VMD-multifractal spectrum method was used in the wear experiment. A method based on VMD variational mode decomposition for nonlinear vibration analysis of cylinder blocks is proposed, which can effectively extract friction vibration signals from piston assemblies and cylinder blocks:(1)
With the proposed method, the nonlinear vibration of the cylinder block is processed, and the wear state of the piston assembly can be preliminarily figured out, which could provide valuable advice for preventive maintenance and replacement of the piston assembly.(2)
With the multifractal method, the wave characteristics of the nonlinear vibration of the cylinder block could be effectively figured out, combined with the general increasing trend of the Hurst index, which shows that the nonlinear vibration of the cylinder block of a reciprocating compressor is persistent.(3)
The experimental results show that the VMD-multifractal spectrum can qualitatively analyze the wear condition of the cylinder piston assembly. It is found that the width of the multifractal spectrum decreases with the wearing process, and the nonlinearity of the cylinder vibration signal can be characterized by the multifractal spectrum and the corresponding indicators.However, the potential impact of other compressor components on non-linear cylinder vibration, such as spindle shingle wear, small and large head shingle wear, cross-head loosening, piston rod steering impact, etc., should be carefully considered in future research. The influence of these factors on the non-linear impact of the cylinder block and the fault diagnosis of the reciprocating compressor may be solved in future research. In addition, for the proposed VMD-multifractal spectrum method used in other types of equipment fault diagnosis, the extraction method for the typical fault features will also be the focus of future research.



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*Source: 2504170-2023-01-17.xml*