A research on rubbing feature extraction based on information fusion and signal decomposition algorithm

Abstract

To effectively identify the rotor–stator rubbing fault, the paper has brought forward a method combining principal component analysis (PCA), intrinsic time-scale decomposition (ITD), and information entropy (IE). Firstly, in considering that the characteristic information of faults extracted from the information collected by single sensor is not complete or comprehensive, the approach blends the vibration signals collected from 4 different positions at the same moment based on PCA algorithm; secondly, regarding that ITD algorithm can effectively avoid the problems of poor adaptivity and end effect, blended signals are broken down based on ITD algorithm; thirdly, calculate the IE of self-correlation function of each PRC based on the fact that the smaller IE is, the less confusion system has and the easier it is to extract fault characteristics, and treat the self-correlation function of PRC related with the minimum IE as optimal component to represent fault characteristics; fourthly, characteristic extraction of rotor–stator rubbing fault and identification are done on the basis of the frequency spectrum of optimal component. To prove the availability of method, vibration signals are subjected to validation and analysis, which are collected from different rotation speeds, casing thicknesses, rubbing positions, and types. The result indicates that the proposed PCA–ITD–IE can equally and effectively extract the characteristics of rotor–stator rubbing faults of aero-engine involved in various conditions.

Keywords

Rotor–stator rubbing information fusion intrinsic time scale decomposition principle component analysis information entropy

Introduction

Aero-motor is the core part of aircraft and stator–rotor rub is a very typical nonlinear fault in aero-motor. When a stator–rotor rub occurs inside aero-motor, its vibration signal will show non-stationary¹ resulting in unstable operation of aero-motor and the risks of personal injury and economic loss. Therefore, it is of special significance to precisely and effectively identify the stator–rotor rub faults of aero-motor.^2–5

To effectively identify rub faults, scholars have made a great number of studies. For example, Norden Huang proposed the new method of combining Empirical Mode Decomposition (EMD) and Hilbert time spectrum analysis.⁶ Chen Xiangmin, et al. brought forward the early diagnosis method of rotor rub faults based on the difference of signal patterns, which analyzes the early rub faults of rotor based on different signal patterns.⁷ Tang Guiji et al. put forward a characteristic extraction method based on complete ensemble robust local mean decomposition with adaptive noise to identify stator–rotor rub faults.⁸ Liu Yang, et al. presented a rub fault diagnosis method based on Fourier decomposition method.⁹

Intrinsic time scale decomposition (ITD) algorithm can break up complicated nonlinear and non-stationary signals into the sum of proper rotation components and residual trend components.¹⁰ This algorithm overcomes the defects of traditional time-frequency domain analysis method and the end effect and low operation efficiency in decomposing and processing signals with classical modes and has been widely applied to fault diagnosis. For example, Liu et al., introduced ITD algorithm into the fault diagnosis of large-sized rotary machines.¹¹ Zhang proposed a diagnosis method based on ITD and probabilistic neural network.¹² Literature¹³ describes the application of ITD algorithm and multi-scale morphological filtering algorithm to extract characteristic frequency of faults in rolling bearings. Chen Yongqi et al., proposed a new characteristic extraction method based on ITD and sample entropy and this method is capable to work with vector machine to make fault diagnosis of rolling bearings.¹⁴ Qin Guilin combined improved ITD and Hilbert envelope analysis to extract and identify the faults of rolling bearings.¹⁵

Principle component analysis (PCA) algorithm is the method to reduce the dimensionality of data with linear mapping. This method maximizes the capacity to retain the variance information of original data while removing the relevance of data. PCA algorithm has its place in the rub fault diagnosis of rotary machines. For example, Chen Guo brought forward a rub fault characteristic extraction method based on the analysis of principal components.¹⁶ Zhu Tianxu invented a method based on PCA-LMD (local mean decomposition) to extract combined characteristics of vibration signal from rolling bearings and make intelligent fault diagnosis. This method conducts PCA noise suppression for actual vibration signal based on Hankel matrix.¹⁷ Li Weiguang, et al. put forward the PCA algorithm based on SVR (singular value ratio) spectrum and concluded through simulated analysis and flexible thin-wall bearing fault experiments that the variation rate of outer-ring fault signal frequency is twice of rotating frequency of bearings while the variation rate of inner-ring fault signal frequency is consistent with rotating frequency of bearings.¹⁸

The concept of information entropy (IE) was firstly proposed by Shannon¹⁹ and has been widely applied in communication theory. Since the 1980s, researchers integrated information theory and signal analysis methods in extracting the characteristics of mechanical fault signals, and introduced IE to fault diagnosis.^20–22 For example, Du Yihao et al., organically combined dynamic analysis and parameter identification to analyze the typical faults of rotor system based on IE characteristic extraction methods.²³ Ai Yanting, et al. combined n-dimensional characteristic parameters distance (n-DCPD) and IE method to make fault diagnosis of rolling bearings.²⁴ Pan Hongxia invented an early fault diagnosis method of automatic ramming system based on IE and Elman neural network.²⁵

For that, the paper combines principal component analysis, intrinsic time scale decomposition algorithm, and information entropy in extracting the characteristics of rotor–stator rub faults of aero-motor. To verify the effectiveness of method, an analysis has been given to the vibration acceleration signal of casing when rotate speed, casing thickness, rub positions, and fault type are different.

PCA–ITD–IE Algorithm

Principal Component Analysis

PCA is the method to reduce the dimensionality of data with linear mapping and can maximize the capacity to retain the variance information of original data while removing the relevance of data. Signal fusion and dimensionality reduction can help to highlight the fault information which may be overwhelmed in single-channel signals. Dimensionality reduction thought of PCA is to map the whole data set to the coordinate axis convenient to represent the data and its decomposition principle is shown in Figure 1.

Figure 1.

PCA schematic diagram of dimensionality reduction.

Given original vibration signal is $X = [x_{1}, x_{2}, \dots, x_{n}]$ and there are n samples and each sample contains m features, then, covariance matrix C_x of x is

C_{x} = \frac{1}{n} \sum_{i = 1}^{n} (x_{i} - E (X)) {(x_{i} - E (X))}^{T} = \frac{1}{\sqrt{n}} \frac{1}{\sqrt{n}} \sum_{i = 1}^{n} (x_{i} - E (X)) {(x_{i} - E (X))}^{T} = A A^{T}

(1)

In which $E (X) = \frac{1}{n} \sum_{i = 1}^{n} x_{i}$ is the mean value of sample space. Set characteristic value of C_x $λ_{1}, λ_{2}, \dots, λ_{m}$ is $λ_{1}, λ_{2}, \dots, λ_{m}$ and feature vector is µ1, µ2 …, µm, namely

C_{x} μ = λ_{i} μ_{i}, i = 1, 2, L, m

(2)

Make and sample space is reconstructed as

Y = U^{T} A

(3)

By formula ref-type="disp-formula">3), sample space X is converted into sample space Y of feature space, in which the sample in is the j-th principal component of sample x_i in x_i. Accumulative contribution rate is defined as

ϕ (L) = \frac{\sum_{I = 1}^{L} λ_{i}}{\sum_{i = 1}^{m} λ_{i}}

(4)

When accumulative contribution rate $ϕ (L) \geq 0.95$ , the $U_{L} = [μ_{1}, μ_{2}, \dots, μ_{L}]$ composed by previous L feature vectors is treated as low-dimension projection space to finish dimensionality reduction.

PCA–ITD

ITD algorithm can break down non-stationary complex signals into the sum of a series of mutually independent proper rotation components (PR) and reference line component (res). Given X_t is the vibration signal after dimensionality is reduced by PCA algorithm. Define a reference line signal operator as L and decompose this vibration signal into the sum of a proper rotation component L_t and a rotation component H_t, and then decompose the signal once with ITD algorithm. The result can be represented as

X_{t} = L X_{t} + (1 - L) X_{t} = L_{t} + H_{t}

(5)

Segmented reference line signal extraction factor L is defined as

L X_{t} = L_{t} = L_{k} + (\frac{L_{k + 1} - L_{k}}{X_{k + 1} - X_{k + 1}}) (X_{t} - X_{k}), t \in (τ_{k}, τ_{k + 1})

(6)

In the formula, X_k is the extreme point of original signal and is the moment of extreme point (k=1, 2, ……, M, M is the number of extreme points). L_k+1 can be represented as

L_{k + 1} = α [X_{k} + (\frac{τ_{k + 1} - τ_{k}}{τ_{k + 2} - τ_{k}})] (X_{k + 2} - X_{k}) + (1 - α) X_{k + 1} (k = 1, 2, & & M - 2)

(7)

Range of linear gain control parameter α is $0 < α < 1$ , usually $α = 0.5$ .

Reference line signal L_t is treated as new original signal. Repeat the above procedure until L_t becomes a monotone function or constant.

Let the number of iterations is P, and the final resolution is

X_{t} = H_{t} + L_{t} = H_{t} + (H + L) L_{t} = H \sum_{k = 0}^{p - 1} L_{t}^{p} + L_{t}^{p}

(8)

H is the extraction operator of proper rotation component; $L_{p}^{t}$ is the residual trend component, a monotone function or constant.

PCA–ITD–IE

IE is usually used to indicate the uncertainty of information. By calculating the relation of IE values, we can determine the perplexity of system in such a way that the smaller IE is, the weaker perplexity of system is and less uncertainty. In actual application, smaller IE is more effective to information processing. Given current sample set D is the rotation components obtained from ITD decomposition in section of PCA-ITD the proportion of type k samples is P_k (k=1, 2, . . ., |y|), and the IE of D is defined as

H (X) = - \sum_{k = 1}^{| y |} P_{k} \log_{b} P_{k}

(9)

In formula ref-type="disp-formula">9), X represents the proper rotation component of one layer after ITD decomposition and its output is represented as H(X). P_k represents probability function, usually b = 2, because binary coding is usually included in computer coding. As the more uncertainty variables are, the larger entropy will be. For that, the smaller H(X) is, the purer X will be.

To effectively extract the characteristics of stator–rotor rub faults, the paper integrates PCA, ITD, and IE. To verify the superiority of proposed method, the proposed method is compared with other methods, ITD–IE and PCA–ITD-Correlation coefficient. Specific process of each method is shown in Figure 2.

Figure 2.

Method Block of ideas.

ITD–IE: ITD and IE are combined based on single-channel casing vibration signals to extract the characteristics of stator–rotor rub.

PCA–ITD-Correlation coefficient: PCA is combined with ITD algorithm to extract the characteristics of stator–rotor rub faults of aero-motor based on correlation coefficient. Firstly, PCA algorithm is used to blend the information of multi-channel vibration signals and reduce its dimensionality; after dimensionality reduction, the signal is decomposed by ITD algorithm to obtain the autocorrelation function of corresponding rotation components; secondly, figure out the autocorrelation function of each proper rotation component and correlation coefficient after dimensionality reduction, and choose the autocorrelation function of PR with maximal correlation coefficient to be optimal PR; finally, characteristics of stator–rotor rub faults are extracted based on frequency spectrum of optimal rotational component.

Specific embodiment procedures of PCA–ITD–IE method

(1) For the vibration acceleration signals collected by sensors from 4 different positions at the same moment, data fusion is performed based on PCA;

(2) blended signals are decomposed by ITD, and the IE of self-correlation function of each PRC is calculated;

(3) choose the self-correlation function of PRC associated with minimum IE as optimal PRC;

(4) the frequency spectrum of optimal PRC is used as the basis for characteristic extraction and fault identification of rotor–stator rubbing fault.

Experiments of rotor–stator rub faults

All experiment data comes from an aero-motor rotor tester shown in Figure 3(a). The tester is the same with motor casing in shape but the size is reduced by 3 times; inner structure is simplified and core part is reduced to 0-2-0 support structure whose rigidity is adjustable to adjust the dynamic characteristics of system; multi-stage compressor is simplified to single-stage disk structure and blades reduced to incline plane, and the number of blades is 32; sealed labyrinth is demountable; axle is solid and rigid design. The maximum working speed of tester is 7000 r/min. However, due to the dramatic vibration in a rubbing fault experiment and for safety purpose, the maximum working speed of tester is 2400 r/min. The paper chooses the rubbing fault data under 3 different rotation speeds ((1000 r/min, 1200 r/min, and 1500 r/min) for analysis. The tester is able to simulate the typical faults which may happen in aero-motor, for example, point rubbing and partial rubbing. Figure 3(b–c) shows the installation positions of acceleration sensors and rubbing positions in single-point rubbing experiment (casing thickness 4 mm). The channel configuration of each sensor is shown in Table 1. Figure 3(d) shows turbine casings of different thicknesses. Figure 3(e) reveals the installation positions of acceleration sensors and rubbing positions in single-point rubbing experiment (casing thickness 7 mm) (the installation positions of sensors on thick casing are completely the same with that of thin casing, except that rub positions are upper right, lower right, lower left, and upper left to casing); Figure 3(f) is partial rubbing experiment in which experimental devices are adjusted to achieve the goal. Limited by the length of paper, we only choose the rubbing experiment on the ends of compressor (experiment results from turbine end are similar to that of compressor). In the experiment, acceleration sensors are still installed on turbine. In modern large-sized rotation machinery, for example, aero-engine, rotor–stator rubbing fault is the secondary fault mainly caused by rotor imbalance, casing distortion, and decentraction of support. As a rotor has large mass and casing is light due to its thin-walled structure, rubbing force is hard to make rotor rebound. Therefore, in aero-engine, rubbing mainly occurs in the fixed position of casing, but rubbing position may vary as a result of casing distortion and direction of support decentraction. For the consideration that an actual rubbing spot may be in any position of the whole circumference, the paper has set 4 evenly distributed sensor positions and rubbing spots to locate the rubbing position to 4 directions and carries out rubbing experiment on this basis. The installation position of sensor is vertically upward, horizontally right, vertically downward, and horizontally left. As for rubbing positions, when a single-spot rubbing fault occurs, rubbing position is upward, downward, left, and right; when a partial rubbing fault occurs, partial rubbing is on the ends of compressor and turbine. To fully extract the characteristic information of rubbing faults, we have chosen the vibration signals obtained by 4 sensors for information fusion. The rub type, sensor installation positions (with the plane toward turbine casing as reference), rubbing positions, casing thickness, and rotation speed of each experiment in this paper are shown in Table 2. Acceleration sensor used in simulation experiment is model 4508 and the sampling frequency is 10,000 Hz.

Figure 3.

Rubbing fault test. (a) Aero-engine rotor tester, (b) the sensor is mounted vertically, horizontally right, (c) sensors are installed in vertical, horizontally left, (d) different thickness casing tester, (e)single point rubbing of thick wall presser casing, and (f)subtraction of casing.

Table 1.

Channel configuration.

Channel	Content
CH1	Turbine casing vertical on
CH2	Turbine casing horizontal right
CH3	Turbine casing vertical under
CH4	Turbine casing horizontal left

Table 2.

Rubbing experiment conditions explanation.

Cases	Rubbing type	Rubbing position	Sensor installation location	Thickness of casing (mm)	Rotation speed (r/min)
Case 1	Single-point	Vertical on	On、under、left、right	4	1515.15
Case 2	Single-point	Vertical on	On、under、left、right	4	1003.25
Case 3	Single-point	Horizontal left	On、under、left、right	4	1532.57
Case 4	Single-point	Right upper	On、under、left、right	7	1220.02
Case 5	Partial	Partial Left	On、under、left、right	4	1242.78

Characteristics extraction of rotor–stator rubbing faults

All the data of fault sources from the aero-motor rotor tester described in section of Experiments of rotor-stator rub faults and rotate speed (r) is a known variable. When rubbing occurs, blades impact rubbing spot one by one. When rotor takes a round, this function has a cycle. Therefore, the characteristic frequency of rubbing is the product of blade number and rotate frequency, namely, the frequency of blades through casing. For that, the characteristic frequency of rubbing fault can be expressed as

F_{n} = n \cdot \frac{r}{60} \cdot N, (n = \frac{1}{2}, \frac{1}{3}, \dots; 1, 2, \dots)

(10)

In the formula (10), N is the number of blades (in this case, the number of blades of rotor tester N = 32). Rotate frequency f_s is defined as

f_{s} = \frac{r}{60}

(11)

As the friction caused by rubbing is nonlinear, vibration frequency contains 2×, 3×, and higher harmonic. When rubbing fault occurs, in the positions of characteristic frequency and its frequency multiplication, due to modulation of rotation frequency, higher characteristic frequency may be in the position of $F_{n} \pm m \cdot f_{s} (m = 1, 2, \dots)$ .

Due to the length of paper, we take the vertical up rubbing against turbine, the rotation speed 1515.15 r/min (the rotate frequency [fs] is 25.25 Hz and the theoretical value of 1 time frequency is 808 Hz.), the casing thickness 4 mm, vibration signals collected by single channel (section of ITD-IE), and 4 channels (section of PCA-ITD-correlation coefficient and Proposed method) as an example. The 2 contrast methods mentioned in section of PCA-ITD-IE(Shown at Figure 2) and the proposed method of paper are taken to extract rubbing characteristics and for comparative analysis.

In the frequency spectrum of optimal components, the characteristic frequency of rubbing fault and its frequency multiplications are indicated by the symbol “” with specific values (the text is red); related rotation frequency and its frequency multiplications are indicated by the symbol “” with specific values (the text is green); irrelevant and inexplainable frequency components to rubbing faults are indicated by the symbol “” with specific values (the text is blue).

Contrast scheme

ITD–IE

Firstly, ITD is combined with IE (ITD–IE) to extract the characteristics of stator–rotor rub faults from single-channel casing vibration signals. Select random an acceleration signal of a channel (the acceleration signal detected by horizontal right sensor in this section) and decompose the signal by ITD algorithm into proper rotation components (PRC); secondly, figure out the autocorrelation function of each PRC and its IE, respectively; thirdly, select the autocorrelation function of PR related with the minimal IE as optimal component; finally, characteristics of rotor–stator rub faults are extracted based on the frequency spectrum of optimal component. The IE results for each PRC autocorrelation function calculated based on ITD-IE are shown in Table 3.

Table 3.

IE index—CH2 of PRCs (ITD–IE).

PR component	IE
PRC1	0.8940
PRC2	0.0033
PRC3	0.1244
PRC4	0.0204

Choose the self-correlation function of proper rotation component (PRC2) related with minimum IE index as optimal component and implement characteristic extraction of rubbing fault based on ITD-IE. The result is as shown in Figure 4. Figure 4(a) is the time domain of raw vibration acceleration signal; Figure 4(b) is the frequency spectrum of Figure 4(a). Figure 4(c) is the time domain of PRCs (PR1-PR4) obtained by ITD algorithm; Figure 4(d) is the autocorrelation function of each PRC shown at Figure 4(c); Figure 4(e) is the autocorrelation function (minimal IE) of PRC2 selected by Figure 4(d) and Table 3; Figure 4(f) is the frequency spectrum of Figure 4(e).

Figure 4.

Feature extraction of rubbing fault (ITD–IE)-CH2. (a) Time-domain acceleration signal of CH2, (b) frequency spectrum of (a), (c) proper rotational components of acceleration signal after ITD, (d) autocorrelation function of PRCs, (e) autocorrelation function of PRC2, and (f) frequency spectrum of (e).

Due to the structure of tester, when rubbing occurs, each blade rubs the point in turn. When rotor finishes a round, this function will cycle. Therefore, the vibration caused by rubbing is very similar to the vibration of gears and its rub frequency is equal to meshing frequency, being the product of number of blades and rotation frequency. When rotation speed is 1515.15 r/min, the rub frequency is F₁ = 808.08 Hz (the product of number of blades and rotation frequency).

It can be found from the analysis of Figures 4(b) and (f) that:

From the frequency spectrum of raw vibration signal and the frequency spectrum of optimal proper rotation component obtained based on ITD–IE, we can find characteristic frequency of rubbing and its frequency multiplication (double and triple frequencies are F₂ = 1616.16 Hz and F₃ = 2424.24 Hz, respectively). However, rubbing frequency is not clear enough and noise component is very large, which is likely to cause a misdiagnosis.

PCA–ITD-Correlation coefficient

For a comparative analysis, the data chosen includes the 4 acceleration signals collected at the same moment described in section of ITD-IE. Firstly, blend the acceleration signals of 4 channels and reduce dimensionality according to PCA algorithm; secondly, resolve the acceleration signals with ITD algorithm after dimensionality reduction to obtain rotation components, and then calculate the autocorrelation function of each rotation component; thirdly, calculate the autocorrelation function of each PRC and the correlation coefficients of signal after PCA dimensionality reduction; choose the proper rotation component of maximum correlation coefficient as optimal rotation component; finally, based on the frequency spectrum of optimal rotation component, extract the characteristics of stator–rotor rub faults of aero-motor. The correlation coefficient results for each PRC autocorrelation function calculated based on PCA–ITD-Correlation coefficient are shown in Table 4.

Table 4.

Correlation coefficient index layer of PRCs (PCA–ITD-Correlation coefficient).

PR component	Correlation coefficient
PRC1	-0.2177
PRC2	-0.1079
PRC3	-0.0027
PRC4	-0.0006

Choose the self-correlation function of proper rotation component (PRC4) related with maximum correlation coefficient index as optimal component and implement characteristic extraction of rubbing fault based on PCA–ITD-Correlation coefficient. The result is as shown in Figure 5. Figure 5(a) provides the time domains of raw vibration acceleration signals acquired by acceleration sensors fixed in 4 directions, vertical up, horizontal right, vertical down, and horizontal left; Figure 5(b) shows the time domains of one-dimensional signals after PCA dimensionality reduction; Figure 5(c) shows the time domains of PR components (PRC1–PRC4) after ITD decomposition; Figure 5(d) shows the autocorrelation function of PRC4 selected by the principle of choosing the maximum correlation coefficient; Figure 5(e) is the frequency spectrum of Figure 5(d).

Figure 5.

Feature extraction of rubbing fault based on PCA–ITD-Correlation coefficient. (a) Time-domain acceleration signal of CH1–CH4 (Rubbing vertical on 1515.15r/min), (b) time domain wave after PCA, (c) proper rotational components of acceleration signal after ITD, (d) autocorrelation function of PRC4, and (e) frequency spectrum of (d).

According to Figure 5(e), in the frequency spectrum of optimal proper rotation component (autocorrelation function of PRC4) selected by PCA–ITD-Correlation coefficient, we cannot find the characteristic frequency of rub faults and its frequency multiplication component. That means, the frequency spectrum of optimal proper rotation component selected by the principle of maximum correlation coefficient cannot effectively identify the stator–rotor rub faults of aero-motor.

Proposed method

To be more precise in extracting characteristics of stator–rotor rub faults, the following procedure is basic on proposed method. For a comparative analysis, the data chosen includes the 4 acceleration signals collected at the same moment described in section of PCA-ITD-correlation coefficient. Firstly, blend the acceleration signals of 4 channels and reduce dimensionality according to PCA algorithm; secondly, resolve the acceleration signals with ITD algorithm after dimensionality reduction to obtain rotation components, and then calculate the autocorrelation function of each rotation component; thirdly, calculate the autocorrelation function of each PRC and the IE of signal after PCA dimensionality reduction; choose the proper rotation component of minimal IE as optimal rotation component; finally, based on the frequency spectrum of optimal rotation component, extract the characteristics of stator–rotor rub faults of aero-motor. The IE results for each PRC autocorrelation function calculated based on proposed method are shown in Table 5.

Table 5.

IE index layer proposed method.

PR component	IE
PRC1	0.8670
PRC2	0.0033
PRC3	0.9992
PRC4	0.0159

Characteristics are extracted from rotor–stator rubbing faults based on optimal rotation components chosen by the proposed method. The result is as shown in Figure 6. Figure 5(a–c) shows the time domain of vibration signals captured by acceleration sensors in 4 directions, vertically upward, horizontally right, vertical downward, and horizontally left, time domain of one-dimensional signal as a result of PCA dimensionality reduction and that of rotation components (PRC1–PRC4) after ITD decomposition of blended signals. Figure 6(a) shows the self-correlation function of PRC2 chosen based on minimum IE principle and Table 5; Figure 6(b) is the frequency spectrum of Figure 6(a).

Figure 6.

Feature extraction of rubbing fault-proposed method. (a) Autocorrelation function of PRC2, and (b) frequency spectrum of (a).

According to Figure 6(b), in the frequency spectrum of optimal proper rotation component (autocorrelation function of PRC2) selected by the principle of least IE (proposed method in this paper), there is outstanding rubbing frequency and its frequency multiplication (F₂ = 1616 Hz, F₃ = 2424 Hz).

That means, the proposed method can effectively identify a stator–rotor rub fault in aero-motor.

Feature extractions in different conditions

To analyze the sensibility and accuracy of proposed PCA–ITD–IE method to characteristic extraction of rotor–stator rub faults, we will study its performance in different conditions, such as different rotation speeds, rubbing positions, casing thickness, and rubbing types.

Different rotation speed

Firstly, we will study the effectiveness of PCA–ITD–IE method in different rotation speeds. Select the vibration data collected at 1003.25 r/min (the rotate frequency [fs] is 16.72 Hz and the theoretical value of 1 time frequency is 535.07 Hz.) with rubbing position upward, casing thickness 4 mm, and the vibration signals collected from 4 channels. According to rotation speed and the number of blades, we can figure out the characteristic frequency of stator–rotor rub faults and its double and triple frequency: F₁ = 534.93 Hz, F₂ = 1069.87 Hz, F₃ = 1604.80 Hz, F₄ = 2139.72 Hz. According to the proposed method, we can figure out the IE of autocorrelation function, which is shown in Table 6.

Table 6.

Information entropy of autocorrelation function of PRCs-1000 r/min.

PR component	IE
PRC1	0.0422
PRC2	0.0118
PRC3	0.0231
PRC4	0.0209

Select the autocorrelation function of PRC2 associated with the least IE as optimal component and extract the characteristics of rub faults based on PCA–ITD–IE. The result is shown in Figure 7. Figure 7(a) is still the time domains of raw vibration acceleration signals captured by the sensors installed in 4 directions, vertical up, horizontal right, vertical down, and horizontal left; Figure 7(b) is the time domain of one-dimensional signals after PCA blend and dimensionality reduction; Figure 7(c) is the time domain (PRC1–PRC4) of rotational components after ITD decomposition; Figure 7(d) is the autocorrelation function of PRC2 selected by the principle of least IE; Figure 7(e) is the frequency spectrum of Figure 7(d).

Figure 7.

Rubbing features extraction in different rotation speed-proposed method (Vertical On, 1003.25 r/min). (a) Time-domain acceleration signal of CH–-CH4 (Rubbing vertical on, 1003.25 r/min), (b) time-domain acceleration signal after PCA, (c) time-domain of PRC1–PRC4 corresponding with (b), (d) autocorrelation function of PRC2, and (e) frequency spectrum of (d).

It can be found from the analysis of Figure 7(e) that contains obvious characteristic frequency of rub faults and its frequency multiplication components (F₁ = 534.7 Hz, F₂ = 1036 Hz, F₃ = 1604 Hz, F₄+2f_s = 2172 Hz).

That means in different rotate speeds, the proposed PCA–ITD–IE method can still extract the characteristic frequency of stator–rotor rub faults and its frequency multiplication components.

Different rubbing positions

Next, we will study the effectiveness of PCA–ITD–IE method in different rubbing positions. Randomly choose the raw vibration data which is collected at 1532.57 r/min (the rotate frequency [fs] is 25.54 Hz and the theoretical value of 1 time frequency is 817.37 Hz.) with rubbing position leftward, casing thickness 4 mm, and the vibration signals collected from 4 channels. According to rotate speed and the number of blades, we can figure out the characteristic frequency of stator–rotor rub faults and its double and triple frequency: F₁ = 817.37 Hz, F₂ = 1635 Hz, F₃ = 2452 Hz. According to the proposed method, we can figure out the IE of autocorrelation function, which is shown in Table 7.

Table 7.

Information entropy of autocorrelation function of PRCs-rubbing position leftward.

PR component	IE
PRC1	0.2609
PRC2	0.0496
PRC3	0.4065
PRC4	0.2563

Select the autocorrelation function of PRC2 associated with the least IE as optimal component and extract the characteristics of rub faults based on PCA–ITD–IE. The result is shown in Figure 8. Figure 8(a) is still the time domains of raw vibration acceleration signals captured by the sensors installed in 4 directions, vertical up, horizontal right, vertical down, and horizontal left; Figure 8(b) is the time domain of one-dimensional signals after PCA blend and dimensionality reduction; Figure 8(c) is the time domain (PRC1–PRC4) of PRC after ITD decomposition; Figure 8(d) is the autocorrelation function of PRC2 selected by the principle of least IE; Figure 8(e) is the frequency spectrum of Figure 8(e).

Figure 8.

Rubbing features extraction in different rubbing position-proposed method (Horizontal Left, 1532.57 r/min). (a) Time-domain acceleration signal of CH1–CH4 (Rubbing horizontal Left, 1532.57 r/min), (b) time-domain signal of (a) after PCA, (c) time-domain of PR1–PR4 of (b) after ITD, (d) autocorrelation function of PRC2, and (e) frequency spectrum of (d).

It can be found from the analysis of Figure 8(e) that contains obvious characteristic frequency of rub faults and its frequency multiplication components (F₁ = 817.9 Hz, F₂ = 1635 Hz, F₃ = 2452 Hz).

That means in different rubbing positions, the proposed PCA–ITD–IE method can still extract the characteristic frequency of stator–rotor rub faults and its frequency multiplication components.

Different casing thickness and rubbing position

Next, we will analyze the effectiveness of PCA–ITD–IE method in extracting the characteristics of rub faults when casing thickness and rubbing position are different. Randomly choose the vibration data which is collected at 1220.02 r/min (the rotate frequency [fs] is 20.33 Hz and the theoretical value of 1 time frequency is 650.687 Hz.) with rubbing position upper right, casing thickness 7 mm (display on Figure 3(d–e)), and the vibration signals collected from 4 channels. According to rotate speed and the number of blades, we can figure out the characteristic frequency of stator–rotor rub faults and its double frequency: F₁ = 650.56 Hz, F₂ = 1301.12 Hz. According to the proposed method, we can figure out the IE of autocorrelation function, which is shown in Table 8.

Table 8.

Information entropy of autocorrelation function of PRCs-casing thickness 7 mm and rubbing position upper right.

PR component	IE
PRC1	0.9268
PRC2	0.0067
PRC3	0.0061
PRC4	0.4893

Select the autocorrelation function of PRC3 associated with the least IE as optimal component and extract the characteristics of rub faults based on PCA–ITD–IE. The result is shown in Figure 9. Figure 9(a) is still the time domains of raw vibration acceleration signals captured by the sensors installed in 4 directions, vertical up, horizontal right, vertical down, and horizontal left; Figure 9(b) is the time domain of one-dimensional signals after PCA blend and dimensionality reduction; Figure 9(c) is the time domain (PRC1–PRC4) of rotational components after ITD decomposition; Figure 9(d) is the autocorrelation function of PRC3 selected by the principle of least IE; Figure 9(e) is the frequency spectrum of Figure 9(d).

Figure 9.

Rubbing features extraction in different casing thickness and rubbing position-proposed method. (a) Time-domain acceleration signal of CH1–CH4 (Rubbing horizontal Left, 1220.02 r/min), (b) time-domain acceleration signal of (a) after PCA, (c) time-domain of PR1–PR4 of (b) after ITD, (d) autocorrelation function-PRC3, and (e) frequency spectrum of (d).

It can be found from the analysis of Figure 9(e) that contains obvious characteristic frequency of rub faults and its frequency multiplication components (F₁ = 655.5 Hz, F₂ = 1271 Hz).

That means in different casing thickness and rubbing positions, the proposed PCA–ITD–IE method can still extract the characteristic frequency of stator–rotor rub faults and its frequency multiplication components.

Partial rub faults

To analyze the effectiveness of proposed PCA–ITD–IE method in extracting characteristics of rub faults when rub fault type is different, we will extract the characteristics of partial rub faults. Randomly choose the vibration data which is collected at 1242.78 r/min (the rotate frequency [fs] is 20.71 Hz and the theoretical value of 1 time frequency is 662.82 Hz.) with rubbing position inclined left to compressor, casing thickness 4 mm, and the vibration signals collected from 4 channels. According to rotate speed and the number of blades, we can figure out the characteristic frequency of stator–rotor rub faults and its triple frequency: F₁ = 662.82 Hz, F₃ = 1988.45 Hz. According to the proposed method, we can figure out the IE of autocorrelation function, which is shown in Table 9.

Table 9.

Information entropy of autocorrelation function of PRCs-partial rubbing.

PR component	IE
PRC1	0.0040
PRC2	0.1343
PRC3	0.0080
PRC4	0.0182

Select the autocorrelation function of PRC1 associated with the least IE as optimal component and extract the characteristics of rub faults based on PCA–ITD–IE. The result is shown in Figure 10. Figure 10(a) is still the time domains of raw vibration acceleration signals captured by the sensors installed in 4 directions, vertical up, horizontal right, vertical down, and horizontal left; Figure 10(b) is the time domain of one-dimensional signals after PCA blend and dimensionality reduction; Figure 10(c) is the time domain (PRC1–PRC4) of rotational components after ITD decomposition; Figure 10(d) is the autocorrelation function of PRC1 selected by the principle of least IE; Figure 10(e) is the frequency spectrum of Figure 10(e).

Figure 10.

Rubbing features extraction in different rubbing types-proposed method (Partial Left, 1242.78 r/min). (a) Time-domain acceleration signal of CH1–CH4 (Rubbing partial Left, 1242.78r/min), (b) time-domain acceleration signal of (a) after PCA, (c) time-domain of PR1–PR4 of 10 (b) after ITD, (d) autocorrelation function-PRC1, and (e) frequency spectrum of (d).

It can be found from the analysis of Figure 10(e) that contains obvious characteristic frequency of rub faults and its frequency multiplication components (F₃ = 1953 Hz).

That means in partial rub faults, the proposed PCA–ITD–IE method can still extract the characteristic frequency of stator–rotor rub faults and its frequency multiplication components.

Conclusion

To efficiently and accurately extract the rotor–stator rubbing fault feature frequency, the efficient identification of rotor–stator rubbing fault is realized. The paper contributes the combined method of PCA, ITD, and IE (PCA–ITD–IE). The proposed method blends the acceleration signals based on PCA algorithm collected from the 4 positions on casing to fully embody the characteristic information of rubbing fault, and then separates the blended signals with ITD; minimum IE is taken as characteristic parameter which can make the option of sensitive PRC showing the characteristics of rubbing fault; at last, rotor–stator rubbing fault can be effectively identified based on the frequency spectrum of optimal PRC. A comparative analysis with other methods has proven the effectiveness and accuracy of proposed method and the following conclusions can be drawn:

(1) As found from the comparative analysis with control method (ITD–IE), the combination of PCA and ITD can solve the problem of single sensor fault which cannot be fully expressed. The combined method of PCA and ITD can further highlight the characteristic components of rubbing fault and help to extract the characteristics of rotor–stator rubbing faults more precisely, and thereby make effective identification of rubbing faults.

(2) It is found from the comparative analysis with control method (PCA–ITD-Correlation coefficient) that it has further proved the minimum IE can serve as a characteristic parameter and make the option of PRC sensitive to characteristic of rubbing fault.

(3) The proposed method (PCA–ITD–IE) can largely reduce noise components and further highlight the characteristics of rubbing faults.

(4) The proposed method is insensitive to rubbing position, rubbing type, casing thicknesses, and rotation speed. In the following 3 different rotation speeds (1000 r/min, 1200 r/min, and 1500 r/min), 4 different rubbing positions (vertically upward, horizontally leftward, upper right, and partial rubbing left), 2 different rubbing types (single-spot rubbing and partial rubbing), and 2 different casing thicknesses (4 mm and 7 mm) cases, the proposed method can achieve the effective extraction of rotor–stator rubbing fault features and effective identification of rubbing fault.

Supplemental Material

sj-doc-1-nvw-10.1177_09574565221093224 – Supplemental material for A research on rubbing feature extraction based on information fusion and signal decomposition algorithm

Supplemental material, sj-doc-1-nvw-10.1177_09574565221093224 for A research on rubbing feature extraction based on information fusion and signal decomposition algorithm by Mingyue Yu, Haonan Cong and Wangying Chen in Noise & Vibration Worldwide

Supplemental Material

sj-docx-2-nvw-10.1177_09574565221093224 – Supplemental material for A research on rubbing feature extraction based on information fusion and signal decomposition algorithm

Supplemental material, sj-docx-2-nvw-10.1177_09574565221093224 for A research on rubbing feature extraction based on information fusion and signal decomposition algorithm by Mingyue Yu, Haonan Cong and Wangying Chen in Noise & Vibration Worldwide

Supplemental Material

sj-docx-3-nvw-10.1177_09574565221093224 – Supplemental material for A research on rubbing feature extraction based on information fusion and signal decomposition algorithm

Supplemental material, sj-docx-3-nvw-10.1177_09574565221093224 for A research on rubbing feature extraction based on information fusion and signal decomposition algorithm by Mingyue Yu, Haonan Cong and Wangying Chen in Noise & Vibration Worldwide

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study is supported by the National Natural Science Foundation of China [grant number:51605309], Natural Science Foundation of Liaoning Province [grant number:2019-ZD-0219], Aeronautical Science Foundation of China [grant number:201933054002], and Department of Education of Liaoning Province [grant number: JYT19042].

ORCID iDs

Haonan Cong

Wangying Chen

Supplemental material

Supplemental material for this article is available online.

References

Meng

Liu

, et al. General synchro extracting chirplet transform: application to the rotor rub-impact fault diagnosis. Measurement 2021; 169: 108523.

Tao

Han

, et al. Time-frequency features of two types of coupled rub-impact faults in rotor systems. J Sound Vibration 2008; 321(3): 1109–1128.

Chu

. Experimental observation of nonlinear vibrations in a rub-impact rotor system. J Sound Vibration 2004; 283(3): 621–643.

Liu

. Some observations of rub-impact fault on Jeffcott Rotor [J]. J Vibration Eng 2001; 01: 100–103.

Zhou

Guo

. Dynamic analysis and intelligent diagnosis for rubbing fault of a rotor-ball bearing coupled system [J]. J Vib Shock 2008; 10: 90–194.

Huang

Shen

Long

, et al. The eimpirical mode dec; miposition and Hilbert spectrum for nonlinear and non-stationaty time series analysis [J]. Proc R Soc Lond Ser A-Mathematical Phys Eng Sci 1998; 454: 903–995.

Chen

Early rub-impact diagnosis of rotors using morphological component analysis [J]. J Vibration Eng 2014; 27(03): 466–472.

Tang

Singh

, et al. Rotor fault diagnosis based on CERLMDAN [J]. J Chin Soc Power Eng 2020; 40(09): 720–727.

Liu

Shan

. Aeroengine rotor fault diagnosis based on Fourier decomposition method [J]. China Mech Eng 2019; 30(18): 2156–2163.

10.

Mark

Ivan

. Intrinsic time scale decomposition: time-frequency-energy analysis and real-time filtering of non-stationary signals [C]. Proc R Soc A 2007; 463: 321–342.

11.

Liu

Zhang

Qin

, et al. Diesel engine fault diagnosis using intrinsic time-scale decomposition and multistage adaboost relevance vector machine [J]. Proc Inst Mech Eng 2018; 232(5).

12.

Zhang

Liu

Luan

, et al. Bearing fault diagnosis based on ITD information entropy and PNN [J]. Coal Mine Machinery 2019; 40(12): 167–169.

13.

Guan

Tian

Zhao

, et al. Fault feature extraction method of rolling bearing based on intrinsic time-scale decomposition and multiscale morphological filtering. Sci Tech Eng 2019; 19(14): 178–182.

14.

Chen

Zhao

Liu

, et al. Fault diagnosis of roller bearing based on intrinsic time-scale decomposition. J Electron Meas Instrumentation 2015; 29(11): 1677–1682.

15.

Qin

Yuan

Wang

. Fault diagnosis of rolling bearings with improved ITD and Hilbert envelope spectrum [J]. Equipment Management Maintenance 2015; S2: 343–345.

16.

Guo

. Rotor-stator rubbing fault testing technique based on one-class support vector machine and primary component analysis [J]. J Vibration Shock 2012; 31(22): 29–33+38.

17.

Zhu

Zang

. Hybrid feature selection and fault diagnosis of rolling bearing signals based on principal component analysis denoising and local mean decomposition (PCA-LMD) [J]. Aeroengine 2020; 46(05): 14–21.

18.

Liu

Lin

, et al. Research on fault diagnosis of flexible thin-wall bearings based on PCA and Hilbert spectrum [J]. Machine Tool&Hydraulics 2020; 48(16): 169–175.

19.

Shannon

. A mathematical theory of communication [J]. Bell System Technical Journal 1948; 27(3): 379–423.

20.

Chaitin

. Information-theoretic computational complexity. IEEE Trans Inform Theor 1974; 20: 10–15.

21.

Dragomir

. An inequality for logarithmic mapping and applications for the Shannon entropy. Comput Maths Appl 2003; 46(8): 1273–1279.

22.

Amadori

Natalini

. Entropy solutions to a strongly degenerate anisotropic convection diffusion equation with application to utility theory. J Math Anal Appl 2003; 284(2): 511–531.

23.

Xie

. Rotor dynamic feature analysis and diagnosis method based on information entropy [J]. J Vibration Shock 2009; 28(02): 77–81+202.

24.

Guan

Fei

, et al. Fusion information entropy method of rolling bearing fault diagnosis based on n-dimensional characteristic parameter distance. Mech Syst Signal Process 2017; 88: 123–136.

25.

Gao

Pan

, et al. Early fault diagnosis of the ammunition supply system based on information entropy and neural network [J]. Machine Des Res 2020; 36(2): 181–188.

Supplementary Material

Please find the following supplemental material available below.

For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.

For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.

0.00 MB

2.04 MB

2.89 MB

2.62 MB