Amplitude filtering characteristics of singular value decomposition and its application to fault diagnosis of rotating machinery

Journal Pre-proofs Amplitude Filtering characteristics of Singular Value Decomposition and Its Application to Fault Diagnosis of Rotating Machinery Mingjun Guo, Weiguang Li, Qijiang Yang, Xuezhi Zhao, Yalian Tang PII: DOI: Reference:

S0263-2241(19)31311-9 https://doi.org/10.1016/j.measurement.2019.107444 MEASUR 107444

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Please cite this article as: M. Guo, W. Li, Q. Yang, X. Zhao, Y. Tang, Amplitude Filtering characteristics of Singular Value Decomposition and Its Application to Fault Diagnosis of Rotating Machinery, Measurement (2019), doi: https://doi.org/10.1016/j.measurement.2019.107444

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Amplitude Filtering characteristics of Singular Value Decomposition and Its Application to Fault Diagnosis of Rotating Machinery Mingjun Guo1, Weiguang Li*1, Qijiang Yang2 ,Xuezhi Zhao1, Yalian Tang3 1

School of Mechanical and Automotive Engineering, South China University of Technology, Guangzhou 510640, China. 2

3

School of Marine Engineering, Guangzhou Maritime University, Guangzhou, 510725, China

Department of Internet Finance & information Engineering, Guangdong University of Finance, Guangzhou, 510521, China

Keywords: Singular value decomposition (SVD); Effective singular values; Amplitude filter; Sliding bearing; Feature extraction; Axis orbit; Fault diagnosis

Abstract: In this paper, two important properties of singular value decomposition (SVD) are deduced theoretically: (1) number law of singular values: one frequency corresponds to two singular values; (2) order rule of singular values: the larger the amplitude of signal is, the greater the corresponding two singular values are. The above two properties are collectively referred as amplitude filtering characteristics of SVD, and a signal separation algorithm (SVD-AF) based on this characteristic is proposed. Research shows that the algorithm shows excellent characteristics in both extracting multiple and single frequency components. What’s more, the purified signal does not contain redundant components, nor does phase deviation occur. Finally, the proposed algorithm is used to purify axis orbits of the rotor of large sliding bearing test bed, the obtained axis trajectories are clear and concentrated, and the misalignment as well as rub impact fault of the rotor are identified successfully.

1. Introduction Tufts et al. [1] first proposed the concept of singular value decomposition filter (SVDF), and it is applied to estimate the useful components of noise signal. Since then, as a signal processing technique, SVD [1,2] has been widely used to data compression [3] face recognition [4] feature extraction [5] signal denosing [6] and fault diagnosis [7-8] and so on. Mcgivney et al. [3] adopted SVD to magnetic resonance fingerprint recognition in time domain, and its low-rank approximation dictionary was

obtained by SVD compression. Guo et al. [4] utilized SVD and KDA for face recognition. Kanjilal et al. [5] extracted fetal electrocardiogram from single channel electrocardiogram of pregnant women using SVD. Yang et al. [6] employed SVD to noise reduction of vibration signals and the number of effective singular values were selected using singular entropy. Aiming at extract the faint fault information from complicated bearing vibration signals, Zhang et al. [7] proposed a new method combing SVD and correlated kurtosis. The selection of effective singular values is involved in the above applications, and a variety of methods have been proposed, among them, the typical ones are singular entropy method [6], clustering method [8] and difference spectrum method [9] and so on . Yang et al. [8] proposed a c-mean algorithm based on dynamic clustering for the selection of effective singular values, but the initial clustering number C is hard to determine using this algorithm. Zhao et al. [9] can realize automatic selection of effective singular values, utilizing the maximum peak position of the difference spectrum of them. It is a pity that these methods only focus on how to obtain some feature points with some algebraic operation of singular values .As a consequence, they are not competent for various complex working conditions in practice. Attempting to overcome this issue, some researchers begin to study the internal relationship between the number of singular values and that of frequencies. Hu et al. [10] believed that a frequency corresponds to a singular value in the sinusoidal signal, but his conclusion is proved wrong later. Zhao et al [11] studied the internal relationship between the number of effective singular values and that of frequencies, and points out that ,under certain conditions, one frequency generates two adjacent singular values, and the larger the amplitude of the frequency component is, the larger the corresponding singular values will be. In another reference, Zhao et al [12] further deduced the algebraic relationship between non-zero singular values and basic parameters of the signal, and proposed the conditions to separate single frequency component using SVD. However, the aforesaid conclusions are not proved theoretically in these two references. Aiming at the shortcomings of the aforementioned researches, the number law and order rule of effective singular values will be deduced theoretically. These two properties are collectively referred as amplitude filtering characteristics of SVD. On the basis of them, a novel algorithm for feature extraction is proposed, called singular value decomposition amplitude filter (SVD-AF). Correctness of the proposed algorithm is verified by the analysis of both the simulation and the real rotor vibration signals. Finally, the proposed algorithm is employed to purify axis orbits of the rotor of large sliding bearing test bed, and the obtained axis trajectories are clear and concentrated, then the misalignment along with rub impact fault of the rotor is identified successfully. This paper is organized as follows: Section 2 introduces the signal processing principle of Hankel matrix-based SVD. Section 3 demonstrates the detailed theoretical derivation processes of the number

law of effective singular values. In Section 4, the order rule of singular values is proved. In Section 5, the conception of amplitude filtering characteristics of SVD is proposed, on the basis of this property, a novel signal separation algorithm called SVD-AF is put forward. At the same time, correctness of the proposed method is verified by both a simulation and several real rotor vibration signals. In Section 6, the proposed method is employed to purify axis trajectories of the rotor, and then the misalignment together with rub impact fault of the rotor are identified successfully. Section 7 draws several conclusions.

2. Signal processing principle of Hankel matrix-based SVD For a real matrix ARmn , its singular value decomposition [1] can be expressed as: A  UDV T

(1)

Where U =(u1, u2 ,, um ) Rmm and V =( v1 , v2 ,  , vn )  R n n

are both orthogonal matrices, called left

singular matrix and right singular matrix ,respectively; ui Rm1 is left singular vector and vi  R n1 is right singular vector; D is a diagonal matrix ， whose elements are descending singular values of matrix A processed by SVD, namely D  diag(1,2,,r ) and 1 2 r 0 , r  min( m , n ) is the rank of matrix A. So we can obtain the values of D after the values in A. By substituting each component of matrix U and V into Eq. (1), it can be rewritten into the following sum of components: T  1  v1    T r 2 T   v2     u v T A = UDV   u1 u2  ur      i 1 i i i    T  r  vr  Where r  min( m , n ) is the rank of matrix A.

(2)

Let Ai   i ui viT( i  1, 2,  r ) , then Eq. (2) can be expressed as:

A=

r

A i1

i

(3)

In the above formulas, matrix A adopts Hankel matrix. After continuous time domain signal x(t) being discretized by sampling interval Ts, discrete series ( X=[x(1),x(2),,x(N)] ) can be obtained, using this sequence, Hankel is constructed as follow:

x(2) x ( n)   x(1)   x(2) x(3) x(n  1)   A          x(m) x(m  1)  x( N  m  1) 

(4)

The matrix A is known as Hankel matrix, also called the reconstructed attractor trajectory matrix. For the construction of Hankel matrix, Zhao et al. [9] proved the parabola-symmetric relationship between noise removing quantity and the column number of Hankel matrix, pointed out that the architecture of Hankel matrix should be determined by following rule (let n as the number of columns and m as the number of rows),if signal length N is even, n =N/2 and m=N/2+1 should be adopted to create Hankel matrix, if N is odd, the Hankel matrix should be constructed with n =(N+1)/2 and m=(N+1)/2. According to the structural characteristics of Hankel matrix, that is, adjacent rows or columns only lag behind one point, a signal component Pi can be obtained from matrix Ai via the direct method [13], as expressed: T ) Pi  ( Ri ,1C i,n

Ri ,1  R 1 n , C i , n  R ( m 1)1

(5)

Where Ri,1 is the first row vector of Ai, and Ci,n is the last column vector of Ai removing the first element, as shown in Fig. 1.  x(1)  x(2) Ai       x ( m)

  

     x(m+1)  x( N  m  1)  x(2) x(3) 

x ( n) x(n+1) 

Ri ,1

Ci ,n Fig. 1: Extraction schematic of Pi from Ai.

The reconstructed signal xˆ can be obtained by adding k components to be extracted together, i.e.: P1 + P2 +  + Pk = xˆ

(6)

Where k represents the number of effective singular values. How to determine effective singular values is always a hot topic in SVD application. Zhao et al [11-12] have proved that there is a twofold relationship between the number of non-zero singular values and that of frequencies in a signal, but this conclusion has not yet been proved theoretically. This paper will prove this conclusion in theory, and combine the relationship between singular value and signal amplitude, further study the essence of singular value decomposition, and then propose an algorithm called singular value decomposition amplitude filter (SVD-AF). According to Eq. 6, in Hankel matrix mode, the process of extracting a certain component from original signal can be realized by simple subtraction operation, which does not affect the phase of each component signal, that is, it has zero-phase offset characteristics [13].

3. Theoretical derivation of the number law of effective singular values The number law of effective singular values is discussed in three cases: the signal includes single frequency, multiple frequency or random noise. 3.1 Single frequency component If the signal contains only one frequency component, the expression is x ( t )  a1 sin( 1t   1 )

(7)

Signal x(t) is discretized by sampling interval Ts, and N points are collected to obtain the discrete sequence . A Hankel matrix H can be constructed according to Eq. (4), as is given as follow:  sin(1Ts  ( n  1)   1 )  sin(1Ts  1   1 )  sin(1Ts  0   1 )   T T     sin( 1 ) sin( 2 )      sin(1Ts  n   1 ) 1 1 s 1 1 s  H 1  a1           sin(1Ts  ( m  1)   1 ) sin(1Ts  ( m  2)   1 )  sin(1Ts  ( N  1)   1 ) 

(8)

Using Euler's formula, following formula can be obtained: sin(1nTs  1 ) 

1 (1nTs  ) j -(1nTs  ) j (e -e ) 2j

(9)

Where j is imaginary unit. Substituting Eq. 9 into Eq. 8 and simplifying it, and the latter can be rewritten:  e 0 Ts1 j e1Ts1 j  e ( n 1)Ts1 j    e 2 Ts1 j  e nTs1 j a1 1 j  e1Ts1 j  e  H1    2j     ( m 1)Ts1 j  e e mTs1 j  e ( N 1)Ts1 j    H 11

e e  e  ( n 1)Ts1 j    e 2 Ts1 j  e  nTs1 j a1 1 j  e 1Ts1 j  e     2j      ( m 1)Ts1 j   mTs 1 j  ( N 1) Ts 1 j  e e  e     0 Ts 1 j

 1Ts 1 j

(10)

H 12

From Eq. (10), it’s obvious that matrix H1 is obtained by summation of matrix H11 and H12. Owing to the linear correlation of the rows, the ranks of them are both 1, namely rank (H11) = rank (H12) =1. According to the study in reference [9], the relationship between the rank of H1 and that of H11 as well as H12 satisfies rank (H1)  (rank (H11) + rank (H12) =2. In order to further determine the rank of matrix H1, the definition of matrix rank is taken into consideration in two cases:

(1) When 1  2k , k  Z , Eq. (8) can be converted into Eq. (11):  sin(( n  1) 1Ts )  sin( 1Ts )  sin( 1Ts  0)    sin( n   1Ts ) sin(1   1Ts ) sin(2   1Ts )  H 1  a1           sin(( m  1). 1Ts ) sin(( m  2)   1Ts )  sin(( N  1)   1Ts ) 

(11)

The second minor M2 of H1 can be calculated as Eq. (12): M2 

sin( 1Ts  0） sin( 1Ts  1) sin( 1Ts  2)

sin( 1Ts  1)

 sin 2 ( 1Ts )  0

(12)

It is obvious that the rank of H1 satisfies rank (H1)≥2. Combined with the aforementioned analysis results (rank (H1)  2), conclusion can be draw that rank (H1) =2. (2) When 1  2k , k  Z The second minor of H1 can be given as Eq. (13): M 2  

a1 1 j sin(1Ts  0+  1） sin(1Ts  1+  1 ) e  sin(1Ts  1+  1 ) sin(1Ts  2+  1 ) 2j a1 1 j e 0 Ts 1 j  e  0 Ts 1 j e  1T  j   1T  j 2j e s 1 e s 1

-

e1Ts 1 j  e  1Ts 1 j e 2 Ts 1 j  e  2 Ts 1 j

(13)

a1 1 j e （  e Ts 1 j  e -Ts 1 j） 0 2j

Then, the rank of H1 meets rank (H1)≥2. Combined with the preceding conclusion (rank (H1)  2), conclusion can be draw that rank (H1)=2. To sum up, when the signal x(t) contains only one frequency, the rank of Hankel matrix H1 constructed from x(t) is equal to 2, namely rank (H1) = 2 . Considering the number of nonzero singular values being equal to rank (H1), after Hankel processed using SVD, the number of non-zero singular values is equal to 2. What’s more, they are closely aligned and independent of the amplitude and phase of x(t). 3.2 Multiple frequency components Supposing that x(t) is a signal with k frequencies, it is displayed as Eq. (14): x (t ) 

k

a i 1

i

sin( i t   i )

(14)

Signal x(t) is discretized by sampling interval Ts, and N points are collected to obtain the discrete sequence X  [x(1), x(2),, x(N)] . A Hankel matrix H can be given as follow: k

H   Hi i1

Where H i  R mn is created from xi (t )  ai sin(it  i ) , as Eq. (16) is written:

(15)

 sin( i Ts  ( n  1)   i )  sin( i Ts  1   i )  sin( i Ts  0   i )   T T          sin( 1 ) sin( 2 ) sin( i Ts  ( n )   i )  i s i i s i H i  ai            sin( i Ts  ( m  1)   i ) sin( i Ts  ( m  2)   i )  sin( i Ts  ( N  1)   i ) 

(16)

Where N is the length of signal xi (t ) , m=N-n+1. Similarly, using Ruler's formula, Hi can be rewritten as the sum of two Hankel matrices:  e 0Tsi j e1Tsi j  e ( n 1)Tsi j    1Tsi j e 2Tsi j  e nTsi j  ai  i j  e Hi  e    2j      ( m 1)Tsi j  mTs i j ( N 1) Ts i j   e e e    H i1

(17)

  e  e e   1Tsi j 2 Ts i j  nTs i j  e e ai  i j  e   e    2j       ( m 1)Tsi j   ( N 1) Ts i j   mTs i j  e e e    0 Ts i j

1Ts i j

 ( n 1) Ts i j

Hi 2

By substituting Eq. (17) into Eq. (15), the latter can be transform into Eq. (18): k

H   Hi i 1

a a   [( i ei j H i1  (  i ei j  H i 2 )] 2j 2j i 1 k

(18)

Judging from Eq. (18), matrix H is the direct sum of Hi constructed from the i-th frequency component of x(t), of which the rank is equal to 2 having nothing to do with frequency i , amplitude ai and phase i . What’s more, 1  2  3    k , so the Hankel matrix Hi ,formed by each frequency signal, is linearly independent. Therefore, the rank of the total matrix H, constructed from signal x(t) consisting of k frequency components ,is equal to 2k. 3.3 The influence of noise on singular values Zhao et al [9] studied the difference singular values between useful signals and noise using difference spectrum, and pointed out that the size of singular values generated by noise signals is uniformly distributed, and it will only affect the size of the singular values of useful signals without changing the number. From the above three cases, the following important conclusion can be drawn: if Hankel matrix is constructed from one-dimensional signal containing k frequency components, after processed by SVD ,2k non-zero singular values can be obtained. In short, a frequency component corresponds to two nonzero singular values, called number law of singular values.

4. Proof of the order rule of singular values In section 3, the number law of singular values has been deduced, that is, each frequency component corresponds to two non-zero singular values. However, it is hard to choose the two non-zero singular values for a certain frequency. Fortunately, it will be proved that the order of singular values depends on amplitude of the corresponding frequency component signal. The larger the amplitude of the signal is, the larger the corresponding two singular values will be. Supposing that the amplitude of a certain characteristic frequency ranks the i-th in amplitude spectrum, then the (2i-1)-th and 2i-th and their corresponding singular vectors will be selected to calculate the approximate matrix, i. e.: H i =  2i -1 u 2 i 1v 2Ti 1 +  2 i u 2 i v 2Ti

(19)

Then, the energy of Hi can be expressed as: m

n

2 H i2 =  2i-1  u 2i-2 1, j  v 2i-2 1, j +  2i2 j 1

j 1

m

n

j 1

j 1

m

n

u v j 1

2

2i , j

j 1

2 2i , j

+

(20)

2 2 i 1 2 i  u 2 i 1, j u 2 i , j  v2 i 1, j v2 i , j

By dint of SVD theory, the singular vectors ui and vi are normal orthogonal, so we can get: m  2 u  u k2 , j  1, l  2 i  1,2i   l i 1  m  2 v  v k2 , j  1, l  2i - 1,2i   l i 1   m  u 2i -1 u 2i   u 2 i 1, j u 2 i , j  0  j 1  m v v  v 2 i 1, j v 2 i , j  0  2i -1 2i  j 1

(21)

Substitute Eq. (21) into Eq. (20), and then the energy of Hi can be given as: H i2 =  2i2 -1 +  22i

(22)

From Eq. (22), it can be seen that the larger the energy of Hi is, the larger the corresponding two singular values will be. According to Eq. (16), the energy of Hi is proportional to square of amplitude of the signal (namely ai2). The bigger ai2 is, the greater the energy of Hi will be. This property of singular values is called order rule of singular values. Therefore, by virtue of the amplitude order of certain frequency in the amplitude spectrum of original signal, the corresponding two singular values can be determined.

5. Amplitude filtering characteristics of SVD 5.1 Conception of amplitude filtering characteristics of SVD It can be known from the preceding analysis that there are two important properties of SVD as follows (1)The number law of singular values: one frequency component corresponds to two non-zero singular values; (2) the order rule of singular values: the larger the amplitude of the frequency components, the greater the corresponding two singular values will be. In this paper, these properties of SVD are collectively referred as amplitude filter characteristics of SVD. This filter is characterized by strong noise resistance [9], no phase shift [12], and the ability to extract and separate single frequency components accurately without affecting adjacent signal components. Compared with other filtering methods, SVD does not cause the loss of useful signals while filtering out interference signals like notch filter, nor does it require reference input like adaptive filter. And unlike wavelet transform or other bandpass filters, whose filtering effect depends on properties of the basis function [14]. 5.2 A novel signal separation algorithm based on amplitude filtering characteristics of SVD Based on amplitude filtering characteristics of SVD, a novel signal separation method is proposed, called singular value decomposition amplitude filter (SVD-AF), whose flowchart is shown in Fig. 2. Begin Collect original vibration signal x(t) Remove the dc component of signal x(t),and then create Hankel matrix A by dint of Eq. (4)

Processed to matrix A with SVD ,getting  , m ) the descending SVars ( 1 ,  2， and their SVers ui , vi（ , i  1, 2, , m）

SCs needed are chosen to calculate approximate matrix Ai

Revover component signal Pi from Ai successively with the direct method

ˆ by adding Obtain the denoising signaL x total k required component Pi

End

Fig. 2: Flowchart of SVD-AF.

Its detailed steps are as follows: (1) The given signal x(t) is discretized by sampling interval Ts. (2) Remove direct component (DC) via zero mean processing, discrete signal can be obtained. And then Hankel matrix A is created by dint of Eq. (4).

 ,  m ) and their (3) After A processed by SVD, the descending singular values (SVars) ( 1 ,  2， corresponding vectors (SVers) ui and vi are obtained. (4) According to amplitude order of the feature frequency in amplitude spectrum, the corresponding singular components (SCs, one singular value along with its corresponding left and right singular vectors called one SC) are selected to calculate approximate matrix Ai. (5) The direct method [13] is adopted to recover component signal Pi successively. (6) The total k required components are added together to obtain the denoising signal xˆ . 5.3 Signal analysis examples (1) Simulation signal In order to verify the effectiveness of the SVD-AF algorithm, a simulation signal is given as follow: (23) x(t) 2.2sin(2 30t 0.8) 1.5sin(2 20t 0.5)   x1 (t)

x2 (t)

Signal x(t) is discretized at a sampling frequency of 1024Hz, and Gaussian white noise with signalto-noise ratio (SNR) of 0.23db is superimposed into it. The noise is so strong that the waveform of x(t) cannot be recognized, as shown in Fig. 3(a). The amplitude spectrum of the original signal is demonstrated in Fig. 3(b). It can be seen that the amplitude of the signal fluctuates a little due to the (a) 10

(b) 2.5

Amplitude/mm

Amplitude/mm

noise. 5 0 -5 -10

500

1000

1500

Sampling order i

2000

( 20,1.488)

1.5 1 0.5 0

0

( 30,2.224)

2

0

10

20

30

40

50

Frequency/Hz

Fig. 3: Original noisy signal: (a) Waveform. (b) Amplitude spectrum.

The proposed SVD-AF algorithm is used to extract characteristic components. Firstly, a Hankel matrix A with column n =513 and row m =512 is created from original noisy signal x(t). Secondly, after this matrix is processed by SVD, 512 SCs can be obtained, and the descending singular values are illustrated in Fig. 4. Thirdly, as can be seen from the amplitude spectrum in Fig. 3(b), the amplitudes of the two components of signal x(t) rank the top two . According to the amplitude filtering characteristics of SVD, the first four SCs are selected to calculate approximation matrix:

Ai 



4 i 1

 i ui v iT

. Fourthly, the direct method [13] is adopted to reconstruct component signal Pi (i=1,

2, 3, 4) successively. At last, the total four components are added together, the result is shown in Fig. 5(a), where the dashed line is the ideal signal x(t), it can be seen that the extracted signal agrees well with the ideal one. The amplitude spectrum is shown in Fig. 5(b), one can easily see that the purified component signals are indeed x1(t) and x2(t) of the original signal, and their amplitudes are 1.476 mm and 2.221 mm respectively. However, in the original noisy signal, they are 1.488 mm and 2.224 mm respectively, indicating that the amplitude of these two components have only slight fluctuation. Singular value

1500 1000 500 0

0

10

20

30

Singular value order q

Fig. 4: Singular value curve (b)

4

Amplitude/mm

Amplitude/mm

(a)

2 0 -2 -4

0

0.5

1

1.5

Time/s The purified result The ideal signal x(t)

2

2.5

( 30,2.221)

2 ( 20,1.476)

1.5 1 0.5 0

0

10

20

30

40

50

Frequency/Hz

Fig. 5: The reconstruction result of the first 4 components: (a) Waveform.

(b) Amplitude spectrum.

Next, we may as well see the extraction effect of single frequency component. According to Fig. 3(b), the amplitude of x1(t) ranks the first, so the front two components (P1 and P2) are selected to added together, the result is depicted in Fig. 6(a), where the dashed line is the ideal x1(t), it can be seen that the extracted result agrees well with the ideal x1(t), so it is indeed x1(t) of raw signal x(t) . Similarly, the amplitude of x2(t) ranks the second, so the third and fourth components (P3 and P4) are selected to added together, the result is depicted in Fig. 6(b), where the dashed line is the ideal x2(t), it can be seen that the purified result agrees well with the ideal x2(t), so it is indeed x2(t) of original signal x(t) .

2

Amplitude/mm

Amplitude/mm

4 2 0 -2 -4

0

0.5

1

1.5

2

1 0 -1 -2

0

0.5

Time/s The purified result The ideal signal x1(t)

1

1.5

2

Time/s The purified result The ideal signal x2(t)

Fig. 6: The reconstruction results of single signal component: (1) x1(t).

(2) x2(t).

In conclusion, the SVD-AF algorithm can reliably extract the main components of the signal, filter out random noise. Moreover, the single frequency component can be extracted accurately while ensuring the precision of phase and amplitude. It is proved that this algorithm has characteristic of zero-phase and strong noise resistance. (2) Extraction of real rotor vibration characteristics The test data is derived from the sliding bearing test bed independently developed by our team (Fig. 7), the eddy current sensors are installed on each side of the rotor perpendicular to each other in an inclined direction of 45 degrees. Two experiments were carried out at different speeds (2760 rpm and 3930 rpm), and 2048 points were collected at the sampling frequency of 1024Hz. The signals of test 1 are demonstrated in Fig. 8, from the amplitude spectrum, it can be seen that the main components of both signals are 1X, 2X and 3X, and there are also high-harmonic, power frequencies, and random noise. The signals of test 2 are displayed in Fig.9, from the amplitude spectrum, one can easily see that the main components of both signal are 1X and 2X, and there are higher harmonic components, power frequencies, and random noise as well. Next, we use the proposed SVD-AF algorithm to extract the main components of all the signals. (b)

(a) Displacement sensor

D2

D1

Acquisition system Console

Shaft (c)

Speed sensor

Fig. 7: Rotor test-bed: (a) Installation of Displacement sensor. (b) Acquisition system. (c) Test-bed.

0.1

(b) 0.1

0.05

0.05

D2/mm

D1/mm

(a)

0 -0.05

(c)

-0.05

0

500

1500

0.015

(d)

1X (46,0.01093) 2X (91.5,0.01075) 3X (137.5,0.01292) Power frequencies

3X

0.01

-0.1

2000

Sampling order i 1X 2X

D1/mm

1000

500

1X 2X

1000

1500

2000

Sampling order i 3X 1X (46,0.01159) 2X (91.5,0.01090) 3X (137.5,0.01331) Power frequencies

0.01

0.005

0.005

0

0

0.015

D2/mm

-0.1

0

0

100

200

300

400

0

500

0

100

200

300

400

500

Frequency/Hz

Frequency/Hz

Fig. 8: Original signals in test 1: (a) Waveform of D1. (b) Waveform of D2. (c) Amplitude spectrum of D1. (d) Amplitude spectrum of D2 (b)

0.1 0.05 0 -0.05 -0.1

500

0 1X

2X

0.01 0.005

4X

0

(d)

Power frequencies

100

200

5X

300

-0.1

2000

1X (65.5,0.01706) 2X (131,0.01005)

0.015

0

1500

1000

Sampling order i

0.02

0 -0.05

D2/mm

D1/mm

(c)

0.1 0.05

D2/mm

D1/mm

(a)

6X

400

500

0

0.02

1X

500

2000

1X (65.5,0.01810) 2X (131,0.008656) Power frequencies

0.015 2X

0.01

4X

0.005

7X

1500

1000

Sampling order i

0

0

100

Frequency/Hz

200

5X

300

6X

400

7X 500

Frequency/Hz

Fig. 9: Original signals in test 2: (a) Waveform of D1. (b) Waveform of D2. (c) Amplitude spectrum of D1. (d) Amplitude spectrum of D2.

Test 1: Firstly, each Hankel matrix Ai (i  1, 2) with column n=513 and row m=512 is created from the signal Di (i  1, 2) in Fig. 8. Secondly, after the matrix is processed by SVD, 512 SCs can be obtained, and the descending singular values are illustrated in Fig. 10. Thirdly, as can be seen from Fig. 8(c) and 8(d), the amplitude of 1X, 2X, and 3X of Di (i  1, 2) rank the top three. According to the amplitude filtering characteristics of SVD, the front six SCs are selected to calculate approximation matrix: Ai* 



6 i 1

 i ui v iT

.Fourthly, the direct method [13] is adopted to reconstruct component signal Pi (i=1,

2,… , 6) successively. At last, the total six components are added together, the results are shown in Fig. 11.

(a)

(b)

20

20

D2

15

D1

25

10

15 10

5 0

5 0

20

40

60

80

100

0

0

20

Singular value order q

40

60

80

100

Singular value order q

Fig. 10: The singular value curves (test 1): (a) of D1. (b) of D2. (a)

(b)

0.04

D1/mm

D1/mm

0.02 0

(c)

500

1000

1500

Sampling order i 0.06

1X (46,0.01073) 2X (91.5,0.01068) 3X (137.5,0.01287)

0.01

0.02 0

0

2000

(d)

D2/mm

0.04

D2/mm

3X 1X 2X

0.005

-0.02 -0.04 0

0.015

0

100

200

300

400

500

Frequency/Hz

0.015

1X

3X 2X

0.01

1X (46,0.01268) 2X (91.5,0.01076) 3X (137.5,0.01339)

0.005

-0.02 -0.04 0

0 500

1000

1500

2000

0

100

200

300

400

500

Frequency/Hz

Sampling order i

Fig. 11: Purified results in test 1: (a) Waveform of D1. (b) Amplitude spectrum of D1. (c) Waveform of D2. (d) Amplitude spectrum of D2

From the spectrum in Fig.11, we can see that both spectrums are very clean, and the transition band between different frequencies is almost completely eliminated. There are no redundant frequency components like higher harmonic components, power frequencies, or random noise. The amplitudes of feature frequencies in Fig. 11 are listed in table 1. From table 1, we can see that there is only a small difference between them, which is caused by the impact of noise and is inevitable. It can be said that the feature components are almost completely extracted using the proposed algorithm. Table 1: Comparison of amplitudes between original and purified signal in test 1.

Frequency(Hz) Raw Signal(mm) Purified Signal(mm)

D1

D2

1X

2X

3X

1X

2X

3X

0.01093 0.01073

0.01075 0.01068

0.01292 0.01287

0.01159 0.01268

0.01090 0.01076

0.01331 0.01339

Test 2: Firstly, each Hankel matrix Ai (i  1, 2) with column n=513 and row m=512 is created from the signal Di (i  1, 2) in Fig. 9. Secondly, after the matrix is processed by SVD, 512 SCs can be obtained, and the descending singular values are illustrated in Fig. 12. Thirdly, as can be seen from Fig. 9(c) and

9(d), the amplitude of 1X and 2X rank the top two. According to the amplitude filtering characteristics of SVD, the front four SCs are selected to compute approximation matrix: Ai*





4 i 1

 i ui v iT

.Fourthly,

the direct method [13] is adopted to recreate component signal Pi (i=1, 2,3, 4) successively. At last, the total four components are added together, the results are shown in Fig. 13. (a)

(b)

20

25 20

D2

D1

15 10

15 10

5 0

5 0

10

20

30

40

0

50

Singular value order q

30

20

10

0

40

Singular value order q

50

Fig. 12: The singular value curves (test 2): (a) of D1. (b) of D2. (a)

(b)

0.04

0.02

0 -0.02 -0.04

(c)

0

500

1000

1500

0

2000

(d)

0.04

D2/mm

0.02

D2/mm

2X (131,0.0102)

0.01 0.005

Sampling order i

0 -0.02

0

100

200

300

400

500

Frequency/Hz 0.02

1X (65.5,0.01755)

0.015 2X (131,0.009339)

0.01 0.005

-0.04 -0.06

1X (65.5,0.01709)

0.015

D1/mm

D1/mm

0.02

0

500

1000

1500

2000

0

0

Sampling order i

100

200

300

400

500

Sampling order i

Fig. 13: Purified results (test 2): (a) Waveform of D1. (b) Amplitude spectrum of D1. (c) Waveform of D2. (d) Amplitude spectrum of D2

From the spectrum in Fig.13, one can see that both spectrums are very clean, and the transition band between different frequencies is almost completely eliminated. There are no other frequency components like higher harmonic components, power frequencies, or random noise. The amplitudes of feature frequencies in Fig. 13 are listed in table 2. From table 2, it can be easily seen that there is only a small fluctuation between them, which is caused by random noise. It can be said that all the feature frequencies are almost completely extracted using the SVD-AF algorithm. Table 2: Comparison of amplitudes between original and purified signal in test 2.

Frequency(Hz) Raw Signal(mm) Purified Signal(mm)

D1

D2

1X

2X

1X

2X

0.01706 0.01709

0.01005 0.01020

0.01810 0.01755

0.008656 0.009339

6. Fault diagnosis of the rotor based on axis orbit Researches show that there is closely connection between the shape of axis orbit and fault type of rotating machinery [15-22], for example, external "8" shape or banana shape corresponds to misalignment fault, “petal” shape corresponds to rub impact fault. Next, the two groups of tested signals in section 5.3 (figure 8 and figure 9) are analyzed. In order to obtain clear axis trajectory, the proposed SVD-AF algorithm is used to purify these axis trajectories. According to the spectrum characteristics of each signals, axis orbits are synthesized by the leading frequency components whose amplitudes are higher than others. In the following, we will specifically analyze the purification process of the axis trajectory of the two tests. Test 1: The axis trajectory synthesized directly by the two signals (D1 and D2) in Fig. 8 are shown in Fig. 14. As can be seen from Fig. 14, the axis orbit is in disorder, so it is impossible to distinguish the fault type of the rotor. In fact, only by extracting the main components of the signal (such as 1X, 2X, 3X) can a clear axis trajectory be obtained. 0.1

D2/mm

0.05 0 -0.05 -0.1 -0.1

-0.05

0

0.05

0.1

D1/mm Fig. 14: The original axis orbit synthesized

by D1 and D2 in Fig. 8.

Then, it is necessary to purify the axis orbit in Fig. 14. Of which the essence is to filter out random noise and other redundant frequencies like power frequencies (50 Hz and 150 Hz) as well as higher harmonic, such as 4X, 5X, 6X etc. In this paper, the purification of signals has been completed in section 5.3 (as shown in Fig. 11).Therefore, we directly use the purified signals in Fig. 11 to synthesize axis orbit, and the result is depicted in Fig.15, it is petal-shaped, indicating that the rotor suffers from rub impact fault.

0.06

D2/mm

0.04 0.02 0 -0.02 -0.04 -0.04

-0.02

0

0.02

0.04

D1/mm Fig. 15: The purified axis orbit synthesized

by D1 and D2 in Fig. 10.

Test 2: The axis orbit manipulated directly by D1 and D2 (Fig. 9) are shown in Fig. 16. We can see from Fig. 16 that the axis orbit is messy, so it is difficult to recognize the fault type of the rotor. In fact, only by extracting the main components of the signal (such as 1X, 2X) can a clear axis orbit be obtained. 0.1

D2/mm

0.05 0 -0.05 -0.1 -0.1

-0.05

0

0.05

0.1

D1/mm Fig. 16: The original axis orbit synthesized

by D1 and D2 in Fig. 9.

Then, the proposed SVD-AF is adopted to purify the axis orbit in Fig. 16. Random noise and other redundant frequencies like power frequencies (50 Hz and 150 Hz) as well as higher harmonic (such as 4X, 5X, 6X etc.) need to be filtered out. The purification of signals has been completed in section 5.3 (as depicted in Fig. 13).Therefore, axis orbit is synthesized by the purified signals, and the result is depicted in Fig.17, it is banana-shaped, implying that the rotor is subject to misalignment fault. Which is caused by leakage of lubricating oil between the rotor and the sliding bearing in the process of rising speed. As the rotor speed continues to rise, the rotor center deviates from that of the bearing.

0.04

D2/mm

0.02 0 -0.02 -0.04 -0.06 -0.04

-0.02

0

0.02

0.04

D1/mm Fig. 17: The purified axis orbit synthesized

by D1 and D2 in Fig. 13.

7. Conclusion In this paper, the signal processing principle of Hankel matrix-based SVD is studied, both the number law and the order rule of singular values are deduced theoretically. The two properties are collectively referred as amplitude filtering characteristics of SVD, and a signal separation algorithm (called SVD-AF) based on this characteristic is proposed. The conclusion can be drawn as follows: (1) The number law of singular values is deduced theoretically. The research shows that: the Hankel matrix is created from one-dimensional signal with k frequencies, after processed by SVD, and then 2k singular values are generated constantly. Moreover, the number of them is not affected by amplitude, phase and random noise. (2) The order rule of singular values, that is, the correspondence between order of singular values and signal amplitude, is studied. It is pointed out that the larger the amplitude of the frequency component is, the larger the corresponding two singular values will be. By dint of this property, the order of singular values can be determined. (3) According to the number law as well as order rule of singular values, it can be seen that SVD is essentially a kind of amplitude filter. In this paper, these two properties are collectively referred as singular value amplitude filtering characteristics. Based on this characteristic, a signal separation algorithm named SVD-AF is proposed, and the effectiveness of the algorithm is verified by the analysis of simulation and several real rotor vibration signals. (4) The SVD-AF algorithm is applied to purify axis orbits of the rotor of large sliding bearing test bed. Both the misalignment and rub impact fault of the rotor are identified successfully. Of course, it is also suitable for fault diagnosis of other rotating machinery. In this research, two important properties of SVD are proved, and then a separation method SVDAF is proposed, what’s more, it is effective in fault diagnosis of rotor. In the future, this method will

be applied to the vibration signal analysis of other types of rotating machinery, such as rolling bearing, gear box etc.

Data Availability The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest The authors declare that they have no conflicts of interest.

Acknowledgments The support from National Natural Science Foundation of China (NSFC, Grant No. 51875205 and 51875216), the Natural Science Foundation of Guangdong Province (Grant No. 2018A030310017 and 2019A1515011780), as well as the Science and Technology Plan Projects of Guangzhou from China (Grant No.201904010133) for this research is gratefully acknowledged.

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Highlights   

Find one frequency corresponds to two singular values The number law of singular values and order rule of SVD are deduced theoretically A signal separation algorithm based on SVD amplitude filter is proposed

The authors declare that they have no conflicts of interest.