Operational transfer path analysis with crosstalk cancellation using independent component analysis

Journal Pre-proof Operational transfer path analysis with crosstalk cancellation using independent component analysis Wei Cheng, Yapeng Chu, Xuefeng Chen, Guanghui Zhou, Diane Blamaud, Jingbai Lu PII:

S0022-460X(20)30055-9

DOI:

https://doi.org/10.1016/j.jsv.2020.115224

Reference:

YJSVI 115224

To appear in:

Journal of Sound and Vibration

Received Date: 11 June 2019 Revised Date:

27 January 2020

Accepted Date: 28 January 2020

Please cite this article as: W. Cheng, Y. Chu, X. Chen, G. Zhou, D. Blamaud, J. Lu, Operational transfer path analysis with crosstalk cancellation using independent component analysis, Journal of Sound and Vibration (2020), doi: https://doi.org/10.1016/j.jsv.2020.115224. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier Ltd.

Operational Transfer Path Analysis With Crosstalk Cancellation Using Independent Component Analysis Wei Cheng1*, Yapeng Chu1, Xuefeng Chen1, Guanghui Zhou1, Diane Blamaud1,2, Jingbai Lu1 1. State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an, 710049, Shaanxi Province, PR China 2. Arts et Métiers Paris Tech, 75000, Paris, France

ABSTRACT To eliminate crosstalk effects between reference signals of operational transfer path analysis (OTPA), a novel crosstalk cancellation method based on independent component analysis (ICA) for OTPA is proposed. Firstly, ICA is used to separate the measured signals at reference points and crosstalk between sources is significantly eliminated. Then, separated signals are identified based on the prior knowledge of the sources. In the next phase, amplitudes and phases of the separated signals are corrected and finally, the transmissibility function matrix is obtained based on the corrected separated signals and the measured signals at the target point. The performance of the proposed method is comparatively studied with the conventional methods, according to numerical case studies for an acoustical radiation system and experimental case studies on a test bed with thin shell structures. Generally, the proposed method can increase the accuracy of transfer path identification and contribution evaluation, which can benefit vibration monitoring, reduction and control of mechanical systems. Keywords: Crosstalk Cancellation, Operational Transfer Path Analysis, Independent Component Analysis, Transfer Path Identification, Path Contribution Evaluation.

1

1 INTRODUCTION In the procedure of design and development of cars and underwater vehicles, vibration and noise level inside is one of the significant testing index for performance evaluation [1, 2]. Therefore, vibration and noise transfer path identification and contribution evaluation methods [3-6] are developed to provide important evidences for vibration and noise monitoring and control. Operational transfer path analysis (OTPA) is an advanced vibration and noise transfer path identification and contribution evaluation method [7, 8]. However, the accuracy of OTPA is still unsatisfied in engineering applications and is seriously influenced by the crosstalk problem, which is the consequence of reference point signals measured at operating conditions. Because all of the sources are working synchronously and all response signals are also measured synchronously, it causes crosstalk components in measured signals. Therefore, it is of great significance to cancel crosstalk to increase the accuracy and practicability of OTPA. Operational transfer path analysis is becoming a research hotspot due to simple and fast modeling in engineering applications. De Sitter [5] applied OTPA to study the NVH problems of the car, and the results showed that no disassembling was required and operational forces also don’t have to be eliminated. Robert [9] applied OTPA to study the vibration and noise problems of a high-speed train bogie, and it was found that OTPA was a faster and often cheaper way than traditional TPA. By estimating the transmissibility functions based on operating data, OTPA significantly improves the efficiency of the conventional TPA. However, the high efficiency of OTPA also causes insufficient accuracy in engineering applications, especially for mechanical systems with complex structures. To improve the accuracy of OTPA, Diez-Ibarbia and Battarra [10] comparatively studied OTPA with the conventional TPA on an electric vehicle, 2

and also studied some factors influencing the accuracy of OTPA. Gajdatsy and Janssens [11] pointed out that the errors of transmissibility function estimation, crosstalk between sources, and effects of neglected paths were major problems of OTPA, especially the crosstalk problem. Tcherniak and Schuhmacher [12] numerically conducted experiments on a simple model system, and one of the conclusions was that the crosstalk problem seriously influenced the transmissibility function estimation accuracy of OTPA. Generally, OTPA is a high efficiency and simple method which is suitable for engineering applications. However, the accuracy is seriously influenced by some factors, especially the crosstalk problem. To improve the accuracy of OTPA, many modifications in OTPA methods are proposed. Yoshida and Tanaka [13] proposed a modified OTPA method by using mode contributions instead of path contributions, and thus the crosstalk problem was improved. Wu [14] combined Operational-X Transfer Path Analysis (OPAX) with OTPA, which can solve the crosstalk and path neglecting problems of OTPA by using the loads rather than response signals at reference points. Roozen [15] replaced operational excitations with hammer excitations, and thus vibrations of wide frequency range were excited and more accurate transmissibility functions were obtained. Furthermore, the proposed method by Roozen [16] was applied to transfer path analysis of a gearbox, and dominating transmission paths were successfully identified. Cheng [17-18] applied Tikhonov regularization, wavelet packet denoising and Welch method to increase the accuracy of estimation of transmissibility functions of OTPA. Both numerical studies and experimental studies showed that ill-conditioned degrees were significantly decreased. Even some methods have proved their significance to improve the accuracy of OTPA, the crosstalk problem is still a challenge for engineering applications of OTPA, especially for mechanical systems with complex 3

structures. For mechanical systems, the crosstalk problem is caused by the transmission and mixing of all the vibration sources rather than the corresponding one. Putner [19] pointed out that the reference points should be closer to the excitation points to reduce the effect of crosstalk, which was being widely used in OTPA applications. Mihkel [20] applied singular value decomposition (SVD) and principal component analysis (PCA) to solve the crosstalk problem of OTPA, which cancelled crosstalk by cutting off small singular values or principal components. Cao [21] proposed a crosstalk cancellation method which improved the accuracy of OTPA by extra tests of transfer functions. Currently, most crosstalk cancellation methods are effective only when the crosstalk components are weak or can be regarded as noises, but lose accuracy in strong crosstalk conditions. Furthermore, extra measurements are also time-consuming. Independent Component Analysis (ICA) aims to recover source signals just by measured mixed signals and the goal of ICA is to maximize the independence of each signal [22-29]. Howard [30] applied ICA to cancel crosstalk in the surface electromyography signals, and successfully recovered the actual EMG signals for specific muscle groups. Li [31] applied ICA to mitigating the crosstalk in mode group diversity multiplexing (MGDM) system, and ICA successfully implemented MGDM de-multiplexing. In this paper, a novel crosstalk cancellation method based on ICA for OTPA is proposed. Firstly, ICA is used to separate the measured signals and the crosstalk between the separated signals is significantly reduced. Secondly, the separated signals are identified corresponding to every exciting sources based on prior knowledge. In the next phase, the amplitude and phase of every separated signal are corrected based on the measured signals at reference points. Finally, the transmissibility function matrix is obtained by corrected separated signals and the measured signals at the target points. 4

By using the separated signals instead of the measured signals at the reference points, crosstalk problem can be effectively solved and more accurate transmissibility function matrix can be obtained without losing efficiency and any extra measurements, and thus the accuracy of OTPA can be greatly improved without losing efficiency. The remainder of this paper is organized as follows: in Section 2, fundamental theories of ICA-based crosstalk cancellation method for OTPA are introduced. In Section 3, numerical case studies on source contribution evaluation for spherically radiating acoustical sources are provided, and the performances of the proposed crosstalk cancelled OTPA model are comparatively studied with that of the conventional OTPA model. In Section 4, a test bed with shell structures is constructed and the performances of the proposed crosstalk cancelled OTPA model are also comparatively studied with that of the conventional OTPA model. In Section 5, the conclusions are summarized. Generally, this study provides a more efficient and accurate solution for crosstalk problem of OTPA, and can benefit vibration and noise monitoring, reduction and control for mechanical systems. 2 THEORIES OF ICA-BASED CROSSTALK CANCELLED OTPA 2.1 Basic Model of OTPA The OTPA model of a linear system [5] can be described as shown in Equation (1) and (2): Y = XTT

 Y1 ( f )   X11 ( f )     M   M  Y r ( f )  =  X1r ( f )     M   M Y g ( f )   X g ( f )    1

(1)

L O

X1i ( f ) L M O

L

Xir ( f ) L

O L

M g i

O

X (f) L

X1n ( f )   T1 ( f )    M  M  Xrn ( f )   Ti ( f )    M  M  X ng ( f )  Tn ( f ) 

(2)

where TT is the transposed matrix of the transmissibility function matrix T, X is the 5

reference point signal matrix, Y is the target point signal matrix, n is the number of the reference points, and g is the number of the experimental operating conditions. OTPA estimates the transmissibility function matrix according to response signals under operating conditions, and it should be noted that [32]: (1) The crosstalk exists significantly in the measured signals at the reference points. Therefore, the reference points should be carefully chosen to reduce the effects of the crosstalk between sources and usually should be close to the exciting points. (2) Operating conditions must be independent from each other to reduce the ill-condition degrees of the OTPA model, usually generated by changing the rotating speeds or loads of the equipment. 2.2 Theory of The Conventional OTPA The conventional OTPA employs truncated singular value decomposition (TSVD) to solve the least square problem of estimating transmissibility function matrix T:

(

min XTT − Y

)

(3)

The key process of the conventional OTPA is as follows: (1) Singular value decomposition of X X g ×n = U g ×g Σ g ×nVnT×n

(4)

where U g×g is an unitary matrix of g rows and g columns, Vn×n is an unitary matrix of n rows and n columns, and Σ g ×n is a singular value matrix. (2) Truncated singular value decomposition of X

O σ1  O     Σ g ×n =  σk   O    O σ n  g ×n 6

(5)

 The contribution rate (CR) of each singular value is CR =  σ k 



n

∑ σ  ×100% i

i=1

(1 ≤ k ≤ n) , σ k is the kth singular value in a descending order, k is the truncation coefficient, and τ is the threshold of the CR set. As CR ≥ τ , k singular values are retained. Furthermore, the residual of X and the ill-conditioned degrees of Equation (3) can be described as follows: σ k +1 = X k − X cond ( X k ) =

(6)

σ1 σk

(7)

(3) The transmissibility function matrix

Tk T = X k+ Y

(8)

where X+k is the pseudo inverse of X k .

 Y1   Tk1  1  L 0    M σ   M k uT Y    1  j T  Tki  = Vn×k  M O M vj  U g ×k  Y r  = ∑ σ   j =1 j      M  M  0 L 1  Y g  Tkn   σ k    k×k

(9)

(4) The source contribution evaluation Assuming that the acceleration signals at the reference points and the target points under

practical

operating

conditions

are

%= [ % X x1 ,K , % xi , K , % xn ]

and

T %=  y%,K , y% ,K , % Y y g  . Thus the source contribution can be denoted as Equation (10), j  1

Equation (11), and Equation (12):

% TT y%k = X k k

7

(10)

x1   Tk1  %  M   M    y%k = % xi   Tki      M   M % x n  Tkn 

T

y%ki = % xi Tki

(11)

(12)

where y%ki is the source contribution of source i . 2.3 Fundamental Theory of ICA The linear instantaneous mixing model of ICA is denoted as [33]: X = A×S

(13)

where S is the source signal vector, X is the measured mixed signal vector, and A is the mixing matrix. Generally, the number of source signals is assumed to be no more than the number of the mixed signals. Furthermore, in ICA model the source signals should be independent, and the final goal of ICA is to recover the unknown source signals based on the mixed signals according to the following relation.

Y = BX

(14)

where Y is the separated signal vector expected as the estimation of the source signals, and B is the separating matrix. In this study, the natural gradient algorithm is employed to update the separating matrix B according to the following equation.

B(k + 1) = B(k ) + µk [I − ψ ( y (k ))yT (k )]B(k )

(15)

where B ( k ) is the separating matrix, k is the recursion sequence number, µk is the step size, y(k ) is the approximate signal, and ψ ( y ( k )) is a nonlinear singular function, such as tanh( y (k )), y 3 ( k ) . The key steps of the ICA, based on natural gradient are as follows: 8

(1) Zero-mean process of the measured mixed signals X. (2) Whitening −

1 2

Z = Λ UT X0

(16)

where Z is the whitening matrix, X 0 is a matrix of X after removing the mean value, Λ is the covariance matrix of X 0 , and U is the eigenvector matrix of X 0 . (3) Separating matrix optimization Randomly generating initial separating matrix B(0) , and iterating it until B(k + 1) is very close to B ( k ) . (4) Recover source signals

Y = BZ

(17)

2.4 ICA-based Crosstalk Cancellation Method for OTPA

For OTPA, the measured signals at reference points are expected to be the response signals of the corresponding exciting forces. However, the signals at reference points are measured under operating conditions and thus all the exciting forces work concurrently. Therefore, the measured signals at reference points are mixed signals actually, contributed by all the sources. F1

X1 H1a

X1a

H1b

F1 X1

X1b

X 2a

F2 X 2

H 2a F2 X 2

H 2b

X 2b

Figure 1: The mechanism of crosstalk 9

X a = X1a + X2 a

Xb = X1b + X 2b

The mechanism of crosstalk is shown in Figure 1. H 1a , H1b , H 2a , H 2b are transfer functions between the exciting points and the reference points, and X 1 and X 2 are the response signals of the exciting forces F1 and F2 at the exciting points

respectively. As the exciting force F1 works at the passive part of the system independently, the response signal at the corresponding reference point a is X1a = H1a X1 , while the response signal at the reference point b is X1b = H1b X1 . As the exciting force F2 works at the passive part of the system independently, the response signal at the corresponding reference point b is X 2 b = H 2 b X 2 , while the response signal at the reference point a is X 2 a = H 2 a X 2 . As the two exciting forces work concurrently, the response signals at reference points are described as Equation (18) and Equation (19), or Equation (20). X a = X1a + X 2 a = H1a X1 + H 2 a X 2

(18)

X b = X1b + X 2 b = H1b X1 + H 2 b X 2

(19)

 X a   H1a  X  = H  b   1b

H 2 a   X1  H 2b   X 2 

(20)

where X a is the measured signal at the reference point a from the exciting force F1 , and Xb is the measured signal at the reference point b from the exciting forces F2 . As the two exciting forces work concurrently in operating conditions, the measured signals at the reference point not only depend on the corresponding exciting force, but also disturbed by other exciting forces and thus the crosstalk problem occurs. With the effects of crosstalk, the measured signal at the reference point is not been able to represent the features of the exciting force accurately, leading to an inaccurate transmissibility function matrix. 10

As the reference points are chosen close to the exciting points, normally the response signals X1a and X 2b are supposed to be approximately equal to X 1 and X 2 , respectively. Therefore, solving crosstalk problem is equivalent to recovering

source signals such as X 1 and X 2 from mixed signals X a and Xb , whose mechanism is similar with the model of ICA. Generally, the exciting sources are assumed to work independently and thus the source signals tend to be independent. Therefore, it is reasonable to apply ICA to crosstalk cancellation of OTPA. However, three inherent properties of ICA make crosstalk cancellation very difficult: sequencing indeterminacy, amplitude indeterminacy, and phase indeterminacy of the separated signals. Sequencing indeterminacy means that the separated signals are not signals with labels, and thus the separated signals cannot be corresponded to the reference points or sources automatically. Amplitude indeterminacy means that the amplitudes of the separated signals are amplified or minified with a proportionality coefficient in full frequency band, comparing to the corresponding source signals. Phase indeterminacy means that the waveforms of the separated signals might be the reversal of source signals, and thus the phase differences between the separated signals and corresponding source signals might be 0° or 180°. Therefore, to recover accurate signals at reference points, correction is necessary for the separated signals. 1) Sequencing correction With prior knowledge, such as spectrum features or frequency ranges of the sources, the separated signals can be corresponded to the reference points, and thus the problem of sequencing indeterminacy can be solved. 2) Amplitude correction The measured signals at the reference points can be described as shown in Figure 2. The blue line represents the components from the source signal at the reference point, 11

and the red lines represents the crosstalk components from other reference points. For the frequency band between ω1 and ω2 , the components of the source signal are not affected by the crosstalk components, and thus the signal at the reference point is supposed to be the source signal in this frequency band. Source signal Crosstalk components

O

ω3

ω1

ω2

ω4

f /Hz

Figure 2: The spectrum structure of the measured signals

x j (ω ) = s j (ω ) (ω1 < ω < ω2 )

(21)

where x j (ω ) is the measured signal at the reference point j , and s j (ω ) is the signal of the source j . Due to amplitude indeterminacy, the amplitudes of the separated signals are amplified or minified with a proportionality coefficient in full frequency band and the proportional relation is given as follows:

α j ⋅ y j (ω ) = s j (ω ) (ω3 ≤ ω ≤ ω4 )

(22)

where y j (ω ) is the separated signal corresponding to the source j , and α j is the proportionality coefficient. According to Equation (21) and Equation (22), the relationship between the separated signals and the measured signals at the reference points is as follows:

α j ⋅ y j (ω ) = x j (ω ) (ω1 < ω < ω2 ) Then amplitude correction process can be described as follows: (1) Choosing a frequency ω0 in the frequency band of ω1 < ω < ω2 . 12

(23)

(2) Calculating the proportionality coefficient:

αj =

| x j (ω0 ) | | y j (ω0 ) |

(ω1 < ω < ω2 )

(24)

(3) Correcting the amplitude:

y jcorrected (ω ) = α j ⋅ y j (ω ) (ω3 < ω < ω4 )

(25)

where y jcorrected (ω ) is the corrected signal separated by ICA corresponding to the source j . 3) Phase correction Due to phase indeterminacy, the phase difference between the separated signals and the source signals might be 0° or 180°. Similar to the model of amplitude correction, phase correction can be carried out as follows: (1) Choosing a frequency ω0 in the frequency band of ω1 < ω < ω2 . (2) Comparing the phase of separated signal with the phase of source signal at frequency ω0 . (3) If the phase difference is 0°, then phase correction is not required. Otherwise, the separated signals in time domain should be multiplied with -1. Generally, the key process of the ICA-based crosstalk cancellation method for OTP can be described as follows: (1) Processing the measured signals at the reference points by ICA, and then obtaining the separated signals. (2) Sequence correction of the separated signals. (3) Amplitude correction of the separated signals. (4) Phase correction of the separated signals. Without extra measurements, ICA-based crosstalk cancellation method has a high efficiency. As the corrected signals are obtained, the measured signals at reference 13

points can be replaced for OTPA, and thus the crosstalk between sources can be effectively removed. The ICA-based crosstalk cancelled OTPA model can be described as shown in equation (26) and (27):

Y = CTT  Y1 ( f )   C11 ( f )     M   M  Y r ( f )  =  C1r ( f )     M   M  g   g  Y ( f )  C1 ( f )

(26)

L O

C1i ( f ) L M O

L

Cir ( f ) L

O L

M

O

g i

C (f) L

C1n ( f )   T1 ( f )    M  M  Cnr ( f )   Ti ( f )    M  M  Cng ( f )  Tn ( f ) 

(27)

where T is the transmissibility function matrix, C is the corrected signal matrix,

Y is the target point signal matrix, n is the number of reference points, and g is the number of experimental operating conditions.

3 NUMERICAL CASE STUDIES 3.1 Introductions of Spherically Radiating Acoustical Sources As shown in Figure 3, there are two spherically radiating acoustical sources and a target point. The parameters are shown as follows: the radius of the sources is

a=b=0.34 m and vibration velocities on the surface of the sources are va = vb = 60 m / s . Generally, the reference points should be close to the sources and

every source should have only one reference point. However, considering engineering applications, the reference points sometimes cannot be too close to the sources. Thus, the distances between the sources and the corresponding reference points are rAa = rBb = 0.1 m and the distance between the sources and the target point is rA = rB = 10 m , and the distance between source A and source B is rAB = 1 m .

14

Source A

rAa

a Pa = PA + PB

Source A rA Target point PT = PA + PB

Crosstalk

rAB rB rBb

Source B

b Pb = PA + PB

Source B

Figure 3: Locations of sources, reference points and a target point In free field, sound pressure P(r, t ) [34] can be expressed as:

P(r , t ) =

Amp =

Amp r

ei(ωt − kr )

ρ0 c0 kr02 1 + (kr0 )2

(kr0 + i)va

(28)

(29)

where r is the distance between the source and a certain point in space, r0 is the radius of the source, ω is the frequency of the source, k = ω / c = 2π / λ is the wave number, ρ 0 is the density of the medium (air, ρ 0 ≈ 1.29 kg/m 3 ), c 0 is the sound velocity (air, c = 340 m/ s ) and va is the vibration velocity on the surface of source A. The sound pressure of the reference points and the target point is the sum of contributions of source A and source B:

A  A P (r , t ) =  ei(ω1t − k1r1 − a ) + ei(ω2t − k2r2 −b )  + noise(t ) r2  r1 

(30)

where noise(t ) is a random noise, generated by Matlab with random number function “rand ()”. The characteristic pressure frequencies of source A and source B are 75 Hz 15

and 250 Hz respectively, which are used to describe the crosstalk phenomenon and to test the performance of the proposed crosstalk cancellation method. In this study, five experimental operating conditions were generated and the measured signals at reference points in experimental operating condition 1 are shown in Figure 4. 5000

8000 Reference point a

Reference point b 4340

250Hz

2500 75Hz

4000

crosstalk

0

0

100

200

300

400

0

500

0

100

Frequency f/Hz 8000 6000 4000 2000 0 -2000 -4000

250Hz

75Hz crosstalk

1338

200 300 Frequency f/Hz

400

500

10000 Reference point a

Reference point b 5000 0 -5000

0

0.05

0.1 Time t/s

0.15

0.2

0

(a) The signal of reference point a

0.05

0.1 Time t/s

0.15

0.2

(b) The signal of reference point b

Figure 4: The measured signals of the reference points The waveforms and spectrums of the measured signals at the reference points a and b are shown in Figure 4 (a) and Figure 4 (b) respectively. As stated before, the characteristic frequencies of the signals at reference points a and b are 75 Hz and 250 Hz respectively. However, it can be clearly seen that there are both 75 Hz and 250 Hz components in the measured signals at reference points a and b, which is caused by the effects of source crosstalk as the two sources are working at the same time. 3.2 The OTPA Model of Spherically Radiating Acoustical Sources The OTPA model of the simulation system can be described as:

16

[T11

T12 ]

T

 x11 x12     M M =  x1r x 2r     M M  5 5  x1 x 2 

+

 y11     M y r   1  M  5  y1 

(31)

where r is the label of operating conditions, y1r (r = 1, 2,K ,5) is the measured sound pressure signal of the target point, x1r ( r = 1, 2, K , 5 ) is the measured sound pressure signal of reference point a, x r2 (r = 1, 2,K ,5) is the measured sound pressure signal of reference point b, T11 is the transmissibility function between the reference point a and the target point, and T12 is the transmissibility function between the reference point b and the target point. With prior knowledge, it is known that source A has only 75 Hz component and source B has only 250 Hz component, the waveforms and spectrums of the separated signals in the experimental operating condition 1 by ICA are shown in Figure 5. 0.5

0.5 Separated signal a

Separated signal b

0.2273

0.25

0.21775

0.25

75Hz 0

0

100

0.4

250Hz 200 300 Frequency f/Hz

400

0

500

0.2

0.2

0

0

-0.2

-0.2 0.05

0.1 Time t/s

0.15

100

0.4

Separated signal a

0

0

0.2

200 300 Frequency f/Hz

400

500

Separated signal b

0

0.05

0.1 Time t/s

0.15

0.2

(a) Separated signal of reference point a (b) Separated signal of reference point b Figure 5: Separated signals of the reference points As shown in Figure 5, the separated signals of reference points a and b only have

17

the characteristic frequencies of 75 Hz and 250 Hz in the corresponding spectrums. Furthermore, the waveforms are also more clear and accurate than that of before ICA processing in Figure 4, which means that the crosstalk between sources is significantly cancelled. Comparing amplitudes and phases of the separated signals with that of the measured signals at the reference points, the proportionality coefficients are α1a = 5886 (1338 / 0.2273 , (all the coefficients are calculated in the same way in the

following paragraph) and α1b = 19931 , and the phase difference is 180° and 0° respectively. With corrected coefficients, the spectrums of the corrected signals are shown in Figure 6. 6500 Separated signal a

5000 2500 75Hz 1338 0

0

100

200 300 Frequency f/Hz

400

500

6500 Separated signal b 4340

5000

250Hz

2500 0

0

100

200 300 Frequency f/Hz

400

500

Figure 6: The spectrums of the corrected signals Comparing Figure 6 with Figure 4, the crosstalk components are significantly cancelled while the characteristic components are well reserved. The characteristic frequency amplitudes of separated signals are equal to that of the signals at reference points, which means that the crosstalk problem can be effectively solved by the proposed OTPA model. With the same processing methods, all the signals at the reference points with 5 different operating conditions and 1 practical operating condition are separated by ICA 18

and further corrected by the proposed method. The ICA-based crosstalk cancelled OTPA model can be described as follows.

[T11′ T12′ ]

T

 c11  M = c1r  M c 5  1

c12   M c 2r   M c52 

+

 y11     M  y1r     M y5   1

(32)

where r is the label of operating conditions, y1r ( r = 1, 2, K ,5) is the measured sound pressure signal of the target point, c1r ( r = 1, 2, K ,5 ) is the corrected sound pressure

signal corresponding to source A, c r2 ( r = 1, 2, K ,5) is the corrected sound pressure signal corresponding to source B, T11′ is the transmissibility function between reference point a and the target point, and T12′ is the transmissibility function between

Condition number

reference point b and the target point.

Figure 7: The condition numbers of the coefficient matrix The condition numbers of the coefficient matrix for the proposed OTPA and the conventional OTPA are shown in Figure 7. The condition numbers of the coefficient matrix for the proposed OTPA are much smaller than that for the conventional OTPA, and the order of magnitudes is reduced from 1012 to 104 , which means that the ill-condition degrees are significantly decreased by cancelling the crosstalk 19

components and the accuracy of the transmissibility functions can be effectively improved. 3.3 Comparisons of Contribution Evaluation Using the truncated singular value decomposition method with the contribution rate (CR) for the singular values of 3%, the transmissibility functions between the reference points and the target point calculated by the proposed OTPA and the conventional OTPA are shown in Figure 8. It can be seen that the transmissibility functions changes significantly only at the frequency of 75 Hz and 250 Hz, while remained the same at other frequencies, which means that the noise effects have been significantly

(a) Reference point a

Transmissibility function (/)

Transmissibility function (/)

eliminated.

the target point (b) Reference point b

Sound pressure P/Pa

Sound pressure P/Pa

Sound pressure P/Pa

Figure 8: Transmissibility functions

20

the target point

(a) The conventional OTPA (b) The proposed OTPA (c) The theoretical study Figure 9: The source contribution of source A Based on the OTPA model and theoretical study, the source contributions obtained by the conventional OTPA, the proposed OTPA and the theoretical study are displayed

Sound pressure P/Pa

Sound pressure P/Pa

Sound pressure P/Pa

in Figure 9 and Figure 10.

(a) The conventional OTPA (b) The proposed OTPA (c) The theoretical study Figure 10: The source contribution of source B The source contributions of source A calculated by the conventional OTPA, the proposed OTPA, and the theoretical study are 55.75, 61.05, and 66.95, and the source contribution accuracies of the two OTPA methods for the characteristic frequency 75 Hz are 83.27% and 91.19%. Furthermore, the source contribution calculated by the conventional OTPA also has a 250 Hz component contribution, whose contribution proportion in the whole signal is 39.34%, which is caused by the crosstalk of source B. Comparing the two results, the crosstalk problem of source B is well solved by the proposed OTPA model, and thus the accuracy is obviously improved. The source contributions of source B calculated by the conventional OTPA, the proposed OTPA, and the theoretical study are 180.80, 198.10 and 216.90 respectively while the source contribution accuracies of the two OTPA methods for the characteristic frequency of 250 Hz are 83.36% and 91.33%. Furthermore, the source 21

contribution calculated by the conventional OTPA also has a 75 Hz component contribution, whose contribution proportion in the whole signal is 5.81%, which is caused by the crosstalk of source A. Comparing the two results, the crosstalk problem is well solved by the proposed OTPA model and thus the accuracy is obviously improved. Generally, with the ICA-based crosstalk cancellation method, the strong crosstalk components can be significantly cancelled and thus, the ill-conditioned degrees and the accuracy of the OTPA can be obviously improved, which makes the proposed OTPA model accurate as TPA and efficient as the conventional OTPA. 4 EXPERIMENTAL STUDIES 4.1 Introductions of the Test Bed A test bed with cylindrical stiffened shell structures is constructed to test the performances of the proposed method, which mainly consists of a thin-walled cylindrical shell structure, 7 uniform distributed reinforcements, 2 eccentric vibration motors (vibration sources), supports, and rubber springs. The physical structure of the test bed is shown in Figure 11.

(a) The physical structure

(b) The photo of the test bed

Figure 11: The physical structure of the test bed

22

Figure 12: Locations of the sensors Acceleration sensors are employed to collect vibration data of the test bed and the locations of the sensors are shown in Figure 12. The testing parameters are listed in Table 1, sensor 1 and sensor 2 are chosen as the reference points of the small motor and big motor named as as reference point 1 and reference point 2 respectively while sensor 3 is selected as the target point. Table1: The testing parameters of the test bed Parameters

Values and units

PCB Acceleration Sensors (352C33)

3

Data acquisition system

COCO80

Sampling frequency

1600 Hz

Data length

10 seconds

Table 2: Operating conditions of the motors Operating conditions

1

2

3

4

5

6

7

8

Small motor (r/min)

600

681

681

850

781

681

850

1044

Big motor (r/min)

756

838

919

994

900

994

838

1000

23

By changing the rotating speeds of the motors, 8 operating conditions are generated and signals at reference and target points are measured. It should be noted that operating conditions 1 to 5 are experimental operating conditions while operating conditions 6 to 8 are practical operating conditions. The rotational speeds of the motors in 10 operating conditions are listed in Table 2. 4.2 The OTPA Model of the Test Bed The OTPA model of the test bed can be expressed as:

[T11 T12 ]

T

 x11 x12     M M =  x1r xr2     M M  5 5  x1 x2 

+

 y11     M y r   1  M  5  y1 

(33)

where r is the label of operating conditions, y1r (r = 1, 2, L , 5) is the data of the target point (sensor 3), x1r

( r = 1, 2, L

, 5) is the data of reference point 1 (sensor 1),

x r2 (r = 1, 2, L , 5) is the data of reference point 2 (sensor 2), T11 is the transmissibility function between reference point 1 and the target point, and T12 is the

Condition number(/)

transmissibility function between reference point 2 and the target point.

Figure 13: Condition numbers of the coefficient matrix The condition numbers of the coefficient matrix for the proposed OTPA and the 24

conventional OTPA are shown in Figure 13. It can be seen that the condition numbers of the coefficient matrix for the proposed OTPA are much smaller than that for the conventional OTPA, which means that the ill-condition problem has been significantly improved by canceling the crosstalk components and the transmissibility functions can be obtained accurately. 1

0.8

0.6 Reference point signal Separated signal 0.4

0.2

0

2

4 6 Operating condition(/)

8

Figure 14: Correlation coefficients between separated signals and reference points The correlation can test the waveform similarity of two signals, which can be used to reveal the severity of the crosstalk problem. The correlation coefficients between two separated signals and two signals at reference points in all 8 operating conditions are shown in Figure 14. Generally, the correlation coefficients between the separated signals are much smaller than that between the signals of reference points. It means that the correlation of the separated signals are significantly reduced because of the crosstalk cancellation. 4.3 Comparisons of The Experimental Studies Based on the truncated singular value decomposition method with the contribution rate (CR) for the singular values of 3%, the transmissibility functions between the reference points and the target point calculated by the proposed OTPA and

25

the conventional OTPA are shown in Figure 15. Comparing with the transmissibility functions by the two methods, it is clear that the characteristic frequency components are well reserved while other components are remarkably reduced, which can help to

Transmissibility function(/)

Transmissibility function(/)

obtain an accurate transmissibility functions and also transfer path analysis.

(a) Reference point 1 → the target point (b) Reference point 2 → the target point Figure 15: Transmissibility functions Based on above case studies, one of the source contribution results (6th practical operating condition) obtained by the conventional OTPA, the proposed OTPA and the measurement are displayed in Figure 16 and Figure 17. The measurement is conducted by turning on every source individually and measuring the response signals at the

Acceleration A/g

Acceleration A/g

Acceleration A/g

target point.

(a) The conventional OTPA (b) The proposed OTPA (c) The Measurement 26

Acceleration A/g

Acceleration A/g

Acceleration A/g

Figure 16: Source contributions of the small motor

(a) The conventional OTPA (b) The proposed OTPA (c) The Measurement Figure 17: Source contributions of the big motor The source contributions of the small motor calculated by the conventional OTPA, the proposed OTPA and by the measurement are 0.003482, 0.003592, and 0.003654 while the accuracies of the two methods for the characteristic frequency 11.35 Hz (681 r/min) are 95.29% and 98.30%. Furthermore, the source contribution calculated by the conventional OTPA also has a 16.57 Hz (994 r/min) component contribution, whose contribution proportion in the whole signal is 45.87%, which is caused by the crosstalk of the big motor. While the source contribution calculated by the proposed OTPA also has a 16.57 Hz (994 r/min) component contribution, whose contribution proportion in the whole signal is 1.37%. Comparing the two results, the crosstalk has been significantly cancelled by the proposed OTPA model. Thus, the accuracy is further increased. The source contributions of the big motor calculated by the conventional OTPA, the proposed OTPA and by the measurement are 0.002639, 0.003857, and 0.004240 while accuracies of the two methods for the characteristic frequency 11.35 Hz (681 r/min) are 62.24% and 90.97%. Furthermore, the source contribution calculated by the 27

conventional OTPA also has a 11.35 Hz (681 r/min) component contribution, whose contribution proportion in the whole signal is 14.54%, which is caused by the crosstalk of the small motor. While the source contribution calculated by the proposed OTPA also has a 11.37 Hz (994 r/min) component contribution, whose contribution proportion in the whole signal is 5.77%. Comparing the two results, the crosstalk has been significantly cancelled by the proposed OTPA model. Thus, the accuracy is significantly increased. 150 Conventional model source 1 Conventional model source 2 Proposed model source 1 Proposed model source 2 100

50

0

6

7 8 Operating condition(/)

9

Figure 18: Proportion of the crosstalk component The proportions of the crosstalk component for the 6th, 7th and 8th practical operating conditions are shown in Figure 18. The small motor and big motor are entitled as source 1 and source 2 respectively. For all 3 practical operating conditions, the contribution proportions of the crosstalk component for the proposed OTPA model are smaller than 20%, while the proportions for the conventional OTPA model are much larger, even more than 80% for the operating conditions 7 and 8 which are mainly caused by the misjudgment of dominating frequency band. Source contributions for all 3 practical operating conditions are shown in Figure 19. For all 3 practical operating conditions, source contributions of the proposed OTPA

28

model are closer to the measurement than that of the conventional OTPA model. Relative errors of the source contributions are shown in Table 3. For the conventional OTPA model, 5/6 of the relative errors are greater than 10%, 3/6 of the relative errors are greater than 20% while 1/6 of the relative errors is even greater than 150%, which means that the conventional OTPA model has large errors in this experimental case study. While for the proposed OTPA model, all the relative errors are smaller than 10%, which means that the accuracy of the source contributions is remarkably improved by using ICA-based crosstalk cancellation method and the proposed OTPA model is more

Acceleration A/g

Acceleration A/g

Acceleration A/g

accurate in practical case studies.

Figure 19: Source contributions With the source contribution results, dominating sources are easy to be identified. For the proposed OTPA model, source 2, source 1, and source 1 are respectively dominating sources for all 3 practical operating conditions and it agrees well with the measurement. While for the conventional OTPA model, it is shown that source 1 is the only dominating source for all 3 practical operating conditions, which shows the dominating source misjudgment for the operating condition 6. 29

Table 3: Relative error comparisons of source contributions Relative error of source contribution Operating condition

Conventional OTPA

%

Proposed OTPA

Source 1

Source 2

Source 1

Source 2

6

4.71%

37.76%

1.69%

9.03%

7

14.19%

30.05%

9.57%

0.35%

8

155.76%

13.54%

0.09%

9.43%

Generally, in the experimental case studies, the proposed crosstalk cancellation method can effectively cancel strong crosstalk of the measured signals at the reference points. With the proposed crosstalk cancellation method, the accuracies of the OTPA model and source contribution evaluation can be significantly improved and the dominating source misjudgment can be avoided. 5

CONCLUSIONS In this paper, a novel ICA-based crosstalk cancellation method for OTPA is

proposed, and a crosstalk cancelled OTPA model is constructed to improve the efficiency (especially accuracy) of the OTPA method. The main conclusions of this study are summarized below: (1) Crosstalk between the sources makes the measured signals at the reference points not accurate to represent the features of the corresponding exciting forces, which can cause inaccurate evaluation of transmissibility functions and source contributions for the conventional OTPA method. To overcome the crosstalk problem, the ICA-based crosstalk cancellation method for OTPA is proposed without any extra measurements, which can correct the sequences, amplitudes and phases of the separated signals based on the prior knowledge and the measured signals at the reference points. The proposed method is efficient as OTPA and also accurate as TPA. 30

(2) Comparing the source contribution evaluation in the numerical case studies, the accuracies of the proposed OTPA are 91.19% and 91.33% without crosstalk effects while that of the conventional OTPA are 83.27% and 83.36% with the crosstalk components of 39.34% and 5.81% in whole signal. In the experimental case studies, the accuracies of the proposed OTPA are 98.30% and 90.97% with relative very small crosstalk components of 1.37% and 5.77% in whole signal, while that of the conventional OTPA are 95.29% and 62.24% with the crosstalk components of 45.87% and 14.54% in whole signal. Therefore, with a significant crosstalk cancellation, the proposed OTPA model performs more accurately than the conventional OTPA without any extra measurements, and the dominating sources are accurately identified based on accurate source contributions. This study can benefit vibration and noise monitoring, reduction and control. ACKNOWLEDGEMENT This research was funded by the Projects of National Natural Science Foundation of China (No. 51775407), the National Key Research and Development Program of China (No.2019YFB1705403), the General Project of Joint Preresearch Fund for Equipment of Ministry of Education (No. 6141A02022121), the Fundamental Research Funds for the Central Universities, and Basic Research Project of Natural Science in Shaanxi Province (No. 2015JQ5183). REFERENCES [1] De Sitter G, Devriendt C, Guillaume P and Erik P, Operational transfer path analysis. Mechanical Systems and Signal Processing, 24(2010): 416-431. [2] Keersmaekers L, Mertens L, Penne R, Guillaume P and Steenackers G, Decoupling of mechanical systems based on IN-SITU frequency response

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An ICA-based crosstalk cancellation method for OTPA is proposed




The sequencing, amplitudes and phases of the separated signals are corrected




The accuracy of OTPA is significantly increased without any extra measurements




This study can benefit transfer path identification and contribution evaluation

Conceptualization: Wei Cheng, Yapeng Chu Data curation: Yapeng Chu, Diane Blamaud Formal analysis: Wei Cheng, Yapeng Chu Funding acquisition: Wei Cheng, Xuefeng Chen Investigation: Wei Cheng, Xuefeng Chen Methodology: Wei Cheng, Yapeng Chu Project administration: Wei Cheng Resources: Wei Cheng, Yapeng Chu Software: Yapeng Chu Supervision: Wei Cheng, Xuefeng Chen, Guanghui Zhou Validation: Yapeng Chu, Jingbai Lu Visualization: Wei Cheng, Yapeng Chu Roles/Writing - original draft: Wei Cheng, Yapeng Chu, Jingbai Lu Writing - review & editing: Wei Cheng, Jingbai Lu

Declaration of interests ☑ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: