DSpace at VNU: A new method for beam-damage-diagnosis using adaptive fuzzy neural structure and wavelet analysis

A new method for beam-damage-diagnosis using adaptivefuzzy neural structure and wavelet analysis a Inha University, Republic of Korea b Laboratory of Applied Mechanics LAM of Ho Chi Minh

A new method for beam-damage-diagnosis using adaptive

fuzzy neural structure and wavelet analysis

a

Inha University, Republic of Korea

b

Laboratory of Applied Mechanics (LAM) of Ho Chi Minh University of Technology, Vietnam

c

Ho Chi Minh University of Industry, HUI, Vietnam

d

Department of Mechanical Engineering, Smart Structures and Systems Laboratory, Inha University, Incheon 402-751, Republic of Korea

a r t i c l e i n f o

Article history:

Received 18 December 2012

Received in revised form

22 March 2013

Accepted 23 March 2013

Available online 16 April 2013

Keywords:

Fuzzy neural networks

Wavelet transform

Damage location

Damage diagnosis

Structure health monitor

a b s t r a c t

In this work, we present a new beam-damage-locating (BDL) method based on an algorithm which is a combination of an adaptive fuzzy neural structure (AFNS) and an average quantity solution to wavelet transform coefficient (AQWTC) of beam vibration signal The AFNS is used for remembering undamaged-beam dynamic properties, while the AQWTC is used for signal analysis Firstly, the beam is divided into elements and excited to be vibrated Vibrating signal at each element, which is displacement in this work, is measured, filtered and transformed into wavelet signal with a used-scale-sheet to calculate the corresponding difference of AQWTC between two cases: undamaged status and the status at the checked time Database about this difference is then used for finding out the elements having strange features in wavelet quantitative analysis, which directly represents the beam-damage signs The effectiveness of the proposed approach which combines fuzzy neural structure and wavelet transform methods is demonstrated by experiment on measured data sets in a vibrated beam-type steel frame structure

& 2013 Elsevier Ltd All rights reserved

1 Introduction

It is well known that the health monitoring of structures is one of the serious public issues which is concerned by many researchers The approach for the health monitoring of structures can be classified into two major groups: model-based and

understanding of failure model progression Since there are many different structures and many different types of failures,

it is difficult to develop accurate models in most practical instances Moreover, in some cases the fault propagation of the structures is quite complex and not fully understood In the second method group, collected condition data is used for building the fault propagation models This group is based on mechanical properties in which some physical characteristics

of the structure such as structure stiffness affect to structure vibration characteristics Damage appearing on structure reduces its stiffness, or reduces horizontal section area, or both of these Hence, the changes of the vibration characteristics

of the structure, such as the natural frequency, displacement or mode shape are signs to observe the damage in the structure These features are very important for structural health monitoring systems including structural damage detection,

Mechanical Systems and Signal Processing

0888-3270/$ - see front matter & 2013 Elsevier Ltd All rights reserved.

n Corresponding author Tel.: +82 32 860 7319.

E-mail addresses: nsidung@yahoo.com (S.D Nguyen), seungbok@inha.ac.kr (S.-B Choi).

been presented[3–8] It is remarked that since the vibration signal of the real structures is the measured signal with noise distortion, the filtering noise and the use of signal-analysis tools are to be carefully treated One of effective approaches to

structure-health-monitor system, which is based on data-driven methods, has to be correlatively established at two times: structure-undamaged time and the checking time This is really difficult if this process is based on traditional ways because

we are not able to exactly repeat an exciting status at two different times Hence mathematical models such as artificial neural networks technique (ANN)[6,12–14] or fuzzy logic (FL)[15], or combination of these models[1,11] are frequently

detection was proposed in which the location of the damage was obtained by using modal energy-based damage index values

The wavelet transform method analyzes the signal in two dimensions: time-frequency or space-frequency In this use, instead of using only a constant width window such as Fourier transform, wavelet transform uses a window-width-variable parameter called scale a which can play a role similar to frequency By this way, wavelet transform is able to locally analyze

or cracks ANN technique and FL are types of artificial intelligence techniques They have the potential to deliver effectively

models can map any complicated functional relationship between independent and dependent variables, they can provide better prediction and identification capabilities than traditional methods[17–19] NF is a fuzzy system built based on ANN Since these mathematical models could combine to get advantages of FL and ANN, this has saliently strong points than ANN

or FL only NF has usefully been used for identification and prediction[1,20,21]

To build NF, data space was classified to establish data clusters having common features[20,21] This process creates the data clusters having a role as a skeleton to build fuzzy sets of the fuzzy insurance system On the other hand, this classifying process could separate alien data samples from the data set to place them in distinct clusters At these distinct clusters membership values of other data samples in the data set are very small, or even are zero As a result, this could reduce influence of these alien data samples on calculated output of data samples to increase accuracy of NF system Hence in this side, the impact of the above data classifying process on system can be seen as a noise-filtering process for the data set This is one of featuring advantages of the NF built based on this way However, the effectiveness of this issue depends on

Nomenclature

að:Þj ; j ¼ 0…n coefficient of A(.)

process

the ANN

beam

of the data set (or the factors relative to the

vibration-exciting statuses)

damage checking process

undamaged time of beam

TΣcheck set of checking data sets

ðxi; yiÞ the ith input (xi)–output (yi) data sample of

training data set

xi¼ ½xi1 xi2:::xin the ith input vector of training data

set (the ith vibration-exciting status)

the surveyed beam

corre-sponding to the jth hyperplane, A(j)

the NF system

k-labeled-fuzzy set

½ℵi ℘i the ith input (ℵi)–output (℘i) data sample of

the ANN

suitable degree of the NF-membership function for data set features As usually, to adjust the role of each data cluster in

it significantly affects the prediction accuracy of the model However it is very difficult to find out appropriate values of this for each real application

Consequently, the main contribution of this work is to propose a novel approach for beam-damage-diagnosis which is easily applicable in practice The proposed approach focuses on the combination of fuzzy neural structure and wavelet transform methods In order to achieve this goal, AFNS is used for building NF system with data clusters typed hyperboxes which are created and joined as a part of input data space of the ANN Subsequently, the role of each data cluster in output-value-calculating process of the net is adjusted by the training process of the ANN By this way, the influence of unsatisfactory-relationship-value quantity between fuzzy sets on calculating result of the fuzzy influence system is compensated by the ANN when calculating the output value This increases accuracy of the model Furthermore, in order

to overcome the difficulty of signal analysis, a wavelet-quantitative analysis method is proposed In this research, the damage positions are investigated using two main steps As a first step, database for the beam health monitor system, which

is based on the change of its dynamic property signals, is established at two times: the structure-undamaged time and the checking time under same vibration exciting status (VES) Usually this is really difficult due to that we can not exactly repeat VES at two different times However, the use of the proposed AFNS could exactly interpolate data at any time when the structure is not damaged The second step is to analyze the created database using the wavelet-quantitative analysis based

on average wavelet transform of vibration signal with a used-scale-sheet to calculate the corresponding difference of AQWTC between two cases: the undamaged status and the status at the checked time Database about this difference is then used for finding out the elements having strange features in wavelet quantitative analysis, which are beam-damage signs This could overcome the difficulty about finding out the optimal scale in wavelet transform for each application The proposed method is experimentally implemented on the beam-typed steel frame structure Experiment results are analyzed

to verify the effectiveness of the proposed method as well as to evaluate the application possibility of the method for actual structures

2 Proposed fuzzy neural system

To well interact to a system its mathematical model needs to be built firstly In structure damage location using data-driven methods as mentioned in the introduction, this model is usually established by identification based on measured

vector,yiis corresponding output of the ith data sample in the set having P data samples There are various ways to identify

and cascade-forward neural networks (ANN) The ANN consists of one input layer, one hidden layer and one output layer Number of input signals depends on feature of training data set and structure of the NF; number of neurons at the hidden layer of the net is adaptively established in training process

(1) Building data clusters of given data set

space and hyperplane classes AðkÞ; k ¼ 1:::M; in output data space

to build pure data clusters, in which each pure cluster contains common-labeled samples

(2)

Building NF system

– Using the algorithm HLM1 for training process to build NF system

(3)

Structure of the ANN

time of the training process and adaptively adjusted in the AFNS training process Where N0is default number In this

neurons The‘purelin’ function, f ðsÞ ¼ s, is used for output of the neuron at output layer Transfer function is used for all of neurons at the hidden layer as follows:

– The ith input–output sample of data set used for training the ANN is signed and established as follows:

½ℵ ℘ ¼ ½ðx yðiÞÞ; y:

This is combined by the ith input–output sample of the data set(1), ðxi; yiÞ, and vector of values of corresponding hyperplanesyðiÞof the NF system trained by this data set (1), yðiÞ¼ ½yðiÞ1 yðiÞ2::: yðiÞM These values are calculated as follows:

yðiÞk ¼ ∑n

j ¼ 1

aðkÞj xijþ aðkÞ

0 ; k ¼ 1:::M; i ¼ 1:::P

whereaðkÞp ; p ¼ 0…n, are coefficients of A(k)created by training process of the NF

(4)

Training the ANN to adjust net parameters

Let W be the weight vector of the ANN, W ¼ ½w1 w2:::wHT The error equation of the ANN can be written as follows:

ErðWÞ ¼1

P ∑P

i ¼ 1ð℘i− ^℘iðWÞÞ2

i ¼ 1

e2

iðWÞ

where

VðWÞ ¼ ½v1ðWÞ v2ðWÞ ::: vPðWÞT

¼ ½e1ðWÞ e2ðWÞ ::: ePðWÞT

is calculated as follows:

Wkþ1¼ Wk−½JTðWkÞJðWkÞ þ μI−1JTðWkÞVðWkÞ

where, I is unit-square matrix, size H;μ is an adaptive index; and J is the matrix Jacobian as follows:

JðWkÞ ¼

∂v 1

∂w 1

∂v 1

∂w 2 ⋯ ∂v 1

∂w H

∂v 2

∂w 1

∂v 2

∂w 2 ⋯ ∂v 2

∂w H

∂v P

∂w 1

∂v P

∂w 2 ⋯ ∂v P

∂w H

2 6 6 6

3 7 7 7

ðW Þ

( )t

y1

i y

( )t M y

[ ]i in

x = 1

(i = 1 P )

11

i

μ

1

1R i

μ

1

ij

μ

j

ijR

μ

M

MR i1

μ

1

iM

μ

( ) 2

b

t yˆ

( ) 1

k b

( )lr k W

( )k 1 = N

( )j= 1 M+n

( )in1

j W

( )in2

i W

( ) i 1 = n

( )in2

i W

( )i= 1 M

Fig 1 Structure of the AFNS based on an adaptive neuro-fuzzy system (NF) and cascade-forward neural networks (ANN).

3 Wavelet transform

3.1 Theory basis

Wavelet analysis provides a powerful tool to characterize local features of a signal Unlike the Fourier transform, where the function used as the basis of decomposition is always a sinusoidal wave, other basis functions can be selected for

function of zero average:

This can be dilated or compressed with a scale parameter a, and translated by a position parameter b as follows:

ψa ;bðtÞ ¼ 1ffiffiffi

a

a

The wavelet transform of f at the scale a and position b is computed by correlating f(t) with a wavelet atom The continuous wavelet transform of f(t) is defined as follows[16]:

Wf ða; bÞ ¼Z þ∞

−∞ f ðtÞ 1ffiffiffi

a

a

dt

When using the wavelet for signal analysis, if the scale parameter a is small, it results in very narrow windows and is appropriate for high frequency components in the signal f(t) Oppositely if the scale parameter a is large, it results in wide windows and is suitable for the low frequency components in the signal f(t) The choice of scale can be based on actual analysis of the demand signal The greater the scale is, the more details of the frequency division are In fact, the advantages

of wavelet transform in signal analysis can be realized by selecting only appropriate wavelet function and wavelet scale 3.2 Average quantity wavelet coefficient

3.2.1 Selection of mother wavelet function

In signal analysis using wavelet transform, selection of the most appropriate wavelet mother function is really essential

In this work, the following considerations are used for this:

– The database used in this work is the displacement signal of the surveyed vibration beam which has a nearly symmetric shape In the database, it can be observed that signal value can usually reach zero at the initial and final points in a period Consequently, the wavelet mother function has to be compactly supported or nearly compactly supported with a finite duration In addition, both of orthogonal and biorthogonal properties are not required in this work since the reconstruction of the original signal is not required

due to their efficiencies Relation to the orthogonal wavelets, to satisfy the orthogonal property, integral of the wavelet function and integral of the square of the wavelet function must respectively be equal to zero and one However, the

proposed, in which an optimization solution is utilized By this approach, derivatives of the wavelet function are required

in order to minimize errors between the model outputs and the actual outputs For this reason, in this use a non-orthogonal differentiable wavelet mother function is suitable

Based on issues abovementioned, it can be seen that the Mexican hat function is an appropriate selection for this work

In fact the Mexican hat wavelet function has several principal characteristics Firstly, it is a rapidly vanishing function[25]

and is a computationally efficient function Secondly, it can be analytically differentiated and can be used conveniently for decomposing multidimensional time series Equation of the selected wavelet is given as follows:

ψa;bðtÞ ¼ 2ffiffiffi

3

p π−1=4 1− t−ba

exp −12 t−ba

ð3Þ

3.2.2 Sampling frequency

Actually, the appropriate sampling solution is usually selected according to the target of each use For example, the relation to digitally recorded noisy data decomposition and reconstruction, sampling at unequally spaced times is usually

sampling rule Differently, the relation to signal analysis to investigate crucial characteristics of the signal source without

reconstruction the initial signal, sampling at equally spaced times can be used In[24], a time-frequency domain wavelet analysis of acceleration signal of earthquake records was accomplished based on this solution

In this study, the goal is to investigate signal characteristics to find out beam damage signs, without initial signal reconstruction Consequently, firstly the initial signal f ðtÞ is sampled at equally spaced times, corresponding to a constant sampling frequency A set of translation parameter b is then taken at the same points where the f ðtÞ is sampled A scale vector a ¼ ½as1 as2⋯asm, which is being depicted in detail in the next section, is then created to achieve an appropriate range

of frequency resolution Subsequently, the created parameters a and b are used to dilate or compress the mother wavelet in order to accomplish a family of waveletψa;bðtÞ Next f ðtÞ is multiplied by the wavelets ψa;bðtÞ at different values of the scale a

created produces Wf ða; bÞ indicates the correlation between the signal and the wavelet functions ψa ;bðtÞ

3.2.3 Building the scale vector

As abovementioned, the selection of the optimal scale for beam-damage diagnosis is very important to achieve best effective result However, the common way presented above is not enough to choose the best value of the scale In addition, due to the analyzed data the measured signals with noise distortion are occurred Consequently, the result of using only one value of the scale which is considered to be suitable for wavelet analysis to obtain information relative to the system is sometimes unsatisfactory Instead of getting the system information we may receive only features relative to noisy

To overcome these, in this work the use of the scale vector a is proposed to build the AQWTC for finding out beam damage signs To create the scale vector a ¼ ½as1as2⋯asm, the modified Gram–Schmidt algorithm of[27]is used, where m is number

of wavelets obtained This work is performed as follows:

signed Ki; i ¼ 1:::Ne; are correspondingly established Relation to building Ki; the process can be summarized as follows Firstly, empty wavelets whose supports do not contain any data are deleted from the wavelet decomposition Result of this work is a nonzero wavelet coefficient frame to be created Next, from this frame the wavelet which gives best approximates to the measured data is selected Subsequently, this wavelet is combined with the remainder of the nonzero wavelet coefficient frame one at a time to determine the best combination This procedure is repeated for all nonzero wavelet coefficients and finally the element scale vector is created as a result of the accomplished process – Establishing the scale vector: The vector ais structured based on the following set:

K ¼ fK1∪K1∪:::∪KN eg

3.2.4 Average quantity wavelet coefficient

beam, signed zji, is calculated as follows:

zji¼ 1

k ¼ 1

where fjiis the function relating to jth element at ith vibration state of the beam In this paper f is displacement function of element

4 Beam-damaged location algorithm (BDLA)

Based on the AFNS and using wavelet analysis as presented above, we propose an algorithm for beam-damaged location

At the beam-undamaged time:

Step 1 Building data sets

data set, corresponding to the jth element, has P data samples ðxi; zjiÞ; in which xi¼ ½xi1:::; xin ; i ¼ 1:::P, expresses the ith vibration-exciting status (VES); n depicts factors relative to the vibration-exciting statuses; zjiis the AQWTC calculated based

on(4)as follows:

zji¼ 1

Bm∑m

k ¼ 1∑ðbÞWfjiðask; bÞ, j¼1…Newhere B is number of sampled points of function f (it is also the size of vector b) Step 2 Identifying elements at the beam-undamaged time

At the checking time:

Step 3 Calculating damage coefficients

identification as presented in Step 1 At each exciting status, vibration signal at elements is measured, filtered to calculate AQWTC (zji; ) based on(4) j ¼ 1:::Ne; i ¼ 1:::Ptest Based onzji;calculate AQWTC at each element for all Ptestvibration exciting statuses as follows:

mBPtestP∑test

i ¼ 1 ∑m

where B is number of sampled points of function f (it is also the size of vector b)

– Based on Ptestvibration exciting statuses above, use Nethe AFNS built in Step 2 for establishing AQWTC corresponding to

Step 1.

Step 2

N N

STOP

- Making vibration of the beam by x k

- Using the AFNS j for calculatingzˆjk

- Calculating z jkbased

on (4),j= 1 N e

Calculating Damage Coefficients (6), (7)

dj rj

c c

Determine condition

of the beam based on ,

dj rj

c c

Continue ?

Set up P test ; k=0

k =: k +1

?

test

kP=

START ( , )

Building jth data sets x i z ji

at undamaged time of the

beam at undamaged time

Step 4

Y

Y

Checking time Step 3

Undamaged time beam, j=1 N e ,i=1 P

by ( ) to identify the x i z ji

Fig 2 Flow chart of the proposed algorithm BDLA.

xi2

xin

xi1

ˆji

z

z ji

(j=1…Ne)

- Measure vibration signal

,

c c

j = 1 N e

The ith vibration exciting status( ,x i i =1 P test)

Calculate

Fig 3 Principle for calculating damage indexes based on the AFNS.

statuses as follows:

zðnotÞj ¼ 1

PtestP∑test

i ¼ 1

^zji; j ¼ 1:::Ne

jth element is calculated as follows:

cdj¼



zj−zðnotÞ

j





max

k ¼ 1 :::N e

cdk

Step 4 Locating damage of beam

Element having crj¼ 1(7)is the one most decreased in flexural rigidity, and degree of this damage is shown bycdj(6)

5 Experimental investigation

The proposed algorithm, BDLA, is used for data sets measured on a vibrated beam-typed steel frame to find out its damaged locations

5.1 Test rig and procedures

different positions The center deviation level of M, Md, is easily varied by varying distance d from position of fixed M to the

the vibration-exciting statuses n is 3 The frame, length L¼3 m, is divided into 12 equal parts by 13 nodes signed Y1,…,Y13 Sensors (1), the vibration signal measuring system LAM_BRIDGE (2), and a computer (3) are used to measure vibration signal at these nodes at different vibration-exciting-statuses (VES)

D, we created P ¼1500 VES to measure corresponding vibration signals of the structure at Y2,…,Y12 By this way, a data set

At the checking time: To make frame-damaged conditions, the frame was cut at one or two positions In each case, the reduction of horizontal section area (HSA) of the frame at the cut positions was performed by a group in 4 levels: a–a (3.56%

by exciting to produce vibration of the frame in order to measure vibration signal at its nodes we established a checking data set, named TΣcheckused for checking process Based on TΣcheckand the AFNSs trained by TΣ, the algorithm BDLA was then used

to find out beam-damaged locations corresponding to these conditions of the frame, upon this the effectiveness of the proposed algorithm was estimated

5.2 Results and discussion

5.2.1 Analysis of sensitivity

this section In this survey, the beam was damaged at Y6with 3 degrees, a–a, b–b, c–c Analysis of vibration signal at this

identifying undamaged status of the beam and a value a¼ 0.5 of the scale used for wavelet transform vibration signal, and (2) using the proposed method with a used-scale vector a In both of these cases, beam-vibration-exciting statuses were the

in the second case Namely cdis 0.014, 0.02, and 0.028, in the first case; and is 0.0444, 0.0509, and 0.0542 in the second case, corresponding to damage degrees a–a, b–b, and c–c, respectively Besides, high increase of zjiinFig 5b after each cycle more

The above results can be analyzed based on the use of the scale parameter as follows When using one value of the scale a

with any frequency component of the signal, wavelet coefficient values are small for both damaged and undamaged statuses

of the structure Hence no strange sign for damage diagnosis appears However, it is different if the proposed scale vector ais used By this way, in the group of scales as1; as2; :::asmbelong to the scale vector a; there is more probability of that at least

In addition, it can be seen that when damage appears such as stiffness reduction of the beam, the natural frequency of the vibration beam decreases Consequently, natural frequencies of the beam at two statuses, damaged and undamaged structure, are different In this circumstance, if the scale parameter a is used for both, there may be two cases The first is that neither of these frequencies correspond with the scale a Hence there is no clear sign for considering In the second case, the scale a corresponds with one of these frequencies or with both of them but different degrees As a result, wavelet coefficients corresponding to vibration signal of the beam at damaged and undamaged statuses are different This different feature is the positive sign for damage diagnosis In this side, it can be clearly seen that if the scale vector a is used, probability of appearing the second case is higher than using the one-scale a Besides, if the scale vector a together with the proposed average-quantity solution to wavelet transform coefficient (AQWTC) is used, the above different feature is accumulated to create the more striking sign for damage diagnosis

Y13 Y12 Y11 Y10 Y9 Y8 Y7 Y6 Y5 Y4 Y3 Y2 Y1

L = 3m

Fig 4 Experimental model: (a) photograph of the model; (b) the dividing nodes.

0.0 0.2 0.4 0.6 0.8

Time (sampled points)

0.0

0.2

0.4

0.6

0.8

Time (sampled points)

c-c, cd=0.028 b-b, cd=0.020 a-a, cd=0.014

c-c, cd=0.0542 b-b, cd=0.0509 a-a, cd=0.0444

Fig 5 The beam was damaged at Y 6 with 3 degrees: a–a, b–b, c–c AQWTC z ji (4) and damage index c d (6) were calculated based on analysis of vibration signal at this position (Y 6 ) of the beam in two cases: (1) using neural networks for identifying with a value of scale a ¼0.5 used for wavelet transform (a); and (2) using the proposed method (b).

Based on the argument and verifying results as abovementioned, it can be observed that the proposed method has advantage compared with the previous method as usually used It is noted that the sensitivity of the signal used for the

5.2.2 Analysis of vibration signal at elements

performed at damaged positions but also at the other along the beam Vibration signal at 6 positions: Y2, Y4, Y6, Y8, Y10and

However, at damaged positions the increase is faster Namely at Y4and Y8AQWTC, zji, get the largest In graphs ofzji, graphs

of z4iand z8i are highest This feature is used for damaged location of the proposed algorithm, BDLA

5.2.3 Single damage and the frame divided into 4 elements

In this experiment, the frame was divided into four elements and it was damaged at only one position Four divided elements were as follows: Y1–Y4, Y4–Y7, Y7–Y10, and Y10–Y13having the equal lengths, L/4 We made single damage positions

on the frame, i.e the frame reduced HSA at only one position, as follows:

– Or the frame was cut at Y4+(the middle of Y4and Y5) with 4 levels a–a, b–b, c–c, d–d;

– Or the frame was cut at Y6+(the middle of Y6and Y7) with 4 levels a–a, b–b, c–c, d–d

Thus, in this test the different cut points all belong to the 2nd element The algorithm BDLA was used in order to determine

proposed algorithm BDLA rightly determines the positions in which HAS was reduced, even at low defected level (3.56%) 5.2.4 Case of double damage

In this case, the beam was divided into 6 elements and it was simultaneously damaged at two positions belonging to 2nd

Result shows that at two damaged elements, damage coefficient gets the largest values This means that the proposed method exactly locates the damaged positions appearing on the beam, even when damages simultaneously appeared and damage degrees are quite small (6.9%)

5.2.5 Single/double damage and the frame divided into 3 elements

The frame was divided into 3 elements as following Y1–Y5, Y5–Y9, and Y9–Y13having the equal lengths, L/3 BDLA and[6]

were used to determine the positions reduced HSA of the frame in two cases, single damage or double damage with three levels b–b, c–c, d–d In the first case, the frame was cut at Y6+ In the second case, the frame was simultaneously cut at two

(Fig 10b) and d–d (Fig 10c) The diagrams show that the BDLA exactly determines the position reduced HSA at all these

0 5 10 15 20 25 30

Time (sampled points)

z2i

z4i

z6i

z8i

z10i

z12i

Fig 6 The beam was damaged at Y 4 and Y 8 , damage degree 11.16% (c–c) AQWTC z ji (4) was calculated based on analysis of vibration signal at Y 2 , Y 4 , Y 6 , Y 8 ,