五 azuhiro Takeyasu Y parameters, paper, deteri Ol ation detection, comparison, Thus, evaluated, vibration, function, kurtosis, moment, far, (Bolleter,

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Title

Comparison of Absolute Deterioration Factor of 6th Order Momen in a B road Sense for the Case of Affiliated Impact Vibration

Author(s)

Takeyasu, Kazuhiro; Higuchi, Yuki

Editor(s)

Citation

大阪府立大學經濟研究. 2009, 55(1), p.13-35

Issue Date

2009-06-30

URL

http://hdl.handle.net/10466/11037

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五azuhiro Takeyasu ・ Y

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ABSTRACT

Among many dimensional and dimensionless amplitude parameters, kurtosis (4th normalized

moment of probability density function) is recognized to be a sensitive good parameter for machine diagnosis. In this paper, simplified ca1culation method of kurtosis and 6thnormalized moment are introduced for the analysis of impact vibration inc1uding affiliated impact vibration. Affiliated impact vibration is approximated by triangle and simplified ca1culation method is introduced. Furthermore absolutedeteriOl・ation factor is introduced. Various models are examined for these two models and it is shown that simplified calculation method of 6th

normalized moment is much more sensitive than those of kurtosis. For further detection, the

concept of absolute deterioration factor of n-thorder moment in a broad sense is introduced and the analysis is executed. From the resu1tof comparison, absolute deterioration factor of 6-th

order moment in a broad sense is better than other factors in the viewpoint of sensitivity and practical use. Thus, n-th order moment in a broad sense was examined and evaluated, and we

obtained practical good results.

Keywords: impact vibration, probability density function, kurtosis, 6thnormalized moment,

deterioration

1. INTRODUCTION

Machine diagnosis techniques play important roles on manufacturing. So far, many signal processing methods for machine diagnosis have been proposed (Bolleter, 1998).As for sensitive parameters, Kurtosis, Bicoherence, Impact Deterioration Factor (ID Factor) were examined (Yamazaki, 1977:Maekawa et a.l1997;Shao et a.l2001;Song et a.l1998;

Takeyasu, 1987, 1989).In this paper, we focus our attention to the index parameters of vibration. Kurtosis is one of the sophisticated inspection parameters which calculates normalized 4th moment of Probability Density Function (PDF). Kurtosis has a value of

14 Comparison of Absolute Deterioration Factor of 6th_{Order Moment in a Broad Sense for the Case of Affiliated Impact Vibration}

3.0 under normal condition and the value generally goes up as the deterioration proceeds. But there were cases that kurtosis values went up and then went down when damages increased as time passed which were observed in our experiment in the past (Takeyasu,

1987, 1989).

In this paper, simplified calculation method of kurtosis and 6thnormalized moment for

the analysis of impact vibration including affiliated impact vibration are introduced.

A旺iliated impact vibration is approximated by triangle and simplified calculation method

is introduced. Furthermore absolute deterioration factor is introduced. Various models are examined for these two models. Comparing them, we show that the absolute

deterioration factor of 6th_{normalized moment is much more sensitive than those of}

Kurtosis.

For further detection, the concept of absolute deterioration factor of n-th order

moment in a broad sense is introduced and the analysis is executed. In this paper, we

consider the case such that impact vibration occurs on the gear when the failure arises. Higher moments would be more sensitive compared with 4-th moment. Kurtosis value is 3.0 under normal condition and when failure increases, the value grows big. Therefore,

it is a relative index. On the other hand, Bicoherence is an absolute index which is close

to1.0 under normal condition and tends to be 0 when failure increases.

In this paper, we deal with the generalized n-th moment. When theoretical value of

n-th moment is divided by calculated value of n-th moment, it would behave as an

absolute index. New index shows that it is1.0 under normal condition and tends to be

o

when failure increases.

In this paper, we introduce a simplified calculation method to this new index and

name this as a simplified absolute index of n-th moment. Furthermore, as Bicoherence

can be considered to be a kindof ふ th order moment, several factors concerning absolute

deterioration factor of 6-th order moment are compared and evaluated. Trying several

n, we search n which shows the most similare百ect to the behavior of Bicoherence.

From the result of comparison, absolute deterioration factorof ふ th order moment in a

broad sense is better than other factors in the viewpoint of sensitivity and practical use. The rest of this study is organized as follows. We survey each index of deterioration in section 2. Simplified calculation method of Kurtosis includinga笠iliated impact vibration

lS S

Comparison of Absolute Deterioration Factor of 6th_{Order Moment in a Broad Sense for the Case ofA伍liatedImpact Vibration 15}

section 5 and corresponding method is summarized in Section 6. Remarks is made in section 7. Section 8 is a summary.

2. FACTORS FOR VIBRATION CALCULATION

In cyclic movements such as those of bearings and gears, the vibration grows larger

whenever the deterioration becomes bigger. Also, it is well known that the vibration

grows la1'ge when the setting equipment to the g1'ound is unsuitable (Yamazaki, 1977).

Assume the vib1'ation signal is the function of time asx(t).And also assume that it is a stationa1'Y time series with mean O. Denote the probability density function of these time se1'ies as

p

(

x

)

.

Indices fo1'vibration amplitude are as follows. XroOI

=

[

f

:

l

x

1

i

p(x刈

X

,.ms

=[ι制)dxT

X

_abs

= に Ixlp{x)めc

X

peak

コ

_{~~~[仁川)dxy}

(1) (2) (3) (4)

These a1'e dimensional indices which are not normalized. They di笠er by machine sizes01'rotation frequencies. The1'efore, normalized dimensionless indices are required. The1'e are four big categories fo1'this purpose. A. Normalized root mean squa1'e value B. No1'malized peak value C. Normalized moment D. Normalized correlation among f1'equency domain A. Normalized root mean square value a. Shape Factor : SF 大阪府立大学経済研究 第55巻 1 (227) [2009.6J

16 Comparison 01 Absolute Deterioration Factor 01 6_'.1Order Moment in a Broad Sense lor the Case 01A伍liatedImpact Vibration

xm

_一一九

F c d ₍₅₎

(X

abs : mean of the absolute value of vibration) B.Normalized peak value b. Crest Factor : CrF X __,,1. CrF=~ニ;ふ Xrms (6) (X_pωk : peak value of vibration) c. Clearance Factor : CIF X _",,1. CIF=~ニニ X_rOOf (7) d. mpulse Factor : IF

IF= 主竺

-Xαbs (8) e. Impact Deterioration Factor : ID Factor

ID= 主笠

-X

_c (9)

(X

c : vibration amplitude where the curvature of PDF becomes maximum) C. Normalized moment f. Skewness: SK

'K= に x

3

_p(x)ゐ

[にの(x)改

、 tI ノ ハ U 1 2 4 / 4 ¥ 大阪府立大学経済研究 第55巻 1 (227) (2009.6J

Comparison of Absolute Deterioration Factor of 6th_{Order Moment i a nBroad Sense for the Case ofA日i!iatedImpactVíbrati臼n 17} g. Kurtosis: KT

い仰づ叫

J Xi 一、ゴ J

川一，内

X 一; 。∞一 X3 cfi 一 F! 日 一「 ili--L T K (11)

D

.

Normalized correlation in the frequency domain h. Bicoherence Bicoherence means the relationship of a function atdi旺erent points in the frequency domain and is expressed as:

Bxxx(んん)

Bic，捌 (1;， fJ=

.

J

S xx

(

1

;

)

•

S

,

x

(

J

2 ). Stx(.月十五) (12) Here x

_T

(1; ).xA.ん ).x; (J;+ ん)

B.ux(んん)=

(13) T2 means Bispectrum and ゃく t

<

T

)

(

e

l

s

e

)

T : Basic Frequency Interval

xT(f)= に XT(t)e-州 d

(14)

人 (/)=jX7(川(J)

(15) Range of Bicoherence satisfies

。 < Bicラxxx (1;いん )<1

(16)

When there exists a significant relationship between frequenciesλand

/

2

'

Bicoherence is near 1 and otherwise comes close to O.

These indices are generally used in combination and machine condition is judged

18ωmparìson of Absolute Deterioration Factor of 6th_{Order Moment n Bra oad Sense for the Case ofA伍liatedImpactVìbratìo丑}

totally. Among them, Kurtosis is said to be superior index (Noda, 1987) and many researches on this have been made (Maekawa et a.l1997; Shao et a.l2001; Song et a.l 1998). Judging from the experiment we have made in the past, we may conclude that

Bicoherence is also a sensitive good index (Takeyasu, 1989, 1989).

Eq.(15) is a power spectrum. Power spectrum is a Fourier Transform of

Autocorrelation function (Tokumaru et al., 1982). Therefore it is a kind of second order

moment in a broad sense. Watching at the denominator of Eq.(12), a square root is taken

for the triple products of power spectrum. Normalization is executed by this item. That

is, Bicoherence is equivalent to the square root of normalized 6-th order moment in a

broad sense. Therefore, Bicoherence can considered to be a kind of an absolute

deterioration factor of normalized 6-th order moment in a broad sense.

In Maekawa et a (1997).l , ID Factor is proposed as a good index. In this paper, we

focusing on the indices of vibration amplitude, simplified calculation method of Kurtosis and 6th_{normalized moment including affiliated impact vibration is introduced.}

Furthermore absolute deterioration factor is introduced. Varying the shape of triangle, various models are examined. An absolute index ofn-thmoment and Bicoherence are compared with this index and analysis is executed.

3.SIMPLI 円 ED CALCULATION METHOD OF KURTOSIS 3.1. Absolute index of n-th moment Mean value X ofx(t)is calculated as:

寸 xp(x)ホ

Discrete time series are stated as follows. xk=x(k!J.t)(k=1

,

2,...) Where !J.

t

is a sampling time interva.l

x

is stated as follows under discrete time series. M

王ニ PE 合 ZXI

Under the following Gaussian distribution 大阪府立大学経済研究第55巻 1 (227) [2009.6)

Comparison of Absolute Deterioration Factor of 6th_{Order Moment in a Broad Sense for the Case of Affiliated ImpactVibra包on 19}

砂(x)口マケーM子r

its moment is described as follows which is well known (Hino, 1977).

子河口 O

(18) x(加)ロ日 (2k -1)σ 211 (19) Ifwe divide Eq. (19) by σ2/1， we can obtain normalized moment.In general, normalized n-th moment is stated as follows.

i∞か王Y' p(x)めE

Qか)=ム

[にい州知y

In discrete time system, it is described as:

立(X

i

一王

_)"

Qい)= .l}m-=f口l

…(よかi-

X

We describeQ(n)asQN い) if it is calculated by using N amount of data

必(n)= JZか2 一王Y'

{~会朴正

Absolute index ofn-th moment is described as follows. 大阪府立大学経済研究 第55巻 1 (227) [2009.6) (20) (21) (22)

20 Comparison of Absolute Deterioration Factor of6'hOrder Moment in a Broad Sense for the Case ofA伍liatedImpact Vibration ZN い)= 日似 -1)

43bz 一王t

(中 -i)

(23)

Under the normal condition

,

Z N (n)

•

1 (N →∞)， and iffailure becomes larger

,

Z N

(n)

•

0

3.2. Several facts on Kurtosis Kurtosis(KT) is a normalized 4-th moment stated as follows.

KT ニにX4p(X)めc

[に川ゆr

J

7

Z

(

X

1

3

Y

1

9

i

h

-

Z

)

2

j

LetKT of N amount of data be stated asKT_N 3.3. Simplified Calculation Method of Kurtosis (24) When there arise failures on bearings or gears, peak value arises cyclically. In the early stage of the defect. this peak signal usually appears clearly. Generally. defects will injure another bearing or gears by contacting the inner covering surface as time passes. When defects grow up, affiliated impact vibrationar色es. Impact vibration including

a百iliated impact vibration occurs in the case that there is a failure of such as bearings' flaking. See Chart1.

Comparison of Absolute Deterioration Factor 01 6'hOrder Moment i a nBroadSe日記 forthe Case ofA悶atedImpact Vibration 21

Chart 1. Example of affiliated impact vibration

These signals can be approximated by triangle model.Hereafter, we analyze these cases by utilizing simplified mode.l

Assume that the peak signal which has p times magnitude from normal signals arises during m times measurement of samplings. As for determining sampling interval,

sampling theorem is well known (Tokumaru et a.l1982). But in this paper, we do not

pay much attention on this point in order to focus on our proposal theme.

Suppose that affiliated vibration can be approximated by triangle and set sampling count asd, then we can assume the following triangle model (Figure 1).

When d = 1, the peak signal which has p times magnitude from normal signals arises.

i 、 p-l

When d

=

i, the peak signal which has p -(i-1)……~ times magnitude from normal

五、ノ q signals arises(i 口 1 ，...，q ) ・ When d:?:q + 1, normal signa.l p q 4一一一一一一一一軒

‘ー

事惨 m Figure 1. Impact vibration and affiliated vibration 大販府立大学経済研究 第55巻 1 (227) (2009.6)

22 Comparison of Absolute Deterioration Factor of6'hOrder Moment in a Broad Sense for the Case of Affiliated Impact Vibration

LetσN stateasσ-;;2 when impact vibration occurs.

As for 4th moment and Kurtosis, let them stateas 苛両)，支古子N int恥he s槌am蹴ew問a勾y

σN2C伺an bec伺alculat怯ed a部s follows.

U= ニトt 一王)

=

2

[

t{pート1)引]廿い吋詰

コ σN2 十品哨十+ヤ -γl}

(25) As for 耳石い)， utilizing :

gf= 件立r

5i4= 会い哨n+ 伽2

+3n-l 店元) canbe 叫culated as follows.

耐=品川

吋ト-(i一明い)引~(4)

十凸吋炉い刷川一吋リず巾

31j炉仰+静蜘祈qイ2均舟削刑引叫一斗イ

4仲

1)中仲仲州)ド凶仲伽州+巾す削判糾叫ヤいド刷

H

川川一斗

4イ叶1)ず付沖巾

y日小2:?j1(

(26) Then we getKT N as:

Eιflμ+ 元凸hい+川

[トiμ+古乙臼凸(q+川2~;10;-1)+2}l

(27)

If the system is under normal condition, we may supposep(x) becomes a normal

Comparison of Absolute Deterioration Factor of 6th_{Order Moment in a Broad Sense for the Case of Affitiated Impact Vibration 23} distribution function. Under this condition.KT is always :

KT=3.0

Therefore. K九三 3.0 (28) As failure increases.KT N value grows up. The absolute deterioration factor such as Bicoherence is easy to handle because it takes the value of1.0 under the normal condition and tends to be 0 when damages increase. Therefore inverse number of the(KT_N -2) would make an absolute deterioration factor.

Z"

=

1

KT

_N

-2

Under the normal condition.ZN is 1 and tends to be 0 when damages increase. Another method for an absolute deterioration factor is considered to be as follows.

広71 」z

KT

_N

4.SIMPLlFIED CALCULATION METHOD OF 6-TH NORMALlZED MOMENT

4. 1.Several Facts on 6-th Normalized Moment (29) (30) 6-th normalized moment is transformed into the one for the continuous time system as

Q

ι (X 一正y p(x)ゐ

[にか-王r防防]

And it is transformed into the onefor・ discrete time system as :

二τ 玄 (X

_i

一王)6

Q=P曳 f 二 γ1

はさ (X

i

_引

大阪府立大学経済研究 第 55巻 1 (227) (2009.6J (31) (32)

24 Cornparison of Absolute Deterioration Factor of6th_{Order Mornent in a Broad Sense for theC蹴 ofAffiliatedIrnpact Vibration}

4.2.Simplified Calculation Method of 6-th Normalized Moment

As for 6-出 moment and 6-th 間四lized moment, letM T_N (6) and QN state asM克明，

QN when impact vibration occurs. As for 耳石布)， utilizing :

十件立r

(33)

zi4= 計十伽+伽2

+3n-l) (34)

zi52 会2(川

(35) 、、 11 ノ 唱_si

+

n 今 J 局、 d n / O 十 必峰 BB n 今コ ，，，，， .E--z ‘、 、、 . . . . ,,,, 唱 'A n 吋 L VH ハ 唱_EEA 十 n /azz ‘、、 々 'b z r l

一位

一一 〆 O n す何一同 (36) 耳石事) can be calculated as follows.

可}=品朴正y

24針。-1)与!?jMPmい)

Here, n M e A ヘノ“ n u e A ぺ，ノ“ Juun ハ H M A ，， az--EE--、、 nut 一 、、， EZFJ 間 一一 ηy iwu 一 1IA …内 JJ “ ! ? 、、 BEg-f n M E A n w A Jvun ハ nw ,. n w e a f h H U L 一一内 /M nut-dι ー ペーム一 /tt ，、一 、 alt--一一 5q iwu 一 P112 汁_MIEn ペ iij 』 n u t vua ハ n U A FIll--'ili--41111 」 守ん一一 一 m l ? 一一 n k

+ザ山 -3q-l)-SJヂヤ1)+押印一i川

(38) QN is stated as follows. 大阪府立大学経済研究 第55巻 1 (227) (2009.6)

Comparison of Absolute Deterioration Factor of 6'h_{Order Moment in a Broad Sense for the Case of Affitiated Impact Vibration 25}

52;仏句9)

Here,

s=トニ(q+恥127い1)+

2

J

}

(40) Under the normal condition, Q is always :

Q

=

15.0 (Yamazaki, 1977) Therefore

QN

~ 15.0 As failure increases, QN value grows up. As stated before in 3.3., absolute deterioration factor forQN is introduced in the same way as follows. 一 A 呼

一一仏

N

W

(41) Another method for an absolute deterioration factor is considered to be as follows. U"

=

-

2三

N

一一一-QN

(42) 5.NUMERICAし EXAMPLE 5. 1. Transition ofKT N Under the assumption of 3.3., letm=12. Considering the case p=2,3,"',6 and q =1スム4 ， we obtain Table 1 from the calculation of Eq. (27). 大阪府立大学経済研究 第日巻 1 (227) (2009.6J

26 Comparison 01 Absolute Deterioration Factor 016'h Order1vI0ment in a Broad Sense lor the Case 01 Affiliated Impact Vibration Table 1.KT N for each case p 1 2 3 4 5 6 1 3.0 4.683 7.740 10.227 11.934 l3.083 2 3.0 4.263 6.090 7.431 8.325 8.964 q 3 3.0 3.960 5.127 5.9l3 6.426 6.753 4 3.0 3.165 4.336 4.932 5.247 5.463

As p increases, F，αand KT" N i...,.1..."-'"...,...,. ncreases. On t...,... ...h...L.e.... o'-J..t...h...e...r.. hJ...\.4..., and.F_ .La and KT" decreases

asq increases when p is the same.

When damages increase or transfer to another place, peak level grows up and

affiliated impact vibration spread. This means thatKT N value shift from the left-hand

side upwards to theright.嶋 hand side downwards in Table1.For example, the following

transition ofKT N can be supposed.

When q

=

1

,

P =1

,

KT_N =3.0

When q=2ラ p=2ラ KT_N =4.263

When q=4ラ p=4

,

KT_N =4.932

When q =4

,

p=6ラ KT_N =5.463

CalculationZ N and V_N in the same way, we obtain Table 2 and Table 3 respectively.

Table2. Transition ofZ N p 1 2 3 4 5 6 1 1.0 0.373 0.174 0.122 0.101 0.090 q 2 1.0 0.442 0.244 0.184 0.158 0.144 3 1.0 0.510 0.320 0.256 0.226 0.210 Table3. Transition ofV_N p 1 2 3 4 5 6 1 1.0 0.641 0.388 0.293 0.251 0.229 2 1.0 0.704 0.493 0.404 0.360 0.335 q 3 1.0 0.758 0.585 0.507 0.467 0.444 4 1.0 0.948 0.692 0.608 0.572 0.549 大阪府立大学経済研究第55巻 1 (227) C2009.6J

Companson of Absolute Deterioration Factor of 6th_{Order Moment in a Broad Sense for the Case ofA伍tiatedImpact Vibration 27} 5.2. Transition ofQN U nder the assumption of 3.3.. letm = 12. Considering the casep = 2.3. ・・・ .6 and q = 1，2，3ラ4. we obtain Table 4 from the calculation of Eq. (39). Table4.QN for each case p 1 2 3 4 5 6 1 15.000 50.612 135.274 215.969 276.246 318.714 2 15.000 38.621 80.455 115.435 140.747 158.784 q 3 15.000 31.414 54.081 71.011 82.813 91.173 4 15.000 26.201 39.159 48.225 54.453 58.880 As p increases.Fband QN increase. On the other hand.Fband QN decrease as q increases when p is the same.

When damages increase or transfer to another place. peak level grows up and

a妊iliated impact vibration spread. This means thatQN value shift from the left-hand side upwards to the right-hand side downwards in Table 4. For example. following transition ofQN can be supposed.

When q = 1ラ p =1

,

K九=15.0

When q=コ 2， p=2ラ KTN =38.621

When q=4ヲ p=4

,

KTN =48.224

When q=4ラ p 二口 6ラ KT_N = 58.880

CalculatingW_N and U N in the same way. we obtain Table 5 and Table 6 respectively.

Table 5. Transition ofW_N p 1 2 3 4 5 6 1 1.0 0.027 0.008 0.005 0.004 0.003 2 1.0 0.041 0.015 0.010 0.008 0.007 q 3 1.0 0.057 0.025 0.018 0.015 0.013 4 1.0 0.082 0.040 0.029 0.025 0.022 大阪府立大学経済研究 第55巻 1 (227) [2009.6)

28 Comparison 01 Absolute Deterioration Factor 01 6'h Order Moment in a Broad Sense lor the Case 01 AffiliatedImpact Vibration Table 6. Transition ofU N p 1 2 3 4 5 6 1 1.0 0.296 0.111 0.069 0.054 0.047 2 1.0 0.388 0.186 0.130 0.107 0.094 q 1.0 0.477 0.277 0.211 0.181 0.165 3 4 1.0 0.572 0.383 0.311 0.275 0.255

Comparing the case ofKT N with QN. we can see thatQN is much more sensitive and

machine troubles can be detected easily. Though the mathematical formulation ofQN is little more complex. both methods enable us to calculate the new indices even on a pocketsize calculator quickly and easily in the industry.

6. SIMPLlFIED ABSOしUTE INDEX OF n-TH MOMENT

To compare n-th moment and Bicoherence with newly introduced index stated above. we show simplified absolute index of n-th moment (Takeyasu et al.. 2003).

6.1. Simplified Absolute ofn-th moent

When the number of failures on bearings or gears arise. the peak value arise cyc1ically. In the early stage of the defect. this peak signal usually appears c1early. Generally. defects will injure other bearings or gears by contacting theinner・ covering

surface as time passes.

Assume that we get N amount of data and then newly get L amount of data. Assume that mean. variance and moment are same withl~N data and N+ トN+L data except for the case where a special peak signals arises.

Let mean .variance and n-th moment calculated by usingl~N data state as:

王N' σ2 Nラ MN(n)

And as forN+l~N+L. let them state as :

王N Il， σ 2 N /l， MNII{n)

Where

Comparison 01Ab則自teDeterioration Factor 016th_{Order Moment in a Broad Sense lor the Case 01A鐙liatedImpact Vibration 29} qJ 4 一X /行い N

す

l-M

1 一 N i M V N M A 斗 4 d 吐 一 x i_円 γ

似す山中

N M Therefore, QN+I{n)is stated as: MN(n)

QN い)=一一

σN (45) Assume that the peak signal which has S times impact from normal signals arises in eachm times samplings.

Letσ 'N /I and MN/I of this case be(j'N/I, MN/I, then we get

, N+I 52N/lz;2; 仏-王)2 . i=NI

l

_

i

ヨつ旦 σ2N+ 子山 2 N ゥ (. S2-1ﾎ =σ “ NI l+一一一一一 i m (46) , N+I

MNII{n)=+ エ仏一王y

. i=N+l 5 つ旦 M川か)+子 S /1M

NIl

(. S/1-1 1 = 11 十一~ IMNII{n) m (47) From these equations, we obtain QN+I(寸 as QNjn) of the above case 大阪府立大学経済研究 第55巻 1 (227) [2009.6J

30 Comparison of Absolute Deterioration Factor of 6'h_{Order Moment in a Broad Sense for the Case ofA節liatedImpact Vibration} ιωω++， (n，λベ小(い体n

(品ん戸古σ2沖

Nイや(卜1+干

γf

s

n

-1 1 十一一一一一一 N+l m

[1+元7Ekly

[1+山守)三

WhileQN+

,

{n)

is Kurtosis when

n

= 4. QNい)=KT MN{n) σ 2_N (48) We assume that time series are stationary as is stated before in 2. Therefore. even if sample pass may di旺ér. mean and variance are naturally supposed to be the same when the signal is obtained from the same data occurrence point of the same machine.

We consider such case when the impact vibration occurs. Except for the impact vibration. other signals are assumed to be stationary and have the same means and variances. Under this assumption. we can derive the simplified calculation method for machine diagnosis which is a very practical one. From the above equation. we obtainKTx+1 in the following way. 1+ _1 . S4-1 十一一一一・一一一一一 一一一… N+l KL.. 三ー x3.0

(

1

1

十一一一一・一一一一ー

S

2

-

7

N 十 1 m Consequently. we obtainZN+I(n)as: 大阪府立大学経済研究 第55巻 1 (227) [2009.6) (49)

Comparison ofAbsol山 Deterior必onFactor of 6th_{Order Moment in a Broad Sense for the Case of} A偲tiatedImpact Vibration 31 日似 -1) ZN+'い)=土L一一

_{- QN+}

_,

_{n)

日(2k

-

1

)

N+1 m

;

;

-

Q

N

{

n

)

[1+古平y

Under the normal condition.

QN い)=

I

1

(

2

k

-

l

)

Therefore. we get

ZN+，(n)七三三)2

6.2. Numerical Examples (50) (51) (52)

Ifthe system is under normal condition. we may suppose p(x)becomes a normal

distribution function. Under this condition .

Q

(

n

)

is as follows theoretically when

n

=

4,6,8

Q

(

4

)

= 3.0 Q和)=15.0

Q

(

8

)

= 105.0 Under the assumption of 3.3. letm=12. Considering the case S

=

2,4,6 for 3.3. and setting N

•

0

.1

•

N. we obtain Table 7. 大阪府立大学経済研究第55巻 1 (227) (2009.6)

32 Comparison 01Absol日teDeterioration Factor 01 6th_{Order Moment in a Broad Sense lor the Case ofA缶liatedImpact Vibration} Table 7.Q.Z. by the variation of S S 2 4 6 Q 3.0 4.32 13.2 21.3 4 Z 1.0 0.69 0.23 0.14 Q 15.0 48.65 450.71 970.89 n 6 Z 1.0 0.31 0.03 0.02(0.016) Q 105.0 956.93 22378.44 62453.16 8 Z 1.0 0.11 0.0047 0.0017 7. REMARKS We introduced the two absolute deterioration factors for each model respectively. We compare them with Bicoherence and evaluate them accordingly. In Maekawa et a.l

(1997). the waveform is simulated in three cases as (a) normal condition. (b) small defect condition (maximum vibration is two times compared with (a)).(c)big defect condition (maximum vibration is six times compared with (a)). They showed the result of Kurtosis in these cases. We showed the relations between those results and QN Cin detail see Takeyasu et a.l(2003)). Subsequently. we examine Bicoherence. We made experiment in the past (Takeyasu (1987). Takeyasu (1989)). Summary of the experiment is as follows. Pitching defects are pressed on the gears of small testing machine. Small defect condition Pitching defects pressed on 1/3 gears of the total gear. 班iddle defect condition Pitching defects pressed on 2/3 gears of the total gear. Big defect condition Pitching defects pressed on whole gears of the total gear. We examined several cases for the1;.

1

2

in Eq.(12). We gotbest-五t result in the following case.

(

五

:

…

切附叫向

n郎叫叩

C句引

y刊O…

p jλ2 : 2 五 We obtained the following Bicoherence values in this case (Table 8). 大阪府立大学経済研究 第55巻 1 (227) (2009.6J

Comparison 01 Absolute Deterioration Factor of6th_{Order Moment in a Broad Sense lor the Case of Affiliated Impact Vibration 33} Condition Bicoherence Normal 0.99 Table 8. Transition ofU N Small defect 0.38 Middle defect 0.09 Big defect 0.02

Thus, Bicoherence proved to be a very sensitive good index. These results can be taken into account, though the definition of defect size does not necessarily coincide. Bicoherence is an absolute index of which range is 1 to O. Therefore it can be said that it is a universal index.

Now, we compare this index with proposed simplified absolute indices. The proposed

methods are the absolute indices of which range are from 1 to 0 similarly as Bicoherence. As for sensitivity, Z N is better than V_N for the group of KT N ' W_N is better than V_N

in sensitivity butW_N falls too fast. Even when p

=

2 , it has the value of less than 0.09. While U N decreases smoothly, which may be appropriate for the practical use. As

a whole, indices ofQ N group (W_N , UN) are sensitive than those of KT N group (ZN'

V_{N )}. Comparing W_N withU N' U N is better thanW_N for the practical use.

Thus, we can get a sensitive index for machine diagnosis in a simple way. These deterioration factors are said to a kindof ふ th order moment in a broad sense. They have the value around Bicoher・ence. Someone is much more sensitive than Bicoherence

and someone is moderate compared with Bicoherence. Therefore proposed one is a

practical good index with simple calculation method.

Next, we compare Bicoherence with proposed simplified absolute index ofn-th

moment. The proposed method is an absolute index of which range is from 1 to 0

similarly as Bicoherence. As for sensitivity, the case ofn

=

6 is quite similar to

Bicoherence, but the proposed one is slightly much more sensitive. The value is already

0.31 at small defect condition and 0.03 at middle defect condition which show quite sensitive behavior.Itis suitable for especially early stage failure detection.

Sensitivity is better inn

=

6 than that ofn

=

4. Sensitivity is much better inn

=

8, but the value falls too fast therefore the judgment becomes hard. Therefore the case

刀工 6 is good for the practical use. As Bicoherence is one of the kind of 6-th order moment in a broad sense, these deterioration factors of 6-th order moment in a broad sense found to be sensitive and practical indicies.

This calculation method is simple enough to execute even on a pocketsize calculator as is shown in Eq. (52). Compared with Bicoherence which has to be calculated by Eq.

34 Comparison of Absolute Deterioration Factor of 6'h_{OrderlvIoment in a Broad Sense for the Case of Affiliated Impact Vibration}

(12)~(15) ， proposed method is by far a simple one and easy to handle on the field defection. Comparing with these indices, simplified calculation method of ふ th normalized moment inc1udes index with much more sensitive than others. J udging from the machine status, suitable method should be selected. For example, sensitive index should be

selected for the machine which requires severe control for machine failure.

8.CONCしUSION

We proposed a simplified calculation method of Kurtosis and 6-th normalized moment for the analysis of impact vibration inc1uding affiliated impact vibration.A狂iliated impact vibration was approximated by triangle and simplified calculation method was introduced. Furthermore absolute deterioration factor was introduced.

Varying the shape of triangle, various models were examined for two models and it

was shown that absolute deterioration factor of 6-th normalized moment was much more sensitive than those of kurtosis. Utilizing this method, the behavior of 6-th normalized

moment would be forecasted and analyzed while watching machine condition and exquisite diagnosis would be executed.

For further detection, an absolute index ofnぺh moment and Bicoherence were

compared with this index and analysis was executed. Focusing that Bicoherence was considered to be a kind of absolute deterioration factor of normalized 6-th order moment in a broad sense, analysis was executed. From the result of comparison, absolute factor

of 6-th order moment in a broad sense was better than other factors in the viewpoint of sensitivity and practical use. The e百éctiveness of this method should be examined in

vanous cases. REFERENCES Bolleter, U.(1988). Blade Passage Tones of Centrifugal Pumps. Vibration 4 (3), 8 回 13 Maekawa, K..S. Nakajima. and T. Toyoda(1997). New Severity Index for Failures of Machine Elements by Impact Vibration(in Japanese).J.SOPE ]apan 9 (3). 163-168. Noda (1987). Diagnosis Method for a Bearing(in Japanese).NSK Tec.J.(647), 33-38. Shao.Y.. K.Nezu. T. Matsuura.Y.Hasegawa. and N. Kansawa (2001). Bearing Fault Diagnosis Using an Adaptive Filter(in ]apanese).J.SOPE ]apan 12 (3). 71-77. Song.J.W..H. Tin. and T. Toyoda (1998). Diagnosis Method for a Gear Equipment by Sequential 大阪府立大学経済研究第55巻 1 (227) (2009.6)

Comparison of Absolute Deterioration Factor of 6'" Order Moment in a Broad Sense for the Case of Affiliated Impact Vibratioll 35

Fuggy NeuralNetwor・k (in Japanese).J,SOPE Japan 10(1), 15-20.

Takeyasu, K.(1987). Watching Method of Circulating Moving Object(in Japanese). Certi怠ed

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Takeyasu, K., T. Amemiya, K.Ino and S, Masuda (2003). Machine Diagnosis Techniques by Simplified Calculation Method. IEMS 2 (1), 1-8

日ino， M. (1977).Spectrum Analysis(in Japanese). Asakura shoten Publishing.

Tokumaru, H., T. Soeda, T. Nakamizo, and K.Akizuki(1982)Measurement and Calculation(in J apanese).Baifukan Publishing.

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Kazuhiro Takeyasu, Yuki Higuchi.(2005). Analysis of The Behavior of Kurtosis by Simplified

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