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TitleSimplified Machine Diagnosis Techniques Techniques Using ARMA Mo

del of Absolute Deterioration Factor with Weight

Author(s) Takeyasu, Kazuhiro; Ishii, Yasuo

Editor(s)

Citation 大阪府立大學經濟研究. 2008, 54(1), p.53-73

Issue Date 2008-06-30

URL http://hdl.handle.net/10466/10999

Rights

Simplified M瀋hine Diagnosis Techniques

Using ARMA Model of Absolute

DeteriorationFactor with Weight

Kazuhiro Takeyasu ・ Yasuo Ishii

Abstract: In order to diagnose machines accurately, the Kurtosis or Bicoherence method was utilized. Calculating system parameter distan鐺 was also utilized to apｭ

ply time series dat� to Autoregressive (AR) model or Autoregressive Moving Averｭage (ARMA) model. In thi,s paper, a simplified method of calculating autocorrelation

function is introduced and is utilized for ARMA model identification. Furthermore, an absohitedeterioration factor such as Bicoherence is introduced. 恥1achine diagnoｭsis can be executed by this simplified method of calculating system parameter disｭ

tance with weight. Useful results were obtained.

Keywords: ARMAmodel , autocorrelation function , cumulants , absolute deteriorac

~ion factor with weight

1. INTRODUCTION

53

In mass production industries such as steel making that have largeequipment, sudｭ

den stops of production process due to machine fa匀ure can cause severe problems such

as shortage of materials to the later process , delays to the due date and the increasing

idling time.

To prevent such situations, machine diagnosis techniques play important roles. 80

far, Time Based Maintenance (TBM) technique has eonstituted the majority the maｭ

chine maintenance. TBM performs checks at previously fixed times, but it performs

checks at scheduled times without taking into account whether theparts.arestill keepｭ

ing in good conditions or not. On theother hand, Condit�n Based Maintenance (CBM)

performs maintenance checks by watching the condition of machines. Therefore, if the

parts are still keeping in good condition beyond their expected life span, the cost of

maintenance may be lowered because machines can be used longer than planned~

54 Simplified Machine Diagnosis Techniques Using ARMA Model

of Absolute Det巴rioration Factor with Weight

Therefore the use of CBM has become dominant, since parts maintenance cost less and

also because it has a lower 士atiooffailure. However, it is'mandatory ,to catch sympｭ

toms of the failure as soon as possible if a transition 、 from TBM to CBM is to be

made. Many metho司s'have been developed f~c~lsing onthissubjé仇 In this paper, we proｭ

pose a method for the early detection � the failure ónr叫ating machìne , which is the

most common theme in the machine failure detection field.

To date, many signal processing methods for machine diagnosis have been proposed

(Bolleter, 1998; Hoffner, 1991).Asfor sensitivityindices, Kurtosis , Bicoherence and Imｭ

pact Deterioration Factor (ID Factor) were examined (Yamazaki, 1977; Maekawa et al.,

1997; Shao et al., 2001, Song et al., 1998; Takeyasu, 1989). Calculating system param。

ter distance was also utilized to apply time series data to Autoregressive (AR) model

or Autoregressive MovingAverage (ARMA) model (Yamazaki, 1977).

In thispåper, a simplified method of caleulatirig theautocorrelati�function is introｭ

duced and utilized for ARMA model identification. Furthermore, an absQlute deterioraｭ

tion factbr such as Bicon鑽ence is introduced. Machine diagnOsis can be executed by

this simplified method of calculating system par疥eter distance.

The rest of the paper isorganized as follows. , In section 2, machine diagnosistech-

niques utilizing system parameter distance are exhibited. The parameter est絈ation

method of ARMA model is stated in section 3. In section 4, moment, Bicoherence and

cumulants are introduced for the fundamental preparation. Utilizing 4th cumulants

and autocorrelation function , the parameter estimation method of ARMA model is ex-

hibited' in' section 5. Utilizing these methods , numerical examples are exhibited in sec-

tion 6. Section 7 summarizes the findings�

2. DETECTION OF FAILURE BY SYSTEM PARAMETER DISTANCE

In the analysis of tim:eseries data, AR model or ARMA model are adoptedfre-

quent1y. In this paper, we adopt ARMA model, because it is a general technique used

in time series analysis ,

Consider the (ρ ， q )th order ARMA model expressed as:

p q

Xn+ 害lGzZn t=en+ 豆島en-j (1)

Here

大阪府立大学経済研究 54 ・ 1(223) (2008.6)

Simplified Machine Diagnosis Te鑄niques .UsingARMA' ModeT of Abso1ute Deterioia土ioàFactor'with Weight. 55

{Xj} : SaIhple process � astationary ιergodicGaussian proceSs x(t) ci =,1, 2,, 3;

-・， 'N， …)

{ι} : Gaussian white noise with mean 0, and variance a; Assume that (1) satisfies the stationary condition.

Now , we calculate the~ystem p~rameter distçmce，、Modern con~r91 thE;lory shows that

ρ 豆 q when state space representation is used (Tokumaru et αl.， 1977). In this paper,

we setρ== q for sim�licity.

( 2 )

(3 )

'/ b1¥l . " ・・ v

b' =1 v2 I ~ system paraﾌneters under normal condltion

¥bA/ I 、 pl

I b1¥1

6 弓 l 川 l 飢ma伽1 of system papram仰、s while I I abnormal contion proceeds

¥bp/ J

¥IIIlli--/¥11111111/

1

2

P

1

2

P

αα:

・

αLAαAα:

・

Aα

,frill--Il--

‘

JIlli---\

一一一一

α

《α

Set

Then, following Z[ is utilized as an evaluation function. Set'the system par瀘eters unｭ

der normal condition asαi' bi , and th� estimation of system parameters as â i , bi・

ZI= ーよ-，(l = 0, 1, 2, 3, 4, 5,…) 問+1

(4 )

Where

( 5 ) 町二 (2ρ)[x 土l{(dt-U+(6z bz)2}

L= きl{(dtG仲 ( 6 )

Watching the behavior of Z[ , we can judge the system becomes abnormal when Z[ falls

below the certain value (Figure 1). Then we can make failure detection. This evaluation

•abnorma1 system

Z[=l

一一一令 time

Detecting Abnorma1 Conditions by the transition of Z[

Z{=o

Figure 1.

'54 ・ 1(223) (2008,6J 大阪府立大学経済研究

56 Simplified Machine DiagnosisTechniques Using ARMAModel

of Absolute DeteriorationFactorwith Weight

function is an absolute deterioration factor such as Bicoherence. This is 1.0 when the

system is under normal condition, and tends to be 0 when the system b.ecomes abnorｭ

mal.

3. ESTIMATION OF SYSTEM PARAMETER BY ARMA MODEL

In estimating ARMA model parameters {的} {bj} , the right hand side of the equaｭ

tion is itself a colored noise. Therefore, it should be noted that a biased estimation is obｭ

tained by using the least squares method. In order to obtain an unbiased estimate, the

extended least squares, sequential maximum likelihood, instrumental variable, and

pseudo linear regression methods have been developed (Tokumaru et α1.， 1982;

Katayama, 1994). The pseudo linear regression method, utilizes the bootstrap method

and makes iterative calculations. However, it has difficulty because there are many

cases where these iterations do not converge (Sagara et αl. ， 1994; Katayama, 1994). We

have proposed an improved method for this problem (Takeyasu èt αl. ， 2002). By utilizｭ

ing this method, the total. calculation time can be reduced.

Introducing the cross correlation function of output and noise, and also using the

theoretical relation, the above improvement was achieved. In this paper, we introduce

a simplified method of calculating autocorrelation function in the cases where shock

waves occur, and propose simplified machine diagnosis techniques utilizing 4th

cumulants and autocorrelation function. Since 4th cumulants are 0 under the normal

distribution , cumulants can be calculated in combination using autocorrelation funcｭ

tion and utilizing the formulaic relation of 4 variables.

We have previously proposed a bootstrap algorithm before in order to get an unbi時

ased estimate. This time, ouraim is to propose a new simplified calculation method

for machine diagnosis. Yule-Walker equation for 4th cumulants is introduced. Knowｭ

ing that we can obtain an unbiased estimate of AR parameters under the condition

that MA items are white noise under a normal distribution (Inoue, 1992a, 1992b), the

equation for ARparameterestimation is introduced. Furthermore, we introduce a MA

parameter estimation equation which can be estimated analytically in the case of 2nd orｭ

der. This is then applied to a simplified calculation method and numerical calculation

is executed. The Proposed method is simple enough to be calculated even on a pocketｭ

size calculator.

大阪府立大学経済研究 54 ・1(223) (2008.6J

Simplif�d Machine Diagnosis Techniques Using ARMA Model of Absolute Deterioration Factor with W鑛ght 57

4. FUNDAMENTAL PREPARATION

4.1. About Moment

In cyclic movements such as those of bearings and gears, the vibrations grow larger

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

grows large when the setting equipment to the ground is unsuitable (Yamazaki, 1977).

Assume the vibration signal is a function of time as, x(t). Also assume that it is a staｭ

tionary time series with mean X. Denote the probability density function of these time

series asρ (x).

The average X is stated as follows:

x= よ ~ρ (x)dx (7 )

Variance σ2 is stated as follows:

σ2 = L:(X-X)2p(x)dx ( 8 )

The 3rd moment MT(3) and the 4th moment MT(4) are stated as follows respecｭ

tively (Hino, 1977):

MT(3) =に(X-X)3灼)dx

MT(4) = に(X-X)4内)dx

( 9 )

(10)

4.2. About Bicoherence

Bicoherence means the relationship of a function at different points in the frequency

domain and is expressed as:

Bxxx(ん fz)Bic,xxx (iI, fz) =

l' n/ J Sx/iJ) . Sx/五) .Sxx(h十五)、}ノ

唱ai

唱EA

/，‘、

Here

XT(五)・ XT(fz) ・ X;(五十五)Bxxx(iJ, fz) = ---------;;

Tτ

(12)

means Bispectrum. Each item of right hand side of Eq.(12) is a Fourier Transform of

the original time series. See Appendix in detail.

Range of Bicoherence satisfies:

0< Bic ,xxx (ん 12) < 1 (13)

大阪府立大学経済研究 54 ・ 1(223) (2008:6)

58 Simplified Machine Diagnosis T゚chniquesUsing ARMA Model

of Absolute Deterioration Factor with Weight

When there exists a significant relationship between frequencies h and J;, Bicoherence

is near 1. Otherwise, the value of Bicoherence corr冾s close to O.

4.3. About Cumulants

The characteristic function 1JI(u) is definedas follows (Hino, 1977):

1JI(u) ==に e引 (14)

The Taylor expansion � the characteristic function is exhibited in(15). c(n) is th� coefｭ

ficient.

lnV(u)=zdn)1Jfun n= ¥ n!

I (-i)ndnln 1JI(u)l c(n) = l' "/ ~

"~" ~ ，~/

I LαU Ju = 0

(15)

(16)

This c(n) is called cumulants. The following relation holds between cumulants and

the moment.

c(l) = x

c(2) = σ2

c(3) = MT(3)

c(4) = MT(4) -3 {MT(2)}2

(17)

(18)

(19)

(20)

It is well-known that when the probability density function is a normal distribuｭ

tion , cumulants over the 3rd order are 0 (Inoue, 1992).

4.4. 2nd Order Autocorrelation Function

Using the following formula concerning the relation of Gaussian 4 variables

(Nakamizo, 1988):

E[αbcdJ = E[αbJE[cdJ + E[αcJE[bdJ +E[αdJE[bcJ (21)

We can get following equation for x(t) , x(t+ τ\) ， x(t+ τ2) ， x(t十月).

E[x(t)x(t+ τ\)x(t+ τ2)x(t+ τ3)J

= R(τ\)R( τ3一 τ2)+R( τ2)R( τ3一 τ\)+R(τ3)R( τ2一 τ\) (22)

Where, RτI(Rτ1 = R(τ\)) is the autocorrelation function of {x(t)}.

Rτ1=E[z(t〉 z(t+τ\)J (23)

Set ττ\， τ3 = 0, then we can obtain

E[X(t)2X (t+τ1)2]=R;+2Rjl(24)

大阪府立大学経済研究 54 ・ 1(223) (2008,6J

Simplified Machine Diagnosis.Techniques Using ARMA Model of Absolute Deterioration Factor with. Weight 59

Set τ1 = -l， τ2 = -l， τ3 = -i, then we can obtain

(25) E[X(t)X(t-l)2X(t-i)J = RoRi+2RIRl-i

This is the correlation function of the correlation function. That is to say that this

is a 2nd order correlation function.

5. ESTIMATION OF SYSTEM PARAMETERS ABOUT ARMA MODEL

First, the AR parameter estimation using Yule-Walker equation concerning 4th

usmg algorithm Next, the ARMA parameter estimation cumulants is exhibited.

autocorrelation function is stated.

5.1. Estimation of AR Parameter

Under a normal distribution , 4th order cumulants becomes O.

(Inoue, 1992), bymultiplying by XnX!-1 As {en} is white under 4th order cumulants

on both sides of (1) and averaging, we can get the following equation:

E[X!X!_IJ + 土lGzE[Znz; 凡 J = O(l ;:::: 1) (26)

Rewriting (26) , we can get the following equation:

C1 + 互角Cl- i = O (lミ1) (27)

where Cl-i is the 4th cumulant.

Cl-i = C(Xn, Xn- l , Xn- l , Xn-)

= E[XnX!-IXn-J

= R( -l)R (lーの +R( ~l)R (l-i)+R( -i)R(O)

= RoRi+2RIRI-i (l ;:::: 1, i = 1 ，…， ρ) (28)

about the 4th (27), following Yule Walker equation 1 toρm Varying 1 from

cumulants can be derived:

C1 αl C1-P C-1 c。

(29) C2

仇μ・

•••

c

α2 -・・ C2-P c。C1

α p c。C~_ ， C p-l "p-2

Rewriting this in matrix form , we can get:

(30)

54 ・ 1(223) [2008.6) 大阪府立大学経済研究

cα = -cs


of Absolute Deterioration Fact� with Weight

Then αis solved as follows:

α=-c ICs (31)

The estimation of Cl-i(l = 1, 2,…, i = 1,…, p) is expressed as �l-i' �l-i is stated as fo1-

10ws:

ê,_, = _1 立ヮ=ム XnX~-IXη-i

l-i N -(Z-i) n= �=i+l (32)

Substituting �l for Cl in each c, cs' ê, �s is obtained. Then the estimate � ofαcan be obｭ

tained as the unbiased estimate by the following equation.

α= 一 c -cs (33)

This is the same meaning that an unbiased estimate of AR mode1 parameters can be

obtained by solving Yu1e-Wa1ker equation.

5.1.1. 2ndorder case

¥1It---f

sA9

白

pup

し

//It--1\

、

γ1lj/

-

A

U

一

c

c

c

c

,,/Ili--\

一一

¥1111/

'in

,“

αα

/,

JI--1\

‘

=-(f?γ??で)R~+2R~ 3R~ / \R~+2R;/

1 ((1 +2ρî)(1 -pD\

4(2+ρî)(1一ρî) \1-2ρi+3ρ;-2ρiJ

5.2. Simplified Calculation Method ofAutocorrelation Function

In a discrete time system, samp1ed signa1 data are expressed as follows:

{XJ : i = 1, 2,…, N

ニr which is the mean of {Xi} , is stated as follows:

N

:i= 子:;- I; x, lV i = 1

For the purpose of genera1ization, :i is not set O.

Variance σ2 of {xJ is stated as follows:

N

t= 一二-:;- I; (xi- :i) 辺N-1 i=l

大阪府立大学経済研究 54'・ 1(223) (2008.6J

(34)

(35)

(36)

Simplified Machine Diagnosis Techniques Using ARMA Model of Absolute DeteriorationFactor with Weight

Autocorre1ation function is stated as follows:

1 N-[

Rl = E[x;x;+J = 一一一-;- L; X;X;牛 1 (l = 0, 1, 2,…) ι Z-ZTL.... N-I i-=-l -Z-Z-rl ，~

61

(37)

As {xi} is assumed to be a stationary time series, we can assume E[xJ = 0 without

10ss of genera1ity (Tokumaru et al., 1982). Therefore, (37) can be re-stated as follows:

R[ = E[(xi-.i) (Xi+l-.t)]

N~l

=一二---;- I; (x;-.t) (x山-.t)N-l i::-¥ , -Z けt(38)

When N is sufficient1y 1arge, the Oth autocorre1ation function is near1y equa1 to the

variance. The autocorre1ation coefficient is expressed as follows:

Rl Pl = Ro (39)

When the number of failures on bearings or gears arise, the peak value arise cycliｭ

cally. In the early stage of the defect, the peak signal usually appears clearly. Generｭ

ally, defects will damage other bearings or gears by contacting the inner covering surｭ

face as time passes. Assume that the peak signa1 which has S times impacts from the

norma1 signa1 arises in each m times of samplings. As for determining the sampling inｭ

terva1 , the well-known samp1ing theorem can be used (Tokumaru et al., 1982), but in

this paper, we do not pay much attention to this point in order to focus on our pro時

posal. Assume that the peak signa1 which has S times impacts from the norma1 signa1

arises in each m times of samplings , and a1so assume that the mean and variance are

the same except for the case where a special peak signal arises. Expressing 02 of this

case as Õ2, we get the following equation:

2 1 N o 一二-， t (x;-.t)2

N-1 i::-¥ '-z

N N-~ー

N

さ一一一竺Lσ2+ __m . S202 N-1 - , N-1 --

ミ (1+与l)σ2 (40)

For the autocorre1ation function , we express Ri in this case as Rパi = 1, 2, ・ぃ) in the

same way. The Oth autocorre1ation function is nearly equa1 variance, so we get:

Ro ミ õ2 == (1十号l)σ2 (41)

大阪府立大学経済研究 54 ・1(223) (2008.6J



5.2.1. Autocorrelation function of 1st order lag

In the next, we examine the feature of autocorrelation function of 1st order lag. By

definition, RI is stated as follows:

1 ペニIE 一一

Al-N-12fプ1 内向+1 (42)

When the peak signal of S times impacts from the normal signal arises in each m

times of samplings, the following ^ parts of {xJ may be considered to have peak valｭues during the calculation of RI'

1, 2,… , m-2, m-1 , m , m十 1 ， m十 2，… ， 2m-l , 2m , 2m十 1，・・

^̂ < <

From these, the following products arise:

1 product with peak value and {(m-1)-l} products with ordinary values up to m

3 products with peak values and {(2m -1) -3} products with ordinary values up to 2m

(2k-1) products with peak values and {(km-1)-(2k-1)} products with ordinary

values up to km

2k products with peak values and {(km + 1-1) -2k} products with ordinary values up

to km+1

Therefore when N = km (case 1)

(2(N/m)-1) products with peak values and {(N- 1) 一 (2(N/m)-1)} products with

ordinary values up to N

WhenN= km+α (1 豆 α 三三 (m -1)) (case 2)

(2(N α)/m) products with peak values and {(N- 1) 一 (2(N α)/m)} products with

ordinary values up to N

When S is large, we make the following simplified calculation.

Suppose

zε < xi < .i+E

ε> 0, ci = 1, 2,… , N)

except for the case when peak signals arise. In that case we get the following relaｭ

tions:

<Case 1>

When.i > 0

大阪府立大学経済研究 54 ・ 1(223) (2008.6J

Simplified Machine Diagnosis Techniques Using ARMA Model of Absolute Deterioration Factor with Weight

合{s(づ l)+(N パづト e)2 < Rj

<合{s(寸小(N- l)-(づ寸山)2When:i < 0

N~l {s(づ小(N- l) 一べ(づ一 1)ト)ドい(ω( :i -e

>tτ1{ド似仲Sベや(←2づすす一 1)ド+刊(N-1ο) 一-(2すす一 1)}山)2 (44)

Rj

ρ 豆J

~ N~l {やす-1) + (N -1) -(づ l)) ;2ー l

= {(去-tTポョ)(s 山}. mぷ-1<Case 2> When:i > 0

N~l {S(2号旦)+山)-(2 N:竺)}(:i-e)2< 昆l

<長τ{S(2 N:旦)+ω- 1) -(2 N:旦)}(:i +e)2When:i < 0

tIK44+山)_(2 N:a )}臼一ε)2> R

>tTK2t旦)+(N-1) -( 2 N:a)}(日)2In the same method as < Case 1> , we get the following equation:

Rj

ρ Ro

=長τ{や号旦)+仰ーパ

= {(去(詰サ)(S一山}. mA1

大阪府立大学経済研究 54 ・ 1(223) (2008.6)

(45)

(46)

(47)

(48)

63

Simplified Machine Diagnosis Techniques Using ARMA Model of Absolute Deterioration Factor with Weight 64

(45) and (48) are the simplified calculations of coefficient of autocorrelation function of

1st order lag. Under normal conditions ， ρ1 = lwhenS = 1. WhenN→∞， S →∞， /5 1 • O.

Generally， αis very small compared with N. Therefore we make simplified calcula-

tion using (45). Under the assumption of 5.1, set m = 12. Considering the case 8=2, 4,

6, 8, 10 and the case N = 100, we obtain Table 1 from the calculation of (45).

10 0.2621

8

0.3372

with varying S

6

0.4573

'rable 1.戸l

|4

|0.6554 I 2

0.9266 S

f!J

/5 1, can be utilized as a deterioration factor. We name this as “Deterioration Factor

of the Autocorrelation Function Type of 1st Order Lag'¥

5.2.2. Autocorrelation coefficients of the 2nd order lag through 4th order lag

Then, we can get the autocorrelation function of 2nd order lag through 4th order

lag using the same method.

10 0.2620 0.2619 0.2618

Table 2. P2 through P4 with varying S

s=ρ二ρ=ρ

5.3. Estimation of MA Parameters about ARMA Model

Generally, the MA part of the ARMA model becomes a non-linearequation as fol-

lows:

P

Xn = Xn + I; aixn-i

8et

(49)

then we can get the following equation from (1):

q

tn=en+FI 鳥en-j (50)

From (49) , (50) , we set 九 as the autocorrelation function of Xn , and set bo = 1. Then we

can get the following relations which are well known (Tokumaru et al., 1982).

54 ・1(223) (2008.6J 大阪府立大学経済研究


、，EBEEtrEEEEEJ

¥/

唱EA

、1JJq

+

<一

q

k>

一

fkL

凡/l¥

.,J +

ル凡

AU

b

nu

t

o

「Z=

E-y

qLe

o

rEEEEE屯EEEEI

、

一一

九

q

久 = a: I; b・j=O

65

(51)

(52)

A recursive algorithm has been developed as for this. The only parameters to be estiｭ

mated are b¥ and b2・ Therefore they can be solved using the following method.

The case of q = 2;

ち =0

ち =σ';b 2

円 =σ;(b\+bん)

九 = a;(1 + bi+ bD

When a; • 1, then we get the following equations:

九= 1 +bi+b~

角 = b¥(1+b2)

ち = b2

Therefore we can obtain b¥ and b2 in the following equations.

r, b , = 一

1+ち

b2 = ち

From (49) , we get the following equations:

内 = E[xnxn+¥]

= E[(xn +a\xnー\+α2Xn-2) (Xn十 I 十 α \Xη+α2Xn-\)]

=αJ1+α2) (Ro+ R2) +α/R\十R3)+ (1 +αi+必 )R\

ち = E [XnXn+2]

= E[(九+α\Xnー I 十 α 2Xn-2) (xn + 2 +a\xn十 l 十 α 2Xn)]

=α\ (1 +α2) (R¥ + R3) +α2(Ro + R4) + (1 +αi+αDR2

b , = -:-丘一 α\ (1 +α2) (Ro+ R2) +α2(R\+R)+ (1 +αi+αDR\ 一一

1+ち 1+α\ (1 +α2) (R¥+ R3) +α2(Ro+ R4) + (1 +αi+α~)R2

b2 = ち =α\ (1 +α2) (R\ 十R3)+α2(Ro十R4)+ (1十αi+必 )R2

(53)

(54)

(55)

(56)

(57)

(58)

(59)

(60)

(61)

(62)

(63)

(64)

As mentioned above, the (2 ,2) of the ARMA model can be solved analytically. We

(65)

大阪府立大学経済研究 54 ・ 1(223) (2008:6J

66 Simplified Machine Diagnosis Techniques Using ARMA MQdel


can solve ARMA mode1 on1y by calcu1ating the autocorre1ation function from the 1st

through 4th order.

6. NUMERICAL EXAMPLES

6.1. Estimation of AR Parameters

Under the assumption of 5.2 , set m = 12. Considering the case S = 2, 4, 6, 8, 10 and

the case N = 100 , we obtain Tab1e3 from the calcu1ation of (34) using the va1ues of

ρ] ， ρ2'ρ3'ρ4 in the Tab1e 1 ~2 of 5.2.

S

。

'rable 3. Estimation of AR parameters with varying S

2

ー0.2380

-0.2373

10

ー 0.1375

-0.1375

We get the estimations â] , �2 of the AR parametersα]， a 2 ・ Ca1cu1ation mode1 data

chart is exhibited in Figure 2.

Figure2. Calculation model data chart

6.2. Estimation of MA Parameters

Under the same assumption of 5.1 , we solve b], b2, which are an estimate of b] , b2 of

MA parameters respective1y , using ρ] ， ρ2'ρ3'ρ4 ， and â], �2 as mentioned above.

As {xJ is assumed to be a stationary time series , we can assume:i = 0, a2 = 1 with日

out 10ss of genera1ity (Tokumaru et al., 1982).

Ro= σ2 = 1

Therefore, from (34) we can get following equation:

R[= ρ[ ， Ro = ρl (66)

Then

大阪府立大学経済研究 54 ・ 1(223) [2008.6)

Simplified Machine Diagnosis Techniques Using ARMA Model ofAbsolute Deterioration Factor with Weight 67

ρ。主主 l (67)

Therefore we get the estimates b1, b2 of the MA parameters b1, b2 using (64) and (65) (Taｭ

ble 4).

Table 4. Estimation of MA paramet目s with varying S

10 'OアO

0.0485

0.0363

6.3. System Parameter Distance

6.3.1. AR parame白m

When S = 1 ，出e system is considered to be under normal conditions. Therefore we

get the following equation by (34).

(1+2ρî)C1 ρD α=

4(2+ρî)C1 -ρî) (68)

l-2ρ~+3ρi-2ρi ao

4(2+ρî)C1 -ρî)

When S = 1, from (45) , we get

P1= ρ1， βρ2 = 1

(69)

When

lim j(x) = 0, lim g(x) = 0 Z • x • O

(70)

utilizing the following relation:

g(x) g'(x) 一… 一一…z• ô j(x) ;• ô j'(x)

we obtain the following results:

日 1-ρa , 11m 一一一一-0 = u Aρl→ 1 ，ρ2 → 12+4ρ;

(71)

(72)

1+2ρil α。 11口1 一一一一一一一uρ1→ 1 ，ρ2 → 1 -2-4ρ; 乙

(73)

6.3.2. MA parameters

When S = 1, the system is considered to be under normal conditions. Therefore we

can calculate the following equation by substituting a 1, a2 of (72) , (73) into (62) , (63).

大阪府立大学経済研究 54 ・ 1(223) (2008.6)

68

bj =

Simplified Machine Diagnosis Techniques Using ARMAModel of Absolute Deterioration Factor with We�ht

3Rj-2R3

4- 2Ro+5R2ー 2R4

1 _ .5_ 1 b? = --::-Rn+~ R?--::-R,

話 2 り 4 乙 2 ~..

(74)

(75)

From (66), (67) and (45) we get

Ro= ρ。 = 1, Rj = ρj = 1, R2 = ρ2 = 1,

R3 = ρ3 = 1, R4 = ρ4 = 1

From (74) and (75) , we obtain the following results.

1 _ 1 b, = -::-. b? = -=:-

5' -" 4

6.3.3. Calculation of the evaluation function

Calculating Zz using (4) , we obtain Table 5 and Figure 3 to 6.

As S grows larger, the evaluation function Zz becomes smaller. As l grows larger Zz beｭ

comes smaller. Therefore we can utilize this index as a machine diagnosis index. Z2

has relatively suitable value for machine diagnosis which is easy to catch a symptom

of machine failure. These results are useful for machine diagnosis in early stage.

Table 5. Zl with varying S

S 1 2 4 6 8 10

1=0 Zo 1.0000 0.8883 0.8791 0.8598 0.8389 0,8204

1 = 1 Zj 1.0000 0.6652 0.6452 0.6053 0.5655 0.5332

1 =2 Z2 1.0000 0.3319 0.3125 0.2772 0.2455 0~2221

1=3 Z3 1.0000 0.1105 0.1020 0.0875 0.07522 0.0666

0.8

0.6

0.4 直函

0.2

。

8=1 8=2 8=4 8=6 8=8 8=10

Figure 3. The Transitionof Evaluation Function Zo

大阪府立大学経済研究 54 ・ 1(223) (2008.6J


0.8

0.6

0.4

0.2

0.8

0.6

0.4

0.2

。

。

匝ヨ

8=1 8=2 8=4 8=6 8=8 8=10

Figure 4. The Transition of Evaluation Function z�.

亙回

8=1 8=2 8=4 8=6 8=8 8=10

Figure 5. The τ'ransition of Evaluation Function Z2

11!1.l1=31

8=1 8=2 8=4 8=6 8=8 8=10

Figure 6. The 'I旨ansition of Evaluation Function Z3

大阪府立大学経済研究 54 ・ 1(223) (2008.6J

69



Now , we compare this new index with Bicoherence. We made experiment in the past

[15, 16]. 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.

Middle 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 the .t;, Jz, in Eq.(l1). We got best-fit result in the follow-

mg case.

l川…y……五: 2五

We obtained the following Bicoherence values in this case (Table 6).

Condition Bicoherence

0.8

0.6

Table 6. Transition of Bicoherence value

Normal

0.99

Small defect

0.38

Middle defect

0.09

Big defect

0.02

0.4 厄白己ohe…ー10.2

。

Normal Small Middle Big defect defect defect

Figure 7 . The Transition of Bicoherence

Thus , Bicoherence proved to be a very sensitive good index. Bicoherence is an absoｭ

lute index of which range is 1 to O. Therefore it can be said that it is a universal inｭ

dex. In those experiment, smal1 defect condition is generally assumed to be S = 2 and

big defect condition is generally assumed to be S = 6 [10].

In the case of l = 2 (Figure. 5) in Table 5, the value is 0.3319 at small defect

大阪府立大学経済研究 54 ・ 1(223) (2008.6J


condition and 0.3125 at middle defect condition and 0.2772 at big defect condition

which show nearly sensitive behavior in the early stage of Bicoherence. The case of

l = 3 would be too much sensitive. It could besaid that the case l = 2 would be sensiｭ

tive enough for the practical use. Therefore, the case l = 2 would be recommended in

this new method. This calculation method is simple enough to be executed even on a

pocketsize calculator. Compared with Bicoherence which has to be calculated by Eq.

(11) , (12) , (76) and (77) , proposed method is by far a simple one and easy to handle on

the field defection.

7. CONCLUSION

In order to make machine diagnosis , the method of calculating Kurtosis or

Bicoherence was utilized in the past. Calculating system parameter distance was a1so

uti1ized to app1y time series data to Autoregressive (AR) model or Autoregressive Movｭ

ing A verage (ARMA) model.

In this paper, a simplified calculation method of autocorrelation function was introｭ

duced and it was utilized for ARMA mode1 identification. Furthermore , an absolute deｭ

terioration factor such as Bicoherence is introduced. Compared with the results obｭ

tained so far , the results of numerical examples of this paper are reasonable and

nearly sensitive of another method such as Bicoherence. Judging from these results ,

our method is properly considered to be effective for the early stage failure detection esｭ

pecially. This calcu1ation method is very simp1e and is very practica1 at the factory of

maintenance site. This can be installed in microcomputer chips and uti1ized as the too1

for the early stage detection of the failure.

Machine diagnosis could be executed by this simplified calculation method of system

parameter distance with weight. Proposed method proved to be a practical index for maｭ

chine diagnosis by numerical examples. The effectiveness of this method should be examｭ

ined in various cases.

APPENDIX

XT(f) is a Fourier Transform stated as follows.

大阪府立大学経済研究 54 ・ 1(223) (2008.6J



r x(t) (0 < t < T) Xr(t) = ~

品 L 0 (else)

T: Basic Frequency Interval

XTC!) = 工二XT(t)e-j2ψdt

SxxC!) is a power spectrum stated as follows.

Sxxc!) = ~ XTC!)X;ω *: conjugate

REFERENCES

(76)

(77)

[ 1 J Bolleter, U.,“'Blade Passage Tones of Centrifugal Pumpsて Vibrαtion 4 (3) , 8-13, 1988

[2 J Box. Jenkins, "Time Series Anαlysis"， Third Edition, PRENTICE HALL, 1994

[3 J Hino, M,“Spectrum Anαlysii;" (in JIαpαnese) Asakura Publishing, 1977

[4 J Hoffner, J , "Preventive mαintenαnce for No-Twist rod mills using vibration signαture

anαlysis"， Iron αnd Steel Engineer 68 (1), 55-61 , 1991

[5 J Inoue鳥， Y. , and T. Ma抗tsui ，

Non-minimum Phαωse"， Proc of the Workshop on Higher-Order Spectral Analysis, 1989

[ 6 J Inoue, Y. , "Signαl processing-1 by Using Higher Order Cumulants", the Institute of Sysｭ

tems, Control and Information Engineers, Vol. 36, No. 2, 1992

[7 J Inoue, Y.,“Signal processing四 n by Using Higher Order Cumulants", the Institute of Sysｭ

tems, Control and Information Engineers, Vol. 36, No. 5, 1992

[8 J Katayama, T. , "Introduction to System Identificαtion" (in JIαpαnese)， Asakura Publishing,

1994

[9 J Krishanamurthy, V. , and H. Vincent Poor,“'Asymptotic Anαlysis of an Algorithm for Paｭ

rαmeter Estimαtion αnd Identification of l-b Quαntized AR Time Series", IEEE Trans. Sigｭ

nalProcess. Vol. 44, No. 1, 1996

口OJ Maekawa, K., S. Nakajima, and T. Toyoda, "New Severity Index for Failures of MαchineEle司

ments by Impαct Vibration" (in Japanese). J. SOPE Japαn 9 (3) , 163-168, 1997

[l1J Nakamizo T.,“Signal Anαlysis αnd System Identification" (in J，α:panese)， CORNA PUBｭ

LISHING, 1988

口2J Sagara, S. , K. Akizuki, T. Nakamizo , T. Katayama, "System Identification", The Society

of Instrument and Control Engineers, 1987

口3J Shao , Y. , K. Nezu, T. Matsuura, Y. Hasegawa, and N. Kansawa,“Bearing F，αult Diαgno

sis Using αnAdα:ptive Filter" (in Japanese) , J. SOPE Japan 12 (3) , 71-77, 2001

大阪府立大学経済研究 54 ・ 1(223) (2008.6)


口4J Swami, A. , and J. M. Mendel,“'ARMA Pαrαmeter Estimαtion Using Only Output

Cumulants", IEEE Trans. Acoust, Speech Signal Processing, Vol. ASSP-38, No. 7, 1990

口5J Takeyasu, K., "Wαtching Method of Circulating Moving Object" (in Japanese) , Certified

Patent by Japanese Patent Agency (1987)

口6J Takeyasu, K.，“Wαtching Method of Circulating Moving Object" (in Japanese), Certified

Patent by Japanese Patent Agency (1989)

口7J Takeyasu, K., T. Amemiya and H. Goto,“'Robust System Identification Algorithm of Bootｭ

strα:p Type", IEMS Vol. 1, No. 1, 2002

日8J Takeyasu, K., Y. Ishii，匂0'stem p，αrameter Distance for Mαchine Diagnosis", IFORS, 2005

口9J Tokumaru,H. , K. Takeyasu,“The expectation of the Spectral Density by the Discrete Time

Lineαr Model", The Society of Instrument and Control Engineers, Vol. 13, No. 2, pp. 148-

153, 1977

[20J Tokumaru, H., T. Soeda, T. Nakamizo , and K. Akizuki，“'Meαsurement αnd Cαlculαtion"

(in J;α:panese)， Baifukan Publishing, 1982

[21J Toresten Sりderstörm ， "An efficient αnd Versalile Algorithm for Computing the Covαriance

Function of αn ARMA Process", IEEE Trans on Signal Processing, Vol. 46 , No. 6, 1994

[22J Xian-Da Zhang, H. Takeda "An Approαch to Time Series Analysis and ARMA Spectral Esｭ

timation", IEEE Trans on Acoustics, Speech & Signal Processing, Vol. ASSP-35 No. 9,

1987

[23J Yamazaki, H,“Failure Detection αnd Prediction" (In Japanese), Kogyo Chosakai Publish>

ing, 1977

大阪府立大学経済研究 54 ・ 1(223) (2008.6J

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