a statisical approach to machinery condition monitoring - blake

8/11/2019 A Statisical Approach to Machinery Condition Monitoring - Blake

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STATISTICAL APPROACH TO MACHINERY CONDITION MONITORING

Randall W. Blake

Predictive Maintenance Staff Manager

St. Johns River Power Park

~acks onville, lorida

_ . ..

:

.::.I

, : Randall W. Blake is the Predictive Maintenance

: Staff Manager for St. Johns River Power Park in

; Jacksonville, Florida. In this capacity, he is

i

responsible for the predictive maintenance program

which includes vibration analysis, oil analysis

and equipment reliability assessment.

Randy has

been a SJRPP employee for eight years and has

twenty-two years experience in electrical power

generation.

Abstract :

This paper shows how to apply straightforward statistical methods

to real world vibration problems. Statistical methods are

effective tools for improving production processes and reducing

unscheduled failures. Statistical tools can lend objectivity,

accuracy and

focus t o your vibration program.

Introduction:

All rotating or reciprocating machinery emits a unique pattern of

vibration characteristics. The pattern of vibration, or

vibration signature, represent the current mechanical condition

of the individual machine. As time passes, the equipment

mechanical condition will change due to internal wearing,

unbalance, misalignment, looseness, and related problems. These

changes

in machinery condition will also affect the machines

vibration signature.

The purpose of a vibration monitoring and

analysis program is to detect changes in equipment signatures and

use this

information t o pinpoint equipment degradation and thus

schedule corrective maintenance or overhaul. The statistical

methods outlined in this paper are effective tools fo r monitoring

machinery condition, improving production processes and reducing

unscheduled failures.

Statistics:

Webster defines statistics as, the mathematics of th e

collection, organization, and interpretation of numerical data,

esp. the analysis of population characteristics by inference from

sampling.


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As Webster s definition points out,

data collection by sampling,

interpretation of numerical data,

and analysis of population

characteristics are all tools used in a vibration program.

The

statistical tools outlined in this paper are more tools to help

monitor machinery condition.

The following FACTORS are presented as an aid to help avoid

problems in the statistical methods presented in this paper.

Factor

DATA COLLECTION

Regardless of type of data collection equipment used,

CONSISTENCY

is of the utmost importance in data collection.

Consistency in

the, data collection techniques, location of data collection

points, equipment operating parameters and, in some cases,

ambient conditions.

It is highly desirable that the same personnel remain in the

program.

It is especially important to have the same personnel

collect data. Consistency in personnel helps to ensure accurate

and consistent data collection and analyses.

Factor

2:

DOCUMENTATION

Once data is collected, various statistical methods may be used

for analysis so that the data becomes a meaningful source of

information.

The data recording methods will vary greatly but

the following points need to be considered no matter what method

is used.

First, the origin of the data must be clearly recorded.

Data

whose origin is not clearly known becomes dead data.

Quite

often, little useful information is obtained despite the fact

that months were spent collecting data, because the date it was

collected or which machines .the data represented was omitted.

Secondly, data should be recorded in such a way that it can be

used easily. Since data is often used later to calculate the

mean, standard deviation, and days t o alarm etc, it is better to

record the data in a manner which will facilitate these

computations.

Statistical Methods:

All of the statistical methods outlined in this paper can easily

be operated with

a

simple calculator or with any number of

inexpensive computer software packages.


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Method 1 HISTOGRAMS

The data obtained from a sample serve as the basis for decisions

on the population. The larger the sample size, the more

information we gain about the population. But an increase of

sample size also means an increase in the amount of data and it

becomes difficult to understand the population from these data,

even when they are arranged into tables or reports. In such

cases, we need a method which will enable us to understand the

population at a glance. A histogram answers that need. By

organizing many data into a histogram, we can understand the

population in a objective manner.

Making a Histogram

Let's assume, for this demonstration,

that we want to set or

adjust the alarm level

for one of the vibration parameters for

pump motor B1.

Table

1

shows the latest vibration data for the pump motor. Let

us make a histogram using the data set PAR 4.

Step 1: Obtain the largest and the smallest of collected values and

calculate

R.

R (the largest value) (the smallest value)

The largest value .048 in/sec.

The smallest value .020 in/sec.

R

.048 .020 .028

Step 2: The class interval is determined so that the range, which

includes the maximum and the minimum of values,

is divided into equal

sizes. Obtain the number of interval by, dividing R by .001,.002

or.005 (or 0.1, 0.2, 0.5:10,20,50, etc.) so as to obtain from 5 to 20

class intervals of equal size. When there are two possibilities, use

the narrower interval.

Thus, the class intervals can be either 0.002 or 0.005, since

0.002 is the narrower of the two intervals, we will use it for this

example.


4/20

TABLE 1

Machine

1: PUMP MOTOR B1

..........................................

DATE TIME SPEED OVERALL PAR l PAR 2 PAR83 PAR 4 PAR 5 PAR 6

1 AFP B1

-MPA

06-NOV-91 11:32

03-DEC-91

09:48

13-DEC-91

13:27

02-JAN-92 16:Ol

29-JAN-92 10: 55

11-FEB-92

14:12

13-FEB-92 10~18

09-MAR-92

08: 49

27-MAR-92 08:48

29-APR-92

08:54

29-APR-92 13:58

30-APR-92 06:34

13-MAY-92

05: 54

01-JUL-92 14:13

15-JUL-92 12:31

22-JUL-92 14: 12

21-AUG-92

15:16

08-SEP-92 08:44

24-SEP-92 14:55

30-SEP-92

09:48

12-OCT-92

15:23

27-OCT-92

10:57

27-NOV-92 14:45

16-DEC-92

14:28

29-DEC-92

10:57

Step 3: Prepare a frequency table, as in Table 2, on which the class,

midpoint, frequency count, frequency, etc., can be recorded.

Step

4:

Determine the boundaries of the intervals so that they

include the smallest and the largest of values,

and write these down

on the frequency table.

Determine the lower boundary of the first class by subtracting

112

of

the class interval size from the smallest sample value. If the lower

boundary is

0

or less, use

0

for the lower boundary. The upper

boundary can be determined by adding 112 of the class interval size to

the smallest sample value.

Lower class boundaries

=

0.02 0.001 0.019

Upper class boundaries =

0.02 0.001= 0.021

Then keep adding the size of the interval to the previous value to

obtain the second boundary, the third, and so on, and make sure that

the last class includes the maximum value.

Boundaries for the

first class =

0.0190

0.0210

Boundaries for the

second class

=

0.0210

0.0230


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Step :

Calculate the mid-point of

each class, and write them on the

frequency table.

Mid-point of

the first class

= 0.0190

0.0210)

2

=

0.02

Mid-point of

the second class =

0.0210

0.0230)

2 0.022

Step : Read the collected values one by one and record the

frequencies falling in each class

Table 2 Frequency table

Step 7: Plotting the Histogram

On a sheet of

squared paper or with one of the software programs,

label the horizontal axis scale based on the class interval. Mark the

left-hand vertical axis with the frequency scale.

The height of the

vertical scale axis should be one unit of measure above the maximum

frequency notation.

Draw a bar whose height corresponds with the

frequency in that class.

refer to figure 1)

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

Step 8: Mean and Standard Deviation

Mid- Point

of class

0.020

0.022

0.024

0.026

0.028

0.030

0.032

0.034

0.036

0.038

0.040

0.042

0.044

0.046

0.048

Class

0.019 - 0.021

0.021

-

0.023

0.023

-

0.025

0.025 - 0.027

0.027

-

0.029

0.029 - 0.031

0.031

-

0.033

0.033

-

0.035

0.035

-

0.037

0.037

-

0.039

0.039 - 0.041

0.041

-

0.043

0.043 0.045

0.045

-

0.047

0.047

-

0.049

Standard Deviation STD) measures the degree to which individual

values in the population vary from the Mean average) of all values in

the population.

The lower the STD, the less individual values vary

from the mean.

Frequency

notation

1

0

0

0

0

0

0

1

3

12

4

2

1

0

Ex

Mean =


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xi= The ,th samples in the population

N = number of samples in the population

Mean = average of samples in the population

In a blank area of the

histogram, note the number of

sample points, the mean, and

the standard deviation.

Using the Histogram:

Now that the data has been

organized into a histogram,

objective decisions can be made as

to the alarm levels. Knowing the

mean and standard deviation of the

population, we can set or adjust

our alarm levels based on the

histogram and statistical data.

For example, draw a vertical line

on the histogram that represents

the mean 2 STD and another that

represents the mean - 2 STD, 95 percent of the data falls between the two

lines. Alarm levels could be set +/- 1 (68 ), 2 (95 ), or

3

(99 ) STD

for multiple alarm levels, as shown in figure 1.

Level 1

=

0.03784

+

(1 0.004531)

=

0.04531 in/sec.

level 2 =

0.03784

+ (2 0.004.531)

= 0.04690 in/sec.

level

3 =

0.03784

+ (3 0.004531)

= 0.05140 in/sec.

Figure 1 Histogram

14

Frepuency

N PS

. .

h

am

YOH

12

BrD-03D Dl

am

Another important point may be more obvious by using t he histogram in this

fashion, that being OUTLIERS. Outliers are data points that, for one

reason or another are in error. As shown in figure data point .020

in/sec. may be an outlier. If it is discovered to be an outlier,

corrections to the histogram, mean, standard deviation and alarm levels

will need to be made.

Method 2: OUTLIERS

.

All data collection systems may produce corrupted data points.

These

points may be caused by variations in process parameters, ambient

conditions or actual variations in the vibration levels. Errors of this

type should not be included as part of the analyses. Such points are

meaningless as test data. All data should be inspected for corrupted data

points as a continuing check in the data collection process.

The effects of these outliers will increase the random error of the

population. A test is needed to determine if a particular point from a

sample is an outlier. The test should consider two types of errors in

detecting outliers:

a m

.

1 M I U

.

-

Vibration

(intsec.)

. . . . .

.am

e

a10

. m

10

01 .

6

(1) Rejecting a good data point

(2) Not rejecting a bad data point

l-.bml'i:

.. -

.

.

4

2 -

J-J

.

-

.o19 . . 7 ' l h .030

A3

MT

. I

.055 .W

m m n m

.

,

0

. .

.


7/20

The probability of rejecting a good point is usually set at

5%.

This

means that the odds of rejecting a good point are 20 to

1 (or less). For

larger populations, (several hundred sample points), almost all corrupted

data points can be identified. For small populations (five or ten),

corrupted data points are more difficult to identify.

A

test commonly used t o

identify questionable data points as outliers is

the GRUBBS1 Method. (ref.1)

Consider the ,th sample of N measurements. The mean (M) and an

experimental standard deviation (S) are calculated using (1) (2).

Suppose the ith observation is the questionable data point; then, the

absolute statistic calculated is:

Using table 3, a value of TN is obtained for the sample size (N) at

the 5% significance level. This limits the probability of rejecting a

good data point t o 5%.

To test for outliers, compare the calculated TN with the table TN .

If

TN calculated is larger than or equal to TN table, is an outlier. If

TN calculated is smaller than TN table,

is not an outlier.

Table

3

Rejection values for

Grubbs' Method

Let's reconsider the data points

in TABLE 1, PAR 4. Th e histogram in

figure 1 giv es reason to suspect data point .02 inlsec. to be an

outlier. By using the Grubbsl Method:

Mean

(M) 0.03784

Exp. STD

(S) 0.00453

Sample

size

N

3

4

5

6

7

8

9

0

11

12

13

14

15

16

17

18

19

Sample

size

N

20

21

22

23

24

25

30

35

40

45

50

60

70

80

90

100

5

(1-side)

1.15

1.46

1.67

1.82

1.94

2.03

2.11

2.18

2.23

2.29

2.33

2.37

2.41

2.44

2.47

2.50

2.53

5

1-side)

2.56

2.58

2.60

2.62

2.64

2.66

2.75

2.82

2.87

2.92

2.96

3.03

3.09

3.14

3.19

3.21


8/20

Sample Size N)

=

25

From TABLE

3

TN,,,

=

2.66,

therefore, since 3.937 is

larger than .2.66, data point

-0 2 in/sec. is an outlier

according to the Grubbsl test

Method data point .048 in/sec

is also an outlier).

Figure 2 shows the corrected

histogram plot with possible

alarm levels.

Note: Its important to do the

Grubbs test and set alarm levels

before the vibration data begins to

show signs of trending up.

-- Figure 2

Histogram

flcqu ncy u-m

Yn.r mn

810. Lk

- . .

Vibration (inlsec.)

m

01

Method

3:

SCATTER DIAGRAMS

It is often essential to study the relationship of two corresponding

sample data sets.

For example, to what extent will the vibration of a

machine change in respect to time.

To study two variables such as

vibration of, the machine and time, a scatter diagram can be used.

Making a Scatter Diagram

A scatter diagram can by made by following these steps:

Step 1: Collect paired data x,

you want to study the relationshi

Step 2: Find the maximum and

minimum values for both the x

and

y.

Make th e scales of

horizontal and vertical axis

so that lengths are approxi-

mately equal. Keep the number

of unit graduations to 3 to 10

for each axis.

Step 3: Plot

the data on

section paper or a computer

.software program. Include all

necessary information: 1.

title of th e diagram, 2. time

interval, 3. number of points,

4 title and units of each

axis, 5. name of person who

made the diagram. See figure 3)

y or time, vibration), on which

?


9/20

In order to understand

the strength of the relationship in quantitative

terms, it is useful to calculate the correlation coefficient.

TABLE 4

x, y) DATA

Outliers * )

The correlation coefficient, r, is in the range -1to +l. When

r

is near

+I, it indicates a strong positive correlation between

x

and

y.

Like-

wise, when

r

is near -1, it indicates a strong negative correlation.

Using the data collected in table 4, overall vibration amplitude and

cumulative days data, the correlation coefficient can be calculated see

table 5)

Sample

Date

06-NOV-91

03-DEC-91

13-DEC-91

02-JAN-92

29-JAN-92

11-FEB-92

13-FEB-92

09-MAR-92

27-MAR-92

29-APR-92

29-APR-92

3 -APR-9 2

13-MAY-92

01-JUL-92

15-JUL-92

22-JUL-92

21-AUG-92

08-SEP-92

24-SEP-92

30-SEP-92

12 0CT-9 2

2 -OCT-9 2

27-NOV-92

16-DEC-92

29-DEC-92

Cumulative

Days

1

27

37

57

84

97

99

124

142

175

175

176

189

238

252

259

289

307

323

329

341

356

387

406

419

Vibration

Amplitude

0.131

0.162

0.149

0.169

0.172

0.107

0.106

0.186

0.190

0.178

0.174

0.172

0.222

0.197

0.154

0.168

0.148

0.167

0.176

0.246

0.253

0.246

0.178

0.256

0.232


10/20

TABLE

5

Method 4: Regression Analysis

NO.

2

3

4

5

6

7

8

9

10

12

13

14

15

16

17

SUM

After establishing a strong correlation between

x

and y,

a regression

analysis can be used t o extrapolate how many days a machine can run before

reaching an alarm point. To realize this analysis and to determine days

to alarm, it

is necessary to comprehend the relation between the vibration

amplitude and cumulative days, quantitatively.

As previously determined, the scatter diagram in figure 3 shows a strong

correlation between vibration amplitude y) and cumulative days x) . From

this diagram, it would seem that vibration and days have a straight-line

relation.

Such a straight line is called a linear regression line. The least

squares regression analysis is the most popular means of curve fitting.

For higher order fitting see attachment A.

Cumulative

Days

dependent

v ri ble

x

27

37

57

84

124

142

175

175

176

189

238

329

3 4 1

356

406

419

3276

X Y

0 . 1 3 1

4.374

5.513

9.633

14.448

23.064

26 .98

31.15

30 .45

30.272

41.958

46.886

80.934

86 .273

87.576

103.936

97.208

720.786

xA2

729

1369

3249

7056

15376

20164

30625

30625

30976

35721

56644

108241

116281

126736

164836

175561

924190

vibration

Amplitude

Y

0 .131

0.162

0.149

0.169

0.172

0.186

0 .19

0.178

0.174

0.172

0.222

0 .197

0.246

0.253

0 .246

0.256

0.232

3.335

YA2

0.017161

0.026244

0 .022201

0 .028561

0.029584

0.034596

0 .0361

0.031684

0.030276

0.029584

0.049284

0.038809

0.060516

0.064009

0.060516

0.065536

0.053824

0.678485


11/20

independent variable

constant

regression c oe f f i c i e n t

This quantitative way of grasping the relationship between

x

and y by

seeking a regression from

x

and y is called

regression analysis.

Using

the data in Table 6 lets calculate the regression line.

TABLE

mean

of x

192.7058

mean o f y

0.196176


12/20

The regression line is expressed by

y

=

0.144782 0.000266~. That is,

for every day of run time,

the

vibration will increase by 0.000266

inlsec.

Figure 4 shows th e regression line

as calculated above.

The points on

the scatter diagram should be

evenly distributed around the

regression line.

Method 5: Goodness Of Fit 1)

There are two quantitative measures

of goodness of fit.

The deviation of data points around the regression

line can be characterized by the standard of error of estimate SEE), the

precision index of residuals.

The smaller the residuals, the smaller the

SEE, the better the fit.

S

=

Yi Yi,fiC)*

i

N c

Where

C

is the number of coefficients of the regression.

For a linear

relation, C = 2. This same equation applies to higher order fits, where C

again indicates the number of coefficients of the regression.

Table

7

and equation demonstrate the goodness of fit for the previous

regression calculations.


13/20

TABLE

7

S 0 003404

0 .015064

15

Method 6: Goodness Of Fit 2)

Another commonly used goodness of fit test is the

coefficient of

determination

The fraction SSR/SST is called the coefficient of

determination and is represented by the symbol r2 , Which varies from -1 to

+l.

The closer r2 is to

1

the better the fit.

S S R

12

ST

S S R y i d r l t - 7

SST

S S R

SS

( 1 5 )


14/20

T BLE

8

The preceding calculation shows goodness of fit r2to be 0.999979, which

was determined earlier to be a very good fit.

YI,M

0.145048

0.151983

0.154650

0.159984

0.167184

0.177852

0.182653

0.191454

0.191454

0.191721

0.195188

0.208256

0.232525

0.235726

0.239726

0.253061

0.256528

SUM

Method 7: Days To Alarm Analysis

Once a regression line has been fitted to the data, goodness of fit has

been determined and alarm levels have been set, Days To Alarm can be

calculated. Days to alarm can be determined in two ways, by calculations

or reading it from a graph.

i

0.131

0.162

0.149

0.169

0.172

0.186

0.19

0.178

0.174

0.172

0.222

0.197

0.246

0.253

0.246

0.256

0.232

3.335

Calculation Method:

Now that the constant and regression coefficient has been

established,

days to alarm can be calculated as follows;

S S

10.17578

10.13159

10.11462

10.08072

10.03505

9.967578

9.937289

9.881879

9.881879

9.880202

9.858418

9.776526

9.625346

9.605498

9.580717

9.498344

9.476986

167.5084

DT

alarm

-

P

-

ct

16)

SSE

0.000197

0.000100

0.000031

0.000081

0.000023

0.000066

0.000053

0.000181

0.000304

0.000388

0.000718

0.000126

0.000181

0.000298

0.000039

0.000008

0.000601

0.003404

DTA = Days To Alarm

C,

=

Cumulative Time Days)

yh = Alarm Point 0.300 in/sec

for exponential regression curve fits;


15/20

DT =

lo Y

-

lo a

lo

t

for'power regression curve fit;

log Y log

DT = 1 0 ~ )Cc

Graphical Method:

The days to alarm can simply be read from the scatter diagram with

a fitted regression line by following these steps (see figure 5).

step 1. Extend the regression line to the y axis alarm

intersect point.

step 2. Read the corresponding x axis, cumulative tim e

(days).

step 3. Subtract the present cumulative days from projected

cumulative days.

Method

:

Confidence Interval

Confidence Intervals are measurements of precision in estimating a

parameter, in this case days to alarm.

A confidence interval around an

unknown parameter is an interval of numbers derived from sample data that

almost assuredly contains th e parameter.

Managers are often interested in how far from the true value an estimated

days to alarm might deviate.

For this example, 95 confidence level will

be used, that is to say, th e calculated interval has a 95 chance of

containing the true parameter.


16/20

DTA

=

0.3 - (0.144782 (2~0.015064) -

ql

1

0.000266

DTA

=

0.3 - (0.144782

-

(2x0 015064) )

-

41g

=

278

0.000266

DTA

= Days to .Alarm upper limit

DTA, = Days t o Alarm lower limit

In an earlier example, the days to alarm was calculated at

165

days. We

can now state with

95

confidence, the days to alarm will be between

51

and

78

days.

Method

9:

Confidence Bands o n th e Regression Line

In Method 8, the confidence interval was calculated for a single point.

In much the same way, upper and lower Confidence Bands ca n be placed on a

regression line. see figure

5.1)

y

=

(a

-

(2 SEE))

+ px

yl = (a (2

x

SEE))

+ px


17/20

ATTACHMENT A

Higher Order Curve ~ i t t i n g

Exponential Curve Fit:

For vibration data, the most often used curve fitting calculation will be

either Exponential or Linear regression. Use the goodness of fit test to

determine which curve fit calculation best fits the data. see figure 6)

TABLE

X

Log

x EXP

fit Y log y og

Y X*Y

27

37

57

84

124

142

175

175

176

189

238

329

341

356

4 6

419

SUM-

s xy)

=

C x

log Y

C

x

C l og Y

N

S x x ) =

Ex2

Ex)


18/20

Power Regression Curve Fit:

Another curve fit that may be useful in analyzing vibration data is the

Power Regression fit.

S xy) C l0g logy ) -

Elog Zlog y )


19/20

REFERENCES

1. Grubbs, F. E., Procedure for Detecting Outlying Observations in

Sample,I1 Technometrics, Vol. 11, no. 1, February 1969.

2. ANSIIASME PTC 19.1-1985, Measurement Uncertaintv. Par t 1,.

3. Dr. Robert

B.

Abernethy, I1Test Measurement Accuracy, Jan uary 1989.

4. Hitoshi

Kume,

Statistical Methods for Qualitv Im ~r ov em en t, une

1990.


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a statisical approach to machinery condition monitoring - blake

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